slurry systems handbook baha e. abulnaga, pe

SLURRY
SYSTEMS
HANDBOOK
BAHA E. ABULNAGA, P.E.
Mazdak International, Inc.
McGRAW-HILL
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In memory of my father, Dr. Sayed Abul Naga,
and in dedication to my mother, Dr. Hiam Aboul Hussein,
who devoted their lives to comparative literature
as authors and translators. May their efforts contribute
to a better understanding among mankind.
And to my children Sayed and Alexander
for filling my life with joy and happiness.

BAHA ABULNAGA, P.E., obtained his Bachelor of Aeronautical Engineering in
1980 from the University of London and his Masters in Materials Engineering in 1986
from the American University of Cairo, Egypt. The first years of his professional career
were devoted to the adaptation of air cushion platforms to desert environments, as well as
the development of renewable energy systems. In 1988, he joined CSIRO (Australia) as a
scientist. There he conducted research on complex multiphase flow for the design of
smelting furnaces.
Since 1990, he has been active in design of rotating equipment, pumps, and slurry
pipelines and processing plants. His career has been a balanced mixture of design of
equipment and consulting engineering.
He has been employed as a design engineer for a number of manufacturers such as
Warman Pumps (now part of Weir Pumps), Svedala Pumps and Process (now part of
Metso Mineral Systems), Sulzer Pumps North America, and Mazdak Pumps and Mixers.
He has also contracted as a slurry and hydraulics specialist for major consulting engineering
firms such as ERM, SNC-Lavalin, Fluor, Bateman, Rescan, and Hatch and Associates.
His involvement in the design, expansion, and commissioning of projects has included
ASARCO Ray Tailings (USA), LTV Steel (USA), Zaldivar Pipeline (Chile), Southern
Peru Expansion (Peru), Lomas Bayes (Chile), Escondida (Chile), BHP Diamets (Canada),
Muskeg River Oil Sands (Canada), Bajo Alumbrera (Argentina), Homestake Eskay Creek
(Canada), and many other engineering projects, feasibility studies, and audits.

PREFACE
The science of slurry hydraulics started to flourish in the 1950s with simple tests on
pumping sand and coal at moderate concentrations. It has evolved gradually to encompass
the pumping of pastes in the food and process industries, mixtures of coal and oil as a new
fuel, and numerous mixtures of minerals and water. Because of the diversity of minerals
pumped, the wide range in sizes [43 �m (mesh 325) to 51 mm (2 in)], and the various
physical and chemical properties of the materials, the engineering of slurry systems requires
various empirical and mathematical models. The engineering of slurry systems and
the design of pipelines is therefore fairly complex. This handbook targets the practicing
consultant engineer, the maintenance superintendent, and the economist. Numerous
solved problems and simplified computer programs have been included to guide the
reader.
The structure of the book is essentially in two parts. The first six chapters form the
first part of the book and focus on the hydraulics of slurry systems. Chapter 1 is a general
introduction on the preparation of slurry, the classification of soils, the siltation of dams,
and the history of slurry pipelines. Chapter 2 focuses on water as a carrier of solids. Chapter
3 progresses with the mechanics of mixing solids and liquids and the principles of rheology.
Chapter 4 presents the various models of heterogeneous flows of settling slurries,
whereas Chapter 5 concentrates on non-Newtonian flows. Due to the importance of open
channel flows in the design of long-distance tailings systems or slurry plants, Chapter 6
was dedicated to a better understanding of these complex flows, which are seldom mentioned
in books on slurry. In Part II, the book focuses on components of slurry systems
and their economic aspects. In Chapter 7, the important equipment of slurry processing
plants is presented, including grinding circuits, flotation cells, agitators, mixers, and
thickeners. Chapter 8 presents the guidelines for the design of centrifugal slurry pumps,
and methods of correction of their performance. Chapter 9 reviews the continuous improvements
of positive displacement slurry pumps in their different forms, such as
plunger, diaphragm, or lockhopper pumps. As slurry causes wear and corrosion, aspects
of the selection of metals and rubbers is presented in Chapter 10. To guide the reader to
the various aspects of the design of slurry pipelines, Chapter 11 presents practical cases
such as coal, phosphate, limestone, and copper concentrate pipelines. This review of historical
data is followed by a review of standards of the American Society of Mechanical
Engineers and the American Petroleum Institute, as they are extremely useful tools for the
design and monitoring of pipelines. Finally, as the big unknown is too often cost, Chapter
12 closes the book by offering guidelines for a complete feasibility study for a tailings
disposal system or a slurry pipeline.
The author wishes to thank the staff of Mazdak International Inc, particularly Ms.
Mary Edwards for providing typing services with great dedication over a period of two
years. The author particularly wishes to thank Fluor Daniel Wright Engineers for allowing
him to use their excellent library in Vancouver, Canada. The author wishes to thank
his former colleagues in a colorful career, particularly Mr. K. Burgess, C.P.Eng. of Warman
International; Mr. A. Majorkwiecz, K. Major, and Mr. Peter Wells of Hatch & Associates;
Mr. I. Hanks, P.Eng. and W. McRae of Bateman Engineering; Mr. R. Burmeister
xvii

xviii PREFACE
H. Basmajian, and Dr. C. Shook, consultants; Mr. C. Hunker, P.Eng, V. Bryant, D.
Bartlett, and W. Li, P.Eng. of Fluor Daniel; and Mr. A. Oak, P.Eng. of AMEC for allowing
him to work on very challenging assignments in Australia and South and North America.
The author wishes to thank the following firms for their contributions in the form of
figures and data to this handbook: The Metso Group (formerly the companies Nordberg
and Svedala), Red Valves, Geho Pumps (Weir Pumps), Mobile Pulley and Machine
Works, Inc., Wirth Pumps, Hayward Gordon, Mazdak International Inc., the BHR Group,
and GIW/KSB Pumps.
The author is grateful to the various publishers and associations who allowed him to
reproduce valuable materials in the book.

CHAPTER 8
THE DESIGN
OF CENTRIFUGAL
SLURRY PUMPS
8-0 INTRODUCTION
The centrifugal slurry pump is the workhorse of slurry flows. Chapter 7 briefed the reader
about some important slurry circuits, and it was explained that the grinding circuits consume
a fair portion of the power of a concentrator. One particular pump at the discharge
of the SAG, ball, or other mills is called the mill discharge pump. Wear in these pumps is
particularly harsh, leading to frequent replacement of impellers and liners, because a fair
portion of the solids remain fairly coarse until recirculated back through the classification
circuit.
The design of centrifugal pumps involves a combination of mathematical and empirical
formulae and models. Although water pumps have been the subject of extensive research
in the past, slurry pumps have been designed based on a compromise of what can
be cast with hard alloys, molded in rubber, and what can meet the hydraulic criteria.
A lot of papers have been published over the years on various aspects of wear in a slurry
impeller or volute, performance corrections and derating, etc. The reader of these papers
is often left with the impression that the design of these pumps is a combination of science
and art. What is often lacking in the literature are guidelines for the design of slurry pumps.
Whereas there are hundreds of manufacturers of water pumps on this planet, the number
of manufacturers of slurry and dredge pumps has been reduced to a handful. This
chapter presents some guidelines for the design of slurry mill discharge pumps. These
guidelines were developed by the author on the basis of the analysis of existing pumps in
the market, throughout his career as a consultant engineer. The designer can vary the
numbers or dimensions presented in the tables of this chapter within a margin of ±15% to
design a pump of his or her choice. These guidelines by themselves must be followed by
proper testing, prototype development, finite element analysis, and ultimately by fieldtesting.
In this chapter, the concepts of expeller, pump-out vanes, and dynamic seal will also
be examined. These are very important aspects of slurry pump design that have suffered
from a dearth of information in the published literature.
Wear remains a concern for the design of a slurry pump. There is no direct correlation
between the best hydraulics and the highest wear life. In fact, the whole activity of designing
a slurry pump is to find an optimum compromise.
8.1

8.2 CHAPTER EIGHT
8-1 THE CENTRIFUGAL SLURRY PUMP
A centrifugal pump is essentially a rotating machine with an impeller to convert shaft
power into fluid pressure. The dynamic energy is then converted into pressure or head in
a special diffuser or casing. The manufacturers of slurry pumps have developed a number
of specialized designs such as
� Dredge pumps with impellers as large as 2.6 m (105 in)
� Mill discharge pumps for milling and grinding circuits
� Vertical cantilever pumps (without submerged bearings)
� Froth handling pumps for flotation circuits
� High-pressure tailings and pipeline pumps
� General purpose pumps
� Low-head slurry pumps for flue gas desulfurization or flotation circuits
� Submersible slurry pumps
The slurry pump may be cased in a hard metal (Figure 8-1) or may be cast in iron, with an
internal liner (Figure 8-2), which may be of hard metal or rubber.
The components of the slurry pump are divided into two groups:
1. The bearing assembly or cartridge and frame
2. The wetted parts forming the wet end
The main components of the wet end are
� The pump casing volute
� The volute liner
� The front suction plate, or throat bush in large pumps
� The rear wear plate
� The impeller
� The expeller
� The shaft sleeve
� The packing rings
� The stuffing box and gland, greas cup, and associated water connections
� In very special cases the mechanical seal
The drive end of the pump consists of
� The pump shaft
� Piston rings or alternative protection against solids penetrating the bearing assembly
� Forsheda seals or O-rings
� Bearings and bearing nuts
� Grease retaining plates, grease nipples. or oil cup

8.3
adjustment
bolt
bearings
cartridge
shaft sleeve
gland plate
casing joint
stuffing box
water connection
packings rings
frame
back wearplate
impeller
discharge joint
casing
suction joint
FIGURE 8-1 Components of an unlined hard-metal pump. (Courtesy of Mazdak International Inc.)

8.4 CHAPTER EIGHT
coverplate
coverplate liner
throatbush
suction flange
impeller
discharge companion flange
backplate liner
backplate
expeller
Stuffing box
shaft sleeve
bearing assembly
pump frame
pump shaft
FIGURE 8-2 Components of a rubber-lined slurry pump. (Courtesy of Mazdak International
Inc.)
� Bearing cartridge and bearing covers
� An adjustable bolt or mechanism to adjust the impeller within the casing by moving
the shaft
� The pump frame
� Couplings or pulleys
The purpose of the pump is to produce a certain flow against a certain pressure. This is
done at a certain efficiency. The optimum point at which the efficiency is at a maximum
is called the best efficiency point. For every size or design of pump, there is a best efficiency
point at a given speed. The performance of the pump is plotted on a curve of head
versus flow (Figures 8-3 and 8-4) By combining different sizes of pumps on a single
chart, a pump tomb chart is produced (Figure 8-5).
Before dwelling on the design of a slurry pump, it is essential to have a basic understanding
of the hydraulics involved. But since the design of slurry pumps must also take
in account the wear due to pumping abrasive solids, many other factors enter into the
equation, such as the ability to pump large particles and the use of special alloys or polymers
for liners or impellers.
Practically all slurry pumps are single stage. Multistage pumps are limited to mine dewatering
applications. Slurry pumps are rubber lined whenever they are designed to handle
particles finer than 6 mm or 1 – 4�. Because rubber is susceptible to thermal degradation
when the tip speed of the impeller exceeds 28 m/s or 5500 ft/min, rubber-lined pumps are
typically reserved for a maximum head of 30 m (98.5 ft) per stage.
White iron is a very hard material. It is used in different forms such as Ni-hard and
28% chrome to cast impellers, casings, and metal liners of slurry pumps. Due to concern
about maximum disk stresses, most white iron slurry pumps are limited to an impeller tip
speed of 38 m/s or 7480 ft/min. Metal-lined pumps are limited to 55 m or 180 ft per stage.

Head (ft)
300
250
200
150
100
50
0
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
Flow rate (L/s)
5
Head vs flow curve
Efficiency Curve
10 15
0 50 100 150 200 250
Flow rate (US gpm)
90
80
70
60
50
40
30
20
10
Efficiency (%)
Head (m)
FIGURE 8-3 Performance of a pump showing head versus flow and efficiency versus flow
at constant speed.
Head (ft)
300
250
200
150
100
50
0
4200
3900
3600
3300
3000
20%
30%
5
Flow rate L/s
10 15
40%
MINIMUM LIMIT OF USE
45%
48%
2700 r/min
2400
2100
1800
1500
1200
45%
Efficiency curve
0 50 100 150 200 250
Flow rate in US gpm
FIGURE 8-4 Composite curve for the performance of a pump showing head versus flow and
efficiency versus flow at various speeds.
90
80
70
60
50
40
30
20
10
Head (m)
best efficiency curve
8.5
speed or rotation (rev/min)

8.6 CHAPTER EIGHT
HEAD (METRES)
90
80
70
60
50
40
30
20
10
0
0
1105
8X6
816
2000
10X8
667
FLOW IN US GALLONS PER MINUTE
4000
903
575
780
12X10
6000
510
RUBBER RANGE
200 400 600 800 1000 1200
White iron should not be confused with steel. Certain grades of steels are used in slurry,
dredging, and phosphate matrix pumps. They are cast at a lower hardness than white
iron and by being more ductile can withstand higher disk stresses. Impellers cast in steel
can be used in slurry pumps up to a tip speed of 45 m/s (8858 ft/min).
These are general guidelines, but the consultant engineer should collaborate closely
with the manufacturer. For example, certain special anti-thermal-breakdown additives are
used with some rubbers to exceed the limit of 28 m/s or 5500 ft/min on tip speed. In certain
situations, a metal impeller may be installed with rubber liners, particularly when
there are concerns about slurry surges (water hammer) in tailings pipelines.
8-2 ELEMENTARY HYDRAULICS OF THE
SLURRY PUMP
14X12
8000
691
450
16X14
10000
609
METAL RANGE
390
The correlation between the tip speed and the head per stage is established from basic hydraulics
of impeller design. There have been two schools in the past for the design of water
pumps—the American school lead by Stepanoff and the European school lead by Anderson.
The Stepanoff method is based on the concept that an impeller is designed on the
basis of velocity triangles, and that an ideal volute for best efficiency is then found using
12000
18X16
340
528
14000
FLOW RATE (LITRES/SECONDS)
FIGURE 8-5 “Tomb chart” for pumps showing size of pump versus flow range and head.
20X18
16000
460
18000
20000
300
250
200
150
100
50
HEAD (FEET)

various empirical factors. The Anderson school is based on the concept that one of the
most important parameters in pump design is the ratio between the throat area of the volute
and the impeller discharge area, and therefore more than one volute design can be
matched to a given impeller.
In the case of slurry pumps, passageways are larger than in water pumps to accommodate
solids and the Anderson area ratio is difficult but useful to use. Unfortunately, many
leading references on slurry pump design written in North America, such as the work of
Herbrich (1991) and Wilson et al. (1992), continue to ignore the area ratio methods and
focus on the Stepanoff school, which believes that the impeller is the main producer of
head and efficiency.
The design of a centrifugal slurry pump is complex. Performance depends on the area
ratio, impeller tip angle, recirculation patterns, change with wear of the impeller, back
vanes, and front pump-out vanes.
The flow in an impeller is fairly complex. A review of the hydraulics is essential to appreciate
wear. In simple terms, a vortex is formed.
8-2-1 Vortex Flow
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
The vortex creates a pressure field related to the radius from the center of the vortex in
accordance with the following equation:
� = C × R m v0
(8-1)
where
� = angular speed of rotating fluids
Rv0 = local radius of vanes
m = exponent
Stepanoff (1993) described various forms of vortices from a free vortex, with angular
velocity � inversely proportional to the square of the radius Rv0, to a super-forced
vortex, in which the angular velocity is proportional to the radius, as shown in Table
8-1.
The general distribution of pressure through a vortex, according to Stepanoff, is
� � = C
� � 2 2(m+1)
P Rv0 ��
� + z (8-2)
� 2(m + 1)g
where
C = constant
P = pressure
� = density
m = exponent
g = acceleration due to gravity
z = liquid elevation above the fixed datum
For a forced vortex, the angular speed is constant and the liquid revolves as a solid
body. Disregarding friction losses, Stepanoff (1993) claims that no power would be needed
to maintain the vortex. The pressure distribution of this ideal solid body rotation is a
parabolic function of the radius. When the forced vortex is superimposed on a radial outflow,
the motion takes the form of a spiral. This is the type of flow encountered in a centrifugal
pump. Particles at the periphery are said to carry the total amount of energy applied
to the liquid.
8.7

8.8 CHAPTER EIGHT
TABLE 8-1 Patterns of Vortex Flow
Angular Peripheral
velocity velocity Pressure
distribution, distribution, distribution,
Case � = C 1 × R m v0 V × R n v0 = C dp = � (�� 2 /g)rdr Type of vortex Remarks
1 –� � = C1 × Rv0 � V × Rv0 = C1
2 P/� = C 1 + z1 � = 0, stationary
2 –5/2 � = C2 × Rv0 3/2 V × Rv0 = C2
2 3 P/� = C 2/(3 · g · Rv0) + z2 � is
3 � = C3 × R –2
v0 V × Rv0 = C3 2 P/� = C 3/(2 · g · Rv0) + z3 Z3 + (P/�) + (v2 /(2 · g) higher
4 � = C 4 × R –3/2
v0
5 � = C5 × R –1
6 � = C6 × Rv0 7 � = C 7 × R 0 v0 V × R v0
8 � = C 8 × R 1/2
v0
9 � = C9 × Rv0 V × R –2
10 � = C10 × R m v0 V × Rv0 After Stepanoff (1992).
The parabola shown in Figure 8-6 is a state of equilibrium for a forced vortex and is
similar to a horizontal plane for a stationary fluid. To maintain a flow outward against the
applied pressure, the energy gradient must be smaller than the energy gradient for no
flow. This is what happens in a pump at near shut-off condition, where maximum static
head is obtained without any flow. As flow increases through the impeller, the head
drops.
In the case of the expeller, the designer tries to reach the parabola for energy gradient
without flow. However, as Case 7 in Table 8-1 shows, the pressure gradient is a square
function of R and inversely proportional to the square of the angular velocity. And in fact,
below a certain angular velocity, there is not enough pressure to overcome the difference
between volute and outside atmospheric pressure. The expeller or dynamic seal then stops
performing and leakage occurs.
8-2-2 The Ideal Euler Head
= constant, free vortex toward
V × R 1/2
v0 = C 4 P/� = –C 4/(g · r) + z 4 center of
v0 V × R0 v0 = C5 2 P/� = [C 5/g] · log Rv0 + z5 V = constant the vortex
–1/2 –1/2 V × Rv0 = C6
2 P/� = C 6 · Rv0/g + z6 V2 /Rv0 = constant
–1 = C7
2 2 P/� = C 7 · Rv0/(2 · g) + h7 = centrifugal force
� = constant, � =
forced vortex constant
–3/2 V × Rv0 = C8 P/� = C82 · R3 v0/(3 · g) + h8 Super forced vortex � is
v0 = C9 P/� = C9 · R 4 v0/(4 · g) + h9 Super forced vortex higher
–(m+1) = C10 P/� = [C 2 2(m+1) Rv0 ]/ General form of super toward
[2(m + 1) · g] + h forced vortex periphery
of vortex
The ideal pressure that a pump impeller can develop is called the Euler pressure. Consider
the flow through a radial impeller between two radii R1 and R2. The impeller is rotating
at an angular speed � (in rad/s) so that the peripheral speeds are respectively:
U1 = R1 · � (8-3a)
U2 = R2 · � (8-3b)
The liquid flows radially at a meridional velocity Cm, perpendicular to the peripheral
velocity U. The value of Cm is determined from continuity equation, It is necessary to take
into account the local area of the flow, which is a function of the radius and the width of

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
FIGURE 8-6 Pressure distribution in an impeller versus radius for condition of flow and no
flow. (From Stepanoff, 1993. Reprinted by permission of Krieger Publishers.)
the channel, minus the blockage area due to the finite thickness and angle of inclination of
the blades.
The channels between the impeller vanes follow a certain profile. At the intersection
with the radius under consideration, the angle between the vane and the tangent to the radius
is defined as �. A component of velocity is in the direction of � and is called the relative
velocity W.
The vector addition of U and W result in the absolute velocity V. Both V and W share
the same component of meridional speed Cm; a vector representation is shown in Figure
8-8.
The Euler “total” head between radii R1 and R2 is defined as
HE = (8-4)
where
(V 2 – V 2 1) = change in absolute kinetic energy
(W 2 2 – W 2 1) = change in relative kinetic energy
(U 2 2 – U 2 1) + (W 2 2 – W 2 (V
1) = change in static energy through the impeller
2 – V 2 1) – (U 2 2 – U 2 1) + (W 2 2 – W 2 1)
����
2g
It is clear that W = C m · cot �. Static head rise is
gH s = (U 2 2 – U 2 1) + (C m2 · cot � 2) 2 – (C m1 · cot � 1) 2 (8-5)
Furthermore because the curvature of the front and back shrouds of an impeller, are different,
the meridional velocity is not uniform and may be higher toward the back shroud.
For a linear variation of the meridional velocity between the front and back shrouds (Figure
8-7), Stepanoff (1993) derived the following equation for theoretical head:
U 2 2
� g
U2Cm2 �
g tan �2
(V2 – V1) 2
��
H t = � � – � 1 + � (8-6)
12 Cm 2
8.9

8.10 CHAPTER EIGHT
FIGURE 8-7 Pressure and velocity distribution for cases shown in Table 8-1. (From
Stepanoff, 1993. Reprinted by permission of Krieger Publishers.)
The term 1 + [(V 2 – V 1) 2 /12 C m 2 ] is greater for the wide impellers encountered in mining
slurry pumps. For slurry pumps, the value of � 1 at the tip diameter of the eye of the
impeller is between 14 and 30 degrees. The value of � 2 at the tip diameter of the vanes is
typically between 25 and 35 degrees. Stepanoff (1993) has indicated that inlet angles as
high as 50 degrees are used on water pumps. This is, however, not the case with slurry
pumps, as prerotation causes tremendous wear of the throat bush.
The vast majority of modern pumps have a discharge angle � 2 smaller than 90 degrees.
They are called impellers and have backward curved vanes. Expellers are often designed
with radial vanes (i.e., � 2 = 90 degrees). Forward vanes with � 2 larger than 90 de-
W 1
C m1
1 1
U1 V1
Inlet velocities at R 1
rotation
W 1
C
m1
1
1
V
U 1
1
R 2
R 1
2
W2
FIGURE 8-8 Ideal velocity profile in an impeller.
U 2
Outlet velocities at R 2
m2
W 2 C
2
V
2
V
C 2
m2
2
2
U
2

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
grees are restricted to very low flow and high-head pumps and to some expellers. Theoretically,
an impeller with forward vanes would give a higher static head rise. Unfortunately,
it is also the largest consumer of power and is considered to be inefficient.
Clay and other slurries can be very viscous. Herbrich (1991) has suggested using discharge
angle � 2 as high as 60 degrees on impellers for very viscous slurries but did not
produce data to support such a suggestion. Stepanoff (1993) recommended the following
design procedures for special pumps. These pumps would be suited to pump viscous liquids,
but their performance may be impaired on water.
1. Use high impeller discharge angles up to 60 degrees to reduce the impeller diameter
necessary to produce the same head and effectively reduce disk friction losses. Consequently,
the impeller channels become shorter and the impeller hydraulic friction is reduced.
2. Eliminate close-clearance wide sealing rings at the impeller eye and provide knifeedge
seals (one or two) similar to those used on blowers. Leakage loss becomes secondary
when pumping viscous liquids.
3. Provide a similar axial seal at the impeller outside diameter to confine the liquid between
the impeller and casing walls. This in turns raises the temperature of the liquid
in the confined space (due to friction) well above the temperature of the remaining liquid
passing through the impeller. Due to the temperature effects, viscosity is artificially
reduced and disk friction losses are trimmed down. In fact, Stepanoff (1993) goes as
far as suggesting injecting a light or heated oil in the confined space to reduce power
loss due to friction.
4. Provide an ample gap (twice the normal) between the casing tongue or cutwater and
the impeller outside diameter. Otherwise, the shrouds of the impeller would produce
head by viscous drag at low capacities, and would decrease the efficiency of pumping.
5. High rotational speed and high specific speed lead to better efficiency and more head
capacity output than low specific speed pumps on viscous liquids.
These recommendations were written with very viscous fluids in mind. Obviously,
points 2 and 3 would not apply to a slurry pump. However, slurry pumps may use pumpout
vanes, which effectively are dynamic seals. These recommendations can be modified
to suit the design of special pumps for viscous slurries. The field of slurry pumps for very
viscous slurries and difficult flotation frothy slurry associated with the oil sands industry
is continuously evolving.
In some cases of pumping oil sand froth, it has been found that injecting 1% of water
or a light oil as a lubricant just at the suction of the pump can improve the efficiency of
the pump.
8-2-3 Slip of Flow Through Impeller Channels
8.11
Due to the curvature of the vane, the flow on the upper surface of a vane is usually faster
than the flow on the lower surface of the vane. If we consider the direction of rotation, the
upper surface is also called the advancing surface or leading surface. The lower surface is
the trailing surface. The pressure being higher on the trailing surface, the fluid leaves tangentially
only at the trailing surface. A certain amount of liquid is attracted by the lower
surface of the following vane and a pattern of flow recirculation develops as shown in
Figure 8-9. To compare this situation with that of an airplane, which many of us have examined,
vortices form behind a flying wing, as air tends to roll from the upper pressure

8.12 CHAPTER EIGHT
R
2 R 1
zone of the lower surface toward the lower pressure zone above the wing. A vortex sheet,
called “horseshoe vortex,” forms behind the airplane wing.
The velocities in a real impeller do not follow the ideal “Euler” impeller pattern, and a
degree of “slip” occurs. The angles of flow and forces deviate from the theoretical values
as shown in Figure 8-10 by a “lag” angle. The slip factor is in fact as the ratio of the measured
absolute velocity to the theoretical Euler absolute velocity at the tip diameter of the
vanes:
� = V2�/V2 (8-7)
Since the average meridional velocity is essentially a function of the ratio of flow rate
to the discharge area at the tip of the impeller, it is not affected by slip. However, a change
in the absolute velocity is accompanied by changes in the relative velocity and of the angles
with respect to the peripheral tangential speed. Various equations have been developed
over the years to evaluate the slip factor. The most famous is Stodola’s formula:
� · sin �2 � = 1 – ���� (8-8)
Z
where Z = number of vanes. Stodola’s formula was originally developed for zero flow, but
has been extensively used for design flows of water pumps even at best efficiency point.
Another equation used to determine slip was developed by Pfeiderer (1961):
a
�
Z
�2 �
60
rotation of impeller
relative recirculation
FIGURE 8-9 Recirculation in pump impellers (after Stepanoff, 1993).
R 2 2
� S
� = 1/� 1 + � 1 + � � (8-9)
theoretical V' 2 measured
W W'
2 2 V2
2
2
U 2
2
2
(with slip)
FIGURE 8-10 Slip of flow in impellers versus ideal velocity profile.

where
S = � R2 R
R dR (8-10)
S is called the static moment and is obtained by graphical integration along the meridional
plane of the vanes. In the special case of a cylindrical vane
S = � R2 R
R dR = |(R 2 2 – R 1 2 )
and the slip factor is
� = 1/�1 + �1 + a �2 2
� ����� (8-11)
2 2 Z 60 1 – (R 1/R 2) In the special case in which R1/R2 is smaller than 0.5, the slip does not increase anymore,
and a ratio R1/R2 = 0.5 should be assumed.
The magnitude “a” depends on the design of the casing. Pfeiderer (1961) established
the following values for the coefficient “a”:
Volute a = 0.65–0.85
Vaned diffuser a = 0.60
Vaneless diffuser a = 0.85 – 1.0
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.13
Most slurry pumps use a volute (Figure 8-11). Vaned diffusers are used in certain
mine dewatering pumps.
Defining the hydraulic efficiency as �H, the head developed by the pump is expressed
as:
H = ��HU2V2 (8-12)
Equation (8-12) establishes the effect of the casing and the impeller on the head developed
at the so-called best efficiency point. Because of the rather simplistic Stodala equa-
volute casing diffuser vane casing
FIGURE 8-11 Volute and vaned diffusers.

8.14 CHAPTER EIGHT
TABLE 8-2 Test Data from Herbich (1991)
Velocity Theoretical Measured
Radial 4.21 ft/s (1.3 m/s) 15.6 ft/s (4.8 m/s)
Tangential 55.80 ft/s (17.0 m/s) 39.6 ft/s (5.2 m/s)
tion (8-8), it is sometimes assumed that the impeller is the main contributor to head. The
equation for the head is also expressed in terms of the discharge angle from the vanes, the
slip factor, and the hydraulic efficiency as:
H = ��H �1 – � Cm2 · cot
�2 ��
(8-13)
U2
Herbich (1991) measured extensively the lag angle and deviation from theoretical angles
in the case of the Essayon dredge pump and reported two cases of impeller tip vane discharge
angle �2 (Table 8-2). In the first case, the vane was designed with a physical tip angle
at the vanes of 22.5°. This would have been theoretically the angle for the relative speed
W. However, test data measured an average angle of 30.5°. In the second case, the vane had
a discharge angle of 35° but test data indicated that the relative velocity was effectively inclined
at an average angle of 12°. In fact there is no definite value. In the case of the first
impeller with a vane angle of 22.5°, at a flow rate of 63 L/s (1000 gpm) the flow between
the channels was measured to have streams inclined between 61° on the lower surface and
25° on the forward surface with various values between 21 47°. A different pattern was noticed
at 38 L/s (600 gpm). The distortion of the flow is therefore a function of the ratio of
flow rate to normalized flow (at best efficiency point). When the experimental angle is
higher than the theoretical, Herbich pointed out that it would mean that the particles tend to
avoid contact, thus minimizing the possibility of scour. On the other hand, if the measured
angle is less than the theoretical, the solids will impact the vanes and cause wear.
Because it is difficult to measure slip, an experimental head coefficient is defined as:
�SI = � 2g
U
HBEP � (8-14)
2 U 2
For some historical reasons, U.S. books drop the numerator 2:
2 2
�
g
�US = � (8-15)
2 U 2
The reader must therefore be careful when comparing pumps manufactured in North
America with those manufactured in Europe.
8-2-4 Specific Speed
gH BEP
The steepness of the curve between the best efficiency capacity and the shut-off point of
the pump depends on the geometrical design of impeller and casing. With so many different
designs of pumps, engineers have used nondimensional specific speeds and other parameters.
In the International System of Units, the specific speed is defined as:
N · �Q�
Nq = � (8-16)
3/4 H

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
where
N = rotational speed in rev/min
Q = capacity in cubic meters per second, at best efficiency capacity
H = differential head in meters at best efficiency capacity
The specific speed in the United States is defined as:
N · �Q�
NUS = � (8-17)
3/4 H
where
N = rotational speed in rev/min
Q = capacity in U.S. gallons per minute, at best efficiency capacity
H = differential head in feet at best efficiency capacity
Some books include the acceleration of gravity g or 32.2 ft/s in the denominator for
the sake of consistency, but for historical reasons Equation 8-17 is used.
Another term sometimes used in international books is the characteristic number:
� · �Q�
Ks = � (8-18)
3/4 [gH]
Most slurry pumps operate at a specific speed smaller than 2000 in U.S. units or 39 in
SI units. In this range, the tip diameter of the impeller may be between 2 to 3.5 folds of
the suction diameter. The shut-off head is then between 150% and 110% of the best efficiency
point head at the same speed (Figure 8-12).
Addie and Helmly (1989) measured the head coefficient (as defined in the United
States) and the efficiency of a number of slurry and dredge pumps. Their results are
shown in Figures 8-13 and 8-14. They pointed out that the slurry and dredge pumps were
on the average between 5% and 9% less efficient than their water counterparts.
Example 8-1
A slurry pump is to be designed for a head at best efficiency of 150 ft at a flow rate of
1200 gpm. Assuming a head coefficient of 0.5 (by U.S. definition), determine the diameter
and the speed of rotation if the specific speed is 1100 (in U.S. units).
Solution in USCS Units
From Equation 8-15:
gHBEP 32.2 × 150
�US = � = 0.5 = ��
2 2 U 2
U 2
U2 = 98.3 ft/s
From Equation 8-17, the specific speed in the United States is defined as:
Ns = N · Q1/2 /H 3/4 = 1100 = N · 12001/2 /1503/4 8.15
N = 889 rpm = 93.1 rad/sec
Since U = R�, then R = 98.3/93.1 = 1.06 ft. The impeller diameter is therefore 2.11 ft or
25.3 inch (643 mm).
Every manufacturer has their proprietary design criteria, and for a given size some
manufacturers may have an impeller design that pumps more than others. In the case of

8.16
FIGURE 8-12 Shape of impeller versus specific speed in USCS units. [From I. Karasik et al. (Eds.), Pump Handbook, reprinted by permission from
McGraw-Hill.]

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.17
FIGURE 8-13 Head coefficient versus specific speed from Addie and Helmly (1989) (reproduced
by permission of Central Dredging Association, Delft, Netherlands). This plot is somewhat
confusing as it uses the U.S. definition of the head coefficient (as per Equation 8-15)
against the specific speed in SI units. The reader should multiply the head coefficient by a factor
of 2 to use the SI definition of head coefficient as per Equation 8-14.
slurry pumps, attention must be paid to the wear life of the pump. Too little flow in a large
pump leads to excessive recirculation, and too much flow would cause rapid wear. The
relationship between the volute shape and the impeller plays a major role, too. These parameters
are refined through detailed engineering and field-testing. A good starting point
for the design of mill discharge pumps is shown in Tables 8-3 and 8-4. These are realistic
values that mills expect from pumps.
The next step is to define the steepness of the curve. Slurry pumps are designed to be
forgiving as processes too often change. Very steep curves are not encouraged, but flat
curves do create overloading problems to the drivers. A shut-off head in the range of
125% to 135% of the best efficiency head is recommended. The slurry pump design engineer
should then establish what is often referred to as a 5-points curve, as shown in Tables
8-5 and 8-6.and Figure 8-15.
As early as 1938, Anderson developed a concept of the ratio of the area of flows between
the vanes of the impeller and the throat area (Figure 8-10) that is basic to the performance
of the pump. His methodology is called the “area ratio” (Figure 8-16). Worster
(1963) demonstrated this to be correct by mathematical derivation. Anderson (1977,
1980, 1984) extended his analysis by statistical analysis of a large number of water pumps
and turbines. Unfortunately, no similar work has been done on slurry pumps and because
slurry impellers are fairly wider than water pump impellers to allow the passage of rocks
and large particles, the Worster curves do not apply well to the design of solids-handling
pumps.
Not all applications of pump slurries require wide impellers. In fact in the last 20
years, grinding circuits have greatly evolved to the point that very fine ores are pumped.
For these applications, narrower and more efficient impellers should designed.

8.18 CHAPTER EIGHT
FIGURE 8-14 Efficiency of large dredge pumps versus specific speed (in SI units). (From Addie
and Helmly, 1989. Reproduced by permission of Central Dredging Association, Delft, Netherlands)
8-2-5 Net Positive Suction Head and Cavitation
When the pressure on the suction flange of the pump is insufficient, the pump starts to
cavitate and becomes very noisy. The net positive suction head (NPSH; see Figure 8-17)
is the absolute head above the vapor pressure at the suction flange of the pump:
P 2
e – PD – PV V e
NPSHA = �� + Z1 – Z2 � (8-19)
�g
�g
where
Pe = pressure at the surface of the liquid in absolute terms on the suction side
PD = pressure losses between the surface of the liquid and the pump, due to friction,
valves, etc.
PV = vapor pressure
Z1 = geodetic elevation of liquid surface above the centerline of the pump impeller
Ze= geodetic elevation of the centerline of the pump impeller
Ve = suction speed at the eye of the impeller

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
TABLE 8-3 Recommendations for Design of Rubber-Lined Mill Discharge Pumps
TABLE 8-4 Recommendations for Design of Metal-Lined or Hard Metal Mill
Discharge Pumps
8.19
Size
suction to
discharge, inch
Flow
_____________
Head
___________ Efficiency
Suction speed
____________
Discharge speed
_____________
(mm/mm) L/s US gpm m ft % m/s ft/s m/s ft/s
8 × 6
200/150
130 2061 30 98.5 70 4.2 13.7 7.2 23.5
10 × 8
250 × 200
220 3487 30 98.5 74 4.5 14.8 6.7 22.1
12 × 10
300 × 250
310 4915 30 98.5 76 4.4 14.4 6.12 20.1
14 × 12
350/300
425 6737 30 98.5 79 4.4 14.5 5.86 19.2
16 × 14
400 × 300
560 8877 30 98.5 81 4.3 14.1 5.64 18.5
18 × 16
450 × 400
685 10859 30 98.5 83 4.3 14.1 5.45 17.9
20 × 18
500/450
875 13870 30 98.5 84 4.3 14.2 5.33 17.5
From Abulnaga (2001). Courtesy of Mazdak International Inc.
Size
suction to
Flow
discharge, inch _____________
Head
___________ Efficiency
Suction speed
____________
Discharge speed
_____________
(mm/mm) L/s US gpm m ft % m/s ft/s m/s ft/s
8 × 6
200/150
176 2797 55 180 70 5.7 18.6 9.7 32
10 × 8
250 × 200
298 4732 55 180 74 6.1 20 9.1 30
12 × 10
300/250
421 6670 55 180 76 6 19.5 8.3 27
14 × 12
350/300
577 9143 55 180 79 6 19.7 8 27.3
16 × 14
400/300
760 12047 55 180 81 5.8 19.3 8 25.1
18 × 16
450/400
924 14647 55 180 83 5.8 19.3 7.4 24.1
20 × 18
500/450
1188 18823 55 180 84 5.8 19.3 7.2 23.8
From Abulnaga (2001). Courtesy of Mazdak International Inc.

8.20 CHAPTER EIGHT
TABLE 8-5 Preliminary Range for Efficiency versus Flow (L/s units) For Mill Discharge
Pump—Rubber-Lined Version
Pump Size (suction/discharge)
8 × 6 in 10 × 8 in 12 × 10 in 14 × 12 in 16 × 14 in 18 × 16 in 20 × 18 in
Ratio of
flows,
Ratio of
efficiency,
200 × 150 250 × 200 300 × 250 350 × 300 400 × 350 450 × 400 500 × 450
mm mm mm mm mm mm mm
Q/QN �/�BEP Flow in L/s
0.25 0.5 32.5 55 77.5 106 140 171 219
0.5 0.8 65 110 155 213 280 342 438
0.75 0.95 97.5 165 232.5 319 420 523 656
1.00 1.0 130 220 310 425 560 684 875
1.15 0.97 150 253 356.5 489 644 787 1006
From Abulnaga (2001). Courtesy of Mazdak International Inc.
Each pump has a minimum required NPSH that is established through testing. It is defined
as the required NPSH or NPSHR. The suction-specific speed is defined at the best
efficiency point as:
N · �Q�
NSS = ��
(8-19)
NPSHR 3/4
The value of NPSH is established at the point where the suction conditions at best efficiency
flow suffer a 3% drop of total dynamic head.
Solids present in slurry do not contribute to the vapor pressure, but they contribute to
the density of the mixture as well as to the friction or pressure losses on the suction. This
could be confusing to the inexperienced engineer who has to handle water vapor pressure
as well as slurry density. One approach is to calculate the pressure on the suction in units
of pressure and then to convert back into units of length.
TABLE 8-6 Preliminary Range for Efficiency versus Flow (L/s units), Metal-Lined or
Hard Metal Mill Discharge Pumps
Pump Size (suction/discharge)
8 × 6 in 10 × 8 in 12 × 10 in 14 × 12 in 16 × 14 in 18 × 16 in 20 × 18 in
Ratio of
flows,
Ratio of
efficiency,
200 × 150 250 × 200 300 × 250 350 × 300 400 × 350 450 × 400 500 × 450
mm mm mm mm mm mm mm
Q/QN �/�BEP Flow in L/s
0.25 0.5 44 74.5 105 144 190 231 297
0.5 0.8 88 149 210 288.5 280 462 594
0.75 0.95 132 223.5 316 315.8 420 693 891
1.00 1.0 176 298 421 577 760 924 1188
1.15 0.97 202 342.7 484 664 874 1063 1366
From Abulnaga (2001). Courtesy of Mazdak International Inc.

H/H N
1.2
0.0
0.0
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
HEAD
1.0 1.0
0.8 0.8
0.6 0.6
EFFICIENCY
0.4 0.4
0.2 0.2
0.5 1.0
0.0
1.5
Q/Q
N
It is often recommended that the available NPSH be at least 0.9 m or approximately 3
ft higher than the required NPSH as shown on the pump curve.
Example 8-2
Slurry with a specific gravity of 1.48 is to be pumped from a pond 3 m lower than the centerline
of the impeller. The pond is situated at a high altitude. The atmospheric pressure is
85 kPa. The friction losses have been determined to be 1.5 m. The vapor pressure of water
is 4.24 kPa. The slurry enters the pump at a velocity of 3.5 m/s. Determine the available
NPSH.
Solution
Pressure due to friction losses is:
�gH = 1480 · 9.81 · 1.5 = 21,778 Pa
The geodetic elevation of the centerline of the pump impeller is 3 m higher than the liquid;
this results in a negative pressure or
�g�Z = 1480 · 9.81 · (–3) = –43,556 Pa
Dynamic head losses due to a velocity of 3.5 m/s are:
1480 · 3.52 /2 = 9065 Pa
Net positive pressure is:
85,000 – 43,556 – 21,778 – 9065 – 4240 = 24,491 Pa
Converting back into head of water:
24,491/(9.81 · 1000) = 2.496 m of water
N
8.21
FIGURE 8-15 Normalized curves of head and efficiency versus values at the best efficiency
point.

8.22 CHAPTER EIGHT
FIGURE 8-16 The area ratio curves for water pumps. No similar curves have been published
for slurry or dredge pumps. (From Worster, 1963. Reproduced by permission of the Institution
of Mechanical Engineers, UK.
This is very low, and since the engineer must avoid cavitations, he or she may consider
the use of a submersible slurry pump or a vertical cantilever pump instead of a horizontal
pump on the shore.
The NPSH can be expressed as the function of suction speed and the eye tip speed at
the suction diameter (Turton, 1994):
2 2 0.9 C m + 0.115 U 1
NPSH = ��
(8-20)
9.81

8.23
Z S
Liquid at
vapor pressure P v
Z
E
Absolute Atmospheric Pressure
P
A
Pressurized gas at surface at
gauge pressure P
B
H
1
FIGURE 8-17 Concept of net positive suction head.
P = P + P
E A B
Pressure due to
friction losses
P
D

8.24 CHAPTER EIGHT
Example 8-3
A pump impeller rotates at 500 rpm to pump 65 L/s through a suction diameter of 200
mm. Using Equation 8-20, determine the required NPSH.
Solution
The velocity Cm is determined by dividing the flow rate by the suction area:
Cm = 0.065/[0.25 · � · 0.22 ] = 2.07 m/s
U = 2�RN/60 = 2 · � · 0.1 · 500/60 = 5.24 m/s
0.9 · 2.07
NPSH = = 0.715 m
2 + 0.115 · 5.242 ���
9.81
In reality, NPSH depends on many other factors, particularly clearances at the impeller
eye, prerotation, the use of inducers, etc. Many empirical studies tend to support
that a low NPSH impeller should have a vane entry angle of 14° to 15°.
A cavitations parameter � is defined as the ratio of required NPSH to the pump total
dynamic head at the best efficiency point at the given speed:
NPSH
� = � (8-21)
TDH
Addie and Helmly (1989) measured the cavitations parameter against specific speed
for a number of dredging pumps. Their work is represented in Figure 8-18. Tables 8-7 and
8-8 also show certain calculations for the design of mill discharge pumps.
FIGURE 8-18 Cavitation factor versus specific speed (in metric units) for slurry and dredge
pumps. (From Addie and Helmly, 1989. Reproduced by permission of Central Dredging Association,
Delft, Netherlands)

8-3 THE PUMP CASING
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.25
TABLE 8-7 Recommendations for Impeller Diameter, Speed, Specific Speed Number,
and Cavitations Parameter for Rubber-Lined Mill Discharge Pumps (U.S. Units)
Head Sigma
Vane d2, d2/dS, Speed, Flow, Efficiency, Head, Specific factor, cavitation
Model inch tip/suction rpm US gpm % ft speed �US factor
8 × 6 in 20.87 3.48 816 2061 70 98.5 1186 0.14 0.14
10 × 8 in 26.77 3.35 667 3487 74 98.5 1261 0.13 0.16
12 × 10 in 31.10 3.11 575 4915 76 98.5 1290 0.13 0.16
14 × 12 in 35.04 3.92 510 6763 79 98.5 1340 0.13 0.17
16 × 14 in 39.37 2.81 450 8877 81 98.5 1357 0.133 0.15
18 × 16 in 45.28 2.8 390 10859 83 98.5 1300 0.133 0.16
20 × 18 in 55.12 2.76 340 13870 84 98.5 1281 0.119 0.16
From Abulnaga (2001). Courtesy of Mazdak International Inc.
TABLE 8-8 Recommendations for Impeller Diameter, Speed, Specific Speed Number,
and Cavitations Parameter Metal-Lined or Hard Metal Mill Discharge Pumps (U.S.
Units)
Head Sigma
Vane d2, d2/dS, Speed, Flow, Efficiency, Head, Specific factor, cavitation
Model inch tip/suction rpm US gpm % ft speed �US factor
8 × 6 in 20.87 3.48 1005 2790 70 180 1186 0.173 0.14
10 × 8 in 26.77 3.35 903 4721 74 180 1261 0.13 0.16
12 × 10 in 31.10 3.11 779 6654 76 180 1290 0.13 0.16
14 × 12 in 35.04 3.92 691 9121 79 180 1340 0.102 0.17
16 × 14 in 39.37 2.81 609 12018 81 180 1357 0.132 0.15
18 × 16 in 45.28 2.8 528 14701 83 180 1300 0.133 0.16
20 × 18 in 55.12 2.76 460 18779 84 180 1281 0.12 0.16
From Abulnaga (2001). Courtesy of Mazdak International Inc.
The pump casing of a slurry pump often takes the shape of a volute. The best hydraulic
design calls for a constant momentum design or a linear increase of the cross-sectional
area from the tongue to the throat (Figure 8-19). In reality, the profile of the volute is often
simplified to two semicircles. The idea is that hard metals are difficult to cast, and if
the shape can be simplified, the casting will flow better during solidification.
Rc in Figure 8-19 refers to the cutwater radius. The difference between Rc and R2 is effectively
the gap at the cutwater. It must be large enough to accommodate the passage of
coarse particles or rocks.
The head developed by the pump at shut-off is the sum of the head due to the rotation
of the impeller and shape of the volute. Turton (1994) summarized the research of Frost
and Nilsen (1991), who concluded that the shut-off head was insensitive to the number of
blades, the blade outlet geometry, and the channel width of the impeller. They determined
that:
HSV = HIMP SV + HVOL SV
(8-22)

8.26 CHAPTER EIGHT
FIGURE 8-19 Parameters for the calculations of the shut-off head of a water pump used in
Equation 8-24. (From Frost and Nilsen, 1991. Reproduced by permission from the Institution
of Mechanical Engineers, UK.)
where
HIMP SV = shut-off head due to the impeller
HVOL SV = shut-off head due to the volute
HSV = total shut-off head
HIMP SV = [1 – (Rs/R2) 2 2 2 R2� � ] (8-23)
2g
and
2
HVOL SV = � ��R2 2 2
R2� R4 – R2
MD ln(R4/R2) – 2RMD(R4 – R2) + �
� RMD – R 2
� 2
/g (8-24)
Equations (8-23) and (8-24) were derived for water pumps, and it is recommended to
confirm the results when designing a new family of slurry pumps.
Referring to Figure 8-20, the width of the volute is defined by two components, X v in
the x-direction and Y v in the y direction, when the volute is in a position for vertical top
discharge. The magnitude of these two components depends on the clearance at the cutwater,
the throat area, the tip diameter of the impeller, and the discharge diameter of the
pump. These are refined through experimental testing and hydraulic analysis. A good
starting point (or rule of thumb) for the design engineer is to use the shroud diameter of
the impeller d t as a reference and to establish
X V = K xd t 1.3 < K x < 1.4 (8-25)
Y V = K yd t 1.2 < K y < 1.3 (8-26)

cutwater
t L
(liner thickness)
R 4
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
R 3
R 2
R 3
X V
R
c
R M
R
4
discharge
8.27
impeller
tip diameter
throat area
Having established a profile of the volute, the thickness of the liner and the thickness
of the casing are then added before locating the bolts for lined casings.
There is no definite rule of thumb for the thickness of rubber or metal liners. The
thickness of the liner is established by the manufacturer on the basis on their experience
with the application. Table 8-9 recommends a liner thickness in the range of 4% to 6% of
the impeller diameter.
Having sized the thickness of the liner, a parameter D for the volute is defined using
the width XV as
D = XV + 2tL (8-26)
For a single-stage pump designed for a pressure of 1035 kPa (150 psi), with a ribbed
casing, the casing thickness is established as
tc � D/41 (8-27)
Equation 8-27 should be complemented by a full finite element analysis, as the ribs
have to be placed correctly. Modern computers are very useful for checking on the size of
the ribs. Burgess and Abulnaga (1991) have recommended the use of the equivalent thickness
approach. It consists of calculating the second moment of area of the ribs and implementing
them in a plate model for the casing. An alternative but much more tedious approach
is to use brick elements. Since 1991, the science of minicomputers has advanced
greatly and it is now possible to implement very sophisticated three-dimensional models.
t c
suction diameter
Y v
(casing thickness)
FIGURE 8-20 Volute shape of a slurry pump simplified for the sake of manufacturing and
casting of hard metal casing or liners to a minimum number of partial circles.

8.28 CHAPTER EIGHT
TABLE 8-9 Recommended Dimensions for a Single Stage Mill Discharge Pump (metric size
example)
Size (mm) 200 × 150 250 × 200 300 × 250 350 × 300 400 × 350 450 × 400 500 × 450
Impeller d2 530 680 790 890 1000 1150 1400
Shroud diameter dt 560 720 830 930 1050 1200 1500
Cutwater diameter dC 657 843 980 1104 1240 1426 1775
Cutwater gap
(dC – dt)/2 49 62 75 87 95 113 138
XV = 1.3 dt 728 936 1073 1209 1352 1560 1950
YV = 1.25 dt 700 900 1031 1163 1300 1500 1875
Liner thickness tL 34 38 41 45 48 51 55
D = XV + 2 · tL 796 1012 1155 1299 1448 1662 2060
Pressure area Ap (m2 )* 0.503 0.82 1.064 1.363 1.70 2.24 3.87
Working pressure kPa 1035 1035 1035 1035 1035 1035 1035
Design pressure kPa 1380 1380 1380 1380 1380 1380 1380
F = Ap · Pdesign (kN) 694 1132 1468 1881 2348 3105 5341
D/t 40 40.7 41.17 40.42 41 41.07 41.2
Casing thickness
tc (with ribs)
20 24 28 31 34 39 50
Number of bolts 12 12 12 12 12 12 12
Load/bolt kN 58 94 122 157 196 259 445
Bolt area mm2 ** 347 563 731 940 1174 1551 2662
Bolt diameter mm 21 27 31 35 39 45 58
Bolt M24 M30 M36 M40 M46 M50 M62
*A p = 0.9[X V + t L][Y V + t L] · 10 –6 .
**Allowed stress on bolt 166 Mpa.
Note: these calculations are preliminary and must be confirmed by finite element analysis. They are for
single stage, 1.38 MPa rating with ductile iron casing.
Having established the thickness of the casing, it is important to establish the size and
number of bolts for radial split casings. An equivalent pressure area is then established using
the following formula:
Ap = 0.9[XV + tL][YV + tL] (8-28)
The design pressure PD is usually established as the maximum operating pressure times a
factor of 1.25. It is then multiplied by Ap to obtain the total force on the casing Fp: Fp = PD · Ap (8-29)
The size and number of bolts is then established using the yield stress of the bolts.
Detailed finite element analysis of multistage tailings pumps has demonstrated that the
maximum stress occurs at the cutwater. Some of the very high pressure pumps feature a
special bolt at the cutwater that is larger than the other bolts (Burgess and Abulnaga,
1991).
Table 8-9 presents some recommendation for average dimensions of a single-stage
mill discharge pump designed for a maximum operating pressure of 1035 kPa (150 psi).
In this example, it was arbitrarily assumed that the number of bolts is 12, to give the reader
an idea of the effect of loads on size of bolts. Obviously, on the larger pumps, the de-

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
TABLE 8-10 Recommended Dimensions for a Single Stage Mill Discharge Pump
(USCS units size)
8.29
Size (in) 8 × 6 10 × 8 12 × 10 14 × 12 16 × 14 18 × 16 20 × 18
Impeller d2 21� 26.8� 31� 35� 39.4� 45.3� 55�
Shroud diameter dt 22� 28.3� 32.7� 36.6� 41.3� 47.25� 59�
Cutwater diameter dC 25.9� 33.2� 38.6� 43.5� 48.8� 56.1� 69.9�
Cutwater gap (dC – dt)/2 1.95� 2.45� 2.95� 3.45� 3.77� 4.43� 5.45�
XV = 1.3 dt 28.6� 36.8� 42.5� 47.6� 53.7� 61.43� 76.7�
YV = 1.25 dt 27.5� 35.4� 40.9� 45.8� 51.6� 59� 73.8�
Liner thickness tL 1.34� 1.5� 1.6� 1.77� 1.89� 2� 2.16�
D = XV + 2 · tL 31.3� 39.8� 45.7� 51.1� 57.5� 65.43� 81�
Pressure area Ap (in2 )* 777 1272 1687 2114 2973 3482 5990
Working pressure psi 150 150 150 150 150 150 150
Design pressure psi 200 200 200 200 200 200 200
F = Ap*Pdesign (lbf) 155400 254389 337400 422800 594600 696400 1198000
D/t 40 40.7 41.17 40.42 41 41.07 41.2
Casing thickness tc (with ribs)
0.78� 0.95� 1.1� 1.22� 1.34� 1.54� 2�
Number of bolts 12 12 12 12 12 12 12
Load/bolt lbf 12,950 21,199 28,117 35,233 49,550 58,033 99,833
Bolt area in2 ** 0.539 0.883� 1.17 1.47 2.06 2.42 4.16
Min Bolt diameter 0.83� 1.06 1.22� 1.38� 1.62� 1.75� 2.3�
Bolt size (in) 7/8� 11/4 1.375 1.5� 1.75� 2� 2.5�
*A p = 0.9[X V + t L][Y V + t L].
**Allowed stress on bolt 24,000 psi.
Note: these calculations are preliminary and must be confirmed by finite element analysis. They
are for single stage, 200 psi rating with ductile iron casing.
signer may increase the number of bolts to keep them within a reasonable size. Table 8-10
is a similar table using USCS units.
The casing pump takes the shape of the volute (Figure 8-21). In addition to the volute
liner, a front wear plate or throatbush (Figure 8-22) is bolted to the casing.
Compared to a water pump, a slurry pump has a much wider gap at the cutwater with respect
to the impeller. This is due to the fact that the slurry pump must move solids that
should not jam at the cutwater. In certain cases, oversized pumps were sold to mines and recirculation
problems developed with excessive wear. Manufacturers have gone back over
their designs and extended the cutwater to cut down the flow by creating a sort of throttling
effect. They call this sort of volute a low- flow volute (Figure 8-23). The advantage of this
approach is that the pattern of the liner can be modified without having to replace the casing
of the pump. Installing a so-called “reduced eye” impeller may also complement this
approach. A “reduced eye” impeller is an impeller with a suction diameter smaller than the
suction diameter of the casing. This provides a way to throttle the suction. The throatbush
of the pump must also be modified to accommodate the reduced eye of the impeller.
In the case of water pumps, the emphasis is to operate as close as possible to the best
efficiency point, where losses are at a minimum. In the case of slurry pumps, the situation
is more complex, as the best efficiency point does not necessarily coincide with the minimum
wear point. Certain designs of slurry pumps do point to minimum wear at 80% of

8.30 CHAPTER EIGHT
FIGURE 8-21 Casting for the casing and cover plate of a vertical sump pump—clearly
showing the volute shape—with an integral cast elbow at the discharge. (Courtesy of Mazdak
International Inc.)
the best efficiency point. This point is too often overlooked when sizing pumps. The consultant
engineer is encouraged to discuss this point with the manufacturer. Certain manufacturers
of pumps have in-house computational fluid dynamics programs to do a wear
performance analysis. Unfortunately, too often these give a two-dimensional profile of
velocity in the volute, but insufficient data about vortices in the corners where gouging
wear may develop.
FIGURE 8-22 Throatbush or suction liner fixed to the pump front casing plate of a horizontal
pump. The casing shape indicates the volute shape of the liner.

solid
passageway
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
original cutwater
extended cutwater
for "low flow" volute
throat
8.31
modified throatbush
reduced
eye impeller
FIGURE 8-23 Restraining the flow by extending the cutwater and modifying the throat of
the volute or liner, or decreasing the suction diameter of the impeller are methods for correcting
oversized pumps.
Example 8-4
A new mine requires a very large pump to handle 1514 L/s (24,000 US gpm), at a total
dynamic head of 43 m (141 ft) and a specific gravity of the mixture of 1.5. Establish some
preliminary parameters of design for the casing prior to conducting a finite element analysis.
The head ratio is assumed to be 0.9 (see Chapter 9). Assume that this is a pump designed
for single-stage operation with a design pressure of 1.4 MPa (200 psi).
Solution in SI Units
The equivalent water head is 43 m/0.9 = 47.8 m. This is therefore an application for an
all-metal pump. Table 8-5 suggests an average suction speed of 6 m/s and a discharge
speed of 9 m/s at a discharge head of 55 m. Since the pump will operate at 47.8 m, the
ratio of tip speeds is �(4�7�.8�/5�5�)� = �0�.8�6�8� = 0.932. The pump will operate at 0.932 of
the maximum allowed speed of 38 m/s for all metal impellers, or 35.42 m/s (or 116
ft/s):
�SI = = = 0.187
The flow suction speed is established as the ratio of tip speed. This ratio is 0.932, and
using the suggested maximum speed of 6 m/s for metal impellers, the suction speed Vs at
the flow rate of 1514 L/s is then 0.932 × 6 = 5.59 m/s.
The suction area = Q/Vs = 1.514/5.59 = 0.271 m2 .
The corresponding inner diameter is 0.587 m or 23.12�.
The discharge speed Vd is 0.932 × 9 = 8.4 m/s.
The discharge area = Q/Vd = 1.514/8.4 = 0.18 m2 gHBEP 9.81 · 47.8
� ��
2 2
2U 2 2 · 35.42
.
The discharge inner diameter is 0.478 m or 18.8�.
These values of suction and discharge diameter will be added to the liner thickness
and to the casing thickness before calculating suction and discharge flanges and their corresponding
bolt circles.
Using Table 8-8 as a reference, the tip-to-suction diameter of the impeller ratio is assumed
to be 2.75, or the tip diameter of the impeller becomes 0.587 × 2.75 = 1.615 m.

8.32 CHAPTER EIGHT
Since U = 35.42 m/s,
� = U/R = 35.42/1.615 = 21.93 rad/s
N = 21.93 · 60/(2 · �) = 209.4 rpm
Let us round it to 210 rpm. From Equation 8-16, the specific speed (in the International
System of Units) is
N · �Q� 210 · �1�.5�1�4�
Nq = � = �� = 14.22
3/4 H
47.8 3/4
Table 8-9 recommends that the shroud diameter dt be about 6% larger than the impeller
vane diameter dV or 1.06 · 1.615 = 1.712 m.
The next step is to establish a preliminary layout of the volute using Equations 8-25
and 8-26. It is assumed that
Kx = 1.35
or
XV = 1.35 · 1.71 = 2.31 m
and
Ky = 1.25
or
XV = 1.25 · 1.71 = 2.14 m
Table 8-9 recommends a liner thickness in the range of 4% to 6% of the impeller
shroud diameter:
tL = 0.04 · 1.712 = 0.0685 m
Let us assume 69 mm. Having sized the thickness of the liner, a parameter D defined in
Equation 8-26 is:
D = 2.31 + 2 · 0.069 = 2.45 m
For a well-ribbed casing, the thickness of the casing tC in a preliminary stage of analysis
is D/40 or 2450/40 = 63.5 mm; let us assume 64 mm. The outer diameter of the suction
nozzle is therefore
587 mm + 2 · (69 + 64) = 853 mm or 33.5�
This suggests further iteration or the installation of a companion flange to 900 mm for European
sizes of pipes or 36� suction pipes for U.S. sizes of pipes.
The outer diameter of the discharge nozzle is therefore
478 mm + 2 · (69 + 64) = 744 mm or 29.29�
These calculations suggest that the pump is effectively a pump with a discharge flange
of 750 mm for metric pipe sizes or 30� for U.S. sizes of pumps. The equivalent pressure
area Ap is then established using Equation 8-28:
Ap = 0.9[XV + tL][YV + tL] = 0.9[2.31 + 0.069][2.14 + 0.069] = 4.13 m2 At a design pressure of 1.4 MPa, the total force that the bolts must retain is:
Fp = Ap · 1.4 MPa = 4.13 · 1.4 = 5.78 MN

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
Since this is a fairly large casing, the design engineer decides to try 24 bolts around the
casing. Each bolt will retain 5.78 MN/24 = 0.241 MN or 241 kN, assuming an allowed
stress on bolt of the order of 166 Mpa. The cross-sectional area of the bolt at the minimum
thread diameter is 0.241/166 = 0.00145 m2 or a diameter of 42 mm. 20 M48 bolts
are therefore recommended.
Solution in USCS Units
The equivalent water head is 141 ft/0.9 = 156.8 ft of water. This is therefore an application
for an all-metal pump. Table 8-5 suggests an average suction speed of 19.7 ft/s and a
discharge speed of 29.5 ft/s at a discharge head of 180.5 ft. Since the pump will operate at
47.8 m, the ratio of tip speeds is �(1�5�6�.8�/1�8�0�.5�) = �0�.8�6�8� = 0.932. The pump will operate
at 0.932 of the maximum allowed speed of 124.67 ft/sec for all metal impellers, or 116
ft/s:
gHBEP 32.2 · 156.8
�US = � = �� = 0.375
2 2
U 2 116
The flow suction speed is established as the ratio of tip speed. This ratio is 0.932, and
using the suggested maximum speed of 19.7 ft/s for metal impellers, the suction speed Vs at the flow rate of 24,000 US gpm (53.47 ft3 /sec) is then 0.932 × 19.7 = 18.36 ft/s.
The suction area = Q/Vs = 53.47 ft3 /18.36 = 2.912 ft2 .
The corresponding inner diameter is 1.926 ft or 23.12�.
The discharge speed Vd is 0.932 × 29.5 ft/s = 27.5 ft/sec.
The discharge area = Q/Vd = 53.47/27.5 = 1.944 ft/sec2 .
The discharge inner diameter is 1.573 ft or 18.9�.
These values of suction and discharge diameter will be added to the liner thickness
and to the casing thickness before calculating suction and discharge flanges and their corresponding
bolt circles.
Using Table 8-8 as a reference, the tip-to-suction diameter of the impeller ratio is assumed
to be 2.75, or the tip diameter of the impeller becomes 23.12� × 2.75 = 63.6 in or
5.3 ft.
Since U = 116 ft/s,
� = U/R = 116/5.3 = 21.9 rad/s
N = 21.9 · 60/(2 · �) = 209.4 rpm
Let us round it to 210 rpm. From Equation 8-16, the specific speed (In the International
System of Units) is
N · �Q� 210 · �2�4�0�0�0�
NUS = � = �� = 734
3/4 H
156.8 3/4
8.33
Table 8-9 recommends that the shroud diameter dt be about 6% larger than the impeller
vane diameter dV or 1.06 · 63.6� = 67.42�.
The next step is to establish a preliminary layout of the volute using Equations 8-25
and 8-26. It is assumed that
Kx = 1.35
or
Xv = 1.35 · 67.42� = 91�
and
Ky = 1.25

8.34 CHAPTER EIGHT
or
Xv = 1.25 · 67.42� = 84.3�
Table 8-9 recommends a liner thickness in the range of 4% to 6% of the impeller
shroud diameter:
tL = 0.04 · 67.42� = 2.69�
Let us assume 2.7�. Having sized the thickness of the liner, a parameter D defined in
Equation 8-26:
D = 91� + 2 · 2.7� = 96.4�
For a well-ribbed casing, the thickness of the casing tC in a preliminary stage of analysis
is D/40 or 96.4�/40 = 2.41�. The outer diameter of the suction nozzle is therefore
23.12� + 2 · (2.7� + 2.41�) = 33.34�
This suggests further iteration or the installation of a companion flange to 36� suction
pipes for U.S. sizes.
The outer diameter of the discharge nozzle is therefore
18.9� + 2 · (2.7� + 2.41�) = 29.12�
These calculations suggest that the pump is effectively a pump with a discharge flange
of 30� for U.S. sizes of pipes. The equivalent pressure area Ap is then established using
Equation 8-28:
Ap = 0.9[XV + tL][YV + tL] = 0.9[91 + 2.7][84.4 + 2.7] = 7345.14 in2 At a design pressure of 200 psi, the total force that the bolts must retain is:
Fp = Ap · 1.4 MPa = 7345.14 · 200 = 1,469,028 lbf
Since this is a fairly large casing, the design engineer decides to try 24 bolts around the
casing. Each bolt will retain 1,469,028 lbf/24 = 61,210 lbf, assuming an allowed stress on
bolt of the order of 24,000 psi. The cross-sectional area of the bolt at the minimum thread
diameter is 61,210 lbf/24,000 = 2.55 in2 or a diameter of 1.8�. 20 1.875� bolts are therefore
recommended.
The design engineer must make allowance for the diameter of washers and the spotfacing
diameter while laying down the design of the casing, as explained in Table 8-11.
To complete this preliminary design exercise, the engineer needs to calculate the width of
the impeller, including the pump-out vanes. This will be the topic of Section 8-4.
8-4 THE IMPELLER, EXPELLER AND
DYNAMIC SEAL
Slurry, like any liquid, tends to find its way of least resistance. When a pressure difference
exists between the volute pressure and the suction pressure at the front of a slurry
pump or the gland and stuffing box pressure (leaking to atmosphere) exits, slurry tends to
flow back. However, as passageways narrow near the stuffing box or near the suction,
solids become entrapped and accelerate abrasive wear.
Leakage of slurry at the stuffing box can be dangerous to the environment, and can
damage bearings. Various methods have been developed over the years to counteract
leaks. One popular method consists of injecting water at the gland. The gland water pres-

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
TABLE 8-11 Size of Metric bolts and Allowance for Spot Facing. Suitable for Slurry
Pump Casing and Stuffing Box
sure is usually 35–70 kPa (5–10 psi) above the discharge pressure of the pump. The water
acts also as a cooling lubricant to the shaft sleeve and packing rings. As time passes, the
abradable packing rings wear slowly, and the operator has to readjust the gland. Thus, the
gland rings are usually split with tightening bolts (Figure 8-24).
Unfortunately, trucking or pumping fresh gland water to remote tailing pump stations
is not always the most economical solution. The pumping cost of gland water is not negligible
for large pumps. In some cases such as pumping ore concentrate, the process engineer
would prefer to avoid diluting the slurry by adding water at the gland. In the mid-
1960s, slurry pump designers started to investigate the concept of a dynamic seal. A
dynamic seal in its most basic concept consists of a ring of vanes on a shroud capable of
creating a vortex. The designer of the dynamic seal tries to create a vortex field strong
enough to prevent flow to the center of the vortex. In fact, when pressure is sufficiently
reduced at the center of the dynamic seal to a magnitude below the outside atmospheric
value, air is sucked in through the gland, and an air ring is formed .
Despite the appearance of expellers, dynamic seals, and pump-out vanes in the mid-
1960s, there is a dearth of technical information of their performance. Various claims
made in sales brochures are difficult to substantiate. Universities research centers have
not paid much attention either. In some respects, the expeller at first look condradicts traditional
thinking. It is in fact an impeller whose purpose is to repel or prevent flow. It
goes against the logic of rotodynamics.
The dynamic seal of a slurry pump consists of:
� Pump-out vanes on the back shroud of the impeller (Figure 8-25)
� Antiswirl vanes between the impeller and the expeller
� one or more expellers with antiswirl vanes between them
8.35
Clearance hole Washer outside Spot facing Erix Back Spot
Bolt size diameter (mm) diameter (mm) diameter (mm) facing diameter (mm)
M5 6 10 12 15
M6 7 12.5 14 15
M8 9 17 19 18
M10 12 21 24 24
M16 18 30 33 33
M20 23 37 41 43
M24 27 44 46 48
M30 33 56 60 62
M36 39 66 70 72
M42 45 78 80 82
M48 51 92 96 108
M56 59 105 110 113
M64 67 115 120 122
The dynamic seal operates only when the pump is rotating at a sufficient speed. When
the pump is stationary, the dynamic seal ceases to perform and liquid may leak through
the stuffing box, unless an additional stationary seal is provided or external water at sufficient
pressure is flushing the gland.

8.36 CHAPTER EIGHT
FIGURE 8-24 Stuffing box of the ZJ slurry pump (made in China) showing piping connection
to inject water at high pressure and two adjusting bolts.
FIGURE 8-25 Two front pump-out vanes of a slurry pump, before painting and testing (left)
and painted with different colors (right), then installed in the pump of a test loop; the discoloration
indicates patterns of wear. (Courtesy of Mazdak International, Inc.)

Flow in an expeller is complex and depends on the difference in relative motion between
the stationary surface of the case liner and the rotating disk of the expeller.
Consider Figure 8-26 showing a closed impeller with pump-out vanes on the back
shroud. The impeller main vane tip radius is R2, but the pump-out vanes extend only to
the radius Rr. A shaft sleeve behind the impeller has Rs as a tip radius. In the front shroud
of the impeller, another set of pump-out vanes extend to the radius Rf and provide dynamic
sealing between the impeller and the throatbush to repel any solids that may tend to
slip toward the suction (where the pressure is obviously lower). As the impeller rotates, a
pressure field develops on the front shroud of the impeller due to the front pump-out
vanes, and another pressure field develops behind the impeller due to the back pump-out
vanes. In an ideal world, both fields should balance each other. In reality, wear of these
vanes and the difference of clearance between the front and the back vanes with respect to
the casing or its liners tend to create an unbalance.
In reference to Table 8-1, Case 7 for a forced vortex we have:
� = C7 × R0 v0
–1 V × Rv0 = C7
P/� = C 7 2 · R 2 v0/(2 · g) + h 7
Stepanoff (1993) stipulated that when a disk is rotating against a stationary surface,
the average angular speed of the liquid between the two is half the angular speed of the
disk. However, when vanes are added to the rotating disk, the rotational speed of the liquid
is expressed as
Hvf
R2
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
Rf
R1
8.37
1 + t/x
�liq = �imp��� (8-30)
2
x f
t f
B
2
FIGURE 8-26 Dynamic pressure distribution due to front and back pump-out vanes of a
slurry pump impeller.
t b
xb
R sl
R r
H vr

8.38 CHAPTER EIGHT
where t is the depth of the pump-out vanes and x is the total gap between the impeller
back shroud and the casing wear surface. x = s + t, where s is the gap between the pumpout
vanes and the back shroud.
Figure 8-27 represents a simplified case of pump-out vanes that extend down to the
shaft sleeve diameter dsL. The average rotational speed of the liquid between the rotating
impeller and the stationary shroud therefore �imp/2. Applying the Euler head to this region,
the head at the radius Rr is therefore:
2 U n – Un–1
�H = ��
(8-31)
2g
2 2 �H = (R2 – R r) (8-32)
Because vanes extend from Rr and Rsl, �
2 2 �H = (R r – R sl) (8-33)
So if H2 is the head at the tip of the impeller vane, then the head at the stuffing box (in the
absence of any expeller) is the head at the sleeve, or Hsl. Because a certain percentage of
the dynamic pressure is converted to static head in the volute, H2 is often assumed to be
75% of the total dynamic head:
2 imp(1 + t/x) 2
�
8g
��
8g
� 2 imp
� 2 imp
Hsl = H2 – � � ([R2 2 � 2 2 2 2 – R r] – (1 + t/x) · (R r – R sl)) (8-34)
8g
The design engineer establishes H2 as a design criterion. Since the worst condition that
a slurry pump may experience happens when it operates at 30% of the B.E.P capacity and
at a head H30, some engineers calculate H2 as:
H2 = H30 – H1 When H s > H atm, the pump-out vanes will be completely flooded and the liquid will flow
to the gland. To prevent this effect, some liquid at a higher pressure than the stuffing box
pressure may be injected or an additional expeller may be added. When H s < H atm, then
the pump-out vanes suck in air and the stuffing box is sealed against loss of slurry (Figure
8-26).
In the back of the impeller, a second smaller disk with vanes facing the bearing assembly
direction is sometimes installed (Figure 8-27). It is called the expeller in the mining
industry and the repeller in the pulp and paper industry. Its diameter is usually smaller
than 70% of the pump impeller. Its purpose is to reduce further the head between the hub
of the impeller H b and the stuffing box.
Equation (8-34) does not describe the effect of the number of vanes, the breadth of the
vanes, or the shape of the vanes. Over the years, different manufacturers have developed
various shapes such as:
� Straight radial vanes
� Radial vanes but split in the middle with a gap
� L-shaped vanes, also called hockey sticks
� J-shaped vanes
� Radial vanes with an outside ring

8.39
c ve
impeller
Ød ho
expeller area
lve
LE
t e
h e
Ød he
FIGURE 8-27 Geometry of an expeller with radial vanes.
Exp
Ød

8.40 CHAPTER EIGHT
� Radial vanes with an outside ring and a middle ring
� Lotus-shaped vanes
These shapes are represented in Figure 8-28.
Equation (8-34) clearly indicates that the head is proportional to the square of the
speed. There is therefore a minimum rotational speed before that the dynamic seal starts
to function.
The consumed power of an expeller is expressed as:
P (kW) = constant · � · D5 · N3 (8-35)
(a) backward curved vanes (b) radial split at midradius (c) L-shaped vanes ( hockey sticks)
(e) simple radial
(d) radial with ring at mid- radius
(f ) lotus vanes
FIGURE 8-28 Different shapes of vanes and rings of expellers and dynamic seals.

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.41
Although various claims have been made in sales brochures about the merits of each
vane type, and numerous patents have been filed, there has been no substantial scientific
data to confirm the claims. Often, the final shape is a compromise between the requirements
for casting in hard metals and the requirements of the hydraulics.
Impellers of slurry pumps must accommodate solids, and this means that the vanes must
be wide enough. Each manufacturer has their own criteria, with dredge and gravel pumps
requiring very wide impellers (Table 8-12). Adding this passageway to the thickness of the
shrouds of pump-out vanes results in the impeller overall width b 2 (Figure 8-29).
In Equation 8-35, it was pointed out that the power consumption from pump-out vanes
is proportional to the diameter raised to the power of five. Instead of trimming the pumpout
vanes to a diameter smaller than the impeller main vanes, they are sometimes tapered
(t b and t f are gradually reduced toward the tip of the impeller; see Figure 8-29).
In Figure 8-29, the pump-out vane thickness at the root is (g f + t fv), whereas at the tip it
is t fv. In the back of the impeller, the pump-out vanes start at a diameter d b, whereas on the
front side they start at d r. These values are plugged into Equation 8-34 to obtain R r in each
case and to calculate axial thrust.
Because slurry pumps are often cast in brittle alloys such as the high-chrome white
iron, it is important to eliminate sharp edges that may act as stress risers. The manufacturers
establish the radii R 3, R 4, R c, R r, R h, and R sv shown in Figure 8-29 to allow a smooth
casting, but also to improve on the hydraulics. The effect of each parameter on the hydraulics
as described in sales brochures is not always well proven.
The vane diameter d 2 shown in Figure 8-29 is smaller than the shroud diameter d t, but
it is the reference diameter for all calculations.
The shaft sleeve with a diameter d sl is used in all thrust calculations. The sleeve protects
the shaft from wear by the packing and solids that may accumulate between the
packing rings.
TABLE 8-12 Recommended Maximum Size of Spheres for the Design of the Width
of Vanes of Slurry and Dredge Pumps
Mill discharge pumps Gravel and dredge pumps
_______________________________________ _______________________________________
Discharge Size Sphere diameter, Discharge Size Sphere diameter,
(mm) (inches) mm (in) (mm) (inches) mm (in)
25 1.5 × 1 13 (1/2�)
38 2 × 1 18 (11/16�)
50 3 × 2 20 (3/4�)
75 4 × 3 22 (7/8�)
100 6 × 4 38 (�1.5�) 100 6 × 4 80 (�3)
150 8 × 6 50 (�2�) 150 8 × 6 127 (�5�)
200 10 × 8 63 (�2.5�) 200 10 × 8 180 (�7�)
250 12 × 10 80 (�3) 250 12 × 10 230 (�9�)
300 14 × 12 88 (�3.5�) 300 14 × 12 240 (�9.5�)
350 16 × 14 100 (�4�) 350 16 × 14 250 (�10�)
400 18 × 16 115 (�4.5�) 400 18 × 16 280 (�11�)
450 20 × 18 127 (�5�) 450 20 × 18 305 (�12�)
500 24 × 20 140 (�5.5�) 500 24 × 20 360 (�14�)
600 28 × 24 150 (�6�) 600 28 × 24 380 (�15�)
650 30 × 26 180 (�7�) 650 30 × 26 450 (�18�)
915 40 × 36 530 (�21�)

8.42 CHAPTER EIGHT
Ø d b
Ø d th
Ø d h
Ø d sl
t
bs
R
h
R sv
t bv
bv
Most slurry pumps use a threaded shaft. The length of the shaft thread L th is used in
calculations of axial load transmitted from the torque. Some pumps use BSW and others
use ACME thread, and some manufacturers have also their own thread designs to make it
difficult to pirate their impellers.
It is important to establish the center of gravity of the impeller. In the absence of data,
it is often assumed to be at a distance L h. It is also assumed in the calculations that the radial
thrust force is applied at the same point.
8-5 DESIGN OF THE DRIVE END
g b
b
2
The hydraulic loads from the pump wet end are ultimately transmitted to the pump shaft
and bearings. Because of the need to access all the pump parts for replacement due to
wear during maintenance, slurry pumps have standardized cantilever designs, with all
bearings well protected from solids ingestion.
The main loads that are transmitted to the pump shaft are:
t
fs
t fv
g
f
R
R
tb fv
R R
R
2
t
c
L th
Ø d 1
� Radial force due to pressure distribution in the volute
� Axial force due to the pump-out vanes and expellers
h
i
R r
fsv
Ø d r
FIGURE 8-29 Cross-section of an impeller for a slurry pump showing different geometrical
parameters.

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
� Weight of the impeller and expeller
� Torque due to speed and power consumption
� Radial force on the drive end from pulleys
8-5-1 Radial Thrust Due to Total Dynamic Head
The radial force is due to the uneven pressure distribution in the pump casing. It is expressed
as:
FR = K · �gHd2 · B2 (8-36)
where
d2 = tip diameter of the impeller vanes
B2 = width of the pump casing
As shown in Figure 8-26,
B2 = b2 + xf + xb (8-37)
Wear can chip at the surface of the impeller or the casing, thus causing an increase of
xf and xb and a reduction of b2 through the life of the pump.
The value of K may be as high as 0.40 near the shut-off head and as low as 0.10 at the
best efficiency point. It is, however, recommended to conduct proper measurements with
proximity probes over the envelope of the flow rate during the design of a new pump. The
proximity probes are used to measure the deflection at the gland. The magnitude of the
force is then calculated from cantilever stress theory.
As shown in Figure 8-30, different shapes of volutes give different values for the radial
load. Stepanoff (1993) clearly indicated that the direction of the radial force reverses
after the best efficiency point, whereas Angle et al. (1997) do not seem to agree with this
supposition. A misunderstanding of the direction of this hydraulic radial force leads to totally
different estimation of the bearing life. A calculation that assumes a zero radial load
near the best efficiency point (following the Stepanoff approach) can lead to a bearing life
ten times as high as another calculation that assumes that the same radial load adds to the
weight of the impeller, creating a large bending moment on the shaft and reaction loads at
the bearings. A smart salesman may try to convince the consultant slurry engineer of the
superiority of his product over the competition in terms of the rigidity of the bearing assembly,
whereas in reality it is a matter of adding or subtracting loads.
Shafts of slurry pumps have broken at the shaft thread, simply because the radial load
was too high and caused rapid fatigue failure. It is therefore strongly recommended to
limit the minimum flow rate to half the best efficiency flow rate at the given speed. Throttling
an oversize pump is not recommended at all. Downsizing or reducing the speed of
the pump is essential to avoid excessive radial load on the pump shaft.
Each manufacturer has their recommended value of K for the calculation of the radial
load and the bearing life.
8-5-2 Axial Thrust Due to Pressure
8.43
The axial thrust is due to the fact that the pressure on the suction side is different from the
pressure on the back of the impeller. There is a difference between plain impellers and
impellers with pump-out vanes, but since pump-out vanes wear out with time due to abrasion
and erosion, the design engineer should conduct his calculations for both cases of im-

8.44
Head
(a) true volute (b) two semi-circle casing (b) circular casing
FR
Q N
Flow rate
After
Angle &
Rudonov
(1999)
After
Stepanoff
(1993)
Head
FR
Q
N
Flow rate
After
Angle &
Rudonov
(1999)
After
Stepanoff
(1993)
FIGURE 8-30 Radial load for different shapes of casing versus flow rate.
Head
F
R
Q
N
Flow rate

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
pellers with and without pump-out vanes. The presence of an expeller or the addition of
pressurized gland water does affect the axial thrust.
Consider in Figure 8-31 a closed impeller without pump-out vanes. The pressure on
the suction side is P s and at the suction diameter d 1. The pressure on the back of the impeller
is P 1. The pressure above d 1 on both sides of the impeller is equal and balances out.
In the back of the shaft sleeve and shaft there is atmospheric pressure P A, so the resultant
force based on the shaft sleeve diameter is:
T SL = 0.25�d 2 SLP A
On the suction side, there is suction pressure P s, so the thrust force is:
T S = 0.25�d 1 2 Ps
8.45
The net thrust is:
2 2 FA = 0.25�{P1[d 1 – d SL] + PAd 2 2
SL – Psd 1} (8-38)
For the first stage, PS is calculated in a very similar way to the NPSH.
Some manufacturers design the bearing assembly to absorb the axial thrust from a single
stage and others standardize on three stages because they anticipate use in a wide
range of applications from mill discharge to tailings disposal.
Because tailings pumps are often used in series, the bearing assembly may be designed
for a suction pressure equal to the discharge pressure of the stage before the last one, i.e.,
if M is the number of stages:
Ps = (M – 1)�g(TDHst) + PA (8-39)
where TDHst is total dynamic head per stage.
Referring to Figure 8-29, when pump-out vanes are added in the back shroud, Equation
8-34 is then used to calculate the value of Pb at the root of the pump out vanes Rb: Pb = P2 – 0.125��2 2 2
imp{[R2 – R b] – [(1 + tb/xb) 2 2 2 · (R2 – R b)]} (8-40)
where P2 = 0.75�g(TDH) + PS. Ps
R1
d A d P A
FIGURE 8-31 Axial loads on an impeller with plain shrouds.
P 1
sl

8.46 CHAPTER EIGHT
The average thrust force on the back shroud of the impeller is
2 2 T2b = 0.5(P2 + Pb) �{[R 2 – Rb] (8-41)
This value of the pressure Pb is transmitted to the expeller box and becomes the pressure
at the expeller tip diameter dexp (Figure 8-27). The pressure at the expeller diameter dhe (which is often equal to the shaft sleeve or the pressure at the gland) is then
Phe = Pb – 0.125��2 imp{[R2 exp – R2 he] – [(1 + te/(te + cve)) 2 · (R2 exp – R 2 he)]} (8-42)
The average thrust force on the back shroud of the expeller is
Tbe = 0.5(Phe + Pb) �{R2 exp – R 2 he} (8-43)
If the expeller hub diameter is larger than the shaft sleeve, there is a component of axial
thrust as
Tesl = 0.5(Phe + PA) �{R2 he – R 2 SL} (8-44)
On the back of the sleeve and shaft, the pressure is essentially atmospheric so that the
thrust is
Tsl = PA�R 2 SL
(8-45)
On the front shroud of the impeller, pump-out vanes are also added with some impellers.
Applying Equation 8-34 to Figure 8-29, the pressure at the front hub Rr is therefore:
Pr = P2 – 0.125��2 2 2
imp{[R2 – R r] – [(1 + tf/xf) 2 2 2 · (R2 – R r)]} (8-46)
The average thrust force on the front shroud of the impeller between R2 and Rr is:
2 2 T2r = 0.5(P2 + Pr) �{[R2 – R r] (8-47)
If the front shroud hub diameter dr is larger than the suction diameter ds, there is a component
of axial thrust as
2 2 Trs = 0.5 (Pr + Ps) �{R r – RS} (8-48)
The thrust due to the suction pressure is then
2 Ts = Ps�RS (8-49)
Total axial thrust equals total thrust on the back shroud minus total thrust on the suction:
FA = [t2b + Tbe + Tsl] – [Ts + Trs + T2r] (8-50)
In multistage applications with a number of pump in series, the total axial thrust can
change direction as the suction pressure is higher than atmospheric pressure, and the expeller
and pump-out vanes’ effectiveness in balancing thrust drops with increasing number
of stages.
Since the flow calculations need to be repeated at various points on the pump curve, a
computer program would be useful. The program AXIAL-RADIAL was developed by
the author in Qbasic, a language easy to understand by most engineers, but experts may
modify it to PASCAL, C+, Fortran, or other languages as it suits their needs. It calculates
both hydraulic and axial loads on the pump impeller.
COMPUTER PROGRAM “AXIAL-RADIAL”
9 CLS
REM calculations of axial and radial loads on a pump impeller
pi = 4 * ATN(1)
Rem Calculations will be done assuming a specific gravity of 1.7

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
sg = 1.7
INPUT “model name “; na$
INPUT “tip shroud diameter dt (mm) “; dt
INPUT “vane tip diameter d2 (mm) “; d2
INPUT “suction diameter ds (mm) “; ds
INPUT “starting diameter for front pump out vanes dr (mm) “; dr
INPUT “starting diameter for back pump out vanes db (mm) “; db
INPUT “back hub diameter dh (mm) “; dh
INPUT “shaft sleeve o.d dsl (mm) “; dsl
INPUT “overall width of impeller B2 (mm) “; bx
INPUT “ vane tip width b2 (mm) “; b2
INPUT “thickness of front shroud tfs (mm)”; tfs
INPUT “thickness of front pump out vanes tfv (mm) “; tfv
INPUT “anticipated front gap (mm)”; gf
sf = gf + tfv
‘INPUT “thickness of back shroud tbs (mm)”; tbs
INPUT “thickness of back pump out vanes tbv (mm) “; tbv
INPUT “anticipated back gap (mm)”; gb
sb = gb + tbv
INPUT “speed for metal version”; n
PRINT “it shall be assumed that pump out vane to gap ratio =0.7”
PRINT
a1 = .25 * pi * (dr/25.4) ^ 2
a2 = .25 * pi * (d2/25.4) ^ 2
a3 = .25 * pi * (dsl/25.4) ^ 2
a4 = .25 * pi * (ds/25.4) ^ 2
a5 = .25 * pi * (db/25.4) ^ 2
c = 25.4
DIM h(10), fa(10), fan(10), nr(10), Q(10), k(10),fr(10),f(10)
Rem assume a typical curve for an all metal impeller
h(1) = 64;k(1)=0.4
h(2) = 62.7;k(2)=0.35
h(3) = 60.5;k(3)=0.25
h(4) = 55;k(4)=0.15
h(5) = 49.5;k(5)=0.10
h(6) = 35;k(6)=0.12
h(7) = 34.2;k(7)=0.15
h(8) = 33;k(8)=0.20
h(9) = 30;k(9)=0.22
h(10) = 27,k(10)=0.25
INPUT “best efficiency flow rate for metal version “; qnm
Q(1) = .25 * qnm
Q(2) = .5 * qnm
Q(3) = .75 * qnm
Q(4) = 1 * qnm
Q(5) = 1.15 * qnm
Rem calculation for rubber
Q(6) = .25/1.354 * qnm
Q(7) = .5/1.354 * qnm
Q(8) = .75/1.354 * qnm
Q(9) = 1/1.354 * qnm
8.47

8.48 CHAPTER EIGHT
Q(10) = 1.15/1.354 * qnm
FOR i = 1 TO 10
h = h(i)
h2 = .8 * h/.3048
PRINT “h2= “; h2
INPUT “hit any key to continue “; l$
IF h(i) > 35 THEN nr(i) = n
IF h(i)

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
TABLE 8-13 Limiting Dimensions of American National Standard General Purpose
Single Start ACME Threads. External Threads (for Shafts), Class 2G
8.49
Nominal Threads Major diameter, Minor diameter, Pitch diameter,
diameter (inch) per inch min/max, in inch min/max, in inch min/max, in inch
1.50 4 1.4875–1.5000 1.1965–1.2300 1.3429–1.3652
1.75 4 1.7375–1.7500 1.4456–1.4800 1.5916–1.6145
2.00 4 1.9875–2.0000 1.6948–1.7300 1.8402–1.8637
2.25 3 2.2333–2.2500 1.8572–1.8967 2.0450–2.0713
2.50 3 2.4833–2.5000 2.1065–2.1467 2.2939–2.3207
2.75 3 2.7333–2.7500 2.3558–2.3967 2.5427–2.5700
3.00 2 2.9750–3.0000 2.4326–2.4800 2.7044–2.7360
3.50 2 3.4750–3.5000 2.9314–2.9800 3.2026–3.2350
4.00 2 3.9750–4.0000 3.4302–3.4800 3.7008–3.7340
4.50 2 4.4750–4.5000 3.9291–3.9800 4.1991–4.2330
5.00 2 4.9750–5.0000 4.4281–4.4800 4.6973–4.7319
For more information consult ANSI standard B1.5-1977.
ACME external thread (Table 8-13) is used for the shaft and the ACME internal thread
(Table 8-14) is used for the impeller. Because the impeller thread is cast, particularly with
hard metals, Class 2G is suggested because it has a wider range of tolerances than the 3G,
4G, and 5G Classes. BSW shaft threads are used on the smallest sizes. Figure 8-32 represents
a typical ACME shaft thread.
In order to determine the shaft stresses and the axial pull due to torque, the first step is
to assess the torque due power:
Tq = 60PW/(2�N) (8-51)
Example 8-5
A pump is sized for 200 m 3 /hr, at a TDH of 36 m and specific gravity of 1.4. The pump
speed is 600 rpm and the hydraulic efficiency is 67%. Determine the power and the torque.
TABLE 8-14 Limiting Dimensions of American National Standard General Purpose
Single Start ACME Threads, Internal Threads (for Impellers), Class 2G
Nominal Threads Major diameter, Minor diameter, Pitch diameter,
diameter (inch) per inch min/max, in inch min/max, in inch min/max, in inch
1.50 4 1.5200–1.5400 1.2500–1.2625 1.3750–1.3973
1.75 4 1.7700–1.7900 1.5000–1.5125 1.6250–1.6479
2.00 4 2.0200–2.0400 1.7500–1.7625 1.8750–1.8985
2.25 3 2.2700–2.2900 1.9167–1.9334 2.0833–2.1096
2.50 3 2.5200–2.5400 2.1667–2.1834 2.3333–2.3601
2.75 3 2.7700–2.7900 2.4167–2.4334 2.5833–2.6106
3.00 2 3.0200–3.0400 2.5000–2.5250 2.7500–2.7816
3.50 2 3.5200–3.5400 3.0000–3.0250 3.2500–3.2824
4.00 2 4.0200–4.0400 3.5000–3.5250 3.7500–3.7832
4.50 2 4.5200–4.5400 4.0000–4.0250 4.2500–4.2839
5.00 2 5.0200–5.0400 4.5000–4.5250 4.7500–4.7846

8.50 CHAPTER EIGHT
major dia
pitch dia
minor dia
p/2
b t
pitch
p
2 = 29˚
Solution in SI Units
power = (200/3600) · 1.4 · 9810 · 36/0.67 = 40,997 W
torque = power/rotational speed = 40,997/(2 · � · N/60) = 652.5 N-m
The helix angle � of the thread is defined as
tan � = � � (8-52)
where
L = length of a full turn = pitch for single-start threads
L = 2 × pitch for double start threads
pitch = distance between two consecutive threads measured at the thread diameter
dm = pitch diameter
The axial load transmitted through the thread from the torque is expressed as
�dm cos �n – fL
Fth = 2 · Tq����� (8-53)
dm( f�dm + L cos �n) tan �n = tan � cos �
For ACME threads it equals 14.5°. For square threads it is nil. For modified square it is
5°. For buttress threads it is 7°. So for an ACME thread:
tan �n = 0.968 cos �
The coefficient of friction f is measured between the shaft and the impeller. In some
pumps, the shaft is of steel but the impeller may be of bronze. Slurry pumps are essentially
steel against iron and the coefficient of friction is considered to be in the range of 0.14
to 0.15:
�dm cos �n – fL
Fth = 2 · Tq����� (8-54)
dm( f�dm + L cos �n) If n is the number of engaged threads, the axial load from this thread pull force creates a
bending stress Sb and a shear stress Ss at the root of the shaft thread:
L
� �dm
h = p/2
FIGURE 8-32 ACME thread for pump shafts.

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
3F thh
� �dmnb t 2
S b = � � (8-55)
Ss = ��� (8-56)
�dmnb t
where
h = height of the thread tooth = (major diameter – minor diameter)/2; in the case of
ACME threads h = p/2
bt = thread width at the root
8-5-4 Radial Force on the Drive End
When pulleys are installed to drive the pump, the torque transmitted through a pulley diameter
D p results in a force. Different equations are available, but the simplest expresses
the resultant pulley force as:
4Tq �
Dp
8-5-5 Total Forces from the Wet End
F p = � � (8-57)
The total radial force transmitted by the impeller to the shaft is due to the combination of
the hydraulic radial thrust and the weight of the impeller:
F1 = FR + Wimp (8-58)
It is assumed that F1 is acting on the center of gravity of the impeller. The total axial load
is:
F 2 = ±F A
as the axial force may change direction as the number of stages exceeds two pumps in series.
The torque, a source of torsion stress, was defined in Equation 8-51. On the drive
side, the pulley force is upward for overhead-mounted motors or sideways for sidemounted
motors. Calculations are often made on the assumption of overhead-mounted
motors:
F3 = Fp – Wp (8-59)
On this basis, the shaft of the pump is designed. Due to fatigue considerations, the maximum
stress should be smaller than the lesser of 18% yield, or 30% ultimate tensile
strength.
Referring to Figure 8-33, the equilibrium of forces shows that the reaction force at the
wet end is RW and at the drive end it is RD: R W – F 1 – R D + F 3 = 0 (8-60)
Taking moments at the point of contact load of the wet end bearing:
–A · F1 + RD · B – F3(B + C) = 0
F th
8.51

8.52 CHAPTER EIGHT
Torque
F A
L th
F + W
R imp
F3(B + C)
RD = � �
F3C RW = � �
A · F1 + � (8-61)
B
F1 ·(A + B)
� + �� (8-62)
B B
The reader should refer to specialized books on machine design that detail all aspects
of the design of shafts, stress concentration, and bearing life calculations from the reaction
forces at both the drive end (outboard) and wet end (inboard) bearings. The manufacturers
of bearings have their own detailed factors for type of lubricant and ratio of axial to
radial force. Some manufacturers of slurry pumps offer grease lubricated bearing assemblies
and reserve the oil version for high-speed and high-thrust loads (as in pumps in series),
whereas some use oil all across their range of pumps.
8-5-6 Flange Loads
d A
d
R
W
�� B
WE
d
F p - Wpulley
(with belt drive)
A common misconception is that the flanges of slurry pumps can take the same loads as
water pumps. The fact that the discharge flange is split radially to allow access to rubber
or metal liners by itself is an indication that this is not the case at all. The casing of a slurry
pump can be distorted by excessive pipe loads on the flange. The consultant engineer is
therefore well advised to contact the manufacturer for allowed flange loads. It is also necessary
to provide proper pipe supports at the discharge of the slurry pump, and not to use
the pump by itself as an anchor block to piping.
The common error is to apply a large expansion at the discharge of the pump, such as
from a 4� pump discharge to an 8� pipe. Doubling the diameter is effectively multiplying
by four the area exposed to the full pressure, of which a quarter is absorbed by the pump,
leaving three quarters to be balanced by a pipe fitting such as a properly supported dead
end bulkhead, an anchor block, or, whenever possible, by soil friction, as is the case with
pipeline pumps.
R
D
A B C
FIGURE 8-33 Loads on the shaft of a horizontal slurry pump.
d k d D.E

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8-6 ADJUSTMENT OF THE WET END
The wear of the impeller front vanes, the throatbush, is believed to cause a drop in efficiency.
The impeller must be readjusted by moving it forward. To move the impeller relative
to the casing, the shaft assembly must be moved relative to the pump frame, as the
latter is bolted to the pump casing. Two different methods are available:
1. A special bolt under the bearing cartridge (Figure 8-34)
2. Push and pull bolts at the drive end (Figure 8-35)
Once the bearing cartridge is moved, it is fixed in place by clamping bolts that are tightened
against the frame.
8-7 VERTICAL SLURRY PUMPS
8.53
The vertical sump (Figure 8-36) complements the horizontal pump. The vertical pump is
particularly suitable for floor sumps in mill discharge areas and in dealing with flotation
circuits. The vertical sump pump may be supplied as:
� A stand-alone pump with double suction impeller (Figure 8-37) to be installed in a
concrete or metal sump, particularly with flotation columns
bearing
cartridge
pump
frame
clamping
bolts
for
bearing
cartridge
adjustment
bolt
for
bearing
cartridge
FIGURE 8-34 Adjustment of the pump impeller by a special bolt between the bearing cartridge
and the frame.

8.54 CHAPTER EIGHT
clamping bolts
bearing cartridge
push bolts
pump frame
FIGURE 8-35 Bearing assembly of the ZJ slurry pump (made in China) with adjusting push
and pull bolts. (Courtesy of AJP Services Inc. The distributor for Canada.)
FIGURE 8-36 Sump pump with double suction impeller. (Courtesy of Mazdak International
Inc.)

(a)
(b)
Rubber-Lined, Acid-Proof Pumps with Double Suction Impellers
Pump
Size Frame Units A B C D E F G H J K L N P
SP2 BV inch 26 11 32 36 12 20 6 16 20 20 20 40 8
mm 660 280 813 915 305 508 152 406 508 508 508 1016 200
SP3 CV inch 32 14 36 48 16 20 8 22 22 26 26 60 12.6
mm 813 356 915 1220 406 508 203 559 559 660 660 1524 320
SP4 DV inch 37 14 48 60 20 26 10 26 26 30 30 72 14.8
mm 940 356 1220 1524 508 660 254 660 660 762 762 1829 376
SP6 EV inch 48 14 60 72 24 35 14 34 34 38 38 84 19.7
mm 1220 356 1524 1829 610 889 356 864 864 965 965 2134 500
SP8 FV inch 52 14 60 84 14 35 16 47 47 52 52 96 23
mm 1321 356 1524 2987 356 889 406 1194 1194 1321 1321 2438 584
*D is the standard depth—other shaft length are available in 12� increments—consult the plant for critical
speed
*E is the minimum priming level
*C and N are typical sump dimensions for the sump
FIGURE 8-37 Dimensions for sump pump and corresponding sump. (Courtesy of Mazdak
International Inc.)
8.55

8.56 CHAPTER EIGHT
� A single suction impeller with an auger or agitator below the impeller to agitate settled
solids in floor sumps (Figure 8-38)
� A top suction pump supplied integrally with a metal conical tank, called a “tank pump”
(Figure 8-39)
The vertical slurry pump is designed to have all its bearings above the baseplate so as to
be well protected from slurry ingestion (Figure 8-40). Due to the depth of the sump, the
design engineer must pay particular attention to the critical speed of the pump. For this
reason, the shaft of these pumps can be as large as 200 mm (8�) to offer the necessary
rigidity.
Vertical slurry pumps are particularly popular in froth handling circuits. To handle
the combination of solids, air, and liquids, a double suction impeller is often recom-
wearplate
casing
impeller
screen
column
agitator
shaft
motor & pulleys
bearing
assembly
baseplate
2" discharge
eyebolt
fig 8-38
FIGURE 8-38 Sump pump with single suction impeller and auger to agitate settled solids
particularly suited for mill discharge floor. (Courtesy of Mazdak International Inc.)

tank
shaft
impeller
casing
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
motor & pulleys
baseplate
inlet
bearing
assembly
eyebolt
discharge
8.57
39 FIGURE 8-39 Tank sump pump with top suction impeller and integral tank for particularly
difficult frothy slurries.
mended with vertical sump pumps. As an alternative, tank pumps with top suction (Figure
8-39) are used. In either configuration, the impeller must be designed to resist air
biting.
A special type of process used to extract gold is based on cyanide leaching. Leached
gold is then separated by adsorption, the property of certain materials such as carbon to
fix gold on their surface. Carbon spheres are used as an adsorption material. This process
is done in special “carbon in leach” or “carbon in pulp” circuits with mixing tanks. The
transfer of these solutions requires recessed or vortex impellers that can pump without
breaking the carbon lumps. The impeller is recessed out of the flow as shown in Figure
8-41.
A design that is gaining popularity in plants for recycling newspaper is the vertical
pump with a recessed impeller and a chopper blade. It is not uncommon that the recycling
bins for paper now found in every suburb of North America end up containing milk cartons,
plastic bottles, toys, and even pieces of wood. These materials are not very good for
conventional stainless steel pumps with mechanical seals. The cantilever sump pump
(Figure 8-41) with the chopper offers the ability to pump long fibers while chopping them
and eliminating the maintenance problems of mechanical seals.

FIGURE 8-40 Components of a vertical slurry pump showing that the bearings are above
the baseplate. 1. Shaft sea. 2. Top bearing cover. 3. Top bearing. 4. Bearing assembly. 5.
Crease nipple. 6. Bearing locknut. 7. Bearing washer. 8. Bottom bearing—spherical roller for
heavy duty. 9. Discharge pipe. 10. Baseplate. 11. Shaft seal. 12. Bottom hub. 13. Pedestal—
Open structure. 14. Top suction strainer. 15. Wear plate. 16. Shaft. 17. Shaft sleeve. 18. Double
suction impeller for minimum thrust loads. 19. Pump casing. 20. Lower suction strainer.
(Courtesy of Mazdak International Inc.)
8.58

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8-8 GRAVEL AND DREDGE PUMPS
Wet end with cutter
stator
Rotor
8.59
FIGURE 8-41 Vertical slurry pump with a recessed impeller. This pump is suitable for carbon
transfer in gold cyanide circuits and for wastewater treatment applications. The addition of
a cutter (rotor and stator) renders this pump particularly suitable for certain applications such
as sewage treatment and newspaper recycling plants.
Hard metal pumps play an important role in dredging lakes and ports. Some sizes are
presented in Table 8-12. A typical construction of a dredge pump is presented in Figure
8-42. Dredge pumps are designed to handle particularly large boulders and lumps of
clay. Some of the largest dredge pumps are designed to handle 6.3 m 3 /s or 100,000 US
gpm.
A special low-pressure, high-flow pump called the ladder pump is designed to be
mounted at the tip of the suction arm. Its purpose is essentially to move the material up to
the boat hopper or up to a booster pump on a hopper. The booster pump is designed for
higher discharge head.
A particular type of pump is the phosphate-matrix-handling pump. It does resemble in
many aspects a sort of dredge pump, but is built of materials to handle both corrosion and

8.60 CHAPTER EIGHT
Precision
machined
heavy
duty shell
Adjusting
bolt allows
for easy
adjustment of
impeller for
proper
operating
clearances
Integral stuffing box
shell mount
Separate
thrust bearing
wear. It is often driven by a diesel engine through a gearbox. The complete baseplate with
driver and pump are relocated from one area to another as mining is done.
8-9 AFFINITY LAWS
Heavy duty bearing
assembly for high power
& loading conditions
FIGURE 8-42 Components of the Marathon dredge pump. (Courtesy of Mobile Pulley and
Machine Works.)
Affinity laws are used to predict the effects of changing the speed of a pump, trimming an
impeller, and extrapolating the performance of a pump from case (A) to case (B). They
state that:
2 2 HA/HB = N A/N B (8-63)
H A/H B = D A 2 /DB 2 (8-64)
H A/H B = N A 2 /N B 2 (8-65)
Q A/Q B = D A/D B
Q A/Q B = N A/N B
One piece shell and engine
side door/liner for minimum
parts replacement
Solid impeller hub eliminates problems
with threaded or bolted inserts
Advanced design impeller. Good
hydraulic performance without
sacrificing spherical clearance
OPTIONAL ONE PIECE DOOR/SIDE
LINER DESIGN AVAILABLE
(8-66)
(8-67)

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8-10 PERFORMANCE CORRECTIONS FOR
SLURRY PUMPS
Slurry pumps are designed and tested for hydraulics using water as a reference fluid.
However, they are designed to handle large spherical rocks. Often, four-vane impellers
are less efficient but outlast other types, particularly on abrasive slurries. Attempts have
been made to quantify and qualify the spacing of vanes on the performance of dredge
slurry pumps. As early as 1932, Fischer and Thoma conducted tests on a pump built of
a transparent material. Although the flow was observed to be close to the designed value
at best efficiency point, it quickly deviated at other values. They observed a large
area of flow separation on the trailing edge of the vane, with reverse flow in certain instances.
Understanding the effects of solids on centrifugal pumps has been a slow process.
Fairbanks (1941) developed a theory to correlate the head developed by a pump for a slurry
mixture with the volumetric concentration and specific gravity of the solids. He explained
that the fundamental Euler equation could be modified to account for the density
and flow rate of the mixture as:
T = �mQm(r2Vt2m – r1Vt1m) The power needed to pump the mixture by an ideal pump (at 100% hydraulic efficiency)
is then expressed as:
T� = �mQmgHm where � is the angular speed . The mixture head is then expressed as two components for
solids and carrier fluid:
Hm = (�/g�m) · [�s · Cv · (r2Vt2s – r1Vt1s) + (1 – Cv) · (r2Vt2L – r1Vt1L) (8-68)
where
Cv = volumetric concentration of solids
�m = density of mixture
�s = density of solids
Fairbanks concluded from his tests on a single pump that :
� The drop in the head-capacity curve varies not only as the concentration increases, but
also as the particle size of the material in suspension increases.
� The fall velocity of the suspended material is the most important parameter for predicting
the effect of solids on pump performance.
� The power input is a linear relationship of the apparent specific gravity of the solids in
suspension
8-10-1 Corrections for Viscosity and Slip
8.61
Viscosity must be taken in account when pumping viscous slurries. Viscosity reduces the
efficiency of pumping and the head developed by a pump (Figure 8-43). The Hydraulic
Institute Standards provides correction curves for viscous fluids pumping, but warns
against extrapolating to other pumps or fluids. The Institute does not publish curves for
viscous slurries.
Duchham and Aboutaleb (1976) derived equations to predict the effects of viscosity

8.62 CHAPTER EIGHT
N
H/H
1.4
1.2
Head (Viscous fluid)
Head (Water)
1.0 1.0
0.8 0.8
0.6 0.6
Efficiency (water)
0.4 0.4
Efficiency
(viscous fluid)
0.2 0.2
0.0 0.0
0.0 0.5 1.0 1.5
Fig 843
Q/Q
N
FIGURE 8-43 Effect of viscosity on the performance of centrifugal pumps.
and density on the flow rate, head, and power consumption by comparing particle
Reynolds number and power factor. Their analysis did not present a definite appreciation
of the effects of viscosity.
Sheth et al. (1987) investigated slip factors for slurry pumps by conducting tests on a
Wilfley pump. The pump had a 267 mm (10.5 in) diameter, 27 mm (1.06 in) blade width,
and a discharge angle of 31°. The following equation was derived by Sheth et al. (1987)
to account for the effects of the slurry mixture carrier densities:
0.12
�s� � = 0.0989 – 0.00157 � � 0.5 ND2 �
�m ��
� �
�L · N · D �L Q
2
where
� s = slip factor
� = dynamic (absolute viscosity) of liquid carrier
D imp = impeller diameter
N = rotating speed of impeller
� m = density of slurry mixture
� L = density of liquid carrier
Q = flow rate
N
(8-69)

The above equation is empirical and the exponents and coefficients may change for different
pump designs. More research work on different designs would have to be published
before a universal formula is adopted.
Example 8-6
A slurry pump is to be designed to pump slurry under the following conditions:
maximum speed at intake 4 m/s (13 ft/s)
flow rate 120 L/s (1858 USGPM)
head 40 m (131 ft)
slurry density 1470 kg/m 30 (SG m = 1.47)
water carrier
slurry viscosity 100 mPa · s
max solid particle size 25 mm (1 in)
Using the Sheth formula, determine the geometry of the impeller.
Solution
suction area = = 0.04 m2 suction diameter = 0.225 m (8.85 in)
suction area at 4 m/s = 0.03 m2 0.120 m
suction diameter = 0.195m (7.7 in)
3 /s
��
3
Or, calling a = N · D 2 :
If A = 30, then:
If a = 40, then:
If a = 20, then:
If a = 15, then:
If a = 10, then:
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
0.12
�s� � = 0.0989 – 0.00157 � � 0.5 ND2 �
�m ��
� �
�L · N · D �L Q
2
� s · 0.331 = 0.0989a 0.12 – 0.0067a 0.62
� s = 0.298a 0.12 – 0.020a 0.62
� s = 0.298 × 1.5 – 0.0202 × 8.23
� s = 0.28
� s = 0.464 – 0.198 = 0.265
� s = 0.427 – 0.129 = 0.297
� s = 0.4124 – 0.108 = 0.304
� s = 0.393 – 0.084 = 0.31
8.63

8.64 CHAPTER EIGHT
Let us assume a = 10, then:
10 = N · D 2
Since the particle size passage is 25 mm (� 1�), assume discharge width = 30 mm or
0.03 m (1.18 in). The head ratio � = 2gH/u 2 (in the United States) is:
� = 2�H�[1 – (cm/u2) cot �2] If we assume D = 0.4m (� 16u ), then:
10 = N × 0.16 ⇒ N = 62.5 rev/s
U = 78.5 m/s
Cm = Q/ADis ⇒ discharge area = � × 0.4??(1 – zt/sin �2 )
If b = 30 mm (1.181�) then:
A = � × 0.4 × 0.03 (1 – zt/sin �2 )
= 0.037 (1 – zt/sin �2 )
If Z = 4 vanes and t = 30 mm then:
A = 0.037 (1 – 0.12/sin �2 )
If �2 = 15° then:
A = 0.0198 m2 Cm = 0.12/0.0198 = 6.05 m/s
Cm/U2 = 0.077
� = 2� H� s(1 – (C m/U 2) cot � 2) = 0.44� H
2gH 2 × 9.81 × 40
� = = �� = 0.127
� U 2 2
78.5 2
0.127 = 0.44 �H ⇒ �H = 0.289
This is not a very efficient pump due to the combination of viscosity and solid density:
consumed power = g�QH/�H = 9.81 × 470 × 0.120 × 40/0.289
� 240 kw or 327 hp
It is recommended to install a 400 hp motor.
8-10-2 Concepts of Head Ratio and Efficiency Ratio When
Pumping Solids
Stepanoff (1969) explained that when pumping solids in suspension, a pump impeller imparts
energy to the carrier liquid. For a homogeneous mixture, he explained, the impeller
will be able to impart as many feet of mixture as it would have been able to impart head of
water. The performance of the impeller is not impaired but the power consumption increases
linearly with the specific gravity of the mixture. In reality, at best efficiency point,
the presence of solids tends to reduce the hydraulic head by the energy wasted to move
them through the impeller passageways. Similarly, the efficiency of the pump when handling
the mixture will be reduced by the presence of solids. Two factors can be defined
head ratio and efficiency ratio.

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.65
The head ratio is
HR = Hm/Hw (8-70)
or the ratio of head developed when pumping slurry to the head developed when pumping
water.
The efficiency ratio is
ER = Em/Ew (8-71)
or the ratio of efficiency developed when pumping slurry to the efficiency developed
when pumping water.
Stepanoff (1969) indicated that at best efficiency:
HR = ER Stepanoff (1969) reported work by Japanese investigators who indicated that tests on carbide
slurries tended to show that the head–capacity ratio may increase or decrease depending
on whether the solids concentration tended to cause the slurry to behave as a
Newtonian or non-Newtonian mixture.
Reviewing published data between 1941 and 1971, Hunt and Faddick (1971) reported
that various tests in different labs and field applications confirmed that:
� The drop in head (in feet of mixture flowing) developed for a given volumetric discharge
rate decreased as the concentration of the solids in suspension increased.
� The required brake horsepower for a given pump operating at a given capacity increased
as the concentration of solid material in suspension increased.
� The efficiency at a given capacity decreased as the concentration of the solid material
in suspension increased.
Hunt and Faddick (1971) simulated the performance of centrifugal pumps pumping wood
chips by tests using rectangular plastic parts with an average specific gravity of 1.02.
They used four different impeller designs in two different volute designs. There was no
consistency in the extent of head drop or efficiency with solid concentration, and the results
indicated that the actual design of the pump was a very important factor. A difference
of head and efficiency of 5 to 7% was noticed for the different designs. The authors
therefore discouraged applying head and efficiency ratio factors for one pump to another
pump of a different geometry, but encouraged further research into the mechanisms of
flow through the rotating passages of these pumps.
It is important to appreciate the work of Hunt and Faddick. Often, a pump vendor will
produce a chart or curve to obtain the head and efficiency ratio. The limitations of such
curves are that they apply only to pumps of similar geometrical design. The discrepancy
of 5–10% between one design and another may have to be absorbed by the motor.
McElvain (1974) published data on the effects of solids on pump performance. He
worked on the concept of the head and efficiency reduction factors defined as:
RH = 1 – HR (8-72)
R� = 1 – ER (8-73)
He tested impellers up to a diameter of 35 cm (13.78 in) and on various concentrations of
silica and one grade of heavy mineral. He developed a set of curves and established a relationship
between volumetric concentration and the head and efficiency reduction factors
as:
RH = R� = 5 · K · CV (8-74)

8.66 CHAPTER EIGHT
N
H/H
1.2
Head (Slurry)
Head (Water)
1.0 1.0
0.8 0.8
0.6 0.6
Efficiency(water)
0.4 0.4
Efficiency (slurry)
0.2 0.2
0.0 0.0
0.0 0.5 1.0 1.5
Fig 8-44
Q/Q
N
FIGURE 8-44 Effect of solids on the performance of centrifugal pumps.
The K factor was then plotted against the d 50 and for solids of various specific gravity
(see Figure 8-45). The assumption that R H = R � was accepted to hold true for slurry volumetric
concentrations smaller than 20%. This covers a substantial number of pump applications.
Example 8-7
Heavy metal oxide slurry is to be pumped at a volumetric concentration of 18%. The specific
gravity of the solids is 5.0 and the d50 is 400 �m. The calculated head on slurry is 35
m. Determine the head ratio and the equivalent water head on the pump performance curve.
Solution
Using the McElvain equation, the value K is determined from the lower curve at 0.38.
Substituting in Equation 8-74, at a volumetric concentration of 18% (less than 20%):
RH = R� = 5 · K · CV = 5 · 0.38 · 0.18 = 0.342
Substituting into Equation 8-72:
HR = 1 – RH = 0.658
Since the calculated head for friction, the equivalent value on water is
Hw = Hm/HR = 35 m/0.658 = 53.2 m
The engineer must therefore select the appropriate pump speed from the pump curve that
would develop 53.2 m.
N

K-Factor
0.1
0.2
0.3
0.4
0.5
Sellgren and Vappling (1986) reported that at high volumetric concentration the efficiency
ratio was smaller than the head ratio, thus indicating a more pronounced loss of efficiency.
Sellgen and Addie (1993) reported losses as low as half of those predicted by McElvain.
The curves of McElvain do not take into account another important factor, namely
the ratio of particle size to impeller diameter. Burgess and Reizes (1976) proposed that
the head ratio and efficiency ratio were a function of three parameters:
1. Weight concentration
2. Ratio of d 50 to impeller diameter
3. Specific gravity of the solid particles
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
10 100
1000
S = 5.0
s
Particle Size d ( m)
8.67
Sellgen and Addie (1993) indicated that there is a size effect and that head and efficiency
losses were less drastic in large pumps than in small pumps (Figure 8-46). This clearly
demonstrates the importance of pump design on performance.
The importance of pump design on the head and efficiency ratio was confirmed by
Czarnota et al. (1996) through tests on ITT-Flygt submersible pumps. Their work confirmed
that high-efficiency pumps suffered from less degradation of performance than
less efficient pumps. Head reduction was confirmed to be a linear function of the volumetric
concentration of solids. Larger particles were found to slip more than smaller particles.
An important factor they reported is that settling or separation can occur due to
centrifugal forces. These forces are proportional to the square of the radius, and in the
presence of large particles can lead to partial blockage, higher water velocity, and more
slip between solids and liquid. A well-mixed particle distribution tended to decrease derating
of pumps.
Russian engineers developed a very advanced mathematical model based on full
screen analysis instead of the average d 50, which has been the focus of most equations in
50
2.65
4.0
1.5
10000
FIGURE 8-45 Correction to the head K factor of centrifugal pumps on the basis of the specific
gravity and particle size S s = specific gravity of solids. (After McElvain, 1974.)

8.68 CHAPTER EIGHT
FIGURE 8-46 Effect of the size of the pump impeller on the correction factor for head R H
for slurry at a weight concentration of 42%. (From Sellgren and Addie, 1992.).
Australia, Europe, and North America. The work of Kuznetsov and Samoilovich (1985,
1986) was summarized by Angle et al. (1997). These advanced mathematical models permit
corrections based on the number of vanes, discharge angle of the vanes, and volumetric
concentration of each range of diameter of solids in the slurry. It would be very appropriate
to explore these models; however, when examining a worn-out impeller, as in
Figure 8-47, the reader may wonder how practical such models may be.
8-10-3 Concepts of Head Ratio and Efficiency Ratio Due to
Pumping Froth
Flotation froth is complex slurry and may contain an important amount of air and gases
(Figure 8-48). The industry uses the froth factor as a measure. Basically, it is determined
by filling a column of flotation slurry and measuring the height H0. It is then left for 24 hr
to rest. The height of the slurry H� is then measured. The froth factor F is defined as:
F = H0/H� (8-75)
Using this concept of froth factor to size pumps must be done very carefully. Different
grades of froth leads to different levels of entrained gases, as shown in Table 8-15. Conventional
centrifugal pumps can not handle excessive amounts of entrained gases.
A very common misunderstanding in the industry is that the flow rate of slurry must
be multiplied by the froth factor to size the pump. This violates a very fundamental principle
that gases or air are compressible fluids. In other words, as the bubbles pass through
the impeller they are compressed and reduced in size. In fact, the proper sizing of slurry
pumps to handle froth must be based on a full examination of the system. For example, if

THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.69
FIGURE 8-47 Worn-out impeller, showing gradual degradation of the impeller tip diameter
and vanes. Deterioration of hydraulics and head efficiency may occur throughout the wear life
of the impeller and pump.
FIGURE 8-48 Flotation slurry froth contains sufficient air bubbles to degrade the performance
of centrifugal pumps. (Courtesy of EOMCO Process Equipment Co.)

8.70 CHAPTER EIGHT
TABLE 8-15 Correlation between Froth Factor and Percentage of Entrained Air
Froth factor % entrained air Example
1.5 2–3% Normal flotation tailings
2.0 3–5% Flotation tailings with minimum retention time
2.5 5–7% Tenacious flotation tailings with minimum retention time
3.0 > 7% Froth with very fine particles
the flotation cells are away from the pump and the froth is transported by gravity in launders,
it may be argued that the surface area of the launders acts as a deaerator for air removal.
In that case, does a 24 hr tube test apply well?
The correct approach is in fact to remove as much of the air as possible before the
froth enters the pump feed sump. This sump may also be designed in a conical shape to
maximum surface area at the top.
Certain forms of froth are very difficult to pump, such as the type associated with tar
sands, in which viscosity plays a major role. Efficiency as low as 10% was reported with
conventional pumps.
Cappelino et al. (1992) presented a very thorough study on the performance of centrifugal
pumps with open impellers with emphasis on pulp and paper flotation circuits
and deinking cells. High-consistency stock (12%) can have as much as 20–28% entrained
air.
At the inlet to the impeller, the pressure drop tends to cause an expansion of the air
and gases, and this indicates well that the concept of the froth factor can be misleading
Example 8-8
The height of the liquid in a froth cell is 30 m above the pump. The depression at the inlet
to the pump is about 6 m. Determine the expansion of gases, assuming a barometric pressure
of atmospheric air at 9.5 m. The pump is designed to deliver a total head of 54 m.
Determine the final volume of the gases.
Solution
The effective absolute head in the sump is:
30 + 9.5 = 39.5 m
Due to the depression of 6 m, the absolute pressure is then:
39.5 – 6 = 33.5 m
The expansion ratio at constant temperature is:
39.5/33.5 = 1.179
The absolute discharge head is:
suction head + TDH + atmospheric barometric height = 30 + 54 + 9.5 = 93.5 m
Ratio of discharge to suction absolute head is:
93.5/39.5 = 2.37
The size of the air or gas bubbles will then shrink by the inverse of this ratio, or 42.2%.
Since the laws of thermodynamics apply, the concept of a constant froth factor is illusive.
It would be a grave error to size piping and equipment based on the suction froth
factor.

The performance of slurry pumps deteriorates in the presence of entrained gases. Cappelino
et al. (1992) have therefore proposed to define appropriate head and power correction
factors as:
or
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.71
FIGURE 8-49 Head and power correction factors for entrained gas due to flotation circuits.
(From Cappellino, Roll, and Wilson, 1992. Reproduced by permission of Texas A&M University.)
head measured with entrained gas
HF = ����
(8-76)
head measured without entrained gas
power measured with entrained gas
PF = ����
(8-77)
power measured without entrained gas
Special pumps are available for handling froth and entrained gases. One interesting
design is the Sulzer–Ahlstrom ART pump. It features holes through the impeller leading

8.72 CHAPTER EIGHT
straight to an expeller at the back of the impeller. This expeller discharges the air through
a separate discharge flange at the back of the casing. In some other designs, the stuffing
box is connected to an external liquid ring vacuum pump that can remove any entrained
air in the slurry. The manufacturers of pulp and paper pumps have also designed special
impeller with protruding vanes that extend into the suction pipe to break up any large air
particles. This concept is gaining popularity in some oil sand applications to handle particularly
thixotropic and viscous froth.
Dredge pumps sometimes face a similar problem. Gases are disturbed or released (particularly
methane) during certain phases of dredging and end up accumulating in the ladder
pump. Herbich and Miller (1970) conducted extensive test work on the effect of air on
the development of head. Herbich (1992) proposed that special air removal systems be installed
on the suction side of the pump with an ejector, as this is better than a vacuum
pump.
8-11 CONCLUSION
In this chapter, some of the important parameters that give slurry pumps their final shape
were examined. It is obvious that slurry pumps are different from water pumps and that
considerable research should be undertaken in fields such as pump-out vanes, expeller design,
and effects of wide impellers on performance.
The successful performance of these pumps depends on their resistance to wear. The
slurry—in terms of its composition and concentration, and in terms of any froth-induced
gases—is the determining factor for power consumption and the final hydraulics across
the impeller and casing. These parameters are extremely important for the successful installation
of these pumps.
8-12 NOMENCLATURE
A Factor to calculate slip, depending on the use of volute or diffuser
Ap Equivalent casing area for stress calculations
bt Thread width at the root
b2 Width of the impeller at the impeller tip diameter
B2 Width of the casing at the impeller tip diameter
BHP Power in bhp
C Constant
Cm Meridional velocity across the impeller
Cv volumetric concentration of solids
Cw Concentration by weight of the solid particles in percent
Cp Heat capacity
D Equivalent diameter of casing
d2 The tip diameter of the impeller vanes
dSL Diameter of shaft sleeve
dm Pitch diameter
ER Efficiency ratio
Fth Force due to thread pull
FA Axial thrust on shaft due to hydraulic forces
Force on casing due to design pressure
F p

Fp Force due to belts at the drive end of the pump
FR Radial thrust
G Acceleration due to gravity (9.78 to 9.81 m/s2 or 32.2 ft/sec)
H Height of the thread tooth = (major diameter – minor diameter)/2; in the case of
ACME threads h = p/2
H1 Head at the medium diameter of the eye of the impeller
H2 Head at the tip diameter of vane of the impeller
H30 Head at 30% of best efficiency capacity
HE Euler ideal head for an impeller
HIMP SV Shut-off head due to the impeller
HR Head ratio
HVOL SV Shut-off head due to the volute
HSV Total shut-off head
HV Vapor head
K Correction factor for the head ratio
Kx Ky L
Coefficient to determine Xv Coefficient to determine Yv length of a full turn in a shaft thread = pitch for single start threads
Lth The length of the shaft thread
m Exponent in vortex equation
M Number of pumps in series
N Rotational speed of the pump in rev/min
NPSH Net Positive Suction Head
Nq Specific speed in SI units
NUS Specific speed in US units
NSS Suction specific speed
PA Atmospheric pressure
Pb Pe Ps PD Pressure at the root of the pump-out vanes Rb Pressure at the surface of the liquid in absolute terms on the suction side
Pressure at the suction diameter d1 Pressure losses between the surface of the liquid and the pump, due to friction,
valves, etc.
PW Pump power in Watts
PV Vapor pressure
Q Flow rate
R1 Root radius of the vanes of the impeller
R2 Tip radius of the vanes of the impeller
R3 Radius of the smaller circle of a twin circle volute
R4 Radius of the larger circle of a twin circle volute
Rb Radius at the root of the pump out vanes
RC Cutwater radius
RH Head correction factor
R� Efficiency correction factor
RMD Meridional radius of the volute at the throat
Rr Tip radius of the pump-out vanes
Rs Tip radius of the shaft sleeve
Rv0 Local radius of vanes
R1 Radius of the root of the impeller vane
R2 Tip radius of the vanes of an impeller
Rv0 Radius of vortex
Reaction force at the drive end bearing
R D
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.73

8.74 CHAPTER EIGHT
RW Reaction force at the wet end bearing
s Gap between the edge of the pump-out vanes and the back wear plate of the casing
S Static moment, obtained by graphical integration along the meridional plane of
the vanes
Sb Bending stress at the root of the shaft thread
Ss Shear stress at the root of the shaft thread
Tq Torque
t Depth of the pump-out vanes
tC Thickness of the pump casing
tL Thickness of the liner
TS Thrust on suction side
TSL Thrust on shaft sleeve
TDH Total Dynamic Head
U Tip speed
V Absolute velocity across the impeller
W Relative velocity across an impeller
Wimp Weight of impeller
Wp Weight of pulleys
X Total gap between the pump-out vanes’ impeller surface and the pump back
plate
XV Width of the volute in the x-direction
YV Width of the volute in the y-direction
Z Number of vanes
Z1 Geodetic elevation of liquid surface above the centerline of the pump impeller
Geodetic elevation of the centerline of the pump impeller
Z e
Greek Letters
� Angular inclination of the vane with respect to the tangent
� Slip factor
� Density of liquid
� Cavitations parameter
� Angular velocity
�liq Angular velocity of the liquid in the gap between the pump-out vanes and the
pump back plate
�imp Rotational velocity of the impeller or expeller
�SI Head coefficient to SI convention
�US Head coefficient to US convention
Subscripts
1 At the root of the vane
2 At the tip of the vane
E Eye of the impeller
F Front shroud
R Rear shroud
Imp Impeller
Liq Liquid
Sl Sleeve

8-13 REFERENCES
THE DESIGN OF CENTRIFUGAL SLURRY PUMPS
8.75
Abulnaga, B. E. 2001. Recommendations for the design of mill discharge slurry pumps. Mazdak International
Inc. Internal Report 02/2001 (unpublished).
Addie, G. R. and F. W. Helmly. 1989. Recent improvements in dredge pump efficiencies and suction
performances. Europort Dredging Seminar, Central Dredging Association, Delft, Netherlands.
Anderson, H. H. 1938. Mine pumps. J. Mining Soc. Durham, United Kingdom.
Anderson, H. H. 1977. Statistical records of pump and water turbine effectiveness. International Mechanical
Engineers Conference on Scaling for Performance Prediction in Rotodynamic Pumps.
September, pp. 1–6.
Anderson, H. H. 1980. Centrifugal Pumps. Trade and Technical Press: UK.
Anderson, H. H. 1984. The area ratio system. World Pumps, 201.
Angle, T. and J. Crisswell (Editors). 1977. Slurry Pump Manual. Salt Lake City, Utah: Envirotech.
Angle, T. and A. Rudonov. 1999. Slurry Pump Manual. Salt Lake City, Utah: Envirotech.
ANSI/ASME B106.1. 1985. Design of Transmission Shafting. American Society of Mechanical Engineers,
New York.
Burgess, K. E. and B. E. Abulnaga. 1991. The application of finite element methods to Warman
pumps and process Equipment. Paper presented to the Fifth International Conference on Finite
Element Analysis in Australia, University of Sydney, Australia, July 1991.
Burgess, K. E. and J. A. Reizes. 1976. The effect of sizing, specific gravity and concentration on the
performance of centrifugal slurry pumps. Proc. Inst. Mech. Eng., 190, 36.
Cappellino, C. A., D. Roll, and G. Wilson. 1992. Design considerations and application guidelines
for pumping liquids with entrained gas using open impeller centrifugal pumps. Proceedings of
the Ninth International Pump Users Symposium, Texas A&M University.
Czarnota, Z., M. Fahlgren, M. Grainger, and S. Saunders. 1996. The effects of slurries on the performance
of submersible pumps. BHR Group Hydrotranport, 13, 643–655.
Duchham C. D. and Y. K. A. Aboutaleb. 1976. Some tests in a single stage semi-open impeller centrifugal
pump handling coal dust slurries. In Proceedings Pumps and Turbine Conferences, Vol
1.
Fairbanks, L. C. Jr. 1941. Effects on the characteristics of centrifugal pumps. Solids in Suspension
Symposium, Proc. Am. Soc. Civ. Eng., 129, 129.
Fischer, K. and D. Thoma. 1932. Investigation of the flow conditions in a centrifugal pump. Transactions
ASME, 54.
Frost, T. H. and E. Nielsen. 1991. Shut-off head of centrifugal pumps and fans. Proc. Inst. Mech.
Eng., 205, 217–223.
Herbrich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill.
Herbich, J. B. and R. J. Christopher. 1963. Use of high speed photography to analyze particle motion
in a model dredge pump. In Proceedings of the International Association for Hydraulic Research,
London England.
Herbich, J. B. and R. E. Miller. 1970. Effect of air content on performance of a dredge pump. In Proceedings
of the World Dredging Conference, Wodcon 70, Singapore.
Hunt, A. W and R. F. Faddick. 1971. The effects of solids on centrifugal pump characteristics. In Advances
in Solid–Liquid Flow in Pipes and Its Application, I. Zandi (Ed.), New York: Pergamon
Press.
Jekat, W. K. 1992. Centrifugal pump theory. Section 2.1 in Pump Handbook, J. Karassik et al. (Eds.),
New York: McGraw Hill.
Kuznetsov, O. V. and D. C. Samoilovich. 1986. Increase of Reliability of Slurry Pumps in Service (in
Russian). Moscow: CINTIchimneftemash, ser.XM-4.
McElvain, R. E. 1974. High pressure pumping. Skillings Mining Review, 63, 4, 1–14.
Pfeiderer, C. 1961. Die Kreiselpumpen. Berlin: Springler-Verlag.
Samoilovich, D. C. 1986. Experimental Study of Slurry Pumps Performances (in Russian). Moscow:
CINTIchimneftemash, ser.XM-4.
Sellgren, A. and L. Vappling. 1986. Effects of highly concentrated slurries on the performance of
centrifugal pumps. Proceedings of the International Symposium on Slurry Flows, FED Vol 38,
ASME, USA, pp. 143–148.
Sellgren, A. and G. R. Addie. 1992. Effects of solids on the performance of centrifugal slurry pumps.

8.76 CHAPTER EIGHT
Paper presented at the 10th Colloquium: Massenguttransport durch Rohrietungen in Meschede,
Germany, May 20–22.
Sellgren, A. and G. R. Addie. 1993. Solids effect on the characteristics of centrifugal slurry pumps.
Paper presented at the 12th International Conference on Slurry Handling and Pipeline Transport,
Brugge, Belgium.
Sheth, K. K., G. L. Morrison, and W. W. Peng. 1987. Slip factors of centrifugal slurry pumps.
A.S.M.E. Journal of Fluids Engineering, 109, 313–318.
Stepanoff, A. J. 1969. Gravity flow of bulk solids and transportation of solids in suspension. New
York: Wiley.
Stepanoff, A. J. 1993. Centrifugal and Axial Flow Pumps. Melbourne, FL: Krieger.
Sulzer Pumps. 1998. Centrifugal Pump Handbook. New York: Elsevier.
Turton, R. K. 1994. Rotodynamic Pump Design. Cambridge: Cambridge University Press.
K. C. Wilson, G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. London:
Elsevier Applied Sciences.
Wilson, G. 1976. Construction of solids-handling centrifugal pumps. In Pump Handbook, J. Karassik
et al. (Eds.) New York: McGraw Hill.
Worster, R. C. 1963. The flow in volutes and effect on centrifugal pump performance. Proc. Inst.
Mech. Eng., 177, 843.
Further Reading
Kazim, K. A. and B. Maiti. 1997. A correlation to predict the performance characteristics of centrifugal
pumps handling slurries. In Proceedings of the Institution of Mechanical Engineers. Part A.
Journal of Power and Energy, 211, A2, 147–157.
Cader, T., O. Masbernat, and M. C. Rocco. 1994. Two phase velocity distributions and overall performance
of a centrifugal slurry pump. Journal of Fluid Engineering, 116, 316–323.
Gandhi, B. K., S. N. Singh, and V. Seshadri. 2000. Improvements in the prediction of performance of
centrifugal slurry pumps handling slurries. Proceedings of the Institution of Mechanical Engineers.
Part A. Journal of Power and Energy, 214, 5, 473–486.

CHAPTER 6
SLURRY FLOW IN
OPEN CHANNELS AND
DROP BOXES
6-0 INTRODUCTION
The design of mineral processing plants and tailings disposal systems often includes
gravity flows in open channels. Such flows are often called slack flows. They involve a
free boundary to the atmosphere. In the past, many launders were designed using rules
of thumb; however, the development of large mines requires a rigorous scientific approach.
Most of the published papers on sediment transportation in open channels dwell extensively
on the geophysics of canals and rivers. The field of open channel hydraulics is
so vast and complex that the reader may have to consult various reference books such
as Graf (1971). One subject of great interest to civil engineers is the carrying capacity
of the channel for sediments. This is often termed the sediment load and is measured as
a function of the flow rate, width of the channel, and sediment concentration. Using
channels to transport solids has limitations due to the fact that no pumps are used to
force the flow.
Many papers have been published over the years on the geophysics of rivers and the
maximum permissible speeds used to avoid scouring and removal of bed materials.
Transferring the knowledge about scouring speeds of canals and rivers into useful information
for a designer of a hydrotransport system is not a straightforward process. In fact
there, is not a single unified mathematical model to represent slurry flows in open channels.
In this chapter, a methodology is presented to estimate the friction losses for slurry
flows in open channels, cascades, drop boxes, and distribution boxes. In the last twenty
years, new developments in thickeners encouraged various operators to develop the concept
of adding flocculants to launders. Tailings and concentrate slurries are thereby allowed
to flow at higher and higher concentrations in gravity modes. The engineer must
take into account the rheology, particularly certain aspects of high yield stress and non-
Newtonian characteristics.
Mineral processing plants often divide or combine flows in drop boxes, distribution
boxes, and plunge pools. The design principles of such entities are presented at the end of
the chapter.
6.1

6.2 CHAPTER SIX
6-1 FRICTION FOR SINGLE-PHASE FLOWS
IN OPEN CHANNELS
The words flume, launder, open channel, and slack flow are often used to express the
same thing. In the following discussion, these words will be used interchangeably. Even
though launders are crucial to mining, very little research on the subject has been published.
Despite the lack of reference material on launders for slurries, it is important to
start from basic principles. The analysis will focus initially on water flows. The reader
will then be introduced to the complexity of slurry flows. The reader should appreciate
that an upper practical limit on these flows is a 65% concentration of solids by weight.
Since the flow does not fill the launder or pipe, the hydraulic diameter is the defined as
the equivalent diameter of flow for an open channel. The hydraulic radius is defined as
the ratio of the area of the flow to the wetted perimeter. It is also called the hydraulic
mean depth in certain European books.
A
RH = � (6-1)
P
FIGURE 6-1 Large concrete structures offer a method of conveying large quantities of slurry.
This structure was built to convey 150,000 tons per day of soft high clay tailings at a Peruvian
copper mine.

And the hydraulic diameter is
4A
DH = � (6-2)
P
Figure 6-2 shows various possible shapes for open launders with methods to estimate
wetted area and hydraulic radius. The most common launders in mining and dredging are,
however, circular and rectangular in shape. Sometimes a circular pipe is opened and vertical
walls are added to produce a U-shape.
The friction loss for a closed channel and steady-state single-phase flow was examined
in Chapter 2. Using the Darcy factor, the head loss for an open launder can be expressed
in terms of the hydraulic radius:
h = (6-3)
Since the Darcy friction coefficient fD is usually accepted as four times the fanning
friction coefficient fN, Equation 6-3 for a slurry may be rewritten in terms of the fanning
factor fN, discussed in Chapter 2.
h = (6-4)
For a fully developed and uniform flow, the slope or energy gradient of an open launder
is established in terms of the head loss per unit of length (Henderson, 1990):
fNU S = = (6-5)
2
fNLU H
� �
L 2gRH
2
fDLU �
2gRH
2
�
2g(4RH)
2�
2�
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
2� < �
A = R 2 (� – sin � cos �)
P = 2R�
R H =
2� = �
R H = D I/4
R(� – sin � cos �)
���
2�
2� > �
� = � – �
A = R 2 (� – sin � cos �)
P = 2R�
R H =
R(� – sin � cos �)
���
2�
FIGURE 6-2 Hydraulic radius for shapes of open channels.
B
2R
H
H
6.3
if H > R
A = R[2(H – R) + �R/2]
R = 2(H – R) + �R
R H =
A = BH
P = 2H + B
R H = BH � 2H + B
R[2(H – R) + �R/2]
���
2(H – R) + �R

6.4 CHAPTER SIX
or
fDU S = = (6-6)
Many models for open channel flows of water are based on the Chezy number and the
Manning number. The Chezy number is inversely proportional to the square root of the
friction factor:
8g
Ch = � (6-7)
�� fD
2 H
� �
L 8gRH
or
or
2g
Ch = � (6-8)
�� fN
The Manning number is a function of both the hydraulic radius and the friction factor:
R H 1/6
n = � (6-9)
Ch
1/6 RH ��2 �f N
g �� n = �
(6-10)
Some experimental values for the Manning number “n” are shown in Table 6-1 as derived
for water flows. These values are not correct for transportation of solids, particularly
solids that introduce a new roughness factor that we will discuss. This table is presented
as a reference for dirty water, very dilute mixtures, or decant water that are present in
mining and tailings circuits but do not constitute real slurries.
Green et al. (1978) summarized the research activities of the U.S. Army Corps of Engineers
who derived the following relationship between the hydraulic radius and effective
roughness of the channel (in USCS units):
R H 0.1667
n = (6-11)
where
RH = hydraulic radius in feet
k = effective linear roughness in feet
n = Manning number in ft –1/3 ���
23.85 + 21.95 log(RH/ks) /sec
The linear roughness ks (Table 6-2) is also used to compute the flow of water in open
channels. The Ministry of Transport (1969) recommends the following equation for flow
of viscous liquids in open channels:
k 1.225(�/�)
V = – �3�2�R� H�S� log� � + ���
(6-12)
14.8 RH RH �3�2�R� H�S�
where
� = absolute viscosity of fluid
� = density of fluid
S = slope or energy gradient
g = acceleration due to gravity

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
TABLE 6-1 Typical Values for the Manning Number “n” for Water Flows (Do Not
Use for Slurries)
Manning factor “n,” Manning factor “n,”
Channel surface ft –1/3 s –1 m –1/3 s –1
Glass, plastic, machined metal surface 0.011 0.016
Smooth steel surface 0.008 0.012
Sawn timber, joints uneven 0.014 0.021
Corrugated metal 0.016 0.024
Smooth concrete 0.0074 0.011
Cement, plaster 0.011 0.016
Concrete culvert (with connection) 0.009 0.013
Glazed brick 0.009 0.013
Concrete, timber forms, unfinished 0.014 0.0208
Untreated gunite 0.015–0.017 0.022–0.0252
Brickwork or dressed masonry 0.014 0.0208
Rubble set in cement 0.017 0.0252
Earth excavation, clean, no weeds 0.020 0.022
Earth, some stones and weeds 0.025 0.037
Natural stream bed, clean and straight 0.020 0.030
Smooth rock cuts 0.024 0.035
Channels not maintained 0.034–0.067 0.050–0.1
Winding natural channels with pools and shoals 0.033–0.040 0.049–0.059
Very weedy, winding, and overgrown natural rivers 0.075–0.150 0.111–0.223
Clean alluvial channels with sediments 0.031 (d 75) 1/6 0.0561 (d 75) 1/6
After Manning (1895) and Henderson (1990).
Having read Chapters 1–3, the reader must have become aware that sizing pipe involves
a straightforward relationship between the flow rate, the cross-sectional area of the
pipe, and the required velocity. In the case of open launders, particularly those of rectangular
and U-shape, the main concern is to avoid spills. At certain bends, around certain
obstacles, or at a sudden reduction of physical slope, the flow may slow down considerably
and even spill out of the launder. For straight runs away from such bends and junctions
of launders, open conduits are designed to be one-third full. When pipes are used as
open launders, they are typically sized to be 50% full, but in the case of tenacious froth
they may be sized to be 25% or 30% full (Figure 6-3).
Designers often prefer to have a steep launder rather than to suffer a loss of time unblocking
settled slurry. As the above equations indicate, the friction loss factor does depend
on the hydraulic radius, and larger launders tend to require less slope than small
launders. Excessively steep launders tend to lose their liners through fast wear. Obtaining
the correct slope without an excessive margin of safety is the correct approach to engineering.
Example 6-1
A slurry of unknown properties is flowing in a half full 457 mm (18 in) pipe with wall
thickness of 9.5 mm (0.375 in). The measured flow rate is 0.189 m 3 /s (3000 US gpm).
The launder is inclined at a slope of 2%. Determine the friction factor and the Chezy and
Manning numbers.
6.5
using d 75 size in feet using d 75 size in m

6.6 CHAPTER SIX
TABLE 6-2 Recommended Values of Absolute Roughness in mm
Classification
Values of roughness ks, mm (*)
Average
effective
roughness
of launder
(assumed clean and new unless otherwise stated) Good Normal Poor mm (**)
Smooth
Drawn nonferrous pipes of aluminum, brass, copper, 0.003
alkathene, glass, Perspex, HDPE
Plastic pipe, welded joints 0.146
Plastic pipe, flanged or coupled joints 2.292
Fiberglass pipe (FRP), flanged or coupled 2.292
Metal
Asbestos cement 0.015
Spun bitumen-lined pipe 0.03
Spun concrete-lined pipe 0.03
Wrought iron pipe 0.03 0.06 0.15
Rusty wrought iron pipe 0.15 0.6 3
Uncoated steel pipe 0.015 0.03 0.06 0.725
Coated steel pipe 0.03 0.06 0.15
Rubber-lined steel pipe 1.35
Plastic-lined steel pipe 0.350
Galvanized iron 0.06 0.15 0.30 0.726
Coated cast iron 0.06 0.15 0.30
Tate relined pipes 0.15 0.3 0.6
Riveted steel pipes (untuberculated)—(good = girth,
riveted only; normal = full riveted, taper, or
cylinder joints; poor = full riveted, butt-strap joints)
Riveted steel pipes (untuberculated)—plates < 6 mm 0.6 1.5 3
Riveted steel pipes (untuberculated)—plates > 6 mm 1.5 3 6
Concrete
Class 4—Monolithic construction against oiled 0.06 0.15
steel surface with no surface irregularities,
smooth-surfaced precast pipelines with no shoulders
or depressions at the joints
Class 4a—Monolithic construction in units of 2 m or 0.15 0.3
over with spigot and socket joints, or ogee joints
pointed internally
Class 3—Monolithic construction against steel, wet-mix, 0.3 0.6 1.5
or spun pre-cast pipes, or with cement or asphalt
coating
Class 2—Monolithic construction against rough 0.6 1.5
texture precast pipe or cement gun surface (for
very coarse textures, take � = size of aggregate in
evidence)
Class 1—Precast pipes with mortar squeeze at joints 3 6 3.63
Smooth trowel led surface 0.3 0.6 1.5 1.35
Lined concrete pipe 0.73
Unlined concrete pipe 1.35

TABLE 6-2 Continued
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
Classification
Values of roughness ks, mm (*)
Average
effective
roughness
of launder
(assumed clean and new unless otherwise stated)
Steel construction
Good Normal Poor mm (**)
Welded sections, unlined 1.35
Rolled sections, unlined 0.73
Rubber lined 1.35
Plastic lined 0.73
Plastic construction, free formed
Clayware
0.73
Pitch fiber
Pitch fiber
0.003 0.03
Glazed vitrified clay, very accurately lined joints 0.06
Glazed vitrified clay in 1 m under 600 mm diameter 0.15 0.3
Clayware, glazed vitrified clay in 1 m under 600 mm
diameter
0.3 0.6
Clayware, glazed vitrified clay in 0.6 m under 300 mm
diameter
0.15 0.3
Clayware, glazed vitrified clay in 0.6 m over 300 mm
diameter
0.3 0.6
Clayware, butt jointed drain tile 0.6 1.5 3 1.35
Clayware, glazed brickwork 0.6 1.5 3
Clayware, brickwork, well pointed 1.5 3 6
Clayware, old brickwork in need of pointing
Mature foul sewers constructed of materials with
roughness � when new, not exceeding those given
for mature sewers
15 30
Slimed not more than 6 mm 0.6 1.5 3
Lime incrustations, grease, or slime not more than
25 mm thick, or even layer of fine sludge
6 15 30
Gritty solids, lying unevenly in inverts (higher figures
relate to shoals of debris at Froude number of order
of 0.3 to 0.5)
Unlined rock tunnels
60 150 300
Granite and other homogeneous rocks 60 150 300
Diagonally bedded slates (use values with design
diameter)
Earth channels
300 600
Straight uniform artificial channels 15 60 150
Straight natural channels, free from shoals, boulders,
and weeds
150 300 600
*Data from the Ministry of Technology, United Kingdom (1969), Hydraulics Research Paper 4,
Tables for the Hydraulic Design of Storm-drains, Sewers and Pipe-Lines.
**Data from Green (1978).
6.7

6.8 CHAPTER SIX
H = R
Solution in Metric Units
Q = 3,000 US gpm × 3.785 = 11,355 L/min = 0.1893 m 3 /s
ID of pipe = 438.15 mm
Area for a half full pipe = �/8 × 0.43815 2 = 0.0754 m 2
Velocity = 0.1893/0.0754 = 2.51 m/s
Wetted perimeter = � × 0.43815/2 = 0.688 m
Hydraulic radius = A/P = 0.0754/0.688 = 0.1095
Slope = 2%
fN(2.51) 0.02 = = 2.93 fN 2
fNV ��
2 × 9.81 × 0.1095
2
�
2gRH
Fanning friction factor, f N = 0.0068
The Darcy factor, f D = f N × 4 = 0.027
The Chezy number, Ch = �� = �� = 53.71 m1/2 /s
The Manning number, n = R H 1/6 /Ch = 0.1095 1/6 /53.714 = 0.0129 m –1/3 /s
Solution in USCS Units
H = Z/3
Z
2R
2g
� fN
Q = 3000
2 × 9.81
�
0.0068
US gpm = 3000/7.4805 = 401.04 ft 3 /min
ID of a pipe = 17.25 in = 17.25/12 = 1.4375 ft
Area of a half full pipe A = (�/8)1.4375 2 = 0.8115 ft 2
Velocity = P/A = 401.04/0.8115 = 494.21 ft/min (8.237 ft/s)
P = �D/2 = � × 1.4375/2 = 2.258 ft
H = Z/3
FIGURE 6-3 Recommended degree of fill for open channel slurry flows (does not apply to
junction boxes).
Hydraulic radius = A/P = 0.8115/2.258 = 0.359 ft
Z
B
H

Slope = 2%
0.02 = = = 2.935 fN The fanning friction factor, fN = 0.0068
The Darcy factor, fD = fN × 4 = 0.027
The Chezy number, Ch = = = 97.2 ft �� �� 1/2 fN × 8.237
2g 2 × 32.2
� � /s
fN 0.0068
1/6 1/6 –1/3 The Manning number, n = RH /Ch = 0.359 /97.2 = 0.0087 ft /s
The reader should be careful when using the Manning roughness “n.” Contrary to
common belief, it is not a nondimensional number and its value changes from SI units to
USCS units by the ratio of conversion from feet to meters to the power of 1/3 or 0.673.
2
fNV ��
2 × 32.2 × 0.359
2
�
2gRH
6-2 TRANSPORTATION OF SEDIMENTS
IN AN OPEN CHANNEL
Determining the Chezy number for slurries is a method of approaching the design of launders.
Julian et al. (1921) measured an average Chezy number of 80 ft 1/2 /s for rectangular
launders (with width = twice the depth) of minimum wetted perimeter for carrying slime
overflow and average stamp-battery pulp in cyaniding gold and silver circuits.
Classical theories of suspended solids in open channels are based on two-dimensional
turbulent flow. Consider a two-dimensional turbulent flow with a velocity U in the horizontal
x-direction and V in the vertical y-direction. Reynolds (1895) defined the shear
stress parallel to x on a plane normal to y as
where
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
� = –� (U�V�) average
� = density of fluid and
(U�V�) average = average of the fluctuations of the turbulent velocities
(6.13)
Boussinesq (1877) developed an equation in the form of
� = ��m (6-14)
where
��m = the eddy viscosity, analogous to the dynamic viscosity discussed in Chapter 2.
�m = the coefficient of exchange of momentum between neighboring streams of the fluid,
expressed in m2 /s or ft2 dU
�
dy
/sec.
Von Karman (1935) developed the following equation:
� = ��V�L mix
dU
�
dy
where
� = correlation coefficient � 1.0 (see Section 6.2.3)
V� = average of absolute values of fluctuations normal to the main flow
L mix = mixing length
6.9
(6-15)

6.10 CHAPTER SIX
By substituting Equation 6-15 into Equation 6-13 it is concluded that
� m = �V�L mix
A rate of transfer of mass of suspended particles per unit area is defined as
dm
dC
� = –�V�Lmix �
dt
dy
(6-16)
(6-17)
But the values of �, V�, and Lmix are not necessarily equal in magnitude in Equations 6-14
and Equation 6-17.
C = concentration of suspended solids
O’Brien (1933) studied the suspension of sediments in an open channel flow. He developed
a theory that the rate of transfer of solids upward must be in equilibrium with the
downward exchange of momentum due to gravitational forces:
dC
VtCy = –�V�Lmix � (6-18a)
dy
VtCy = �s (6-18b)
where
Cy = volume concentration of solids at level y
y = distance from the lower boundary
�s = mass transfer coefficient for sediments, similar to �m but not necessarily equal to it.
Solving Equation 6-18 yields
loge = � y
dC
�
dy
C dC
� � (6-19)
Ca a dy
where C a is the concentration of solids at an arbitrary reference plane of height “a.”
If � s is constant over the depth, then � s(y) = constant. Equation 6-18 is then solved to
give
C
� Ca
= e –J (6-20)
where J = (y – a)(V t/� s).
The correct procedure consists of establishing a relationship between � s and the vertical
coordinate y before solving Equation 6-18. In the absence of detailed information
about the relationship between � s and � m, they are assumed to be equal so that
�
�
�dU/dy
� m = –�V�L mix = (6-21)
Substituting Equation 6-21 into Equation 6-18 yields the concentration of distribution
loge = –�Vt � y
C dU/dy
� � dy (6-22)
Ca a �

In a uniform open channel with a large ratio of width to depth, the shear stress is expressed
as
ym – y
� = �w� � (6-23)
where
�w = shear stress at the wall
ym = distance from the boundary to the liquid surface
Substituting Equation 6-23 into Equation 6-21 yields
loge = –�Vt � y
C
dU/dy
� �� dy (6-24)
Ca a (1 – y/ym)�w The velocity gradient is expressed in the form of the universal defect law as
= log e� � (6-25)
For a pipe, K x = 0.4 and y is the distance from the internal wall at the bottom of a horizontal
pipe. Keulegan (1938) demonstrated that Equation 6-24 applied to open channels.
For a wide-open channel, the value of y m, or depth of flow, is used:
In Chapter 2, the friction velocity U f was defined as
= log e� � (6-26)
�w fN Uf = � = U � �� � ��2
Substituting Equation 6-6 yields
Uf = �(g�R� H�S�)� (6-27)
where
fN = the fanning friction factor
U = the mean velocity of the flow
S = slope
�w = �gRHS (6-28)
Equation (6-28) is called the DuBoys equation (Wood, 1980). It clearly establishes that
for a slurry to move at a density �, a minimum level of shear stress must be available:
where
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
C
� Ca
U – Umax �
�(��w/ ���)�
U – Umax �
�(��w/ ���)�
V t
� �KxU f
log e = log e�� � � (6-29)
C
� Ca
� ym
1
� Kx
1
� Kx
h
� ha
= � � Z
2y
� DI
y
� ym
ym – 1
�
y
a
�
ym – a
6.11
(6-30)
Vt Z = ��
(6-31)
�Kx�g�ym�S�

6.12 CHAPTER SIX
h = � – 1 (6-32)
y
ym – a
ha = � (6-33)
a
Equation 6-30 establishes that the relative concentration of solids depends on their
vertical position and on the factor Z, which is a function of the ratio of the terminal velocity
of the particles Vt to the group �KxUf. It is therefore a measure of the intensity of turbulence.
6-2-1 Measurements of the Concentration of Sediments
y m
Determining the magnitude of the plane “a” is the starting point to solve Equation 6-28
and 6-29. Rouse (1937) suggested the height “a” be equal to the height of the roughness
elements k s. He suggested using Equation 6-19, assuming an interval from y = 0 to y = k s,
and by assuming that �(y) = �(k s) = constant. Rouse indicated that at y = 0 the solids concentration
corresponding to the bed of sediments should be used.
Richardson (1937) reported test results for a 305 mm × 305 mm × 1830 mm (1 ft × 1 ft
× 6 ft) flume and indicated that in the boundary region the concentration of sediments was
inversely proportional to the vertical coordinate y, but that in open streams it conformed
better with Equation 6-30.
Vanoni (1946) conducted a series of tests in an 838 mm (33 in) wide by 18.29 m (60
ft) long flume to validate Equation 6-29 (Figure 6-4). Average velocity was noticed to occur
at 0.368 y m or the depth of the liquid. The velocity profile followed a logarithmic
function of depth.
The following results were obtained by Vanoni (1946).
� The sediment concentration profile followed the pattern set by Equation 6-30. However,
the exponent was smaller than the value of Z expressed by Equation 6-31 when the
sediments became coarser.
� A random turbulence was observed and slip between fluid and sediment was suspected
as the sediment accelerated. Thus, the assumption that mass and momentum transfer
were equal was not satisfied, as the theory did not account for slip and random turbulence.
� For fine materials, the coefficient of sediment mass transfer was smaller than the coefficient
of momentum transfer.
� For coarse materials, the coefficient of sediment mass transfer was larger than the coefficient
of momentum transfer.
� Suspended load decreased the coefficient of mass transfer. The reduction was more
important with fine solids than with coarse solids.
� Suspended load reduced the value of the Von Karman constant. Values of K x between
0.314 and 0.342 were measured (by comparison with 0.4 for full pipes). The reduction
of the Von Karman coefficient indicated a reduced level of mixing and a tendency by
the sediments to suppress turbulence.
� The suspended load tended to reduce the resistance to flow. Sediment-laden water
moved faster than clear water and the Manning roughness number decreased with the
sediment load, as shown in Figure 6-5.

FIGURE 6-4 Velocity profile for the flow of a sand–water mixture in a rectangular open
channel. (From Vanoni, 1946, by permission of ASCE.)
6.13

6.14 CHAPTER SIX
FIGURE 6-5 Variation of the equivalent roughness, Von Karman coefficient, and Z 1 with
the weight concentration of the sand–water mixture. (From Vanoni, 1946, by permission of
ASCE.)

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
� Suspended load tended to cause a flow to become unevenly distributed. Unsymmetrical
sediment distribution within the flow caused secondary circulation.
� The velocity distribution near the center of the flume followed Von Karman universal
defect law.
Example 6-2
Iron sand is flowing in a rectangular open channel that is 300 mm wide × 120 mm high at
a speed of 3 m/s. Assume the Von Karman coefficient K = 0.33. The average particle size
is 0.3 mm. Using Einstein’s approaches, it is assumed that the reference layer “a” is to be
twice the particle size diameter. The flume is one-third full (i.e., y m = 40 mm). The slurry
weight concentration is 45%, the dynamic viscosity is 1 cP, and the specific gravity of
sand is 4.1. Calculate C/C a if the slope is 3% at depth with 2% increments of depth. Ignore
any dunes. Assume � = 1.0.
Solution in Metric Units
The first approach is to determine the terminal velocity of the sand. The Particle Reynolds
number is:
d pV m�/� = (0.3 × 10 –3 × 3 × 1000/1 × 10 –3 ) = 900
This is turbulent flow. By Newton’s law (Equation 3-13):
V t = 1.74[g(� s – � L/� L)] 0.5 d p 0.5
V t = 1.74[9.81 × 3.1 × 0.3 × 10 –3 ] 0.5
Vt = 0.167 m/s
Using Equation 6-31:
Z = 0.167/[0.33 × (9.81 × 0.04 × 0.03) 0.5 ]
Z = 4.664
Applying Equation 6-29:
C/Ca = (h/ha) 4.664
As it will be explained later in this chapter, Einstein proposed that the value of “a” be
equal to twice the grain diameter, or in this case 0.6 mm = twice the particle size of sand.
Let us calculate the concentration of solids at 2% of the depth of the flume.
2% of depth = 0.02 × 40 = 0.8 mm:
y m
h = � – 1 = h = (1/0.02) – 1 = 49
y
ym – a
ha = �
a
ha = (40/0.6) – 1 = 65.67
h
� ha
C
� Ca
= 49/65.67 = 0.735
= 0.735 4.664 = 0.2378
6.15

6.16 CHAPTER SIX
4% of depth:
C
� Ca
h
� ha
h = (1/0.04) – 1 = 24
= 24/65.67 = 0.3655
= 0.3655 4.664 = 9.15 × 10 –3
So most of the solids will be in the bottom 4% of the launder.
Example 6-3
Iron sand (SG = 4.1) is flowing in a rectangular channel 600 mm wide × 120 mm high.
The liquid level is 60 mm high. The slope is 1% and the liquid is flowing at 1.5 m/s. The
particle average diameter is 0.5 mm. The specific gravity of the solids is 4.1 and the dynamic
viscosity of the mixture is 1.5 cP. Calculate C/C a at 4% intervals. Assume Von
Karman K x = 0.33. Ignore any dunes. Assume � = 1.0.
Solution
The particle Reynolds number is :
0.5 × 10 –3 × 1.5 m/s × 1000/1.5 × 10 –3 = 500
This is transition flow. From Chapter 3, Allen’s law would apply:
Vt = 0.20�g � 0.72
�
�L
V t = 0.20(9.81 × 3.1) 0.72
Vt = 1.116 mm/s
Using Equation 6-30:
Z = 1.116 × 10 –3 /[0.33 (9.81 × 60 × 10 –3 × 0.01) 0.5 ]
Z = 3.3822 × 10 –3 /0.0767 = 0.044
The magnitude of “a” is assumed to be twice the particle’s diameter:
a = 2dp = 2 × 0.5 mm = 1 mm
C
� Ca
� s/� L
= (h/h a) 0.044
d � 1.8
� (�/�) 0.45
(0.5 × 10 –3 ) 1.8
���
(1.5 × 10 –6 ) 0.45
60 mm
ha = �� = 59
1 mm – 1
Let us calculate the concentration at 2% depth:
y = 0.02 × 60mm = 1.2 mm
1
h = � = 49
0.02 – 1

4% depth:
8% depth:
12% depth:
16% depth:
20% depth:
24% depth:
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
C
� Ca
C
� Ca
C
� Ca
C
� Ca
C
� Ca
= 49/59 = 0.83
= 0.83 0.044 = 0.99
y = 2.4 mm
1
h = � = 24
0.04 – 1
h
= 24/59 = 0.4067
= 0.4067 0.044 = 0.96
y = 4.8 mm
1
h = � = 11.5
0.08 – 1
C
� Ca
h
� ha
C
� Ca
� ha
= (11.5/59) 0.044 = 0.931
y = 7.2 mm
= (7.33/59) 0.044 = 0.912
y = 9.6 mm
= (5.25/59) 0.044 = 0.899
y = 12 mm
= (4/59) 0.044 = 0.888
y = 14.4 mm
= (3.17/59) 0.044 = 0.879
6.17

6.18 CHAPTER SIX
28% depth:
y = 16.8 mm
= 0.871
32% depth:
y = 19.2 mm
= 0.864
Examples 6-2 and 6-3 show the effect of slope on the distribution of solids. In the case
of Example 6-2, the slope is higher at 3% and most of the solids move at the bottom of the
channel. In Example 6-3, the slope is low at 1% and all the sand is mixed with the water.
Graf (1971) reviewed the experiments conducted by various researchers and a tendency
developed to measure an empirical Z1 as a substitute for Z. He summarized the work of
Einstein and Chien (1955) who developed an approximate relationship between Z and Z1: Z
Z1 = �����
(6-34)
exp(–L2Z 2 2ZL
/�) + �� � �(
2� ��)�
LZ�(2���)�
exp(–x
0
2 C
�
Ca
C
�
Ca
/2)dx
where
x = log e y
L = log e(1 + RK x)
The best fit occurs when RK x = 0.3.
Einstein (1950) called the flow layer right on top of the bed the “bed layer,” and indicated
that it would be impossible to have suspension of the solids there. He measured a
thickness of layer t = 2d p. The material within this layer was the source of the suspended
load and established the lower limit for C a. Einstein then proceeded to derive very complex
equations that require numerical integration. It would be beyond the scope of this
book to dwell on such equations.
6-2-2 Mean Concentrations for Dilute Mixtures (C v < 0.1)
Celik and Rodi (1991) published data suggesting that Einstein’s equations were overestimating
the concentration at a = 2d �. For fairly dilute suspensions with a volumetric concentration
of less than 10%, which is common in a lot of applications, they proposed a
more simplified approach, defining the suspended sediment load q bs as
where
q bs = flow rate of sediment per unit width
q b = total flow rate of mixture per unit width
C = time averaged (mean) concentration
C T = mean transport capacity concentration
U = time averaged velocity in x-direction
R H = hydraulic radius or mean depth of liquid
qbs = qbCT = � RH �LUCdy (6-35)
� a

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
�a = depth of bed load layer
y = vertical coordinate above bottom of channel
Defining parameter Cm as the depth-averaged concentration, Celik and Rodi (1991)
used their results from their previous paper (1984) to conclude that Cm/CT � 1.13, and derived
the following equation:
CT = � ��
(6-36)
where � is a constant of proportionality (� 0.034 from tests on sand).
The exercise consists of calculating C T. For this purpose, these two authors discussed
the problem of the effective shear stress. Reviewing the work of other authors and their
own, Celik and Rodi (1991) pointed out that there are certain important factors in the turbulent
regime, such as:
� When there are elements causing increased roughness, the flow separates, and only a
part of the shear stress determined from the energy slope is effective in moving the
particles in suspension. The work of gravity is then used partly to overcome friction at
the bed.
� The turbulent energy, which is produced in a turbulent regime, is related to the total
shear stress, but the drag force is then not effective in maintaining the particles in suspension.
� The turbulence energy of the upper layers needs to be transferred by convection and
diffusion first to the region above the bed. Smaller quantities of energy are then available
to suspend the bed in the presence of large amounts of roughness.
� The mean velocities and the wall shear stresses in separated regions are much smaller
than in the areas where the flow is still attached.
� The presence of separated regions in the flow and stagnant areas cause the particles to
settle in dead water zones and it becomes very difficult to resuspend them.
� The permeability of a bed increases the resistance of the bed. (Zippe and Graf, 1983).
� For rivers, a typical value of the ratio of friction velocity to average velocity (U f /U) is
0.05.
Tests conducted by Van Rijn (1981) and by Apmann and Rumer (1967) indicate that
the flow over an approximately flat bed of loose sand shows great similarity to the characteristics
of flow over a rough surface.
To take in to account all these effects in the turbulent regime, Celik and Rodi (1991)
proposed an equation for the effective shear rate:
�e = [1 – (ks/ym) � ]�w (6-37)
where
ks = the equivalent resistance parameter (in most cases the roughness height or absolute
roughness)
� = empirical constant (� = 0.06 in tests obtained by these authors)
ym = average depth of liquid in flume
In conclusion, Celik and Rodi (1984) established a simplified relationship between
roughness and friction velocity for dilute mixtures as
k s
� ym
�w Um �
(�s – �L)gym Vt
6.19
= Er exp� –1 – K Um x �� (6-37)
Uf

6.20 CHAPTER SIX
where
E r � 30
K x = the Von Karman constant
Substituting Equation 6-37 into Equation 6-36 and using the effective shear stress
yields
CT = �[1 – (ks/ym) � ] ��
(6-39)
Equation 6-39 is plotted in Figure 6-6.
From test data, Celik and Rodi (1991) obtained a value where � = 0.034 and � = 0.06
for flow over a flat bed of loose sand without large dunes or antidunes. The particle diameter
was between 0.005 mm and 0.6 mm and volumetric concentration was limited to
10%. On a logarithmic scale, the slope he obtained was 0.034 in the range of C T from
k s
�
F = �1 – � � ym
�� � w
(�s – � L)gy m
FIGURE 6-6 The volumetric capacity C T from Celik and Rodi (1991). (Reprinted by permission
of ASCE.)
Um �
Vt
U m 2
��
(�s�� L – 1)gy m
U m
� Vt

0.00001 to 0.10. Nevertheless the authors pointed that the value of � is very empirical and
strongly dependent on the friction velocity U f. Unfortunately, the did not provide the correlation,
and this leaves the engineer facing a situation of measuring such a value or iterating
from case to case.
Example 6-4
Fine sand with an average particle diameter dp of 0.3 mm (0.000984 ft) and SG = 2.625 is
transported at a volumetric concentration CV = 7.91%. The volume flow rate is 1200
m3 /hr (42,378 ft3 /hr). The launder is 600 mm (1.97 ft) wide and the height of the liquid is
200 mm. Determine the slope of the launder using the Celik–Rodi method. Assume Kx =
0.33, � = 0.06, and � = 1.5 cP (or 3.13 × 10 –5 lbf-sec/ft2 ).
Solution in SI Units
Ct = = 0.07
The average speed is
Um = = (1200/3600)/(0.2 × 0.6) = 2.79 m/s
Assuming the roughness of the bed to be equal to twice the average particle diameter,
ks = 0.6 mm
From equation 6-38:
= Er exp� –1 – K Cv �
1.3
Q
�
A
ks Um � x �� ym
Uf
Using Equation 6-27:
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
2.79
= 30 exp � –1 – 0.33 �
Uf �
ln(0.001) = 1 – � 0.9207
�
U
U f = 0.112 m/s
U f = �g�R� H�S�
The hydraulic radius for this rectangular channel is
(0.6 · 0.2)
RH = �� = 0.12 m
(0.6+0.2+0.2)
U f = 0.1395 m/s = (9.81 × 0.12 × S) 1/2
S = 0.0107 or 1.07%
f
6.21

6.22 CHAPTER SIX
Solution in USCS Units
The average speed is
Cv Ct = � = 0.07
1.3
Area of flow = 1.97 × 0.656 = 1.293 ft 2
U m = = [(42378 ft 3 /hr)/3600]/1.293 = 9.1 ft/sec
Assuming the roughness of the bed to be equal to twice the average particle diameter,
If the depth is 8 in:
From Equation 6-38:
Using Equation 6-27:
Q
�
A
k s = 0.0236 in
= 0.0236/8 = 0.003
= Er exp� –1 – K Um x �
9.1
= 30 exp � –1 – 0.33 �
Uf �
ln(0.001) = 1 –
U f = 0.38 ft/sec
U f = �g�R� H�S�
The hydraulic radius for this rectangular channel is
6-2-3 Magnitude of ��
k s
� ym
k s
� ym
R H = (1.987 · 0.65)/(1.987 + 0.65 + 0.65) = 0.391 ft
U f = 0.38 ft/sec = (32.2 × 0.391 × S) 1/2
S = 0.011 or 1.1%
3.003
� Uf
Carstens (1952) demonstrated that � never exceeds unity (1.0). For fine particles, � � 1.0
or � s � �. For coarse particles, � < 1.0 or � s < �.
� Uf

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
Brush et al. (1962), Matyukhin an Prokofyev (1966) and Majumdar and Carstens
(1967) have confirmed that � = 1.0 or � s � � for fine particles. For coarse particles, � <
1.0 or .� s < �.
6-3 CRITICAL VELOCITY AND CRITICAL
SHEAR STRESS
6.23
Graf (1971) established a relationship between the forces for the incipient movement of a
set of loose, cohesionless solid particles and the angle of repose as
FT tan � = �
FN
where
FN = force normal to angle of repose
FT = force tangential to angle of repose
These two forces are the resultants of the lift and drag forces (discussed in Chapter 3) referred
to in Figure 6-7:
FN = W cos � – L
FT = W cos � + D
W cos � + D
tan � = ��
(6-40)
W cos � – L
The surface area resisting motion is expressed in terms of a shape factor � 1 and the
particle diameter d �. The surface area associated with lift is expressed in terms of a shape
factor � 2 and the particle diameter d �:
L = 0.5C L�U b 2 �2d � 2 (6-41)
D = 0.5C D�U b 2 �1d � 2 (6-42)
where U b is the bed velocity.
The submerged weight of the particle is expressed as a shape factor and the diameter
of the particle:
W = � 3gd � 2 (�s – �) (6-43)
Substituting Equations 6-40, 6-41, and 6-42 into 6-43 establishes the relationship between
the critical velocity and the actual shape and density of the particle:
U 2�3(tan � cos � – sin �)
= ���
(6-44)
CD�1 + CL�2 tan �
2 bc
��
(�s/�L – 1)
where U bc is the the critical bed velocity to start the motion of the particles.
Graf (1971) defined the right-hand part of Equation 6-44 as the sediment coefficient �.
Fortier and Scobey (1926) conducted extensive experiments on permissible canal velocities
to understand the erosion and transportation of sediments. Their results are presented
in Table 6-3. Their main conclusions which are still valid today, were

6.24 CHAPTER SIX
Drag
Lift
� The laws governing the transport of silt and detritus in open channels are very distantly
related to the laws governing scouring of the canal bed and are not directly applicable.
� The material of seasoned canal beds consists of solids of different shapes and sizes.
When the fines fill the interstices between the coarser solids, they form a dense and
stable mass that is more resistant to the erosion of water.
� The velocity required to scour the bedded canal is much higher than the velocity required
to suspend particles outside the bed.
� Colloids tend to cement clay, sand, and gravel in such a way that the compound mixture
resists erosion.
� The grading of materials ranging from fine to coarse, coupled with the adhesion between
colloids and these solids, makes it possible to operate at high velocities without
appreciable scouring effects.
Neill (1967) derived the following equation to estimate the critical velocity for coarse
particles in launders:
U 2 bc
��
(�s/� – 1)gd P
Weight
= 2.50 � � –0.20
(6-45)
The term d P/y m is sometimes called “relative sand roughness.”
In Chapters 3 and 4 we examined definitions of velocity during all periods of movement
from settling to deposition, etc. Similarly, in open channels the critical scour velocity is well
above the sedimentation velocity (or equal to the terminal velocity in full pipe flow). Sometimes
engineers confuse these two terms, although they are quite different in magnitude.
The critical shear stress is at the point of the incipient motion, or at which the motion
starts, and is expressed as
� cr
��
(�s – �)d �
Velocity
FIGURE 6-7 Lift and drag forces on sediments in an open channel.
= � (6-46)
where � is the the sediment coefficient.
Thomas (1979) argued that sand particles finer than 0.15 mm would be completely enveloped
in the viscous sublayer so that the critical shear stress could be simplified to
dP �
ym

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
6.25
�cr = 1.21 �m[g�(�s/�L – 1)] 2/3 (6-47)
With � = kinematic viscosity. Wilson (1980) indicated that this correlated well with the
work of Ambrose (1953), who had indicated a change of resistance as the grains of sand
became of the order of magnitude of the roughness of the flume.
Substituting Equation 6-47 into the DuBoys equation (6-28), the critical slope to initiate
motion of a bed of small sand particles in an open channel is
�w = �mgRHS = 1.21 �m[g�(�s/�L – 1)] 2/3
TABLE 6-3 Permissible Canal Velocities (after Fortier et al., 1925)
Velocity, after aging, of canals at a depth
of 910 mm (3 ft) or less
Water
transporting
noncolloidal
Water silts, sands,
Clear water, transporting gravel, or
no detritus
____________
colloidal silts
____________
rock fragments
_____________
Original material excavated for the canal m/s ft/s m/s ft/s m/s ft/s
Fine sand (noncolloidal) 0.45 1.5 0.84 2.5 0.45 1.5
Sandy loam (noncolloidal) 0.54 1.75 0.84 2.5 0.61 2.0
Silt loam (noncolloidal) 0.61 2.0 0.91 3.0 0.61 2.0
Alluvial silts when noncolloidal 0.61 2.0 1.07 3.5 0.61 2.0
Ordinary firm loam 0.84 2.5 1.07 3.5 0.69 2.25
Volcanic ash 0.84 2.5 1.07 3.5 0.61 2.00
Fine gravel 0.84 2.5 1.52 5.0 1.15 3.75
Stiff clay (very colloidal) 1.15 3.75 1.52 5.0 0.91 3.0
Graded loam to cobbles, when noncolloidal 1.15 3.75 1.52 5.0 1.52 5.0
Alluvial silts when colloidal 1.15 3.75 1.52 5.0 0.91 3.0
Graded, silt to cobbles, when colloidal 1.22 4.0 1.67 5.5 1.52 5.0
Coarse gravel (noncolloidal) 1.22 4.0 1.83 6.0 1.98 6.5
Cobbles and shingles 1.5 5.0 1.67 5.5 1.98 6.5
Shales and hard pans 1.83 6.0 1.83 6.0 1.5 5.0
TABLE 6-4 Effects of Sediment Load on Equivalent Roughness
(after Vanoni, 1946)
Ratio of equivalent roughness
Average sediment load in grams/liter to size of bottom sand
0 0.328
0.17 0.282
3.21 0.190
7.36 0.110
16.2 0.072

6.26 CHAPTER SIX
The critical slope to start motion is
1.21[�(�s/�L – 1)]
Scrit = (6-48)
2/3
���
RHg1/3 Equation (6-47) confirms the correlation between the average depth of the slurry, the
weight of solids, the viscosity and density of the mixture, and the minimum slope.
Example 6-5
A slurry mixture of fine particles and water has a specific gravity of solids 4.1 and is
flowing in a partially filled rectangular launder 600 mm wide. Determine the critical slope
if the launder is to flow one-third full. The density of the mixture is 1250 kg/m3 (2.42
slugs/ft3 ) and the dynamic viscosity is 20 mPa · s (4.17 × 10 –4 lbf-sec/ft2 ).
Solution in Metric Units
The kinematic viscosity of the mixture is 20 × 10 –3 /1250 = 0.000016 m2 /s. If the launder
is one-third full, the hydraulic radius is
From Equation 6-47:
0.6 × 0.2
RH = � = 0.12 m
0.6 + 0.4
1.21[16 × 10
Scrit = = 0.0063 or 0.63%
–6 (4.1 – 1)] 2/3
���
0.12 × 9.811/3 1.21[�(�s/�L – 1)] 2/3
��
RHg1/3 is the minimum slope to start motion of the slurry in these conditions.
Solution in USCS Units
The width of the launder is 1.97 ft, the height of the liquid would be 0.66 ft, and the hydraulic
radius is
The kinematic viscosity of the mixture is
From Equation 6-47
4.17 × 10 –4 lbf-sec/ft 2
���
2.42 slugs/ft 3
1.97 × 0.66
RH = �� = 0.395 ft
1.97 + 0.66
= 1.723 × 10 –4 lbf-sec-ft/slugs
1.21[1.723 × 10
Scrit = = = 0.0063 or 0.63%
–4 (4.1 – 1)] 2/3
����
0.395 × 32.21/3 1.21[�(�s/�L – 1)] 2/3
���
RHg1/3 is the minimum slope to start motion of the slurry in these conditions.
This analysis was extended by Wilson (1980) for sand flowing in partially filled pipes.
Wilson defined three regimes for sand flowing in open launders with particle diameters in
the range of 0.02 mm to 4 mm (mesh 625–5):

1. Homogeneous flow occurs at a slope S < 0.0006/D I
2. Optimum slope for heterogeneous flow S = 0.10(d p/1000) 1.5 D I –0.7
3. Fully blocked pipe occurs at S > 0.42
For rocks with solid specific gravity between 1.5 and 6.0, Wilson (1980) extended the
analysis and stated, for homogeneous flow:
or heterogeneous flow:
for blocked flow:
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
S1 � × 10 –3 0.6 [�(�s/�L – 1)]
(6-49)
0.6
���
DI 1650
31.6 × 10
S1 < S2 � (6-50)
–3 1.5 d p [(�s/�L – 1)] 1.2
����
0.42[(�s/�L – 1)]
S3 � (6-51)
0.35
��
Example 6-6
A slurry mixture of fine particles and water of d85 = 0.08 mm with a density of solids
4100 kg/m3 is flowing in a partially filled circular pipe with an inner diameter of 438 mm
(17.25 in). The pipe inner diameter is 438 mm (17.25 in). Determine the minimum slope
for flow as a heterogeneous mixture and the slope for a blocked pipe.
Solution from Equation 6-50
For heterogeneous flow:
S 2 �
S 2 =
S 2 = 0.086 or 8.6%
The slope for a blocked pipe is determined from Equation 6-51 as 52.4%.
6-4 DEPOSITION VELOCITY
D I 0.7 1.65 1.2
1.65 0.35
31623d p 1.5 [(�s/� L – 1)] 1.2
���
D I 0.7 1.65 1.2
316230.00008 1.5 [(4.1- 1)] 1.2
���
0.438 0.7 1.65 1.2
6.27
The terrains over which tailing flumes are built do not always have an appropriate slope.
If the flow operates in a subcritical flow regime, the engineer must calculate a realistic estimation
of the deposition velocity. Dominguez et al. (1996) published an equation based
on experimental data measured at Codelco and the Chilean Research Center of Mining
and Metallurgy. For cases where the viscosity effects are negligible

6.28 CHAPTER SIX
� � 0.158
VD = 1.833� � 1/2
8gRH (�S – �m) d85 ��
(6-52)
However, in cases where the dynamic viscosity of the carrier liquid is instrumental,
such as with alkaline water, Dominguez et al. (1996) derived the following equation:
VD = 1.833� � 1/2
8gRH (�S – �m) ��
where
J = R H(gR H) 1/2 /� m
� m = the absolute viscosity of the mixture
� � 0.158 d85 1.2 (3,100/J) (6-53)
A comparison between the deposition velocity as calculated by Equation 6-52 and experimental
data is presented in Figure 6-8.
Example 6-7
A slurry mixture of coarse particles of d 85 = 12 mm with C w = 40%, density of solids 4100
kg/m 3 , and a specific gravity 4.1 is flowing in a circular pipe with an inner diameter of 438
mm (17.25 in). Assuming that the pipe is to be half full, determine the deposition velocity.
Solution
Using Equation 6-51:
�m
�m
� RH
� RH
R H = D/4 = 0.438/4 = 0.1105 m (4.35 in)
� m = 1000/[1 – 0.4(4100 – 1000)/4100] = 1433kg/m 3
FIGURE 6-8 Correlation between calculated and experimental measurements of the deposition
velocity of coarse slurries. (From Dominguez et al., 1996, reprinted by permission of BHR
Group.)

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
V D = 1.833[8 × 9.81 × 0.1105 (4100 – 1433)/1433] 0.5 (0.012/0.1105) 0.158
VD = 1.29 × 4.0174 = 5.183 m/s
Green et al. (1978) and the ASCE and WPCF (1977) proposed that Camp’s equation
for the self-cleaning speed of sewers be used in the design of slurry launders:
Vsc = � dpg� �� 1/2
8Ke �s – �m � �
fD �m
(6-54)
where
Ke = an experimental constant (the ASCE recommends a constant of 0.06 for grit chambers,
whereas Green recommends a constant of 0.8 for sewers)
Vsc is expressed in m/s (ft/s)
d� is expressed in m (ft)
g = 9.81 m2 /s (32 ft/s)
To use this equation, one must first determine the Darcy friction factor and then solve
by iteration. Because the average speed is a logarithmic distribution of depth, it would be
wise to design launders with a mean speed equal to twice the self-cleaning speed.
Example 6-8
Calculate the self-cleaning speed of Example 6-1, assuming a sewer flow where K e = 0.8,
solids at a SG of 3.1, C w = 20%, and d p = 2 mm.
Solution in Metric Units
In Example 6-1, the Darcy factor was calculated to equal f D = f N × 4 = 0.027.
� m = 1000/[1 – 0.2(3100 – 1000)/3100] = 1157 kg/m 3
V sc = [8 × 0.8 × 2 × 10 –3 × 9.81(3100 – 1157)/(1157 × 0.027)] 1/2
V sc = 2.79 m/s
Solution in USCS Units
In Example 6-1, the Darcy factor was calculated to equal fD = fN × 4 = 0.027.
dp = 6.56 × 10 –3 ft
Sm = 1/[1 – 0.2(3.1 – 1)/3.1] = 1.157
Vsc = [8 × 0.8 × 6.56 × 10 –3 × 32.2(3.1 – 1.157)/(1.157 × 0.027)] 0.5
V sc = 9.2 ft/s
6-5 FLOW RESISTANCE AND FRICTION
FACTOR FOR HETEROGENEOUS
SLURRY FLOWS
6.29
Equations 6-6 to 6-10 established the parameter for friction loss of single-phase Newtonian
flows in open channels. Equation 2-25 established the correlation between friction
factor and friction velocity. Two approaches are often used by designers of launders; one
is based on the friction factor and the other on the velocity.

6.30 CHAPTER SIX
6-5-1 Flow Resistances in Terms of Friction Velocity
Liu (1957) and Acaroglu (1968) indicated that the ratio of the friction velocity to the settling
speed of a particle were implicitly related and proposed the following function:
Uf d�Uf � = fct��� (6.55)
Vt �
Graf (1971) proposed including the Froude number to reflect the degree of turbulence:
Uf d�Uf Uav � = fct� � ’ �
� �(g�ym�)� �
Vt
where y m is the average depth of the fluid. This is confirmed in studies by Garde and Dattari
(1963) and Bogardi (1965).
The presence of sand dunes at the bottom of a channel with a typical wavelength � is a
function of the Froude number (Kennedy, 1963). As shown in Figure 6-9, antidunes are
formed in the regime of critical flow (0.8 < Fr < 1.5).
The presence of dunes increases the effective wall shear stress in the form of profile
drag. Graf (1971) proposed to establish a hydraulic radius R H� due to the grain roughness
and a separate value R H�� based on the bed forms so that:
� w = �gS(R H� + R H��)
where S is the physical slope. He defined a friction velocity U f as a combination of the
component due to grain roughness U f� and due to the bed form as U f��:
Froude numbe number Fr
F
2.8
2.4
2.0
1.6
1.2
0.8
0.4
U f 2 = Uf� 2 + U f�� 2
antidunes
dunes
0.0
0 2 4 6 8 10 12 14 16 18
� = wavelength
2 * * D/
FIGURE 6-9 Wavelength of dunes and dunes versus the Froude number in open channel
flows. (After Kennedy, 1963.)

6-5-2 Friction Factors
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
6.31
The previous paragraph demonstrates that the presence of the bed forms (dunes and antidunes)
tends to increase the friction velocity and therefore the friction factors.
6-5-2-1 Effect of Roughness
There is a dearth of information on the effect of roughness factors on slurry flow in closed
or open conduits. The Ministry of Technology of the United Kingdom (1969) recommended
modifying the Colebrook equation by using the hydraulic radius for the single-phase fluids.
Green et al. (1978) concurred with this approach and proposed that the Darcy friction
factor be obtained from the following equation by replacing the diameter with four times
the hydraulic radius and using the Reynolds number based on the hydraulic radius:
1
ks 2.51
� = –2 log10� � + �� (6-56)
�fD�
14.8RH Re �fD�
where
ks = the linear roughness (measured in the same units as the hydraulic radius, e.g., meters)
Re = 4RHV�/�, the Reynolds number expressed in terms of the hydraulic radius
This definition of the linear roughness is difficult to calculate. In a fast flow, the
roughness of the pipe or channel wall may be used. Attempts have been made to define a
(Nikuradse) sand roughness for closed conduits, such as the ratio of the particle diameter
to the inner diameter of the conduit (dp/DI) but very little has been published for open
channels. The problem is far from simple, and the roughness is often taken as twice the
grain diameter. The presence of dunes at low Froude number tends to complicate the picture
by introducing another parameter for roughness.
Example 6-9
Considering Example 6-1, assume that the roughness is 0.0045 mm. Reiterate the friction
factor using Equation 6-56. Assume �m = 1350 kg/m3 and � = 2.8 cP.
fD = 0.027
RH = 0.1095 m
V = 2.51 m/s
Re = �V × 4RH/� = 1350 × 2.5 × 4 × 0.1095/2.8 × 10 –3
Re = 131,987
Iteration 1
0.027 –0.5 = –2log10[0.0045/(14.8 × 0.1095) + (2.51/(131,987 · 0.0270.5 )]
6.085 � 5.077
Iteration 2
Assume fD = 0.038, then
0.038 –0.5 = –2log10(2.777 × 10 –3 + 3.707 × 10 –6 )
5.13 � 5.11
Therefore the Darcy factor is iterated to 0.038.
6-5-2-2 Effect of Particle Concentration on Slurry Viscosity
Green et al. (1978) proposed to incorporate the effect of particle concentration in the form
of increased effective viscosity by using the Einstein–Thomas equation (Equation 1-9) to

6.32 CHAPTER SIX
define an effective viscosity. This effective viscosity is then used to compute the Darcy
factor by using the hydraulic radius in the Colebrook equation. This approach is, however,
essentially limited to Newtonian slurries in pseudohomogeneous flow well above the
deposition velocity. This approach has been covered by Equation 6-53. It does not take in
account any dunes or partial deposition at the bottom of the bed . It is, however, a useful
and straightforward approach for flows at supercritical Froude number.
6-5-2-3 Effects of Particle Sizes on the Chezy Coefficient
Richardson et al. (1967) derived the following equations. For a plane with little or no sediment
transportation:
= 5.9 log� � + 5.44 (6-57)
For a plane with appreciable sediment transport:
= 7.4 log � � (6-58)
For ripples (in English units):
= �7.66 – � log Ch
ym � �
�g� d85
Ch
ym � �
�g�
d85
Ch 0.3 ym 0.13
� � � � � + � + 11 (6-59)
�g� Uf d85 Uf
For dunes and antidunes:
Ch
�
�g�
ym �
d85
�RHS �
RHS
where �RHS is the increase of RHS due to the form roughness.
= 7.4 log � ��� 1� –����� (6-60)
Example 6-10
A launder is designed to transport appreciable coarse sediments over a plane with d85 =
4 mm. The height of the slurry must be limited to 150 mm. Using Equation 6-60, determine
the required slope if the flow rate is 850 m3 /hr and the width of the channel is
450 mm.
Solution
Ch/�g� = 7.4 log(0.15/0.004) = 11.65
Ch = 36.5 m1/2 /s
Area of flow = 0.15 × 0.45 = 0.0675 m2 Q = 850 m3 /hr = 0.236 m3 /s
V = 3.498 m/s
fD = 8g/Ch2 = 8 × 9.81/36.52 = 0.0589
Slope = fDV 2 /8gRH RH = A/P = (0.15 × 0.450)/(0.450 + 0.15 + 0.15)
RH = 0.793 m
S = 0.0589 × 3.4982 /[8 × 9.81 × 0.793] = 0.0116 or 1.16%

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
6-5-2-4 Effect of Bed Form on the Friction
Graf (1971) discussed the importance of Equation 6-52 and proposed to write the total
friction factor for flow in a channel in the presence of dunes or bed forms. He suggested
the following equation for the overall friction factor:
fD = f D� + f D�� (6-61)
where
f D� = Darcy friction factor for the channel without bed forms
f D�� = Darcy friction factor due to bed forms
In concordance with Silberman (1963) and Vanoni and Hwang (1967), Graf (1971) indicated
that f D� can be estimated from conventional pipe equations by substituting the diameter
of the pipe with four times the hydraulic radius. This has already been presented in
Equation 6-54. A similar approach was developed by Lovera and Kennedy (1969).
From lab tests, Vanoni and Hwang (1967) derived the following equation for f D��:
= 3.5 log � – 2.3 (6-62)
e�Hav
where
�Hav = mean height of the bed form
e = A/Ab = (where A = total area and Ab = the horizontal projection of the lee face of
the bed forms)
When the magnitude of “e” cannot be determined, Equation 6-62 can be written in
terms of the wavelength of the bed form as:
1
1
�
�f��� D�
�
�f��� D�
= 3.3 log � – 2.3 (6-63)
(�Hav) 2
where � is the length of the dune. In this section, the importance of dunes was well emphasized.
The designer of a slurry flume should avoid these troublesome regimes by designing
for supercritical flows wherever the topography allows it.
6-5-3 The Graf–Acaroglu Relation
Starting from basic principles of lift and drag forces and buoyancy and weight on a solid
particle, Acaroglu (1968), Graf and Acaroglu (1968) proceeded to develop a methodology
that applies equally well to both closed conduits and open channels. They considered
that the drag force was a function of the Reynolds number based on the bed velocity Ub and the shape factor �1: 2 2
D = 0.5CD�LUb�1d �
However, they chose a different shape factor �D for the drag coefficient:
Ubdp CD = f1� � , �D� (6-64)
�
The bed velocity for the solids-water mixture is expressed as
Uf ks �
�
R H
�R H
y
6.33
Ub = Uf f2� , CV, �� (6-65)
ks

6.34 CHAPTER SIX
where
ks = the Nikuradse’s equivalent sand roughness (d�/DI) CV = the volumetric concentration of solids
The shear friction velocity is expressed as
Uf = � = �R� �� H�S�g� (6-66)
�
By assuming that the absolute roughness of the bed is equal to the particle diameter, Acaroglu
and Graff proceeded to define the shear intensity parameter as
(�s – �L)d� �A = ��
(6-67)
At a critical value of the shear intensity parameter � Acr, the shear stress is equal to the
critical shear stress previously discussed. When � A > � Acr or � w < � cr, no movement of
sediments occurs. When � A < � Acr or � w > � cr, movement of sediments occurs.
The power consumed with friction or head losses in the open channel is expressed in
terms of the energy slope (head loss per unit length) and a nondimensional transport parameter
is derived as
�A = ��
(6-68)
By examining data from various authors and by regression analysis, Graf and Acaroglu
extrapolated the following relationship:
�A = 10.39 (�A) –2.52 (6-69a)
or
C VU avR H
��
�(��s/��� L�–� 1�)g�d� � 3 �
�LSR H
CVUavRH 3 �(��s/��� L�–� 1�)g�d� p� = 10.39� � –2.52
(�s – �L)d� ��
�LSR H
(6-69b)
Equation 6-66 was obtained for finely graded sand with a particle diameter between
0.091 mm and 2.70 mm (0.0036 – 0.1063 in) and was studied in rivers and open channel
flumes. This equation applied to both closed conduits and open channels (Figure 6-10).
Graf (1971) pointed out that this equation was based on extensive data that was often difficult
to analyze. For some unknown reasons, there has not been much research since
1970 to refine the Graf–Acaroglu equation. This is probably due to the fact that practically
all research on slurry flows tends to limit itself to full pipes.
Example 6-11
A mine is located at high altitude. The tailings need to be transported by gravity over a
long distance. The estimated particle size is 6 mm. The volumetric concentration is set at
25%. The specific gravity of the solids is 3.1. The carrier liquid is water. The channel is
rectangular in shape with a width of 750 mm. Assuming an average speed of 2 m/s, determine
the minimum slope for a flow rate of 1100 m 3 /hr using the Graf–Acaroglu method.
Solution in Metric Units
Q = 1100 m3 /hr or (1100/3600) = 0.3055 m3 /s
For a speed of 2 m/s, the height of the liquid would be:
0.3055/(2 × 0.75) = 0.204 m
� w

A
10.0
1.00
0.10
S
L
100.0
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
S R
0.01
The hydraulic radius is
RH = (0.75 × 0.204)/(0.75 + 2 × 0.204) = 0.132 m
From Equation 6-68:
�A = = = 0.188
From Equation 6-69a:
–2.52 �A = 10.39 � A or �A = 4.911
From Equation 6-67:
4.911 × 0.132 = 3.1 × 6 × 10 –3s –1
0.25 × 2 × 0.132 0.0662
��� �
�[( �3�.1� –� 1�)�9�.8�1� ×� 6� ×� 1�0� 0.3515
–3 �]�
S = 0.0287 or 2.87%.
The Graf-Acaroglu method is very useful to determine the bed depth of a full pipe
with saltation. Equation 4-43 of Chapter 4 uses this method.
6-5-4 Slip of Coarse Materials
L
H
d
p
0.1 1.0
6.35
Kuhn (1980) conducted velocity measurements on transportation of coarse coal in
trapezoidal flumes under controlled laboratory conditions. He reported slip between the
coarse solids and liquid. At the inclination of 2.5°, the speed of the coarse material was
of 10% slower than the liquid speed of 4.5 m/s, but it decreased gradually to a slip of
8.5% of the speed of 6 m/s at an inclination of 6°. Such a degree of slip is reminiscent
10
A
100
C U R
V
S L
FIGURE 6-10 The Graf–Acaroglu relationship for flow of sand in open and closed channels,
assuming a particle roughness equal to the grain size. Adapted from Graf and Acaroglu, 1968.
H
1000
g d
3
p

6.36 CHAPTER SIX
of the two-layer models of heterogeneous flow extensively discussed in Chapter 4 for
full pipe flows.
6-5-5 Comparison between Different Models
Blench et al. (1980) conducted a comparison between the different equations to calculate
the sediment discharge versus the water discharge in open channels and plotted them as in
Figure 6-11. It is obvious that different equations yield different results. The sediments
are transmitted in different patterns. When the flow is tranquil (Froude number Fr < 1),
two kinds of sand waves may develop: dunes and ripples. They are similar in shape, with
an upstream surface and a gentle and gradually varying slope, finishing with an abrupt
downstream slope (Figure 6-12). Although similar in shape, ripples are independent of
the magnitude of flow, whereas dunes are strongly dependent on flow.
At Froude number larger than unity (sometimes referred to as supercritical flows), the
flow suppresses the formation of bed forms. Anti-dunes, which are more symmetrical than
ripples, form. They move in the same direction as the flow or even opposite to the flow.
FIGURE 6-11 Comparison between the different models for transport of sediments in open
channels. (From ASCE, 1975. Reprinted by permission of ASCE.)

y m
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
Flow direction
wavelength
liquid level
FIGURE 6-12 Sand dunes at low Froude number (Fr < 1).
Anti-dunes accentuate the deformation of the free surface (Figure 6-13). They do not occur
in closed conduits and their motion is at a lower speed than the fluid.
Tournier and Judd (1945) reported that the specific gravity of the ore is an important
factor to consider. Heavier ores require more slope to be transported in an open channel, as
shown in Figure 6-14. Tournier and Judd (1945) reported that the size of the particles play
an important role, and that larger particles require more slope, as shown in Figure 6-15.
Figures 6-12 and 6-13 clearly demonstrate that it may be erroneous to use conventional
Manning formulae for water flow depending on the roughness of the pipe, as these ignore
the resultant roughness due to sand dunes and anti-dunes. Figures 6-14 and 6-15 clearly in-
y m
Flow direction
wavelength
liquid level
FIGURE 6-13 Sand anti-dunes at high Froude number (Fr > 1).
6.37

6.38 CHAPTER SIX
specific gravity = 3.8
specific gravity = 2.7
slope of launder (%)
0 20 40 60 80
Weight concentration (%)
FIGURE 6-14 The slope is a function of the specific gravity of the ore as well as the weight
concentration (after Tournier and Judd, 1945). The magnitude of the slope is not shown here as
it depends also on the hydraulic radius or shape of the launder.
0 4 8 12 16 18 20
particle size (mm)
slope of launder (%)
FIGURE 6-15 Larger particles require more slope (after Tournier and Judd, 1945). The
magnitude of the slope is not shown here as it depends also on the hydraulic radius or shape of
the launder.

dicate that the density of the ore as well as the size of the solids do increase the slope. These
important factors are unfortunately too often ignored by some engineers who rely on the
Manning equation.
6-6 FRICTION LOSSES AND SLOPE FOR
HOMOGENEOUS SLURRY FLOWS
Green et al. (1978) attempted to estimate the effect of particle sizes and concentration in
the form of increased effective viscosity. In this approach, however, they essentially limited
their studies to Newtonian slurries and did not take into account the effects of yield
stress and plasticity, effects often encountered at high concentrations of fine particles.
Geophysicists prefer to talk about cohesive bed forms when exploring the movement
of clays in rivers. For certain soils, cohesive forces develop between the solid particles.
Graf (1971) indicated that Equation 6-43 should be modified to include X0, a coefficient
of cohesion of the material.
For a Bingham slurry, the yield stress is added to Equation 6-45 to express the critical
shear stress at the point of the incipient motion:
�cr �� = � + X0 (6-70)
(�s – �)gd� Cohesive (soil) materials include clay-sized (colloidal) particles, silt-sized particles,
and sometimes sand-sized particles. Graf (1971) classified clays into the following three
main categories:
1. Kaolinites
2. Montmorillonites
3. Illites
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
6.39
Other more minor clays include halloysites, chlorites, and vermiculites.
Clay materials have residual electrostatic forces that attract cations and anions. This is
measured as a cation exchange capacity in milliequivalents per 100 grams. Grim (1962)
stated that Kaolinites have an exchange capacity of 3–15 milliequivalent per 100 grams,
whereas illites rated higher at 10–40, and montmorillonites at 80–150.
In very simple terms, Grim explained that there are two main structures for clays. One
structure consists of two close sheets of packed hydroxyl molecules, in which aluminum,
iron, and magnesium atoms are embedded in an octahedral coordination equidistant from
the six oxygen atoms in the hydroxyls.
The second structure consists of silica tetrahedrons. In the tetrahedron, the silicon
atom is equidistant from the hydroxyls. The silica tetrahedral groups are arranged to form
a hexagonal network, which is repeated from sheet to sheet.
It would be beyond the scope of this book to discuss the physical aspects of clays. The
slurry engineer should appreciate that these electrostatic forces and arrangement of chemical
groups influence the yield stress at certain concentrations that may be at either the
lower end or the higher end.
Cohesive soils have the ability to absorb water and develop plasticity; they also have
limit liquid. All three characteristics were briefly defined in Chapter 1. Cohesive colloids
can form lumps. Forces on such lumps are shown in Figure 6-16.
Graf (1971) summarized the test work of Karasev (1964) who proposed that the erosion
of cohesive material beds in rivers is due to aggregate-to-aggregate rather than by

6.40 CHAPTER SIX
Drag + Cohesion forces
particle-to-particle contact. Karasev (1964) derived the following equation for the average
scouring velocity:
Ucr = 0.142 Ch� �1.2 + 8 �� 0.5
2d50(�s – �L) + 3� ym ���
(6-71)
where
� = information about cohesion
Patm = atmospheric pressure
A comparison between the computed and measured values of a scouring velocity is
presented in Table 6-5. This value of critical speed should be considered the speed for
minimum transportation of clay by hydrotransport.
The flow of water in canals containing sand, cohesive soils, and cohesive banks was
examined by Graf (1971) on the basis of the work of Simons et al. (1963). The graph in
Figure 6-17 suggests that the hydraulic radius can be correlated to the flow rate by the following
equation:
RH = 0.43 · Q0.361 6-6-1 Bingham Plastics
�L
Lift Force
Weight Force
Whipple (1997) developed numerical models for open channel flow of Bingham fluids
but did not provide a methodology to calculate friction losses. This paper is important for
a geophysicist but provides no useful tools for the slurry engineer.
For a homogeneous slurry, there are two important numbers to calculate: the Reynolds
number and the plasticity number (defined in Chapter 5). In the absence of well-defined
models for friction losses of Bingham slurries, Abulnaga (1997) proposed a methodology
to modify some of the equations of full-pipe flows by expressing the Reynolds and Hedstrom
numbers in terms of the hydraulic radius. At high shear rate, the coefficient of
rigidity is taken as the viscosity for a Bingham plastic:
Re = 4RHV�/� (6-72)
� Patm
Cohesion Force
FIGURE 6-16 Forces on a colloidal lump moving in an open channel.

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
TABLE 6-5 Scouring Velocity of Clays (after Karasev, 1964)
6.41
Scouring velocity in streams
______________________________
Diameter of Karasev’s
aggregate Adhesion � Experimental formula
______________ _____________ _______________ _____________
# Material mm in kPa psi m/s ft/sec m/s ft/sec
1
Aggregate materials
Medium loam 2.2 0.087 225.6 32.7 1.74 3.28 1.83 6
2 Rammed clay 3.0 0.118 550 80 2.16 7.09 2.82 9.3
3 Rammed clay 4.0 0.157 202 29.3 2.06 6.76 2.36 7.74
4 Heavy loam 2.7 0.106 373 54 2.04 6.69 2.22 7.28
5 Heavy loam 4.0 0.157 225.6 32.7 1.20 3.94 1.70 5.58
6 Clay 4.0 0.157 245 35.5 2.20 7.22 1.75 5.74
7 Clay 4.0 0.157 285 41.3 1.30 4.26 1.86 6.1
8 Clay 6.0 0.24 196 28.4 1.43 4.69 1.45 4.76
9 Clay 4.0 0.157 285 41.3 1.54 5.05 1.86 6.10
10 Compacted clay 0.8 0.03 510 73.9 2.23 7.32 3.20 1.05
11 Heavy loam
Dispersed materials
1.0 0.04 304 44.1 2.14 7.02 2.35 7.71
12 Medium loam 4.0 0.157 746 108 1.91 6.27 2.88 9.45
13 Clay 1.5 0.059 706 102 3.06 10.04 2.68 8.79
14 Clay 3.2 0.126 785 114 2.35 7.71 2.4 7.87
15 Clay 4.0 0.157 432 62.6 2.87 9.42 1.93 6.33
16 Clay 0.8 0.03 785 114 2.40 7.87 2.42 7.94
17 Clay 4.0 0.157 549 79.6 1.54 5.05 2.50 8.2
FIGURE 6-17 Correlation between the hydraulic radius and the discharge flow rate. (From
Graf, 1971, reprinted by permission of McGraw-Hill.)

6.42 CHAPTER SIX
And the Plasticity number is written in terms of the hydraulic radius as
� 04R H
PL = � (6-73)
�V
The Hedstrom number is the product of the Plasticity number and the Reynolds number
and is calculated as
16R H 2 ��0
He = � (6-74)
2 �
In Chapter 3, the different categories of non-Newtonian flows were reviewed. Methods
to calculate friction losses for Bingham slurries and power law slurries were presented
in Chapter 5. Modified Reynolds and Hedstrom numbers for pseudoplastics were also
introduced in Chapter 5.
In Chapter 5, the Buckingham equation was introduced as
He
= – + (5-5)
Modifying it for an open channel in laminar flow yields
4
1 fNL He
� � � � 2 3 8
ReB 16 6 ReB 3 f NLReB � –
� – (6-75)
The Darby equation for the friction factor in turbulent regime is
fNT = 10a 2 16RH��0/� � fNL �0 � � �2 4RHV� 16 6V �m
b ReB 2
��
6(4RHV�m/�) 2
� fNL � �
4RHV�m 16
where
a = –1.47[1 + 0.146 exp(–2.9 × 10 –5 He)]
b = –0.193
The values of the parameters “a” and “b” were based on empirical data for closed conduits.
They may be tentatively modified to open channels to yield
4R HV� m
fNT = 10a� � b
(5-10)
where
a = –1.47 [1 + 0.146 exp{–2.9 × 10 –5 2 (16RH �m�0/�2 )}]
b = –0.193
Equations for the empirical parameters “a” and “b” should be confirmed by extensive
testing in open channels. It is unfortunate that very little research is conducted in this extremely
important field of fluid dynamics.
Darby et al. (1992) proposed to combine the laminar and turbulent fanning friction
factors into the following equation:
fN = (f m NL + f m NT) (1/m) �
�
(5-11)
where
40,000
m = 1.7 + � (5-12)
ReB

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
For an open channel this becomes
10,000�
m = 1.7 + (6-76)
� RHV� m
6.43
Example 6-12
The soft high clay tailing from a copper concentrator develops a Bingham viscosity of
400 mPa · s at a weight concentration of 45% as well as a yield stress of 5 Pa. The slurry
density is 1350 kg/m3 . The tailings flow rate is 1600 m3 /hr. If the maximum allowed
slope is 1.5%, determine the suitable size for a half-full smooth HDPE pipe (i.e., ignore
roughness factor).
Solution in SI Units
Iteration 1
Assume a speed of 1.8 m/s. Required area for flow is
A = (1600/3600)/1.8 = 0.247 m2 Pipe ID = [(8/�)A] 0.5 = 0.789 m (31.09 in)
� = 0.4 Pa · s
RH = DI/4 = 0.789/4 = 0.19725 � 0.198 m
Re = 4RH�mV/� = 4 × 0.198 × 1350 × 1.8/0.4 = 2673
He = 16 × 0.1982 × 1350 × 5/0.42 = 26,463
fL � �1 + � � �1 + �
fL � 0.0158
a = –1.47[1 + 0.146 exp(– 2.9 × 10 –5Re)] a = –1.7
fT = 10aRe0.193 = 0.00435
m = 1.7 + 40,000/Re = 16.67
m m 1/m fN = ( f L + f T ) = 0.016
Using Equation 6-5:
S =
S = (0.016 × 1.82 fNV )/(2 × 9.81 × 0.198) = 0.0132
S = 1.32%
This approach was used by Abulnaga (1997) at Fluor Daniel Wright Engineers to design
a tailings launder (see Figure 6-1) to transport tailings rich in soft high clay for a Peruvian
copper mine. The tailings system functioned well.
The Wilson–Thomas method for full flow in closed channels does not rely on empirical
coefficients such as the Darby method, but is based on the assumption that a thick sublayer
lubricates the wall surface of the pipe. It has not been modified yet for open channel
flows. This is a topic well worth further research.
2
16 He 16 26,463
� � � �
Re 6Re 2673 6 × 2673
�
2gRH

6.44 CHAPTER SIX
6-7 FLOCCULATION LAUNDERS
Hydraulic flocculation is sometimes conducted in launders feeding a thickener. The G
gradient is a measure of the average shear rate for a flocculation tank. For a tank flocculated
using a mixer, Camp (1955) defined the G gradient as:
G = (6-77)
��
where
P = power applied by the mixer
Vol = volume of the liquid in the tank
� = average viscosity of the solution
Camp (1955) as well as the American Society of Civil Engineers (1977) modified this
equation for launders, conduits, and flocculation basins by proposing that the power term
P be replaced by the power due to friction in the launder:
P = �Qg�H (6-78)
where
� = density of the mixture
g = acceleration due to gravity
Q = flow rate
�H = head loss due to friction
As a measure to control �H, ASCE proposed to install a baffle system in the flume.
They defined this process as hydraulic flocculation:
G = �� = (6-79)
��
Assuming that no baffles are installed in the launder, it would be possible to express
the volume V as the product of the wetted area by the length of the launder:
Vol = AL
Assuming that the slope S of the launder is equal to the energy gradient (as per Equation
6-5), or friction loss per unit length (or S = �H/L), then
G = �� = P
�
Vol�
P �Qg�H
� �
Vol� V�
�Qg�H �QgS
� � (6-80)
V�A �� A�
Equation 6-80 is valid only if no baffles or other artificial means are added to the launder
to increase friction losses.
Example 6-13
A tailing is flowing to a thickener. In-line flocculation is applied to raise the viscosity to
20 mPa · s. The flow rate is 1800 m3 /hr. The density of the mixture is 1420 kg/m3 . The
launder is rectangular with a slope of 1.2%. The chemical process requires that the G gradient
be smaller than 100. Determine a suitable size for the launder.
Solution
From Equation 6-79:
100 = [1420(1800/3600) 9.81 × 0.012/(A × 0.02)] 0.5

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
100 = 64.66 (1/A) 0.5
A0.5 = 0.646 m
A = 0.4179 m2 If the depth of the liquid is one-third the width of the launder,
A = w2 /3
W = 1.12 m
Depth = 0.373 m
Velocity = Q/A = 0.5 m3 /s/0.4179 m2 = 1.196 m/s
6-8 FROUDE NUMBER AND STABILITY OF
SLURRY FLOWS
A measure of the stability of a flow in an open channel can be expressed in the form of the
Froude number (Fr), a ratio of inertia to gravity forces:
V
Fr = � (6-81)
�g�ym�
In the case of slurry flows, it is important to avoid subcritical flows (Fr < 0.8), as they
often cause settling problems. Critical flows (0.8 < Fr < 1.5) may be associated with a degree
of instability and wavy motion, leading in some cases to working problems and overflows.
Kennedy (1963) reported test work on sand and suggested that antidunes occur at a
Froude number of 0.8 to 1.4 (critical regime). Green et al. (1978) recommended that slurry
launders be designed for Froude numbers in excess of 1.5, to avoid regimes of instability
of flow.
On the other hand, excessively high Froude numbers (Fr > 5) are associated with
steep slopes. Steep slope instability was discussed by Niepelt and Locher (1989). Slug
flow is reported at high Froude numbers, causing working instability in the form of roll
waves.
Although economics may often dictate maintaining gentle slopes of < 2% on many
long-distance tailings projects, the design of plants must too often accommodate tight
spaces. The use of steep slopes (> 8%) may often cause high speeds, in excess of 6 m/s
(20 ft/s). To avoid premature wear of liners or pipes, excessive slopes in plants should be
avoided. If it is not possible to avoid steep slopes, drop boxes, pressurized boxes, chokes,
or full-flow closed conduits should be used wherever wear is a major concern.
There are certain conditions in an open channel that may lead to a sudden change from
a supercritical regime to a subcritical regime. This is often associated with the so-called
“hydraulic jump,” an increase in the depth of the liquid due to the lower speed of motion.
6-9 METHODOLOGY OF DESIGN
6.45
The design of open launders is complex, with many implicit functions. Abulnaga (1997)
suggested starting the calculations assuming a speed of 2 m/s (6.56 ft/s). For many types
of slurries, this is a good starting point. The flow rate is divided by the assumed speed to

6.46 CHAPTER SIX
obtain an area of the flow A. If a pipe is selected, the flow area A is taken as � 45–55% of
the cross-sectional area of commercial pipes to obtain a pipe diameter.
1. If a rectangular launder is to be manufactured, the width is used to compute the height
of the liquid. For these launders, the height of the liquid is assumed to be 1/3 the width.
2. For the area of the flow, the perimeter and hydraulic radius are then obtained.
3. The Froude number is then calculated. If it appears that the flow will be in a subcritical
or critical regime, the speed should be increased.
4. Repeat steps 1 to 4 until the Froude number is larger than 1.5.
5. Input the rheology of the slurry to obtain the plasticity, Reynolds, or Hedstrom number
based on the hydraulic radius of the flow.
6. Compute the friction factor in accordance with one of the equations listed.
7. Obtain the friction loss per unit length and equate it to the slope, as per Equation 6-5.
8. If the energy gradient or slope from Step 9 exceeds the physical contour of the terrain
where the launder is to be installed, reiterate assuming a slower flow.
9. Check on the deposition velocity or self-cleaning abilities of the solids. If the deposition
velocity is more than 50% of the average velocity, speed up the flow by changing
the cross section of the launder or the physical slope.
The following computer program, “Non-Newt-Channel,” uses the Darby method to
design open channel flow for non-Newtonian fluids on the basis of modifications to the
Darby method.
Computer Program “Non-Newt Channels”
CLS
PRINT “CHANNEL FLOW PROGRAM FOR NON-NEWTONIAN FLOWS
PRINT “****************************************”
PRINT
c = 0
pi = 4 * ATN(1)
DEF fnlog10 (x) = LOG(x) * .43242944#
DEF FNASN (x) = x + x ^ 3/6 + 3 * x ^ 5/(2 * 4 * 5) + 15 * x
^ 7/(2 * 4 * 6 * 7) + (15 * 7)/(48 * 7 * 8) * x ^ 9
DEF fnacos (x) = pi/2 – (x + x ^ 3/6 + (3/(2 * 4 * 5)) * x ^ 5 +
15/(8 * 6 * 7) * x ^ 7 + 15 * 7 * x ^ 9/(48 * 7 * 8))
g = 9.81
INPUT “PROJECT “; proj$
PRINT
INPUT “DATE “; d$
CLS
INPUT “NAME OF ENGINEER “; e$
‘e$
CLS
5 INPUT “AREA”; a$
INPUT “LINE NUMBER “; li$
CLS
INPUT “DO YOU INTEND TO USE US UNITS (y/n)”; U$
GOSUB CONVERSION
INPUT “INITIAL FLOW RATE (m3/s)”; q
‘INPUT “INITIAL FLOW RATE (m3/HR)”; QH
‘q = QH/3600

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
q = q * qul
PRINT USING “flow rate = ##.### m3/s”; q
PRINT
INPUT “DESIGN FACTOR “; SF
c = 0
6 IF c = 1 THEN GOTO 19
PRINT
INPUT “specific gravity of carrier liquid”; sgl
INPUT “specific gravity of solids”; sgs
PRINT
PRINT “ please choose between input of weight or volume
concentration”
PRINT “ 1- weight concentration”
PRINT “ 2- volume concentration”
PRINT
12 INPUT “ 1 or 2”; cwe
IF cwe = 1 THEN INPUT “weight concentration in percent”; cwin
IF cwe = 2 THEN INPUT “volume concentration in percent”; cvin
IF cwe = 0 OR cwe > 2 THEN GOTO 12
PRINT
IF cwe = 1 THEN cw = cwin/100
IF cwe = 1 THEN sgm = sgl/(1 - cw * (1 - sgl/sgs))
IF cwe = 1 THEN cv = (sgm - sgl)/(sgs - sgl)
IF cwe = 1 THEN cvin = 100 * cv
IF cwe = 1 THEN PRINT USING “specific gravity of mixture = ##.##,
cv = #.###”; sgm; cv
IF cwe = 2 THEN cv = cvin/100
IF cwe = 2 THEN sgm = cv * (sgs - sgl) + sgl
IF cwe = 2 THEN cw = cv * sgs/sgm
IF cwe = 2 THEN cwin = cw * 100
IF cwe = 2 THEN PRINT USING “specific gravity of mixture = ##.##,
cwin = ##.##%”; sgm; cw
dens = sgm * 1000
PRINT
INPUT “DO YOU KNOW THE VISCOSITY (y/n)”; ZS$
IF ZS$ = “Y” OR ZS$ = “y” THEN INPUT “VISCOSITY (mPa.s)”; vu1
CLS
IF ZS$ = “N” OR ZS$ = “n” THEN KRAT = 1 + 2.5 * cv + 10.05 * cv ^ 2
+ .00273 * EXP(16.6 * cv)
‘ASSUMED VISCOSITY OF WATER 1 mPa · s
IF ZS$ = “N” OR ZS$ = “n” THEN vu = KRAT * .001
IF ZS$ = “Y” OR ZS$ = “y” THEN vu = vu1/1000
CLS
PRINT USING “VISCOSITY = ##.##### Pa.s”; vu
INPUT “hit any key”; t$
CLS
INPUT “yield stress in dynes/cm2 “; y1
y = y1/10
CLS
PRINT “you can let the program iterate for itself or you can input
an initial speed”
PRINT
v1 = 2
PRINT “iteration starts at 2 m/s (6.6 ft/s)”
INPUT “do you prefer to input an initial speed (Y/N)”; b$
6.47

6.48 CHAPTER SIX
IF b$ = “N” OR b$ = “n” THEN GOTO 18
17 PRINT USING “ current speed = ##.## m/s “; v1
IF us = 1 THEN INPUT “ initial speed in ft/s “; v1
IF us = 2 THEN INPUT “ initial speed in m/s “; v1
v1 = v1 * leg
18 INPUT “maximum allowed slope in percent “; s1
19 IF c = 1 THEN q = q * SF
IF c = 1 THEN GOTO 50
21 PRINT “choose shape of launder from following list “
PRINT
PRINT “1- rectangular “
PRINT “2- circular”
PRINT “3- U shape”
INPUT “which choice (1,2,3 etc.....)”; ck
50 IF ck = 1 THEN GOSUB rect
IF ck = 2 THEN GOSUB circ
IF ck = 3 THEN GOSUB ushape
PRINT “v1”
re = v1 * 4 * rh * dens/vu
he = (4 * rh/vu) ^ 2 * dens * y
PRINT USING “Reynolds No = #########, Hedstrom No = ######## “; re;
he
GOSUB friction
IF c = 0 THEN GOSUB settling
IF v2m > (v1/2) THEN PRINT “to avoid settling, flow should be
speeded up”
IF v2m > (v1/2) THEN GOSUB increase
IF sm > s1 THEN PRINT “slope exceeds maximum allowed slope”
IF sm > s1 THEN GOTO 17
‘INPUT “do you want to print out these results as a minimum slope”;
min$
‘IF min$ = “Y” OR min$ = “y” THEN min = 1
‘IF min = 1 THEN GOSUB print1
‘IF min = 1 THEN GOTO 999999
GOSUB gradient
INPUT “do you want a hard copy (Y/N)”; mv$
IF mv$ = “y” OR mv$ = “Y” THEN GOSUB print1
999999 IF SF > 1 THEN c = c + 1
IF c = 1 THEN GOTO 19
END
increase:
v1 = v1 * 1.03
PRINT USING “velocity = ##.## m/s”; v1
area = q/v1
RETURN
decrease:
v1 = v1 * .97
area = q/v1
RETURN
are:
area = q/v1
PRINT “flow area “; area
RETURN

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
CONVERSION:
IF U$ = “Y” OR U$ = “y” THEN us = 1
IF U$ = “n” OR U$ = “N” THEN us = 2
IF us = 1 THEN PRINT “for us units use foot for length, US gallon
for flow”
IF us = 1 THEN PRINT “speed in ft/s”
ft = .3048
gal = 3.785
inch = .0254
IF us = 2 THEN GOTO 888
leg = ft
qul = gal/60000
GOTO 890
888 leg = 1
qul = 1
890
RETURN
6.49
rect:
CLS
IF c = 1 THEN GOTO 1600
PRINT “you have chosen a rectangular launder - do you want to
continue (Y/N)”; mre$
IF mre$ = “n” OR mre$ = “N” THEN GOTO 21
14 IF us = 1 THEN INPUT “width of channel (ft) “; w
IF us = 2 THEN INPUT “width of channel (m) “; w
IF w = 0 THEN GOTO 14
w = w * leg
PRINT
INPUT “do you want the program to calculate height of walls (Y/N)”;
hr$
IF hr$ = “N” OR hr$ = “n” THEN INPUT “height of walls “; hl
IF hr$ = “y” OR hr$ = “Y” THEN hl = .5 * w
hl = hl * leg
1600 area = q/v1
dep = area/w
IF c = 1 THEN GOTO 1606
IF p$ = “y” OR p$ = “Y” THEN GOTO 1606
INPUT “what ratio of fill is acceptable (0.333, 0.5, 0.75)”; fill
1605 IF hr$ = “N” OR hr$ = “n” THEN hl = dep/fill
hfill = hl * fill
1606 IF (dep > hfill) THEN PRINT “depth of liquid exceeds preferred
fill ratio”
IF dep > hfill THEN v1 = v1 * 1.01
IF dep > hfill THEN area = q/v1
IF dep > hfill THEN dep = area/w
IF dep > hfill THEN GOTO 1605
‘calculation of hydraulic radius
rh = area/(w + 2 * dep)
mhd = dep
GOSUB froude
IF nf < = 1.5 THEN PRINT “froude number too low at “; nf
IF nf < = 1.5 THEN INPUT “flow is unstable do you want to stabilize
the flow “; p$
IF p$ = “N” OR p$ = “n” THEN GOTO 1612
IF nf > 1.5 THEN GOTO 1612
v1 = v1 * 1.01

6.50 CHAPTER SIX
area = q/v1
GOTO 1600
1612 RETURN
circ:
rf = 0
PRINT “calculation for a circular launder”
‘RINT “ITERATION STARTS FOR 1/2 FULL PIPE”
INPUT “degree of fullness as a ratio of area of flow to pipe area
(.5,.6 etc..)”; full1
1999 PRINT “speed (m/s)”; v1
INPUT “do you want to change speed (Y/N)”; lk$
IF lk$ = “y” OR lk$ = “Y” THEN INPUT “new speed in m/s “; v1
‘ IF lk$ = “n” OR lk$ = “N” THEN GOTO 2001
2095 area = q/v1
IF c = 1 THEN GOTO 456
dia = SQR(area * 4/(full1 * pi))
IF us = 1 THEN diapus = dia/.0254
IF us = 1 THEN PRINT USING “recommended inner pipe diameter = ##.##
in “; diapus
IF us = 2 THEN PRINT USING “recommended inner pipe diameter =
###.####m”; dia
2001
IF us = 2 THEN GOTO 2100
IF nff = 0 THEN GOTO 2002
PRINT USING “present pipe i.d = ##.### m”; id
IF us = 1 THEN idus = id/.0254
IF us = 1 THEN PRINT “present pipe id = ###.##inches”; idus
2002 INPUT “ pipe outer diameter in inches”; dout
IF rf > 0 THEN GOTO 2004
INPUT “pipe thickness in inches “; thickus
INPUT “pipe liner thickness in inches “; linus
2004 idin = dout - 2 * thickus - 2 * linus
PRINT USING “pipe i.d = ###.### in “; idin
id = idin/12
PRINT USING “pipe id = ##.## ft”; id
GOTO 2105
2100 INPUT “pipe outer diameter in mm”; d2m
IF nff = 1 THEN GOTO 2101
IF rf > 1 THEN GOTO 2101
INPUT “pipe thickness in mm”; thick
INPUT “pipe liner thickness in mm”; lin
2101 idm = d2m - 2 * thick - 2 * lin
id = idm/1000
2105 id = id * leg
r1 = id/2
a2 = pi * r1 ^ 2
456 IF area < a2 THEN GOSUB depth1
IF area > a2 THEN GOSUB depth2
IF a2 = area THEN PRINT “DEPTH = RADIUS”
RATIO1 = dep/(2 * r1)
INPUT “HIT ANY KEY TO CONTINUE”; l$
RATIO1 = dep/id
457 PRINT “RATIO OF LIQUID DEPTH TO DIAMETER”; RATIO1
IF RATIO1 < 1.05 * full1 AND RATIO1 > .95 * full1 THEN GOTO 470
IF RATIO1 < .2 THEN GOTO 490

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
IF RATIO1 > .85 THEN GOTO 492
IF RATIO1 < .48 THEN INPUT “ DO YOU WANT TO INCREASE THE DEPTH OF
LIQUID TO REDUCE SLOPE (y/n)”; n$
IF RATIO1 > .52 THEN INPUT “do you want to decrease liquid depth “;
dp$
IF RATIO1 < .48 AND n$ = “N” OR n$ = “n” THEN GOTO 470
IF RATIO1 < .48 AND n$ = “y” OR n$ = “y” THEN GOTO 465
IF RATIO1 > .52 AND dp$ = “N” OR dp$ = “n” THEN GOTO 470
IF RATIO1 > .52 AND dp$ = “Y” OR dp$ = “y” THEN GOTO 467
GOTO 470
490 PRINT “ please reduce pipe diameter as depth is less than 20%
of diameter”
INPUT “do you want to decrease the pipe diameter “; kjl$
IF kjl$ = “n” OR kjl$ = “N” THEN GOTO 456
IF kjl$ = “Y” OR kjl$ = “y” THEN GOTO 2001
492 PRINT “please increase pipe diameter as depth is more than 90%
of diameter”
INPUT “do you want to change pipe diameter (Y/N)”; qg$
IF qg$ = “Y” OR qg$ = “y” THEN GOTO 465
IF qg$ = “n” OR qg$ = “N” THEN GOTO 470
465 GOSUB decrease
GOSUB are
GOSUB depth1
GOTO 456
467 GOSUB increase
GOSUB are
GOSUB depth2
GOTO 456
470 PRINT “speed (m/s)”; v1
RATIO1 = dep/id
INPUT “hit any key to continue”; l$
GOSUB angle
2120 mhd = dep
mhd = id * (fnacos(1 - 2 * RATIO1) - (2 - 4 * RATIO1) * SQR(ABS
(RATIO1 - RATIO1 ^ 2)))/(8 * SQR(ABS(RATIO1 - RATIO1 ^ 2)))
mhds = mhd/.0254
PRINT USING “mean hydraulic depth = ##.##m ##.### in”; mhd; mhds
GOSUB froude
PRINT “FROUDE NUMBER”, nf
INPUT “HIT ANY KEY TO CONTINUE”; lk$
IF nf < = .8 THEN PRINT “a new diameter is recommended”
IF nf < = 1.4 THEN GOTO 1999
IF nf < = .8 THEN GOSUB increase
IF nf < = .8 THEN GOSUB are
IF nf < = .8 THEN PRINT “flow is subcritical”
3000 IF (nf > .8) AND (nf < 1.5) THEN GOSUB increase
IF (nf > .8) AND (nf < 1.5) THEN GOSUB are
IF (nf > .8) AND (nf < 1.5) THEN PRINT “flow is critical”
IF nf < 1.5 THEN nff = 1
IF nf > = 1.5 THEN nff = 0
IF nf < 1.5 THEN GOTO 456
30001 GOSUB angle
PRINT “perimeter”; per
PRINT “area “; area
rh = area/per
PRINT “hydraulic radius”; rh
6.51

6.52 CHAPTER SIX
INPUT “HIT ANY KEY TO CONTINUE”; l$
RETURN
ushape:
RETURN
froude:
nf = v1/SQR(g * mhd)
IF nf < .8 THEN PRINT “flow is subcritical”
IF (nf > .8) AND (nf < 1.5) THEN PRINT “flow is critical”
PRINT “froude number = “; nf
RETURN
friction:
a = -1.378 * (1 + .146 * EXP(–.000029 * he))
PRINT “reynolds “; re
m = 1.7 + 40000/re
PRINT USING “factor a = ###.###### and exponent m = ##.###”; a; m
PRINT
INPUT “hit any key to continue “; kkkkkkk$
FTU = (10 ^ a) * re ^ (-.193)
PRINT “ft = “; FTU
PRINT
fl = (16/re) * (1 + he/(6 * re))
PRINT “fl = “; fl
ff = (fl ^ m + FTU ^ m) ^ (1/m)
fd = 4 * ff
IF c > 1 THEN GOTO 666
PRINT USING “in absence of roughness fanning = #.###### and darcy =
#.######”; ff; fd
[A section of the program here lists all types of materials and
their roughness as explained by table 6-2, it is not reproduced
here to save space
em refers to absolute roughness in meters and emf in ft]
PRINT USING “estimated roughness for new system = ##.##### m ##.###
ft”; em; emf
666 FOR i = 1 TO 20
fd2 = fd
ro = (em/(3.7 * 4 * rh) + 2.51/(re * SQR(fd)))
h = -2 * fnlog10(ro)
fd = h ^ -2
NEXT i
dg = fd2 - fd
PRINT “revised darcy factor to account for roughness”; fd
PRINT
PRINT “iteration error on darcy “; dg
ch2 = SQR(8 * g/fd)
n2 = rh ^ (1/6)/ch2
PRINT USING “Chazy No = ###.## and Manning number = #.#####
(including roughness)”; ch2; n2
s2 = fd * v1 ^ 2/(8 * rh * 9.81)
sm = s2 * 100
PRINT USING “recommended slope = ##.### % “; sm
PRINT
RETURN
settling:
REM check for any coarse particles being transported in a Non-New-

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
tonian mixture
PRINT “iteration on settling speed for particles using Camp
equation”
INPUT “particle size (mm) “; dp
dp2 = .001 * dp/ft
v2 = SQR((8 * .8 * 32 * dp2 * (dens/1000 - 1))/fd)
v2m = v2 * ft
PRINT USING “SETTLING SPEED = #.## m/s ##.## ft/s”; v2m; v2
IF v1 < (v2m * 2) THEN PRINT “warning settling speed is higher than
half of average speed”
RETURN
gradient:
‘grad = (2 * vu/dens) ^ (–.5) * (((fd/(4 * rh)) ^ .5) * v1 ^ 1.5)
grad = (dens * q * 9.81 * s2/(area * vu)) ^ .5
PRINT USING “velocity gradient = ###.## sec-1”; grad
RETURN
depth1:
d2 = .1 * r1
777 LE = r1 - d2
beta = fnacos(LE/r1)
PRINT “angle beta”; beta
‘INPUT “hit any key to continue”; lllll$
A3 = r1 ^ 2 * (beta - SIN(beta) * COS(beta))
IF A3 < (.975 * area) THEN d2 = d2 + .01 * r1
IF A3 < (.975 * area) THEN GOTO 777
IF A3 > (1.025 * area) THEN dpf = 1
IF A3 > (1.025 * area) THEN GOSUB depth2
PRINT “DEPTH OF SLURRY”; d2
dep = d2
‘INPUT “hit any key to continue”; k$
RETURN
depth2:
IF dpf = 1 THEN GOTO 778
d2 = .9 * r1
778 LE = d2 - r1
beta = FNASN(LE/r1)
REM next line changed for rev 1.02 - pi in front of beta removed
A3 = pi * r1 ^ 2/2 + beta * r1 ^ 2 + r1 ^ 2 * SIN(beta) * COS(beta)
IF A3 > 1.025 * area THEN d2 = d2 - .01 * r1
IF A3 > 1.025 * area THEN GOTO 778
IF A3 < .975 * area THEN GOSUB depth1
dep = d2
depus1 = dep/.0254
PRINT USING “depth = ##.### m ###.### in”; dep; depus1
INPUT “hit any key to continue”; k$
RETURN
angle:
IF dep < r1 THEN theta = fnacos((dep - r1)/r1)
IF dep > r1 THEN theta = F NASN((dep - r1)/r1)
IF dep = r1 THEN theta = pi/2
IF dep < r1 THEN theta = 2 * theta
IF dep > r1 THEN theta = 2 * theta + pi
per = theta * r1
RETURN
6.53

6.54 CHAPTER SIX
Flow may accelerate at bends due to the formation of centrifugal forces. The velocity
profile is then distorted (Einstein and Hardner, 1954).
6-10 SLURRY FLOW IN CASCADES
Cascades are important mechanisms for the transportation of slurry. They are steep open
channels and are associated with a high Froude number and steep gradients. Stricklen
(1984) suggested that cascades be used on slopes between 5% and 65% with velocities in
excess of 10 m/s (33 ft/sec). At these magnitudes of speed, excessive wear would occur
on the walls of the open channel cascade.
There are three types of boxes to consider for reducing the speed:
1. Cascade feed box (Figure 6-18)
2. Cascade receiving sump (Figure 6-19)
3. Siphon feed box (Figure 6-20)
Stricklen (1984) suggested that under certain conditions the localized solid concentration
may exceed 65% by volume and may cause a pattern of “slug” flow with considerable
localized wear. To mitigate against this problem, while controlling the speed, he suggested
that the launder be designed as wide as possible to reduce the hydraulic radius and
depth of the flow, but still narrow enough as to avoid slug flow.
Two parameters need to be computed in order to check for localized slug flow.
1. The Vedernikov number Ve:
U
Ve = ��
(6-82)
(gym cos �) 1/2
2 bw � �
3 Pw
Worn-out mill liner
used to absorb wear
Worn-out pump liner
used to absorb wear
Fig 6-19
Low entry slope
Side ventilation window
(recommended for deep drops)
Nappe of slurry
D
Minimum D/3
Steep outlet cascade
FIGURE 6-18 Entry into a cascade feed box from a low-slope launder.

worn out mill liner
used to absorb wear
worn out pump liner
used to absorb wear
feed pipe
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
steep cascade at inlet
FIGURE 6-20 Siphon feed pipe drop box.
nappe of slurry
(ventilation window not shown)
low slope for outlet launder
FIGURE 6-19 Entry into a cascade receiving sump from a steep launder.
Fig 6-21
pipe tee fitting
6.55
discharge pipe
2. The Montuori number M:
M 2 = (6-83)
where
bw = bottom width of the channel
Pw = wetted perimeter of the channel
� = tan –1 (h/L) = tan –1 U
S
L = length of the channel
Figure 6-21 shows a linear limit between the Vedernikov and the Montuori numbers.
Below the line, no slug flow occurs and the flow is stable. Above the line, slug flow occurs.
2
��
gSL cos �

6.56 CHAPTER SIX
FIGURE 6-21 The correlation between the Vedernikov number and the square of the Montuori
number squared is used to differentiate between slug and no-slug flows. (From Stricklen,
1984.)
If the calculations of the Vedernikov and Montuori numbers indicate that the flow is
of a slug type, it will be necessary to determine the intermediate points from which unstable
rolling waves would be generated.
Niepelt and Locher (1989) as well as Stricklen (1984) proposed to compute a shape
factor for the chute:
x =
y m
� Pw
where
Pw = wetted perimeter
ym = average depth of the slurry in the channel
Steep launders may cause the formation of roll-waves that are associated with instability.
The Vedernikov number may be used as a design guide to determine these areas. Niepelt
and Locher (1989) extended the analysis to slurries and showed a marked difference with
water flows (Figure 6-22).
6-11 HYDRAULICS OF THE DROP BOX AND
THE PLUNGE POOL
Certain remote mines in mountainous regions have chosen over the years to dispose of
their tailings at sea level and sometimes to submerge them in the sea. The drop box has
been found to be an effective method to achieve energy dissipation during transportation.
There are particular design criteria that the drop box or receiving sump must meet to
avoid rapid wear of its walls:

6.57
1
� M 2
= gsL cos (�)
��
U 2
FIGURE 6-22 The Vedernikov number is used as a design guide to determine roll waves associated with steep cascades. There is,
however, a marked difference between water and slurries. (From Niepelt and Locher, 1989, reprinted by permission of SME.)

6.58 CHAPTER SIX
� The incoming liquid or nappe should impact the slurry liquid surface in the drop box
and not the bottom surface or walls.
� The sump should be sized sufficiently large for its walls to be outside the computed
area of impingement or high turbulence.
� If slug flows or flows at high Froude numbers are allowed to enter the receiving sump,
the sump should be fairly long to cope with the fluctuations of flows.
� A weir may be installed in the receiving sump to reduce the length of the hydraulic
jump.
� Froth arresters are recommended for frothy slurries.
� The area of high turbulence or the exit from the receiving sump may have to be covered
to avoid overfills.
The design of such sumps is far from easy. In the next section, the mathematics of the
slurry fall will be presented to the reader in a brief practical approach. Excellent books on
the engineering of small dams are available for further reading.
One question often asked is what is the recommended depth of a plunge pool. The
rule of thumb in the case of water is that the plunge pool should be one-third the depth
of the waterfall. That means that for a waterfall drop of 30 m one would need to provide
an additional depth of 10 m to absorb all the turbulence. This is not always possible
to achieve, and energy dissipaters are then introduced to absorb the turbulence. In
mining, these energy dissipaters are often worn-out mill liners, pump liners, or impellers
that are put at the bottom of the plunge pool to wear away as they absorb the impact of
abrasive slurry fall.
In this chapter, we shall consider the more common drop box found in many mining
plants. The economics and the size of many projects, as well as wear considerations, often
reduce the problem to rectangular or circular drop boxes. Other forms of energy dissipaters
such as ogees and ski jumps that are discussed in certain books on civil engineering have
not found application in mining because of the problem of lining such complex shapes.
For a rectangular entry into the fall, the analysis of this problem is based on dividing
flow rate Q by the width of the launder before the fall:
Q
qb = � (6-84)
w
The following analysis assumes a constant width of the launder starting well upstream
from the fall.
If y is the depth of the liquid well upstream of the fall, and V is the velocity of the liquid,
as in Figure 6-23, the total energy is
V
H = y + (6-85)
If the flow is subcritical well upstream from the fall, it will tend to accelerate near the
fall. Rubin (1997) demonstrated that the minimum energy head for a waterfall occurs
when the flow prior to the drop is in a critical regime with a Froude number of 1.0. Under
such conditions, the flow accelerates toward the brink of the fall, thus reducing the depth
Yb, which according to Fathy and Shaarawi (1954) would be
2
�
2g
Y b
� Y0
= 0.716 (6-86)

Y
subcritical flow
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
flow per unit length q =Q/b
b
Total Energy Line
2
(V /2g)
3
2
Y = Q /b
0
5 Y 0
The Froude number of 1.0 occurs five times the critical depth upstream of the brink. The
critical depth is defined as
Y0 = � � 1/3
(6-87)
For water flow, the critical slope is expressed in terms of the critical depth and the
Manning roughness factor as
1
S0 = � (6-88)
[Y0] 1/3
gn2 �
g
�
Fr
But since Fr = 1.0, Equation 6-88 is also expressed as
S 0 =
q b 2
gn 2
� [Y0] 1/3
6.59
Obviously, for slurries with different roughness values due to the deposition of sediments
or formation of antidunes, Equation (6-88) is not readily applicable.
From the point of view of the designer of a slurry drop box, it is important to determine
the area of impingement of the jet, the depth of the backwater, and the area of the
still water, in order to provide proper liners and protection from wear. The nappe must be
properly ventilated, as in Figure 6-24; otherwise the slurry may tear the structure apart.
It may appear strange to the reader that the author is focusing on the case of minimum
energy with entry in a subcritical flow, although we have reiterated in previous sections of
this chapter the need to maintain a supercritical flow for slurries in launders. The minimum
energy entry is a case of reference used to understand more complex flows at high
Froude number in which the projection of the nappe is even further away. There are cases
in which entry is at minimum energy, such as from a lake into a river, or from a large tailings
pond into an open channel, or from a relatively horizontal channel into a large drop
box used for sampling the tailings. In fact, entering the fall at minimum energy allows for
a better capture of samples for analysis (Figure 6-25).
Y 0
flow Q
VENTILATION AIR
FIGURE 6-23 Entering a waterfall with minimum energy gradient.
width "b"
Y = 0.716

6.60 CHAPTER SIX
FIGURE 6-24 This drop box for a large tailings flow features three 24� ventilation windows
in each side wall to permit ventilation under the nappe.
The energy dissipation at the bottom of the fall was discussed in detail by Moore
(1943) and Rand (1955). The hydraulics of such a fall will therefore be summarized here
for practical design considerations, with focus on the main equations.
Rand observed three different flows for a waterfall with a well-ventilated nappe,
which are depicted in Figures 6-26 to 6-27. In the first case, Case A (Figure 6-27), the
flow approaches the crest of the waterfall in a subcritical regime. The flow is characterized
by a nonsubmerged nappe at the point of impingement with the apron. Rand indicated
without definite proof that the height of the liquid at the crest is 0.715 of the critical
depth. The region between the wall and the nappe is called the under-nappe. It has a depth
d f which is higher than the flow downstream of the point of impingement. In the undernappe,
the flow is recirculating.
As the nappe hits the apron, it turns smoothly into supercritical regime at a distance L d
from the wall. This distance L d is called the drop distance. At the point of impingement,
the depth of the stream reaches a minimum with a depth d 1 at L d from the wall. After d 1,
the flow depth increases smoothly while remaining in a supercritical regime until a certain
distance L j and a depth d b, where a stationary hydraulic jump occurs between the supercritical
and subcritical flows. The depth of the flow increases until a steady level is
reached, d 3, called the tail water depth.
Case B (Figure 6-28) is described by Rand as a borderline case. By comparison with
Case A, the flow is critical or slightly supercritical before the crest of the fall. There is no
relative distance between d 1 and d 3, and the hydraulic jump occurs practically at the region
of the impingement with the apron and extends over a distance L until a steady-state
d 2 is reached for the tail water. The nappe is not submerged, but there is no supercritical
flow over the apron, so the distance between the region of impingement and the tail water
is considered the shortest of the three cases.

6.61
subcritical flow
flow per unit length q =Q /b
b
Total Energy Line
2
(V /2g)
3
2
Y = Q /b
0
5 Y 0
Y 0
flow Q
Ventilation air
Sample of slurry
FIGURE 6-25 Sampling tailings with a moving bucket crossing the nappe in a tailings drop box.
travel of sampling bucket

D d
D d
Fig 6-30
y c
ventilation
ventilation
subcritical flow
d f
subcritical flow
d f
C
L p
L p
B
L d
L
j
A
L L
d
r
L j
d 1
6.62
d 3
L r
L d < L j Lr < L b
Case (A)
d
1
L < L
L > L
d j r b
d < d 2
d < d 3
FIGURE 6-27 Patterns of flow with free fall with entry in a subcritical regime. (After Rand,
1955.)
d
3
FIGURE 6-26 Geometry of the nappe from a waterfall.
d
d

D d
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
In Case C (Figure 6-28), the nappe is submerged, and the depth of the tail water is
higher than d 2. Compared with the previous two cases, this turbulent under-nappe region
is the deepest of the three cases.
The turbulent roller extends much further and is less intense than the hydraulic jump.
Referring to Figure 6-29, if D d is the depth of the fall from the bottom to the brink of the
bottom of the drop box, a drop number Dr can be defined as
2 q b
�3 gD d
6.63
Dr = (6-89)
In real life there is always a sort of churning area of liquid under the nappe, but from a
theoretical point of view, which ignores this pool of liquid, the location of the centerline of
the nappe intersecting with the bottom of the apron or the drop box would be expressed as
= 1.98[Dr1/3 + 0.357 Dr2/3 ] 1/2 Lp � (6-90)
Dd
This point is also called the toe of the nappe.
The hydraulic jump occurs at a distance Ld, which may be smaller, equal to, or even
larger than Lp (Figure 6-26) The ratio of Ld/Lp is maximum at 1.87 when Dr = 1.
Tests reported by Rand (1955) indicate that the value of
= 4.30Dr0.27 Ld � (6-91)
Dd
In cases where the hydraulic jump starts at the toe of the nappe, the experimental work
of Rand (1955) indicates that the reference depth d2 for the tail water can be expressed as
a constant:
Ld � = 2.60 (6-92)
d2
ventilation
df
L d
d 1
FIGURE 6-28 Geometry of nappe from a free fall with entry in a critical regime. (After
Rand, 1955.)
L r
L d = L j Lr = L b
Case (B)
d = d 2
d 2

D d
6.64 CHAPTER SIX
ventilation
d f
Under the nappe, a region of still water develops to a depth d f. The intersection of this
rotating water with the nappe is at point B of Figure 6-29. The height is d f, expressed as
The height of the liquid d 1 is expressed as
= Dr 0.22 (6-93)
= 0.54Dr 0.425 (6-94)
The height of the liquid d 2 in case (b) for entry in a critical regime is expressed as
= 1.66Dr 0.27 (6-95)
And the length to the intersection can be expressed by length L p or
L pB = 1.98[Y 0(D d + 0.357Y 0 – d f)] 1/2 (6-96)
The drop length or the length between the drop wall and the location of minimum
depth of the liquid at the jump d j in Figure 6-26 at point A is expressed as
L d
� Dd
L j
d 1
� Dd
d 2
� Dd
1.98(1 + 0.357 Y0/Dd)�(Y� 0/ �D� �d)
= ����
(6-97)
�[1� +� 0�.3�5�7�(Y� �D� 0/ d) � –� (�d� f/D� �]� d)
Finally, the total length of the hydraulic jump from the point d j to the point where the
tail–water has stabilized can be expressed as
L r
� Dd
d
Ld Lr d > d2 Case (C)
d f
� Dd
L r > L b
FIGURE 6-29 Free fall with a submerged nappe (after Rand, 1955).
d2 d1 = 6� � – �� (6-98)
D Dd
d

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
These equations are based on proper ventilation of the nappe. If the nappe is not properly
ventilated, it becomes semiattached or totally attached to the drop box wall. This
leads to a condition where flows may cause vibration of the drop box, which may tear it
apart if it is not structurally designed to handle the vibration.
The equations of Walter Rand were developed for waterfalls. They are a good reference
for designing drop boxes. Unfortunately, very little has been published over the
years to examine the effect of solids on the level of turbulence at the toe of the nappe and
on the magnitude of the various parameters.
Example 6-12
A mass of liquid approaches a free fall at a Froude number of 1.0. The height of the liquid
at the brink is measured to be 1.2 m (3.94 ft). The fall is 6 m (19.48 ft) deep. It is assumed
that the width of the channel and drop box remain uniform. Determine the geometry of
the hydraulic jump at the apron.
Solution in SI Units
From Equation 6-86:
Yb � = 0.716
Y0
or Y0 = 1.2/0.716 = 1.676 m (or 5.499 ft).
The Froude number of 1.0 occurs five times the critical depth upstream of the brink.
2 1/3 The critical depth is defined as Y0 = [q b/g] , so
qb = �(1�.6�7�6� 3 �·�9�.8�1�)�=� 6�.8�1� m� 2 �s� /
From Equation 6-88, the drop number Dr is
q b 2
6.81 2
6.65
Dr = = = 0.0219
The toe of the nappe is determined from Equation 6-90:
= 1.98 [Dr1/3 + 0.357 Dr2/3 ] 1/2 = 1.98 [0.02191/3 + 0.357 (0.02192/3 )] 1/2 = 1.098
Lp = 1.098 × 6 = 6.6 m
This point is also called the toe of the nappe. The location of the hydraulic jump is obtained
from Equation 6-91:
= 4.30Dr0.27 = 4.3 × 0.02190.27 = 1.533
Ld = 1.533 × 6 = 9.195 m
The hydraulic jump occurs after the toe of the nappe. Under the nappe, a region of still
water develops to a depth df, expressed by Equation 6-93 as
= Dr0.22 = 0.02190.22 � ��
3 3 (gDd) (9.81 · 6 )
Lp �
Dd
Ld �
Dd
df � = 0.4314
Dd
df = 0.4314 · 6 = 2.59 m
If this were slurry, it would be recommended to line this area to a height of 3 m by the
length of Lp (6.59 m).
The height of the liquid d1 is expressed by Equation 6-94:

6.66 CHAPTER SIX
d 1
� Dd
= 0.54Dr 0.425 = 0.54 × 0.0219 0.425 = 0.1064
d1 = 0.1064 × 6 = 0.6386 m
The height of the liquid d2 is expressed by Equation 6-95:
d 2
� Dd
= 1.66 Dr 0.27 = 1.66 × 0.0219 0.27 = 0.5916
d2 = 0.5916 × 6 = 3.55 m
The distance between d1 and d2 or length of the hydraulic jump is
= 6 (d2/Dd – d1/Dd) = 6(0.5916 – 0.1064) = 2.911
Lb = 2.911 × 6 = 17.47 m
This length should be lined to the height of d2 + 10% or approximately 4 m.
Solution in USCS Units
From Equation 6-86:
= 0.716
or Y0 = 3.94/0.716 = 5.499 ft.
The Froude number of 1.0 occurs five times the critical depth upstream from the brink.
2 1/3 The critical depth is defined as Y0 = [qb/g] , so
qb = (5.4993 · 32.2) = 73.17 ft2 Lr �
Dd
Yb �
Y0
/sec
From equation 6-89, the drop number Dr is
q b 2
Dr = = 73.172 /(32.2 · 19.483 ) = 0.022
The toe of the nappe is determined from Equation 6-90:
= 1.98[Dr1/3 + 0.357 Dr2/3 ] 1/2 = 1.98[0.0221/3 + 0.357 (0.0222/3 )] 1/2 = 1.099
Lp = 1.099 × 19.48 = 21.4 ft
This point is also called the toe of the nappe. The location of the hydraulic jump is obtained
from Equation 6-91:
= 4.30Dr0.27 = 4.3 × 0.0220.27 � 3 (gDd) Lp �
Dd
Ld � = 1.53
Dd
Ld = 1.53 × 19.48 = 29.80 ft
The hydraulic jump occurs after the toe of the nappe. Under the nappe, a region of still
water develops to a depth df, expressed by Equation 6-93 as
d f
� Dd
= Dr 0.22 = 0.022 0.22 = 0.432

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
df = 0.432 · 19.48 = 8.42 ft
If this were slurry, it would be recommended to line this area to a height of 10 ft by the
length of Lp or approximately 21.6 ft.
The height of the liquid d1 is computed from Equation 6-94:
d 1
� Dd
= 0.54Dr 0.425 = 0.54 × 0.022 0.425 = 0.1064
d1 = 0.1064 × 19.48 = 2.07 ft
The height of the liquid d2 is computed from Equation 6-95:
d 2
� Dd
= 1.66 Dr 0.27 = 1.66 × 0.022 0.27 = 0.5916
d2 = 0.5916 × 19.42 = 11.49 ft
The distance between d1 and d2 or length of the hydraulic jump is
Lb � = 6 (d2/Dd – d1/Dd) = 6(0.5916 – 0.1064) = 2.911
Dd
Lb = 2.911 × 19.48 = 56.71 ft
This length should be lined to the height of d2 + 10% or approximately 12.6 ft.
6-12 PLUNGE POOLS AND DROPS
FOLLOWED BY WEIRS
In nature, the scouring depth of a waterfall may be typically one third of the depth of the
waterfall. An example of an engineering exercise along these lines was the construction
of Mossyrock spillway on the Colwitz River near Tacoma, Washington (U.S.A.). The
spillway was created to handle a 183 m (600 ft) drop.
In the case of slurries, the wear is accelerated by the very nature of the abrasive and
erosive particles. Spent mill liners, spent mill balls, steel grading, and spent pump liners
are installed at the bottom of drop boxes to prevent wear. It is not always cost effective to
design for a scouring depth equal to one third of the free fall.
A drop box can be expensive to construct. One of the largest slurry drop boxes was
built by Fluor Daniel for the Caujone mine owned by the Southern Peru Copper Corporation
in Peru. It was designed to handle a tailing flow of 7.3 m 3 /s (116,000 gpm). The drop
was 10 m (32 ft) (Figures 6-24 and 6-30) deep and the slurry had to be redirected under an
existing truck road. The author was the hydraulic engineer on the project.
To reduce the length of the pond, it is recommended to add a weir (Windsor, 1938).
This alternative method is included in the discussion of the paper of Moore (1943) by L.
S. Hall (1943). On the basis of the work of Blackhmereff (1936), Hall developed an approach
to reduce the length of the transition region at the toe of the nappe by adding a
weir. The weir raises the water level and causes the nappe to impinge water at a higher
point of intersection.
Referring to Figure 6-30, the length of the pond can be reduced to L�. If D d is the depth
of the drop, an energy line E 0 is defined as
E 0 = D d + 1.5Y 0
6.67
(6-99)

6.68 CHAPTER SIX
E 0
Fig 6 - 32
Y
0
/2
D d
D d
Y 0
The level of the liquid over the weir Z 0 can be expressed graphically as in Figure 6-32
or mathematically as in the following equation:
(Y0/d1) d1 = + – 1.5 (6-100)
2
�2 2�
(Y0/d1) = – � + 1.0 (6-101)
2d1
3
�2 2�
= 1 + – � –1 + ��1� +� 1� 6� �� 2��� +��� –� 1������
where � is determined from the following cubic equation:
(6-102)
– �2 � + 3� + 2�2 Z0 3Y0 d1 Dd 3Y0 � � � � �
Dd 2Dd 2Dd
d1 2d1
Y0 2Dd = 0 (6-103)
Y 0 3
� d 1 3
D d
� Y0
D d
� d1
� d1
� Y0
Depending on the amount of energy dissipation before the location of d 1, � may be assumed
to be 1.0 for no dissipation at all (Bakhemeteff, 1932) or as low as 0.95 for some
dissipation before the jump (Bobin, 1934):
3Y 0
steep cascade at inlet
Z0 = Dd + � – d2 – hw (6-104)
2
where hw is the height of the weir that controls the plunge pool relative to the apron.
The length of the plunge pool is expressed as:
Y0 L� = C���� �
Dd
� Y0
3Y 0
1� +���� Y� 0D� d��
(6-105)
d 2
Z 0
h w
L' L'
2 L'
FIGURE 6-30 A weir to control the flow of slurry from the nappe of a drop box. (After Hall,
1943 in his discussion of Moore, 1943.)

Z / D
0 d
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
where C can equal 1.7 for low spray but can also equal as high as 2.0 for significant spray.
Standish Hall (1943) proposed that length L� be followed by an equal transition.
Example 6-13
Referring to Example 6-12, determine the length of the plunge pool if a controlling weir
is added. Determine the level of the liquid Z 0.
Solution in SI Units
The critical depth was determined to be 1.676 m. The drop is 6 m. Assuming C = 2.0,
Referring to Figure 6-25:
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
L� = 2�[1�.6�7�6� ·� 6� +� 1�.6�7�6� 2 � ] = 7.17 m
Z 0
� Dd
Z / D
0 d
Y 0
� Dd
d / D
1 d
=
1.676
��
6 = 0.279
� 0.84 or Z 0 � 0.84 × 6 = 5.04 m
2.0
1.8
1.6
1.4
1.2
1.0
0.8
6.69
0.0
0.0
0.4 0.8 1.2
Y / D
0 d
1.6 2.0
FIGURE 6-31 Curves to determine the height of the weir in a plunge pool.(After Hall, 1943
in his discussion of Moore, 1943, by permission of ASCE.)
Since Z0 is measured from E0, and
E0 = Dd + 1.5 Y0 = 6 + 1.5 × 1.676 = 8.51 m
the liquid level is 8.51 – 5.04 = 3.47 m above the apron.
If the engineer builds a weir 2 m high (hw) it will be submerged by a depth of 1.47 m,
corresponding to the value of d2. 0.6
0.4
0.2
d / D
1 d

6.70 CHAPTER SIX
Solution in USCS Units
The critical depth was determined to be 5.5 ft. The drop is 19.48 ft. Assuming C = 2.0,
Referring to Figure 6-25,
FIGURE 6-32 Walls of a weir showing sediment coating.
Z 0
� Dd
L� = 2�[5�.5� ·� 1�9�.4�8� +� 5�.5� 2 � ] = 23.44 ft
Y 0
� Dd
5.5
= � = 0.279
19.48
� 0.84 or Z 0 � 0.84 × 19.48 = 16.36 ft
Since Z0 is measured from E0, and
E0 = Dd + 1.5 Y0 = 19.48 + 1.5 × 5.5 = 27.73 ft
the liquid level is 27.73 – 16.36 = 11.1 ft above the apron.
If the engineer builds a weir 6.56 ft high (hw) it will be submerged by a depth of 4.82
ft, corresponding to the value of d2. The flow of slurry in flumes and through drop boxes is fairly complex and under certain
conditions hydraulic jumps occur with considerable turbulence. For fairly abrasive
slurries, wear is a concern. In other situations such as copper mines, the presence of lime
in the slurry may actually end up coating the flume with deposited lime that consolidates
with time. This deposition of lime or similar sediments coats the flume, but does completely
change the roughness of the wall (Figure 6-33). In some cases the designer must
try to avoid break up the transported solids such as coal (Kuhn, 1980).

Values of y/y a
-3.0
+1.0
+0.5
0.0
-0.5
-1.0
-1.5
-2.0
Special transition areas may be lined with abrasion resistant steel or with rubber. The
rubber is glued to steel plates that are bolted to the concrete (see Figure 6-1).
The analyses of Hall (1943) and Moore (1941,1943) are based on the assumption that
the liquid enters the fall from a subcritical regime, with minimum energy, and accelerates
at the brink. The projection of the nappe and contact with the apron is even more complicated
when the jet approaches the brink at supercritical flows. Rouse, in his discussion of
Moore (1943), discussed the changes in Froude numbers of 1–14 (Figure 6-30).
6-13 CONCLUSION
SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
Slurry flows in open channels are fairly complex but they follow many of the principles
of closed conduit flows discussed in the previous two chapters. When the speed is insufficient
or the Froude number is low, deposition occurs and dunes or a stationary bed form.
Since most books on slurry flows are focused on pipe flows, this chapter presented an
exhaustive review of the mathematics of open channel slurry flows and design of drop
boxes. The practical engineer should find in the worked examples a methodology to apply
such complex equations. It is hoped that new generations of academicians and students
will enrich the understanding of such complex flows. The design of open channel flows
requires frequent iterations for slope, stability (Froude number), roughness, etc. The use
of modern personal computers with the appropriate equations allows the engineer to optimize
the hydraulic design.
On a note of caution, the design engineer should not apply data from small to large
flumes. The change of the hydraulic radius and the ratio of particle size to depth of flow
affect the magnitude of the slope of the launder.
NOMENCLATURE
-2.0 -1.0 0.0 1.0 2.0 3.0 4.0
-2.0 -1.0 0.0 1.0 2.0 4.0
Values of x/y a
Fr = 2.18
Fr = 1.8
Fr = 1
a Nondimensional parameter and function of Hedstrom number
a Reference depth for concentration calculations
Fr = 3.02
Fr=4.12
6.71
FIGURE 6-33 Effect of the Froude number at the entry to the waterfall on the shape of the
nappe. [After Rouse (1943) in his discussion of Moore (1943).]
6-14

6.72 CHAPTER SIX
Ab Area of the horizontal projection of the lee face of the bed forms
b Nondimensional parameter
bw Wetted width
C Time-averaged concentration of suspended solids
Ca Concentration at height “a”
CD Drag coefficient of particles for a heterogeneous slurry
Ch Chezy number
CL Lift coefficient
Cm Depth-averaged concentration of solids
CT Mean transport concentration of solid particles in the slurry mixture
Cv Volume fraction of solid particles in the slurry mixture
Cw Weight fraction of solid particles in the slurry mixture
Cy Volume fraction of solid particles in the slurry mixture at level “y”
d Depth
db Depth at which a stationary hydraulic jump occurs between the supercritical and
subcritical flows on the apron after a free fall
df Depth of under nappe liquid between drop wall and nappe
dj Depth at the hydraulic jump on the apron from a free fall
dp Diameter of the particle
dt Final depth of the tail water after the hydraulic jump due to fall
d1 Depth at the toe of the nappe for a free fall and drop
d2 Reference depth for subcritical tail water after the free fall in the case of a hydraulic
jump occurring at the toe of the nappe
d3 Depth of supercritical flow at beginning of the hydraulic jump downstream of the
nappe
d50 Particle diameter passing 50% (m)
d85 Particle diameter passing 85% (m)
Dd Depth of drop box of free-fall drop
DH Hydraulic diameter
DI Conduit inner diameter (m)
Dr Drop number for free fall
Er Coefficient correlating relative roughness to friction and average velocity
E0 Total energy level for a free-fall problem of a liquid relative to the apron
fD Darcy friction factor
fD� Darcy friction factor for the channel without bed forms
fD�� Darcy friction factor due to the bed forms
fDL Darcy friction factor for liquid
fN Fanning friction factor
f1 Mathematical function
f2 Mathematical function
fNL Laminar component of fanning friction factor
FN fluid force normal to the direction of flow
Fr Froude number
fT Turbulent component of fanning friction factor
FT Fluid force tangent to the direction of flow
g Acceleration due to gravity (9.81 m/s2 )
G Flocculation gradient
h Head due to friction losses
ha Depth ratio defined by Equation 6-31
hw Height of weir in a plunge pool with a weir
He Hedstrom number

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
6.73
J Nondimensional parameter to account for dynamic viscosity in deposit velocity
ks Linear roughness (m)
Ke Experimental constant
Kx Von Karman coefficient
L Length of conduit
L� Length of drop pool with a controlling weir
Lb Distance between the point of impingement of the nappe and the tail water depth
Ld Distance between drop wall and toe of the nappe for a free-fall drop
Lj Distance between the wall of the free fall and the hydraulic jump on the apron
Lmix Mixing length for eddies
Lp Theoretical distance to intersection of the center of the nappe and the bottom of the
drop box with under-nappe pool (see Figure 6-17)
Lr Total length to the stable tail water
m Exponent from the Darby equation
M Montuori number
n Manning roughness number
qb Flow rate per unit width of launder (m2 /s)
qbs Flow rate of sediments per unit width
Q Flow rate (m3 /s)
P Power
Patm Atmospheric pressure
PL Plasticity number
Pw Wetted perimeter
R Radius
Re Reynolds number
Rep Particle Reynolds number
RH Hydraulic radius (m)
RH� Hydraulic radius due to grain roughness
RH�� Hydraulic radius due to bedforms
S Slope
Sm Specific gravity of mixture
U Horizontal component of velocity
U� Horizontal component of velocity due to turbulence
Uav Average speed
Ub Bed velocity
Ubc Critical velocity to start the motion of the bed
Ucr Critical velocity to start the flow of cohesive elements
Uf friction velocity
Uf� Friction velocity due grain roughness
Uf�� Friction velocity due to dunes or bedforms
Um Average speed
Umax Maximum speed
V Average velocity of flow (m/s)
V� Average vertical velocity due to eddies
VC Camp minimum self-cleaning velocity for a sewer (m/s)
VD Deposit velocity in a launder (m/s)
Ve Verdinokov number
Vm Mean vertical velocity component
Vsc Self-cleaning velocity of a launder
Vt Particle terminal velocity
Vol Volume

6.74 CHAPTER SIX
w Width of launder
x Local horizontal ordinate
X0 A coefficient of cohesion of the material
y Local vertical coordinate in the launder
ym Average depth of the slurry in the launder
Y Depth of launder
Y0 Critical depth of the liquid at Froude number of one
Z Function of the height above the bed of a launder
Z0 Depth of liquid surface in a plunge pool over the weir
Z1 Empirical function of grain distribution above bed
Greek letters
� Angle of inclination of flow with respect to particle
� Constant of proportionality
� Constant of proportionality in Celik’s equation
�m Coefficient of exchange of momentum between neighboring streams of the fluid
�s Mass transfer coefficient
� Angle of slope
� Factor of energy dissipation before the hydraulic jump in a free fall
� A
Graf–Acaroglu function
� Coefficient of rigidity
� Data about cohesion
� tan –1 S
� Wavelength of deposited dunes and antidunes
� Absolute (or dynamic) viscosity
�m Absolute (or dynamic) viscosity of mixture
� Dynamic viscosity
� Shear stress
�cr Critical shear stress
�L Fluid shear stress
�0 Yield stress for Bingham plastics and pseudoplastics
�w Shear stress at the wall
� Density
�L Density of carrier liquid
�m Density of slurry mixture (Kg/m3 )
�s Density of solids in mixture (Kg/m3 )
� Exponent for effective shear stress � 0.06
� Sedimentation coefficient
�A Graf–Acaroglu function
�D Shape factor
�1 Shape factor
�2 Shape factor
�3 Shape factor
6-15 REFERENCES
Abulnaga, B. E. 1997. Channel 1.0 Computer Program for Open Channel Slurry Flows. Developed
for Fluor Daniel Wright Engineers. Internal report.
Acaroglu, E. R. 1968. Sediment Transport in Conveyance Systems. Ph.D. diss., Cornell University.

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
6.75
Ambrose H. H. 1953. The transportation of sand in pipes with free surface flow. In Proceedings of
the Fifth Hydraulics Conference. Ames: State University of Iowa, pp. 77–88.
The American Society of Civil Engineers. 1975. Sedimentation Engineering. Manuals and Reports
on Engineering Practice. No. 54. New York: ASCE.
The American Society of Civil Engineers and the Water Pollution Control Federation. 1977. Wastewater
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Apmann, R. P., and R. R. Rumer, Jr. 1967. Diffusion of Sediments in a Non-Uniform Flow Field. Report
prepared for the Department of Civil Engineering, Faculty of Engineering and Applied
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Bakhmeteff, B. A. 1932. Hydraulics of Open Channels. New York: McGraw-Hill.
Blench, T., V. J. Galay, and A. W. Peterson. 1980. Steady fluid-solid flow in flumes. Paper C-1, presented
at the 7th Annual Hydrotransport Conference, Sendai, Japan. BHR Group.
Bobin, P. M. 1934. The design of stilling basins. Transactions of the Scientific Research Institute of
Hydrotechnics, XIII, 79–123.
Bogardi, J. L. 1965. European concepts of sediment transportation. Proc. Am. Soc. Civil Engineers,
91, HY1, 29–54.
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Sci. Hydr., Commiss. Land Erosion, No. 59.
Camp, T. R. 1955. Flocculation and flocculation basins. Transactions Am. Soc. of Civil Engineers,
120, 1 1–16.
Celik, I., and W. Rodi. 1984. A Deposition-Entrainment Model for Suspended Sediment Transport.
Internal Report prepared by the University of Karlsruhe, Germany. Report No.
SFB210/T/6.
Celik, I., and W. Rodi. 1991. Suspended sediment-transport capacity for open channels. Journal of
Hydraulic Engineering, 117, 2, 191–204.
Chien, N. 1954. The present status of research on sediment transport. Proc. Am. Soc. Civil Engrs.,
80, No 565, 33.
Cooper, R. H. 1970. A study of bed Material Transport Based on the Analysis of Flume Experiments.
PhD. thesis, Department of Civil Engineering, University of Alberta, Canada.
Dominguez, B., R. Souyris, and A. Nazer. 1996. Deposit velocity of slurry flow in open channels.
Paper read at the symposium, Slurry Handling and Pipeline Transport. Thirteenth annual International
Conference of the British Hydromechanic Research Association, Johannesburg, South
Africa.
Einstein H. A. 1950. The Bed-Load Function for Sediment Transportation in Open Channel Flows.
Technical Bulletin No. 1026. U.S. Deptartment of Agriculture Soil Conservation Service.
Einstein H. A. and J. A. Hardner, 1954. Velocity distribution and boundary layer at channel bends.
Am. Geophysical Union Trans., 35, 114–120.
Einstein, H. A., and N. Chien. 1955. Effects of Heavy Sediment Concentration Near the Bed on Velocity
and Sediment Distribution. MRD Sed. Ser. Berkeley: University of California.
Fathy, A., and M. A. Shaarawi. 1954. Hydraulics of free overfall. Proc. Am. Soc. Civ. Eng, 80, 564,
1–12.
Fortier, S., and F. C. Scobey. 1925. Permissible canal velocities. Trans. Am. Soc. Civil Engrs, 51, 7,
1397–1413.
Garde, R. J., and J. Dattari. 1963. Investigation of the total sediment load of streams. Res. J. University
of Roorkee. Internal report.
Graf, W. H. 1971. Hydraulics of Sediment Transport. New York: McGraw-Hill.
Graf, W. H., and E. R. Acaroglu. 1968. Sediment transport in conveyance systems. Part I. Bulletin.
Intern. Association of Sci. Hydr., 2.
Green, H. R., D. H. Lamb, and A. D. Tylor. 1978. A new launder design procedure. Paper read at the
Annual Meeting of the Society of Mining Engineers, March, Denver, Colorado.
Grim, R. E. 1962. Applied Clay Mineralogy. New York: McGraw-Hill.
Guy, H. P., R. E. Rathbun, and E. V. Richardson. 1967. Recirculation and sand-feed flume experiments.
Paper 5428. Am. Soc. of Civil Eng., 93 HYS, 97–114, Sept.

6.76 CHAPTER SIX
Hall, S. L. 1943. Discussion to paper by W. L. Moore. 1943. Energy loss at the base of the free overfall.
Transaction of the A.S.C.E., 108, 1378–1387.
Henderson, F. M. 1990. Open Channel Flow. New York: Macmillan.
Ismail, H. M. 1952. Turbulent transfer mechanism and suspended sediments in closed channels.
Trans. ASCE, 117, 409–447.
Julian, Smart and Allan. 1921. Cyaniding Gold and Silver Ores. Internal report presented to J. B.
Lippenicott Co., U.S.A. Reported by Tournier and Judd (1945).
Karasev, I. F. 1964. The regimes of eroding channels in cohesive materials. Soviet Hydrol. (Am.
Geophysics Union), Vol. 6.
Kennedy, J. F. 1963. The mechanics of dunes and antidunes in erodible bed channels. Journal Fluid
Mech., 16, 4.
Keulegan, G. H. 1938. Laws of turbulent flow in open channels. Journal of Research (National Bureau
of Standards, U.S. Dept of Commerce), 21, 707–741.
Kuhn, M. 1980. Hydraulic Transport of solids in flumes in the mining industry. Paper C3 read at the
7th International Conference of the Hydraulic Transport of Solids in Pipes, Sendai, Japan.
Cranfield, UK: BHRA Fluid Engineering, pp. 111–122.
Liu, H. K. 1957. Mechanics of sediment—Ripple formation. Proc. Am. Soc. Civil. Eng., 83, HY2,
Paper 1197.
Lovera, F., and J. F. Kennedy. 1969. Friction factor for flat-bed flows in sand channels. Proc. Am.
Soc. Civil Eng., 95, HY4, Paper 6678, pp. 1227–1234.
Majumdar, H., and M. R. Carstens. 1967. Diffusion of Particles by Turbulence: Effect of Particle
Size. Water Res. Center, Report WRC-0967, Georgia Inst. Techn., Atlanta, U.S.A.
Manning R.1895. On the flow of open channels and pipes. Transactions, Institution of Civil Engineers
of Ireland, 10, 14, 161–207.
Matyukhin, V. J., and O. N. Prokofyev. 1966. Experimental determination of the coefficient of vertical
turbulent diffusion in water for settling particles. Soviet Hydrol. (Am. Geophys.Union), No
3.
Ministry of Technology of the United Kingdom. 1969. Charts for the Hydraulic Design of Channels
and Pipes. London: Ministry of Technology of the United Kingdom.
Moore, W. L. 1943. Energy loss at the base of the free overfall. Transaction of the A.S.C.E., 108,
1343–1392.
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Association of Hydrology Research, 12th Congress. Fort Collins, CO.
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41, 12, 1204–1209.
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Richardson, E. V., and D. B. Simons. 1967. Resistance to flow in sand channels. Paper read at International
Association Hydrology Research, 12th Congress, Fort Collins, Colorado.
Rouse, H. 1937. Modern conceptions of the mechanics of fluid turbulence. Transactions of the Am.
Soc. Of Civil Engrs., 102, 536.
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Hydraulics, 123, January, 82–84.
Silberman, E. 1963. Friction factors in open channels. Proc. Am. Soc. Civil Engrs., 89, no. HY2,
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Am. Soc. Civ. Eng., 128, 1.
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Research Association, Johannesburg, South Africa.
Shook, C. A. 1981. Lead Agency Report II For Coarse Coal Transport. MTCH Hydrotransport Cooperative
Programme. Saskatoon, Canada: Saskatchewan Research Council.

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES
6.77
Stricklen, R. 1984. Slurry handling considerations. Paper read at the 1984 Annual Meeting of the
American Institute of Mining Engineering, Denver, Colorado, U.S.A.
Thomas A. D. 1979.The role of laminar/turbulent transition in determining the critical deposit velocity
and the operating pressure gradient for long distance slurry pipelines. Paper read at the 6th
International Conference of the Hydraulic Transport of Solids in Pipes. Cranfield, UK: BHRA
Fluid Engineering, pp. 13–26.
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Ore and Industrial Minerals. New York: Wiley.
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Soc. Civil. Engrs. 93, no. HY3,
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Cranfield, UK: BHRA Fluid Engineering, pp. 101–110.
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Zippe, H. J., and H. Graf. 1983. Turbulent boundary-layer flow over permeable and non-permeable
rough surfaces. J. Hydr. Res., 21, 1, 51–65.
Further readings
Bagnold, R. A. 1955. Some flume experiments on large grains but little denser than the transporting
fluid and their implication. Part 3. Proc. Inst. Civil Engrs, 4. 174–205.
Gilbert, G. K. 1914. Transportation of Debris by Running Water. Paper no. 86. U.S. Geological Survey.
Guy, H. P., D. B. Simons, and E. V. Richardson. 1966. Summary of Alluvial Channel Data From
Flume Experiments, 1956–1961. Paper No. 462-I. U.S. Geological Survey.
Khurmi, R. S. 1970. Hydraulics and hydraulic machines. Delhi: S. Chand & Co.
Lacey, G. 1930. Stable channels in alluvium. Paper no. 4736. Proc. Inst. Civil Engs., 229, 529–384.
Lacey, G. 1934. Uniform flow in alluvial rivers and canals. Paper no. 237. Proc. Inst. Civil Engs.,
237, 421–544.
Lacey, G. 1947. A general theory of flow in alluvium. Paper no. 5518. Journal Inst. Civil Engs., 17,
1, 16–47.
Nino, Y., and M. Garcia. 1998. Experiments on saltation of sand in water. Journal of Hydraulics,
124, 10, 1014–1025.
Turton, R. K. 1966. Design of slurry distribution manifolds. Engineer, 221, 641–643.
Wilson, K. C. 1980. Analysis of slurry flows with a free surface. Paper C4 read at Hydrotransport 7,
Sendai, Japan. Cranfield, UK: BHRA Group, pp 123–132.

CHAPTER 11
SLURRY PIPELINES
11-0 INTRODUCTION
Engineering as a science must balance sophisticated mathematical models with practical
field experience. In this chapter, some specific cases of slurry pipelines will be examined
for coal, iron sand, clay, phosphate, limestone, and other materials. The practical
experience with these minerals in different forms, particles sizes, and volumetric concentrations
is very useful for the design of new pipelines or the modification of existing
systems.
As in the case of coal, which can be pumped in very fine sizes—smaller than 100–125
�m (mesh 140–120) and as coarse as 50 mm (2 in), all kinds of schemes and equipment
have been developed. The economics of preparing the slurry at the feed point of the
pipeline, and the cost of building and operating the pipeline, the capital cost of dewatering
the slurry to recover the minerals in a solid state combine to define the design criteria of
the pipeline and the physics of the slurry.
Corrosion, erosion, and abrasion are very expansive problems that must be taken into
account when designing slurry pipelines. Water hammer can be very destructive and appropriate
protection is recommended during the design as well operation phases.
Instrumentation of pipelines by the use of modern SCADA systems has become very
important in the last 20 years to control stations, measure speed of flow of slurry, monitor
leakage, and detect problems of sedimentation, surges, and transients.
There was great interest in the 1980s following the energy crisis of the 1970s to
convert thermal plants for the direct combustion of coal–water and coal–oil slurry mixtures,
particularly when economists were forecasting prices of $50 per barrel. Unfortunately,
this interest in burning slurry mixtures dwindled in the 1990s. By the year 2000,
some plants had started to burn tar–water emulsions such as the Venezuelan Orimulsion
and some old coal thermal plants were converted to burn to this interesting mixture.
There remain many concerns about transporting slurry mixtures in ships, as the solids
would not float and their recovery would be almost impossible in the case of an accident.
11-1 BAUXITE PUMPING
Bauxite is a prime source of aluminum. Pumping of bauxite slurries was discussed in Section
5-11-1, particularly the work of Want et al. (1982), with a worked example.
11.1

11.2 CHAPTER ELEVEN
11-2 GOLD TAILINGS
Gold tailings at high concentration flow as non-Newtonian mixtures. The work of Sauermann
(1982) in this field was examined in Section 5-12. Gold tailings pipelines are of a
small diameter, because of the limited quantities of materials to pump. In 2001, a new
tailings pipeline was commissioned at Homestake Eskay Creek, in the north of British
Columbia, Canada. The pipeline is 6500 m long (4 miles). The inner diameter of the
pipeline was sized at 2.5� and the flow was in the range of 20 m 3 /hr (88 US gpm). The
flow was of a homogeneous Bingham plastic rheology. A Wirth positive displacement
pump was installed and sized at 14 MPa (2000 psi).
11-3 COAL SLURRIES
Coal is an important fuel for power plants. Its transportation in the form of slurry has received
considerable attention since the successful construction of the Black Mesa Pipeline
(Figure 11-1). In fact, one of the longest slurry pipelines is the ETSI coal pipeline, built in
1979. It spans a distance of 1670 km (1036 miles), uses a 965 mm (38 in) pipe, and transports
23 million metric tons/year (25 US tons/year). In Russia, the Siberian coal pipeline is
260 km (163 mi) long and transports 4 million tons of coal a year from Siberian mines.
11-3-1 Size of Coal Particles
There has been considerable research on coal transportation in the form of slurry. The European
researchers favored transporting coarse slurry over relatively short distances. The
FIGURE 11-1 The Black Mesa pipeline was one of the first long-distance coal slurry
pipelines.

SLURRY PIPELINES
U.S. engineers favored fine and well-ground coal over much longer distances. As a result,
all kinds of schemes ranging from coal as coarse as 50 mm (2 in) to very fine with a diameter
smaller than 1 mm (0.039 in) have been studied.
Brookes and Dodwell (1985) reported that coal up to a diameter of 150mm (6 in) at a
weight concentration of 35% was pumped to the Hammersmith Power Station. Coal with a
top size of 50 mm (2 in) was pumped at a weight concentration of 60% and a speed of 2.5
m/s (8.2 ft/sec). This technique using water as a carrier was widely used for short haulage,
such as at the Loveridge mine in the United States and the Hansa mine in Germany, for distances
up to 6.5 km (4 miles) and for vertical lifts of 600 m (2000 ft). This reduces the dewatering
cost, but is achieved at a high power consumption of 0.8 to 1.0 kWh/tonne-km.
Hydromechanically extracted coal (hydrocal) may be transported in a natural state,
without grinding, over short distances. Particle sizes may range from 0–60 mm (0–2.4 in).
The density of coal varies depending on the moisture content. An average specific gravity
of 1.35 is often used in calculations.
Leninger et al. (1978) classified the types of coal transported as slurry:
� Power plant coal
� Coking coal
� Flotation tailings
� Hydrocoal or hydromechanically extracted coal
In long-distance pumping of coal, Lenninger et al. (1978) indicated that the feeding
plant and discharge dewatering plant might exceed 30% of the cost of the pipeline. This is
due to the equipment associated with screening, crushing, thickening, filtering, and mechanical
dewatering.
Power plant coal is usually ground and dried before being fed to a boiler for burning.
Moisture content must be reduced to less then 10% for proper handling and burning of
coal. This is not an easy task, as coal has a natural tendency to retain water. To avoid pollution
problems, the fly ash content must be low.
Klose and Mahler conducted tests on the critical speed of coal with a top size of 10
mm (� 0.4 in) as a function of weight concentration and pipe diameter. Results are presented
in Figure 11-2.
11-3-2 Degradation of Coal During Hydraulic Transport
11.3
Hydrotransport of coking coal must be done in such a way as to avoid excessive degradation
and oxidation. Lenninger et al. (1978) indicated that the finer the particles of coking
coal, the worse the deterioration by oxidation. They suggest that coking coal be transported
with the top size not exceeding 3.15 mm (0.124 in).
Flotation slimes from coal circuits typically have a top size of 0.75 mm (0.03 in).
One feature of coarse coal is that it tends to break up into smaller particles as it is
pumped over long distances. The degradation of coal during hydraulic transportation
must be taken in account when calculating the deposition velocity as well as the pressure
drop. Friable and coarse lignite degrades rapidly. Particle degradation is accentuated
when there are numerous bends and valves in the pipeline. Shook et al. (1979) reported on
lab tests on coal degradation. They concluded that:
� Particle degradation in recirculating loops is similar to rod and ball milling processes.
� There is a time factor to consider. There may be a substantial initial degradation with
friable coals that decreases exponentially with time.

11.4 CHAPTER ELEVEN
Critical Speed (m/s)
2.0
1.8
1.6
1.4
1.2
1.0
0.3 0.4 0.5
Weight Concentration
� Lignite coal degrades at higher impeller diameter tip speeds.
� For bitumous coal, degradation is linear with relatively small particles, but degradation
is lower with coarser particles.
� Therefore, degradation is not uniform from one grade to another. Degradation can result
in a bimodal distribution of coarse and fine particles
The degradation of coal in a pipeline is an interesting phenomenon. Its physics was
used to understand the degradation of oil sands. There are other slurries in nature, such as
the Canadian oil sands, that are sometimes pumped in sizes as large as 50 mm (2 in).
Mixed with a special solvent and pumped over long distances, the tar–sand balls break up
during transportation and the tar separation is facilitated. This ingenious approach, developed
in Canada, has allowed a shift in technology from hot steam treatment to warm water
and much less expensive processes. It is not farfetched to say that the work on coal
degradation started by Pipelin et al. (1966) on coal as early as the 1960s and continued by
a number of scientists such as Shook et al. (1979) was the basis of a new technology for
oil–sand separation.
Pipelin et al. (1966) reported that the degradation of coal in pumps was a function of
the impeller tip speed or tangential speed at the tip diameter raised to the power of 2.5.
Corrosion control is important in the case of coal slurry pipelines because of the presence
of sulfur in coal.
Ercolani et al. (1988) conducted extensive closed-loop tests on fine coal and water
mixtures at shear rates of 20/sec, 50/sec, and 80/sec. They concluded that pumping at
power rates in the range of 0.1 to 2.0 W/kg (0.045–0.90 W/lb) did not seem to affect the
degradation of coal under mechanical stress.
11-3-3 Coal–Magnetite Mixtures
NB400 (16")
NB300 (12")
NB200 (8")
FIGURE 11-2 Critical speed of coal with top size of 10 mm (0.394 in) as a function of
weight concentration and pipe diameter. (After Klose and Mahler, 1982.)
Buckwalter (1977) investigated the transportation of very coarse coal (>50 mm or >2 in)
in a magnetite-based water mixture. The magnetite consists of very fine particles but they
7
6
5
4
3
Critical speed (ft/sec)

are heavier than coal. When magnetite is mixed with water, this mixture becomes the effective
carrier liquid in which the >50 mm (>2 in) coal particles can float.
In Chapter 4, it was clearly explained that the difference in density (or specific gravity)
between the carried solids and the carrier liquid was an important parameter in friction
loss calculations. This is, in basic terms, the concept of using a heavy medium (water and
magnetite) as a carrier for coarse solids.
A circuit that uses magnetite must have a recovery system at the end of the pipeline.
Since magnetite has very strong ferromagnetic properties, it is first screened from coarse
coal, and then separated from crushed coal (that resulted from deterioration during pumping),
using magnetic separators. The recovered magnetite is then mixed with water at a
high volumetric concentration and pumped via a dedicated pipeline and a positive displacement
back to the starting feed station of the slurry pipeline. It is then stored in special
storage tanks with agitator mixers.
To avoid the use of many booster stations in long pipelines. The lockhopper may be
used for coarse coal (>50mm or >2 in; see Figure 9-17).
11-3-4 Chemical Additions to Coal–Water Mixtures
Special chemicals such as xanthan gum in levels of 200–600 ppm can be used as a stabilizer
to prevent settling of coal slurries and to prevent the formation of hard-packed beds
during hydrotransport. Miller and Hoyt (1988) recommended Pfizer Flocon 4800C as a
very economical additive to coal–water mixtures.
Morway (1965) obtained a patent for using a hydrocarbon oil with a small percentage
of an imidazoline surfactant to coat coal particles uniformly. After adding this mixture,
the coated coal can be mixed with water. The water weight concentration can be reduced
to 20%. This slurry with low overall moisture is easier to heat at the final discharge point
prior to combustion than plain coal–water mixtures.
Bomberger (1965) proposed hexametaphosphate and sulfite as corrosion inhibitors for
coal slurries in steel pipelines.
11-3-5 Coal–Oil Mixtures
SLURRY PIPELINES
11.5
The vast majority of slurries consist of water and solids; however, variations on this are
being implemented, especially in the transportation of coal. In the case of thermal plants,
slurries of water and coal are difficult to burn, and a complete process of dewatering is
needed to separate the coal from water. To rectify this situation, proposals have been
made to use heavy crude oils instead of water to transport and burn coal.
The viscosity of a heavy oil combined with the degradation of coal during pipeline
transportation ultimately leads to non-Newtonian flows. The rheology of such mixtures depends
on particle size, temperature, concentration, and the quality of the coal and the carrier
oil. With the worsening political situation that started in 2001, coal–oil mixtures may see
more and more applications. Kreusing and Franke (1979) recommended the use of coal
particles smaller than 5 mm (0.2 in) as a fuel for the blast furnace. To maintain the viscosity
of a coal-oil mixture in a range that is pumpable, the slurry may have to be warmed to
50° C (122° F) [fig 11-3]. By using heat, Kreusing and Franke (1979) managed to maintain
a slurry viscosity in the range of 13–40 Pa·s (whereas the viscosity of water is 0.001 Pa·s).
A viscosity of 40 Pa·s is an upper limit allowable for use in centrifugal pumps.
In the case of carbonized lignite, Kreusing and Franke (1979) achieved a maximum
weight concentration of 34%, and with brown coal they achieved a maximum weight concentration
of 41–46%, depending on the solid particle size. Brown coal is often a low-

11.6 CHAPTER ELEVEN
calorific coal and is difficult to export. Kreusing and Franke achieved a maximum weight
concentration of 60% with mineral coal. They concluded that it was possible to use coal
for a maximum energy substitution of oil of 52%. Since coal is cheaper than oil, this is not
a negligible result (Figures 11-3 and 11-4).
11-3-6
Dewatering Coal Slurry
At the discharge point of a coal slurry pipeline, water must be removed because coal cannot
be burned at such high water contents. It is essential to establish certain criteria for the
design of a dewatering station:
� Size distribution
� Water content
� End use
� Rheology
� Volumetric concentration
� Suitability of recovered water for further use
� Storage, stockpiling, or ship loading
� Coal degradation during transport
FIGURE 11-3 Viscosity of coal–oil slurry mixture. (From Kreusing and Franke, 1979.
Reprinted with permission of BHR Group.)

SLURRY PIPELINES
11.7
FIGURE 11-4 Viscosity as a function of temperature for coal–oil mixture. (From Kreusing
and Franke, 1979. Reprinted with permission of BHR Group.)
Dewatering may be done mechanically via filter presses, centrifuges, and screens. Because
the efficiency of dewatering often depends on the particle size distribution, screening
is strongly recommended prior to dewatering.
Leninger et al. (1978) reported that in the case of coal with a top size of 10 mm (0.4
in), it was very difficult to use mechanical dewatering devices to reduce moisture below
10.6 %, even using steam with vacuum filtering.
For coals with a top size of 3.15 mm (0.125 in), Leninger et al. (1978) suggested using
two-stage cyclones with the second stage connected to the overflow of the first. The underflow
from the first stage as well as from the second stage should be fed into solid-bowl
centrifuges to reduce the residual moisture content to 17.3%. The use of hydrocyclones
was also discussed by Abbot (1965).
Coal with a top size of 2.4 mm (0.09 in) can be dewatered using two-stage cyclones as
well as solid-bowl centrifuges, but the overall moisture is only reduced to 19%.
For ultra-fine coal with a top size of 1.4 mm (0.055 in), as in the case of the Ohio and
the Black Mesa pipelines, mechanical dewatering must be followed by thermal drying.
One way to reduce the cost is to use the waste gas of the thermal plant to dry the coal.

11.8 CHAPTER ELEVEN
11-3-7 Ship Loading Coarse Coal
The endpoint of a coal slurry pipeline may be a thermal power plant or a ship loading facility
(Figure 11-5) for export. Faddick (1982) reviewed some concepts for loading coarse
coal onto ships:
� Submarine pipeline
� Vertical riser
� Single-point mooring system
� Flexible hose system
The single point mooring (SPM) system is the most economical and feasible concept.
In 1971, the Marcona Corporation installed at Walpipe, New Zealand the first successful
SPM for ship loading of iron sand slurry. Reaching a large vessel with slurry is not always
easy. Many ships require very deep ports or must stay out in deep waters to be
loaded from a mooring point, as is the case with oil tankers. In United States ports, large
carriers with drafts greater than 18–20 m (65–70 ft) have to be reached at great distances
due to the relative shallowness of most American ports.
Submarine pipelines are widely used in the oil and gas industry. For slurries, difficult
accesses with possible sedimentation of coarse particles tend to be discouraging. Submarine
slurry pipelines are typically limited to nonsettling slurries for tailings disposal, or to
relatively short distances.
High-density polyethylene (HDPE) pipes are lighter than water and can be floated.
There are no records of using HDPE pipes with coarse coal and it is unknown whether
they can be used as submarine pipes with this slurry. Flexible hoses are very expensive in
large diameter sizes in excess of 250 mm (10 in) NB (normal bore). Adams (1986) did indicate
that the use of polyethylene pipe is limited to solids with a maximum diameter of
9.5 mm [3/8 in], which would certainly not make these pipes suitable for fairly coarse
solids.
In Chapter 4, the Russian’s work on coarse coal friction losses and deposition velocity
was presented in Section 4-4-3. The Russian equations are useful as an alternative to complex
stratified models.
The density of coal varies depending on the moisture content. An average specific
gravity of 1.35 is often used in calculations.
11-3-8 Combustion of Coal–Water Mixtures (CWM)
In the 1980s, considerable research was conducted in the United States on converting
diesel engines to burning coal or its derivatives. The interest in coal fired diesel engines
died away in the 1990s when the price of oil dwindled to $12 a barrel. With the threat
of a new energy crisis at the turn of the 21st century, interest in coal fired diesel engines
may revive. The most promising schemes required gasification of coal into a combustible
gas. In an effort to bypass the gasification, plant schemes were proposed to
burn coal as slurry in a diesel engine. Tests showed that coal would wear out piston
rings, engine liners, etc. The concept required special construction materials, as is often
the case with slurries.
In an effort to bypass the problems of using pulverized coal in a slurry form with solid
pistons, researchers have investigated liquid piston engines. The idea of a liquid piston
engine was pioneered in the 19th century in the United Kingdom by Humphrey Engines.
In an effort to bypass the problems of wear due to exploding a slurry mixture against sol-

Product Distribution Unit
Slurry Marine Hoses
FIGURE 11-5 Ship loading of coal slurry. (From Faddick, 1982. Reproduced with permission
of BHR Group.)
11.9
Seal Water Wash System
Slurry Pipeline
End Manifold
Slurry Loading Arm

11.10 CHAPTER ELEVEN
id pistons, Abulnaga (1990) developed a liquid piston engine and obtained an Australian
patent.
The Abulnaga engine features a special Darrieus or Savonius rotor between two cylinders.
Essentially, this means that the liquid pistons are formed as a column of liquid that
oscillates between two cylinders. The great advantage of the Darrieus or Savonius rotors
is that they maintain the same direction of rotation irrespective of the oscillation direction
of the column.
Typical coal–water mixtures (CWMs) for direct combustion consist of a weight concentration
of 70% coal and 30% water. Prior to combustion, it is important to degrade the
coal slurry mixture by applying hot air to accelerate the evaporation of water (Garbett and
Yiu, 1988).
The concept of direct combustion of CWM in gas turbines has been proposed in the
literature. There remain many unknowns, particularly as to the erosion of the blades from
fly ash.
11-3-9 Pumping Coal Slurry Mixtures
Hughes (1986) described the development of positive displacement pumps for the Siberian
coal pipeline Belovo-Novosibirsk. This pipeline is 256 km (160 mi) long. It transports
heavily concentrated water–coal slurry. This pipeline features one main pump station and
two booster stations. Each pump station features single-acting triplex Ingersoll Dresser
pumps. Special 100 bar (1,470 psi) gate valves were manufactured in sizes of 200 mm (8
in) and 350 mm (14 in).
Vanderpan (1982) recommended the use of Ni-hard as a material to cast the impellers
and liners of coal handling slurry pumps. For certain high pH applications due to acidic
water, or in the case of high-salt mixtures, special high-alloy irons may be used instead of
Ni-hard.
The use of centrifugal pumps in series is usually limited to a discharge pressure of 4.2
MPa (or 600 psi). This may be suitable for puming coarse coal up to a distance of 50 km
(30 mi).
11-4 LIMESTONE PIPELINES
Limestone is an important material. It was used thousands of years ago to build the colossal
pyramids of Egypt and is used today to manufacture cement and concrete. Many derivatives
of limestone are used as fertilizer, for alkalination of chemical processes, and as
a pollution control substance used to absorb sulfur dioxide pollutants in flue gas desulphurization.
In Chapter 1, a number of limestone pipelines are listed. The pipeline focused on in
this segment is the Gladstone pipeline of the Queensland Cement and Lime Company in
Australia, which started operation in 1979. Venton (1982) described the pipeline in great
detail and diagramed examples of a practical design for a cement plant.
The Gladstone pipeline is 24 km long and is located 400 km north of Brisbane. The reserves
of limestone are overlaid by deposits of clay suitable for a clinker cement plant. A
ship loading facility was built in the Gladstone harbor in order to transport the lime to
Brisbane (Figure 11-6).
At the discharge point of the limestone pipeline, the slurry is dewatered by a thermal
drying processes. The lime is then transported in a powder form. To reduce the relatively

SLURRY PIPELINES
11.11
FIGURE 11-6 The Gladstone limestone pipeline in Queensland, Australia. (From Venton,
1982. Reprinted with permission of BHR Group.)

11.12 CHAPTER ELEVEN
high cost of thermal drying, mechanical dewatering with filter presses is used. The lime is
therefore delivered in a semiwet state, but this is acceptable for a cement plant. The filter
presses reduce the water content from 36% to 17% moisture. In the actual cement plant,
waste heat from the kiln off-gases is then used for further drying.
In a limestone pipeline project, the slurry plant is located at the quarry and needs to be
fitted with a milling or grinding circuit. Water also must be available on-site to be blended
with the ground limestone. Some of this water may be available locally; however, if
the pipeline is relatively short, it may be possible to return water from the pipeline discharge
point.
Processes of slurry preparation are described in Chapter 7. Limestone pipelines typically
operate in a range of 50–60% weight concentration. If other ingredients such as clay
are in the slurry, a small pump test loop is recommended on-site to monitor the composition
and concentration.
In the case of the Gladstone pipeline, the viscosity was in the range of 10 mPa·s (1 cP)
at a weight concentration of 56%, 20 mPa·s (2 cP) at a weight concentration of 60%, but
rose sharply toward a viscosity of 70 mPa·s (7 cP) at a weight concentration of 68%. The
yield stress was in a range of 8–14 Pa (Figure 11-7). The laminar to turbulent velocity in a
200 mm (8 in) pipeline was predicted to be in the range of 1–1.3 m/s at a weight concentration
of 62–64%.
The Gladstone pipeline uses Wilson-Snyder positive displacement pumps. At the
weight concentration of 60–64%, the slurry acted as a Bingham mixture, with non-Newtonian
viscosity characteristics. However, it did feature clay, sand, and iron, as the materials
were formulated for the manufacture of clinker cement. The pipeline operation speed
was maintained at 2 m/s and the pressure drop was around 300 kPa/km. API 5LX steel
with a high yield strength was used. Corrosion rates as high as 0.25 mm/year were measured
during the initial phase of operation of the pipeline.
Pertuit (1985) reported that during the first two years of operation problems of operation
included:
� Severe knocking and vibration of mainline pumps
� Short life of gland packing and piston scouring of the positive displacement pumps
These problems were eventually solved.
The extremely high rate of corrosion was unexpected, since the lab reports suggested
a design for a low corrosion rate of 0.076 mm/year. Venton (1982) reported that the
operators brought down the rate of corrosion by adding inhibitors to the slurry composition.
11-5 IRON ORE SLURRY PIPELINES
Iron ore is critical to our modern economy. A number of iron ore slurry pipelines have
been constructed since the 1960s (see Table 1-9 for specific examples). One of the most
famous is the SAMARCO pipeline in Brazil, which is 390 km (245 mi) long.
In order to understand its rheology, Thomas (1976) conducted tests on iron ore at a
volumetric concentration of 24% and with a solids diameter d50 = 40 �m. The tests were
conducted in 150 mm (6 in) and 200 mm (8 in) pipe. The head loss in meters of water per
meter of pipe length was derived as
� = KV x y DI (11-1)

SLURRY PIPELINES
11.13
FIGURE 11-7 Rheology of the Glasdtone limestone slurry. (From Venton, 1982. Reprinted
with permission of BHR Group.)

11.14 CHAPTER ELEVEN
where
K = 5228
x = 1.77
y = –1.18
The equation of Thomas does not agree well with experimental data published by
Hayashi et al. (1980). However, this may be due to the difference in particle size distribution.
Lokon et al. (1982) conducted further tests at the Melbourne Institute of Technology
in Australia and derived the following equation for the pressure loss gradient in Pa/m:
i m = KV x D I y Cv z (11-2)
where
im = friction gradient of the mixture
k = 54.9
x = 1.63
y = –1.42
z = 0.35
Lokon et al. indicated that their power law compared well with commercial pipelines.
Their data was based on iron ores pumped with solids in a size range of 30–60 �m (mesh
325–250). Obviously, this was the range of nonsettling slurries. Pressure losses are presented
in Figure 11-8.
Example 11-1
Using the Lokon equation, determine the pressure for an iron ore pipeline under the following
conditions:
� Pipeline inner diameter is 175 mm
� Volumetric concentration is 33%
� Flow rate is 48 L/s
� Pipeline length is 50 km
Solution in SI Units
flow area A = 0.25 × 0.175 2 = 0.024 m 2
flow speed V = Q/A = 0.048/0.024 = 2 m/s
im = KV x y z DI Cv
i m = 54.9 × 2 1.63 × 0.175 –1.42 × 0.33 0.35
im = 54.9 × 3.095 × 11.88 × 0.6784
im = 1369 Pa/m
Klose and Mahler (1982) measured the critical speed of iron ore slurry with particles
size in the range of 1 to 2 mm (0.04–0.08 in). However, due to the high density of iron oxide
(SG = 5.0) critical speed as high as 3.5 m/s (11.5 ft/sec) were recorded (Figure 11-9).
To design economic pipelines, Klose and Mahler suggested the addition of special chemical
additives that can reduce the critical speed of the mixture, despite a slight rise in the
pressure drop at these lower speeds (Figure 11-10).

5000
4000
3000
2000
1000
300
600
500
SLURRY PIPELINES
PRESSURE GRADIENT (kpa/Km)
� IRON ORE C V = 24.9
� IRON ORE C V = 26.6
� IRON ORE C V = 28.1
� IRON ORE C V = 30.2
� IRON ORE C V = 31.3
� WATER
VELOCITY (m/s)
0.5 1 2 3 4 5
11.15
FIGURE 11-8 Pressure losses for iron ore oxides in the range of 30–60 �m (mesh 325-250).
(From Lokon et al., 1982. Reprinted with permission of BHRA Group.)
Taconite is a very important source of iron in the United States. It is a form of iron
sand found in the Mesabi range of Minnesota, as well as in Manitoba and Ontario, Canada.
The Shilling Mining Review (1981), in an editorial article, reported on the pumping of
taconite tailings using 20 in × 18 in (500 mm × 450 mm) Warman tailings pumps sized to
a pressure of 350 psi. The pumps were installed in six stages.
Taconite tailings are considered coarse sand and must be pumped in a range of speeds
of 3.4–4.3 m/s (11–14 ft/s). Rubber-lined pipes are used. HDPE pipes are subject to very
fast wear and are not used for tailings disposal. Taconite tailings are typically pumped at a
weight concentration of 35%. The use of special flocculants in modern, efficient thickeners
allows pumping up to a weight concentration of 45%.
The SAMARCO pipeline in Brazil is one of the longest ever built to transport iron ore
oxides and features 500 mm (20 in) and 457 mm (18 in) pipe sections over a distance of
400 km (250 miles). Start-up occurred in 1977 and it is expected to remain in operation
for 40 years (Weston, 1985).
Another long pipeline to transport iron ore oxide is the La Perla-Hercules pipeline in
Mexico, with an overall length of 382 km (239 miles). The pipeline features one main and
one booster pump station with single-acting triplex plunger pumps (Thompson, 1995).

11.16 CHAPTER ELEVEN
Velocity, v (m/s)
Concentration, c v
FIGURE 11-9 Critical speed of iron ore oxides with particle size in the range of 1 to 2 mm
(0.04–0.08 in). (After Klose and Mahler, 1982. Reprinted with permission of BHR Group.)
11-6 PHOSPHATE AND PHOSPHORIC ACID
SLURRIES
Phosphate is a very important source of fertilizer for agriculture and is mined in large
quantities in the United States, Morroco, Egypt, South Africa, China, and other countries.
Phosphate rock is sometimes transported in a pre-milled state over a relatively
short distance—a few kilometers or miles. In Florida, a method of transporting phosphate
rock while mining it in a very similar fashion to dredging a river using a pump
has been developed. Large phosphate matrix pumps driven by diesel engines are available
on baseplates. These are relocated from site to site as the phosphate matrix field is
mined out.
Tillotson (1953) described the phosphate matrix in Polk and Hillsborough counties.
About 5120 km 2 (2000 mi 2 ) contained high grades of phosphate. Eight million short tons
of saleable phosphate pebbles were produced annually. Tillotson described the phosphate
matrix as an unconsolidated mixture of clay smaller than 1 mm (0.04 in), silica sand, and
phosphate rock of a much larger size. This mixture ranged in size from rocks as large as
150 mm (6 in) to as small as 38 �m (400 mesh). Because phosphate matrix pumps have to
handle lumps as large as 150 mm (6 in), they tend to be large with 500 mm (20 in) suction
and discharge sizes.

Pressure loss Dp (10 5 Pa/1000 ml)
22
20
18
16
14
12
10
8
6
4
2
vv c
DN 200
c = v = 0.64
c v
v vc
c
Once the phosphate matrix is pumped or transported, it is processed in a special phosphate
rock treatment plant. Nordin (1982), of the Phospnate Development Corporation
Ltd., described how each year a South African plant produces approximately three million
tons of phosphate rock from foskorite and pyroxenite ores. The ore was then classified
through flotation, thickening, and filters before being stockpiled. The result was fine
gray-white crystalline powder of mineral apatite, with a 36.5% P 2O 5 content, a solid specific
gravity of 3.17, and d 50 � 106 �m. The hardness of the apatite was measured at 5.0
on the Mohr scale.
Nordin (1982) reported that milled phosphate rock is easy to pump in a weight concentration
of 30–70%. He conducted tests on a 100 mm (4 in) loop and obtained the values
of critical velocity shown in Table 11-1, where d 70 � 75 �m. He recommended
pumping at 0.3 m/s (1 ft/s) above critical speed (Table 11-2).
11-6-1 Rheology
4%
4%
3% 3%
v vcc
SLURRY PIPELINES
2% 2%
0
0 1.0 2.0 3.0 4.0 5.0 6.0
Velocity, v (m/s)
11.17
FIGURE 11-10 Reduction of critical speed of iron ore oxides with particle size in the range
of 1 to 2 mm (0.04–0.08 in) by the use of special chemical additives. (After Klose and Mahler,
1982. Reprinted with permission of BHR Group.)
Landel et al. (1963) investigated the rheology of a bimodal (fine and coarse) distribution
of phosphate ore. They reported that in certain cases the finer particles act as a carrier for

11.18 CHAPTER ELEVEN
TABLE 11-1 Critical and Recommended Speed of Pumping Phosphate Rock with d 70
� 75 �m (data from Nordin, 1982)
the coarser solids and that for all intents and purposes the slurry may be considered non-
Newtonian. They proposed the following equation for the consistency factor:
Cv �
Cmax
K = � L� 1 – � –2.5
where
K = fluid consistency index (Pa·s)
C max = maximum solids volumetric concentration
C v = volumetric concentration of solids
Peterson and Mackie (1996) proposed the following equation for phosphate ore:
(11-3)
�0 = B���� (11-4)
Cmax – Cv where B = 13.3.
The data presented by Peterson and Mackie (1996) on the critical speed is consistent with
the data from Nordin (1982).
Anand et al. (1986) indicated that the corrosion rate due to Maton phosphate is of the
order of 0.3 mm/year in steel pipes. A total corrosion and wear allowance of 0.4 mm/year
is suggested by Peterson and Mackie (1996). It is, however, recommended to assume
more wear in the initial dozens of kilometers (miles) in a long pipeline, as particle attrition
and degradation often occur in the initial portion of the pipeline. As the particles become
less sharp, their abrasion of the pipeline decreases (Table 11-3).
11-6-2 Materials Selection for Phosphate
The Miller number of phosphate ore is smaller than 50 (Abulnaga, 2000). This means that
phosphate is suitable for pumping with piston reciprocating pumps.
C v 3
Weight concentration, %
30 40 50 63.5 67 70
Critical velocity m/s 1.30 1.10 0.90 0.85 1.1 1.35
Recommended pumping velocity, m/s 1.60 1.40 1.20 1.15 1.40 1.65
TABLE 11-2 Power Consumption 100 mm (4 in) ID Pipe for Phosphate Rock with
d 70 � 75 �m
Weight concentration, %
30 40 50 63.5 67 70
Power consumption, kW/km 0.196 0.151 0.128 0.112 0.109 0.113
After Nordin (1982).

Pumping phosphoric acid slurries represents a challenge to the manufacturing of slurry
pumps due to the combination of corrosion and wear in some of the critical circuits
such as flash cooling, filter feed, and gypsum removal. Walker (1993) reported that the
wear life of these pumps can be as low as a few thousand hours.
Traditionally, pumps were lined with rubber or manufactured out of stainless steel.
Rubber linings proved less than optimal. Tearing problems occurred during flash cooler
applications, lowering the life of some components to 3000 hrs. Erratic tearing also decreased
the wear life on filter feeds to as low as 1900 hrs, and local holing (formation of
holes in the liner) decreased wear life to 2300 hrs.
In some respects, installing pumps made out of stainless steel is an attractive option
since they can be repaired by welding, but stainless steels are not as hard as abrasionresistant
white iron. A special alloy, which offers as good resistance to corrosion as stainless
steel and hardness as white iron, is Hyperchrome, developed in Australia. Hyperchrome
was derived from hardfacing weld deposit materials defined in the Australian
Standard as AS2576-1982 Type 2. Walker (1993) described an important improvement of
wear life components over comparable stainless steel components.
high chromium content similar to hyperchrome. The new alloy achieved a service life of
2.5–3 years for an impeller in gypsum tailings service in a phosphoric acid environment,
whereas the Cd 4MCu material impeller was badly worn out after 3 months of operation.
Both Weir-Warman and KSB-GIW, the largest manufacturers of slurry pumps, have
experimented with the use of high chrome alloys with chromes in excess of 30% for slurry
pumps handling phosphate rock.
The Chevron Pipeline
SLURRY PIPELINES
TABLE 11-3 Example of Phosphate Ore Properties
Property Fines Coarse Product
Solids specific gravity 3.2 3.2 3.2
Freely settled particles packing (%) 40 44 51
Coefficient of sliding friction (�p) 0.53 0.58 0.58
d10 Particle size (�m) 60 50 14
d50 Particle size (�m) 22 92 75
d90 Particle size (�m) 74 150 145
After Paterson (1996).
11.19
Tian et al. (1996) reported the development of a new alloy, a white iron with a very
11-6-3
About 280 km (175 mi) southeast of Salt Lake City, U.S.A. at Vernal, Utah, there is a
large layer of phosphate ore that covers an area of 38,400 km 2 (15,000 mi 2 ) with an estimated
reserve of 700 million short tons (640 million metric tons). Between 1961 and
1986, the ore was transported by truck over a distance of 216 km (135 mi) and then
loaded onto railroad cars. In 1986, a pipeline for the phosphate concentrate was commissioned.
The diameter of the pipeline is 250 mm (10 in). A pipeline feed station with a test
loop and a pump station is located at Vernal, and a booster is located at Richard’s Gap in

11.20 CHAPTER ELEVEN
Wyoming. After covering an overall distance of 150 km (94.3 mi) the pipeline terminates
at Rock Springs, Wyoming. At Vernal, the concentrate is thickened in a special thickener,
then conditioned in three agitator tanks. These large tanks are 15.24 m × 15.24 m (50 ft ×
50 ft), each holding 2000 tons of phosphate. The slurry has a weight concentration of
53–60%. Weston and Worthen (19xx) indicated that each agitation tank is fit with a 200
hp mixer. The slurry is tested in a 91 m (100 ft) long pump loop prior to being fed to the
pipeline. The pipeline was designed for a flow of 86 L/s (1370 US gpm) and a pressure of
17.915 MPa (2600 psi) per pump station. The booster station is located at 77 km (48 mi),
halfway along the pipeline.
Two positive displacement Wilson-Snyder pumps were installed at the main pump
station. These were driven by 746 kW (1000 hp) direct current motors. The booster station
is connected to a water pond and draws water on demand to avoid slack flow by providing
additional back pressure in low flow conditions. Choke stations are also provided
for additional back pressure. According to Weston and Worthen, the pipeline used highyield-strength
steel rated at 413 MPa (60,000 psi). An allowance of 2.5 mm (0.1 in) for
corrosion/erosion over a lifetime of 25 years was factored into the design. The thickness
of the pipeline varied between 6.4–12.7 mm (0.25–0.5 in).
The pipeline crossed the Rocky Mountains, so the elevation varied between a low
point of 1676 m (5500 ft) and a high point of 2499 m (8200 ft). To minimize freezing
problems, the pipeline was buried to a depth of 1.8 m (6 ft). There are 12 monitoring or
testing points along the pipeline to monitor for pressure. If freezing or sedimentation develop,
the resultant increase in pressure is automatically detected.
Slurry is pumped at an average speed of 1.5–1.67 m/s (5–5.5 ft/sec) and it takes about
26 hours for the material to be transported from start to finish. The pipeline was designed
to transport 2273 million metric tons (2500 million short tons) of phosphate concentrate
per year. Initially, it operated on a special batch mode with 12 hours of water and 12
hours of slurry.
To monitor corrosion, the three following methods are used:
1. Corrosion spools (sacrificial thickness loss)
2. Ultrasonic testing (to measure pipe thickness)
3. Corrosion probes (to measure corrosivity)
Corrosion of phosphate pipelines is reduced by the use of special inhibitors or by raising
the pH to the alkaline range (alkalination).
11-6-4 The Goiasfertil Phosphate Pipeline
Pertuit (1985) described the Goiasfertil phosphate pipeline. It was constructed in the state
of Goias in Brazil to transport phosphate ore along difficult terrain over a distance of 14.5
km (9 miles) from a mine to an existing railway station in the town of Cataloa. The
pipeline was designed to ship 900,000 metric tons of concentrate per year over a period of
6750 hours. The slurry consisted of solids at a weight concentration of 63% to 66%. The
particle distribution consists of 20–25% + 150 microns, and about 25–35% minus 45 microns.
The start-up was in August 1982. The pump station consists of charging centrifugal
pumps, a safety and test loop, and two mainline Wilson-Snyder positive displacement
pumps. They are controlled by variable speed drivers and the speed is adjusted according
to the flow rate.

11-6-5 The Hindustan Zinc Phosphate Pipeline
Pertuit (1985) described the Hindustan zinc phosphate pipeline, which was commissioned
in late 1983 near the town of Udaipar in India. This pipeline was designed to ship on an
intermittent basis 400 metric tons of phosphate per year via a 73 mm (2.875 in) diameter
pipe over a distance of 10.5 km (6.5 miles). Phosphate is shipped at a weight concentration
of 65–68% by weight. The pump station consists of charging centrifugal pumps and
Worthington plunger pumps. At the terminal station, thickeners and agitated tanks were
installed.
11-7 COPPER SLURRY AND
CONCENTRATE PIPELINES
SLURRY PIPELINES
11.21
Venton and Boss (1996) described the wear of the OK Tedi copper concentrate pipeline
in Papua New Guinea. This pipeline is 155 km (95 mi) long. Severe localized wear
caused replacement of certain section of the pipeline. The pipeline was commissioned in
1987. The mine is 156 km (96 mi) from the seaport of Kiunga. About 96 km (60 mi) of
the pipeline uses gravity flow. A booster station was installed to promote flow over the
remaining 60 km (37.5 mi). The normal flow rate was in the range of 85–88 m 3 /hr
(374–388 US gpm) with a weight concentration of 55–60%. The speed of flow was in the
range of 1.22–1.4 m/s (4–4.6 ft/sec). The wall thickness was in the range of 5.6–11 mm
(0.22–0.433 in) (Table 11-4). The pipeline operated in batches of water and concentrate.
The water was not neutralized by an oxygen scavenger.
Venton and Boss (1996) indicated that an initial pipeline failure occurred in 1991.
They attributed this failure to accentuated wear due to coarser particles, not the design
of the valves. The most severe wear occurred at the change of pipe thickness from
5.6–6.4 mm (0.22–0.25 in). Corrosion was also a factor as no oxygen scavenger had
been used. The operators installed corrosion-meter probes in 1992 on the top and bottom
section of the pipeline to monitor wear as loss of wall thickness. Wear of 0.37
mm/year (0.0145 in/year) was measured for the bottom section of the pipe, and wear
of 0.18 mm/year (0.007 in/year) for the top section with continuous slurry water/batching.
Venton and Boss (1996) reported that there were four batches of slurries at 500 m 3
(17,657 ft 3 ) and four batches of water of 40–50 m 3 (1,413–1,766 ft 3 ) per day. To mitigate
against wear, the top size (+106 microns) was cut down to 1%, water batching was eliminated
altogether, and the pipeline was allowed to shut down and restart with slurry. Unfortunately,
60 km (37.5 mi) of pipe had to be replaced with thicker walled pipe to continue
operation over its anticipated life of 15 years.
TABLE 11-4 Velocity of Flow of Copper Concentrate Pipelines
Pipeline Nominal flow rate Nominal velocity
OK Tedi DN150 (6 in OD) 85 m3 /h 374 USGPM 1.24 m/s 4.07 ft/s
Bonguinville DN 150 (6 in OD) 65–109 227–479 1.23–2.04 4.03–6.7
Freeport DN 100 (4 in OD) 1.1–1.6 3.6–5.25
DN 125 (5 in OD) 1.2–1.5 3.95–4.1
After Venton and Boss (1996).

11.22 CHAPTER ELEVEN
Venton and Boss (1996) described in great detail the operating problems of the OK
Tedi pipeline. Their recommendations for better operation included the following:
� Installing tracking modules on each pig for pigging the pipeline.
� Replacing the Rockwell Nordstrom valves, which often fail due to inadequate lubrication
in remote valve stations, by other valves. Tests were run on Audco full bore
valves, Mogas metal-seated valves, and Larox high-pressure hydraulically activated
punch valves. The Mogas valves did not require lubrication and lasted 140 operating
cycles whereas the Rockwell Nordstrom plug lasted 35 cycles, the Audco ball lasted
20 cycles, and the Larox pinch valve lasted 45 cycles. The Mogas valve was therefore
the most appropriate for this copper concentrate pipeline.
One of the largest copper mines in the world is operated by Minera Escondida Ltd. in
Chile. The mine is located at an altitude of 3100 m (10,170 ft) above sea level. To transport
the copper concentrate a pipeline was constructed. The pipeline uses a single pump
station at the beginning of the pipeline and gravity throughout the remainder. The pipeline
spans 165 km (103 mi) of mountainous terrain and transports the slurry at a cost of 1–1.5
dollars per metric ton. This style of transportation is considerably cheaper than the alternative
option of trucks and railroads. Nordstrom valves were used on the pipeline (Boggan
and Buckwalter, 1996).
Bajo Alumbrera is located in northwest Argentina near Catamarca. The plant processes
90,000 tons of ore a day. Copper concentrate is shipped to a port via a 152 mm (6 in)
pipeline over a distance of 320 km. Geho positive displacement diaphragm pumps in the
main pipeline and a couple of booster stations provide the power to pump over such a
long distance.
11-8 CLAY AND DRILLING MUDS
Sellgrem et al. (2000) conducted tests on sand as well as sand–clay mixtures pumped by
centrifugal pumps. The phosphate clays had a diameter d 50 between 1 �m and 50 �m.
The sands were much coarser with d 50 of 0.64 mm (0.025 in), 1.27 mm (0.05 in), and 2.2
mm (0.09 in). The presence of clay and other particles finer than 75 �m and a concentration
smaller than 20% had a beneficial effect by reducing the head loss and efficiency derating
factor. The data recorded by Sellgrem et al. (2000) should not be applied to a higher
concentration of clays because the viscosity effect introduces a new component to the
equation.
Drilling muds and bentonite are pumped at high concentration in the oil industry using
positive displacement pumps. Certain important minerals such as bauxite for the aluminum
industry are found in bentonite.
Soft high clay is present in certain copper ores. In a diluted form at weight concentration
smaller than 40%, it can be handled fairly well. However, particular attention must be
paid in milling circuits when the concentration may approach 50%, as the viscosity affects
flow and recirculation loads.
Codelco in Chile is one of the largest producers of copper in the world. To dispose
tailings to the Ovejeria Tailings Dam, slurry had to be piped from an elevation of 4000 m
above sea level down to 700 m via a 57 km (36 mile) long pipeline. At the dam, the coarse
and fine are separated using a cyclone station. Three Wirth positive displacement pumps
are then used to pump the coarse material around a 4 km loop at a flow rate of 140 m 3 /hr
(616 US gpm) and at a pressure of 4.0 MPa (580 psi) (Figure 11-11).

11-9 OIL SANDS
SLURRY PIPELINES
11.23
FIGURE 11-11 Tailings solids segragation station and pumping facility for copper tailings
at Codelco Andina, Chile, featuring the use of diaphragm pumps. (Courtesy of Wirth Pumps,
Germany.)
In 2001, Canadian oil sands were the most pumped slurries in the world. Due to this large
amount of slurry (1.9 m 3 /s) (ranging up to 30,000 US gpm), pump manufacturers developed
new technologies, including technology for froth treatment.
At the Summit Meeting at Quebec during April 2001, Canada encouraged the United
States to invite U.S. corporations to invest billions of dollars in the oil sand fields of Alberta.
Even without new U.S. investments, an estimated 20 billion will be invested between
2000 and 2020. This is a continuously growing industry that will require sophisticated
slurry systems.
The process of extracting oil from sand is a vast topic and only a few aspects will be
touched on in this chapter. In basic terms, Alberta, Canada sits on layers of tar-rich sands.
The shallowest layers, which are most accessible for open pit mining, are in the north of
Alberta near the Athabasca River and Fort McMurray. Between the first discoveries in the
1930s and the end of the century, a number of technologies were developed to the extract
oil from the sand. The initial approach was to heat slurries of tar sand to a temperature
that reduced the viscosity of the oil and separated it from the sand. Other technologies developed
coannular flows that separated the oil from the sand by degradation of the natural
lumps of oil and sand. More recently, solvents were developed that dissolve the tar or oils
out of the sand. The latest solvent-based technologies use lower temperatures, reducing
energy costs.
Outside the Athabasca region, the oil sands are located in deeper layers. The proposed
extraction method pumps hot steam down approximately 100 m (300 ft) of pipe to the oil
sand bed. The steam would then resurface carrying the oil and tar. This technology was

11.24 CHAPTER ELEVEN
originally proposed to recover oil from oil shale in Colorado, U.S., and Queensland, Australia.
Syncrude and Suncor-Muskeg River Shell in conjunction with a Canadian Research
Institute, the Saskachewan Research Institute, the University of Alberta, and the University
of Toronto, developed this new technology for oil sand slurries pumping. The concept
of stratified two-layer flows was extensively investigated by these companies and institutes
to handle 63 mm (2.5 in) lumps. By 1999, the price of synthetic oil produced from
processed tar sand by Syncrude and Suncor dropped low enough to compete with natural
oil from Texas and the Middle East. A dedicated pipeline from Edmonton, Canada to
Chicago, U.S.A. became the longest pipeline for synthetic fuel.
In a recent paper, Sanders et al. (2000) discussed the effects of bitumen on sand hydrotransport
and conducted tests on a number of grades of oil sands. They reported that
pipeline pressure losses due to friction at cold or warm temperatures increased with the
length of the pipe. A time dependency developed, which was attributed to the formation
of a thin coating of bitumen at the wall of the pipe. They defined an equivalent pipe
roughness in the presence of bitumen of 650–1150 �m, which is much higher than normal
steel at a roughness of 63 �m. In a 250 mm (10 in) pipe, the presence of fines in the
oil sand slurry reduced the deposition velocity to 1.1 m/s, whereas the absence of fines increased
the deposition velocity to 2.7 m/s. Due to the change in speeds, different approaches
are used in the designs of pipelines for coarse grade and fine grade ores. The
lower-grade ores, with less bitumen, do not necessarily exhibit this phenomenon of wall
coating and therefore require higher pressure for pumping.
Another pipe coating focused on in numerous tests is the tar coating of pipes in froth
treatment plants. Under certain conditions, the injection of water through a ring just a few
diameters before the pump suction reduces power consumption and improves the efficiency
of pumps. It is not known whether tar deposits on the impeller end causes a degradation
of pump performance. The physical properties of oils in oil sand would defy any
designer of centrifugal slurry pumps and there are no standard methods to account for derating
of performance.
McKibben et al (2000) conducted tests on water–oil mixtures (without sand) with
crude oils of a viscosity of 5300–11,200 Pa·s. They found evidence against two popular
theories. First, they showed how the injection water did not form a layer at the wall to reduce
pressure losses as was commonly thought. Instead, it formed slug around the oil and
transported at lower pressure losses. Secondly, they also indicated that the viscosity of the
oil was of no consequence. Therefore, it must be said that the flow of oil, sand, and water
as a mixture is fairly complex.
The Canadian oil sand projects have encouraged the manufacturers of slurry pumps to
develop special mechanical seals for slurry pumps (Swamanathan et al., 1990).
11-10 BACKFILL PIPELINES
A backfill is essentially a mine residue or tailing pumped back to fill excavated or mined
pits. A backfill can be mixed with other low-permeability materials such as clays to help
seal the area. Particularly in the case of underground backfilling, the water content should
be minimized to avoid costly dewatering. Backfill slurries are therefore dense, with a high
weight concentration (around 50–65%). Multistage centrifugal pumps or positive displacement
pumps are used to transport them (Figure 11-12).
Steward (1996) conducted an empirical study of vertical and horizontal pipelines. He
demonstrated how backfill consisting of fine and coarse material could be classified

through cyclones in order to separate them. The cyclone underflow was then drained by
gravity. In the South African gold and uranium mines, Steward (1996) reported flow
speeds as high as 12 m/s (40 ft/s). These extremely high speeds are the cause of rapid
wear and erosion–corrosion.
To support and reinforce an underground excavated area after the ore has been removed,
the backfill (including both fine and coarse material) is thickened. The product is
called full plant tailings. Steward (1996) defined the particle sharpness as the rate of directional
change in the particle perimeter.
Pipe wear is measured as the rate of mass loss per unit of time (kg/s or slugs/s). Wear
in a pipeline is an exponential function of the flow speed:
dw/dt = KV n
Since it is also a function of other parameters, Steward (1996) proposed the following
function:
where
f = function of
S m = specific gravity of mixture
SLURRY PIPELINES
dw/dt = f (S m, V, D I, d 90, SI, M)
11.25
FIGURE 11-12 Backfilling of very dense slurry using diaphragm pumps. (Courtesy of Wirth
Pumps, Germany.)

11.26 CHAPTER ELEVEN
V = velocity of flow
D I = pipe inner diameter
d 90 = 90% passing diameter of particles
SI = sharpness index
M = mass of solids per unit length of pipe
From his tests, Steward (1996) derived the following empirical equation (in SI units)
(Table 11-5):
where
m 1 = –6.31374
m 2 = 0.3186193
m 3 = –0.131869
m 4 = 0.0054758
m 5 = 1.7709578
m 6 = 0.6162088
m 7 = 6.5961888
log 10(dw/dt) = m 1S m + m 2V + m 3D I + m 4d 90 + m 5SI + m 6M + M 7
Coetzee (1990) determined that one-third of the loss of pipeline wall thickness associated
with pumping mine water is due to corrosion because mine water is often acidic.
Since corrosion is an important contributor to wear of backfilled pipes, it became evident
that lining the pipes was necessary. To compare piping materials, tests were conducted by
Steward (1996) and indicated that a polyurethane rubber at a Shore hardness 55 Shore A
provided the best pipeline protection in a test with slurry pumped at a speed of 3 m/s. By
comparison, ASTM steel 106 grade B wore seven times faster than polyurethane 82 Shore
A, or high-density polyethylene.
Steward (1996) indicated that the mixing of cementitious binders with slurry could reduce
wear considerably in backfill applications.
Backfill paste is formed by dewatering slurry of tailings (thickening and filtering).
Mixing dewatered slurry with cement (3%–5%) produces a stiff backfill (1.5–3.5 MPa
strength, or 218–508 psi). Coarse aggregates (

11-11 URANIUM TAILINGS
Uranium is considered an important material for nuclear energy production. The pumping
of uranium slurries is more complicated than pumping most slurries due to its radioactivity.
As a result, uranium tailings disposal systems should involve a health specialist and an
environmental engineer.
Uranium ores vary dramatically from one site to another. In certain areas (for example,
Saskatchewan, Canada, Wyoming, U.S.A., and South Australia, Australia), the ore is
rich. In other areas (for example: the mineral sands of Egypt, or the South African
gold/uranium mines), uranium is essentially a by-product.
Due to radioactivity, the design of the tailings disposal system must be left to a lining
specialist. It is extremely important that no seepage be released. Four main types of liners
are usually selected:
� Geological liners
� Clay liners
� Synthetic liners such as synthetic membranes, gunite, asphalitic concrete, or sprays
� Compacted soil or soil cement
Some of the slimes from the tailings can be used to form a lining later. Clays, especially
bentonites, are very good liners with a permeability as low as 10 –7 cm/s (4 × 10 –8 in/sec).
To dissolve uranium in milling, two processes are used:
1. Acid leaching
2. Alkaline leaching
SLURRY PIPELINES
In acid leaching, sulphuric acid is used in a complex ion-exchange or solvent extraction
process to produce yellowcake of very high purity. Various metals (such as vanadium,
arsenic, nickel, iron, copper, etc.) may be leached in this process. Chemicals involved
in this process include sulphuric acid, ammonium nitrate, sodium chloride, amines, alcohols,
kerosene, and ammonia. Considerable process water has to be derived from reclaim
water of the tailings and returned to the mine for preparing the slurry.
The alkaline process is less common than the acid process and provides lower recovery
rates of uranium. For both processes, the ore must be first reduced to a size smaller than 75
�m (mesh 200). Important radioactive pollutants from either process are 226Ra (radium
226) and 222Rn (radon 222). By adding barium chloride in settling ponds or lagoons, radium
is treated. Radium-containing slurries and sludges are the subject of extensive research.
Radium 226 decays to form radon gas from the tailings and is dangerous to health.
The pH of solutions from the solvent-extraction process is typically low, generally
1–2. Conventional carbon steel pipes may be inferior to high-density polyethylene pipes
in resisting combined erosion and corrosion.
11-12 CODES AND STANDARDS FOR
SLURRY PIPELINES
11.27
Safety is of paramount importance with some of the modern pipelines. Slurry is heavier
than water, and the rupture of a pipeline can have disastrous effects on the environment

11.28 CHAPTER ELEVEN
and may cause accidental death. For this reason the engineer and contractor should follow
certain codes and standards. The ASME B31.11 is basically the only standard specific
to slurry pipelines. However, the American Petroleum Institute has many useful
guidelines.
The ASME (American Society of Mechanical Engineers) developed in 1989 a special
section to its piping code B31 called ASME B31.11-1989—Slurry Transportation Piping
Systems. This standard provides useful guidelines for operation and maintenance of slurry
pipelines and piping systems. Because the code came into existence at the end of the
1980s, it is still not well known. Consultant engineers need to refer to it, particularly in
the case of all-metal piping.
It is the ultimate responsibility of the engineer and the contractor to provide safe procedures
for the operation and maintenance of the slurry piping system. The code provides
some appropriate guidelines. The code requires an established plan for operation and a
maintenance procedure. The plan must be written with instructions to operate and maintain
the pipeline. It must include provisions for control of external as well as internal corrosion
and erosion of new and existing pipes. It must describe emergency procedures to
follow in case of system failures, accidents, and other hazards. Employees need to be
trained in these emergency procedures. Because pipelines eventually cross rural or urban
areas and could have an impact on the environment, the local authorities should be consulted
when establishing such a plan.
The plan must detail procedures to review and reflect on important changes in the conditions
affecting the safety and integrity of the piping systems. A methodology must be in
place to patrol and report on changes in construction activities, railroad and highway
crossings, and urban and commercial activity. Obviously, no one should come with an excavator
and damage a buried pipeline because he did not know that it existed. An abandonment
plan must include procedures for shutting down and abandoning a pipeline, and
must include appropriate cleanup procedures before abandonment.
The operating pressure includes the steady-state pressure to overcome friction as well
as the static head. The code requires that the level of pressure rise due to surges not exceed
at any point the internal design pressure by more than 10%.
Markers showing the existence of the pipeline must be installed on each side of a public
road, railroad crossing, or crossing of navigable water. Markers are not required for
off-shore pipelines such as those used in subsea tailings disposal. The following standard
should be consulted: API RP1109: Marking Liquid Petroleum Pipeline Faculties 1993.
This code provides guidelines for the permanent marking of liquid petroleum pipeline
transportation facilities. It covers the design, message writing, installation, placement, inspection,
and maintenance of markers and signs for inland waterway crossings and onshore
crossing. It is a short standard of 12 pages that the consultant engineer should modify
to suit the slurry pipeline.
Patrolling pipelines on a periodic basis is recommended, particularly to ensure that all
valves remain accessible (and are not lost between growing vegetation), to ensure that
emergency diversion ditches remain free of debris, grass, and leaves. The code recommends
limiting patrols to a maximum of one per month. Underwater crossing should be
inspected particularly after a flood or a heavy storm that may have caused mechanical
damage.
Pipeline repairs need to be performed by qualified personnel. ASME B31.11 recommends
using the following publication of the American Petroleum Institute as a guideline:
API RP1111: Design, Construction, Operation and Maintenance of Off-shore Hydrocarbon
Pipeline and Risers (Third Edition 1999).
Safety guidelines for a pipeline crossing are covered by the following document: API
RP1102: Steel Pipelines Crossing Railroads and Highways (Sixth Edition, April 1993).

SLURRY PIPELINES
11.29
API RP1102 covers design, installation, and testing to ensure safe crossing of steel
pipelines under railroads and highways.
High-pressure slurry pipelines, particularly some of the newer 2500 psi pipelines, must
be safely welded. Some of these use API Grade 5LX65 or 5LX70 steel. It is recommended
to implement API Standard 1104: Welding of Pipelines and Related Facilities (Nineteenth
Edition, 1999.) This standard was obviously written for the oil and gas industry. It covers
gas and arc welding to complete high-quality welds of carbon steel as well as low-alloy
steel piping. It covers different types of welding processes, such as shielded metal arc welding,
submerged arc welding, gas–tungsten arc welding, gas–metal arc welding, as well
methods to test and inspect welds by radiography. When a pipeline is constructed, a standard
procedure is to conduct hydrostatic tests. The American Petroleum Institute developed
the following recommended practice document: RP1110: Pressure Testing of Liquid Petroleum
Pipeline (Fourth Edition, March 1997). This document is of interest to the slurry
pipeline engineer because it descries the minimum procedures to be followed as well as
some of the equipment used during hydrostatic testing. Because slurry pipelines are now
built in lengths from few kilometers (or few miles) to hundreds of kilometers (or miles) and
are often unpatroled and buried in remote and isolated regions, control engineers have developed
methods to monitor pipelines for pressure or leakage. One such system is called
SCADA (system control and data acquisition). The American Petroleum Institute has developed
a number of useful documents to consult. Publication 1113: Developing a Pipeline
Supervisory Control Center (Third Edition, February 2000) presents six lists of general
considerations to design a center to monitor and control a pipeline.
With the advent of modern computer-based system for monitoring pipelines, API developed
an appropriate standard: Std 1130: Computational Pipeline Monitoring (First
Edition, October 1995). This standard is particularly important in detecting anomalies,
which can be attributed to leaks, rupture of the line, etc. It covers algorithmic monitoring
tools to help the person in charge of control and monitoring detect such anomalies.
With the importance accorded to early detection of pipeline leaks, the American Petroleum
Institute formed a taskforce in 1989 with the University of Idaho to evaluate commercial
software for leak detection. A report was published as Publ. 1149: Pipeline Variable
Uncertainties and Their Effects on Leak Delectability (First Edition, 1993). This
document was followed by a second and rather useful publication, Publ. 1155: Evaluation
Methodology for Software-Based Leak Detection Systems (First Edition, 1995). These
publications and standards should not be imposed by the consultant engineer without reference
notes and modifications as related to slurry. They have been listed as useful tools
because most high-pressure pipelines are transporting oil and gas and these industries
have contributed greatly to the safety of systems similar to the ones the slurry engineer is
designing. The ASME Code B31.11 recommends certain permanent repair procedures
when particular problems have been detected. When hoop stress is detected exceeding
20% above the minimum yield stress, followed by a reduction of thickeners, the code recommends
making the following permanent repairs:
� Remove all gouges and grooves having a depth in excess of 12.5% of the nominal pipe
thickness.
� Remove dents that affect the pipe curvature at the pipe seam or at any girth weld.
� Remove dents that exceed 7.5% of the nominal pipe diameter.
� Remove or repair all arc burns.
� Remove or repair all cracks.
� Remove or repair all welds considered to be imperfect by the stipulations of the code.

11.30 CHAPTER ELEVEN
� Repair areas where the thickness is smaller than the recommended thickness for pressure
minus the corrosion allowance.
� Remove or repair all sections containing leaks.
Although the code recommends taking out of service defective sections of the pipeline, it
accepts that this is not always possible. It does, therefore, allow for certain methods of repair
while the pipeline is in service, such as the use of mechanically split sleeves, hot tapping,
encirclement of welds, etc. The engineer should, however, be aware that these methods
may not apply to rubber, polyurethane, or HDPE-lined steel pipes as they would
damage the internal lining by heat.
New replacement sections of a pipeline must be subject to a pressure test after their installations,
when the pipeline is operating at a hoop stress of more than 20% of the specified
minimum yield strengths.
The control of corrosion and erosion of slurry pipelines is covered by Chapter VIII of
the ASME Code B31.11. The code correctly points out that in certain cases erosion is an
accelerating factor in the internal corrosion of pipelines, by effectively removing scales,
oxides, films, and lining.
Buried steel pipes can be subject to corrosion. The National Association of Corrosion
Engineers developed the following standard: RP0196-96: Control of External Corrosion
on Underground or Submerged Metallic Piping Systems. This standard provides a
methodology for determining the need for corrosion control, cathodic protection and design,
installation of cathodic protection systems, and control of interference currents.
Sacrificial anodes, such as magnesium blocks, are installed in difficult soils such as
those containing chlorides in certain deserts. Due to the difference between magnesium
and steel on the galvanic corrosion chart, the system is designed so that the established
current causes corrosion to the magnesium blocks rather than to the steel pipe. Monitoring
galvanic corrosion is important and the reader may consult the CEA Report 54276, “Cathodic
Protection on Buried Pipelines,” as well as TPCII, “A Guide to the Organization of
Underground Corrosion Control Coordinating Committees.”
Monitoring cathodic protection is emphasized by ASME B31-11, which recommends
testing at two-month intervals or less of all cathodic protection rectifiers and connected
protective devices. Erosion–corrosion of slurry pipelines may be reduced using special
corrosion inhibitors, by avoiding sharp corners and short elbows, and providing some
lining such as HDPE, rubber, or polyurethane. Monitoring of the pipeline on a yearly
basis and at intervals that do not exceed 15 months is recommended by B31.11. Areas
prone to more rapid localized erosion–corrosion should be monitored more frequently as
dictated by experience. Corrective measures should be established on the basis of historical
leaks.
Records must be kept for location of cathodic protection facilities, repairs of localized
wear problems, and for frequency of such repairs.
11-13 CONCLUSION
Extensive experience has been gained over the years on the pumping of different minerals
and tailings. Some particles are ground to a very fine range and flow as non-Newtonian
mixtures, and some are as coarse as 50 mm (2 in), as in the coal and oil sand industries.
Despite all the promise of coal, only one long coal pipeline has been built in the United
States—the Black Mesa Pipeline. Another long pipeline that transports coal is the
Novo-Siberski pipeline in Siberia, Russia. Other minerals such as copper concentrate,

iron oxide, limestone, and phosphate have become a normal part of the engineering of
mines in very remote and isolated regions such as Escondida, Bajo Alumbrera, Antemina,
Samarco, etc. The science of slurry pipeline engineering has therefore blossomed into
very practical schemes.
Instrumentation and remote monitoring have made slurry pipelines safer, better protected
against leakage, freezing, and surging.
Great progress has been made with pumping sets. Centrifugal pumps are now available
up to a design pressure of 7 MPa (1000 psi). Positive displacement pumps push the
limit of use to 17.5 MPa (2500 psi). Lockhoppers are available to pump very coarse solids
(>50 mm or >2 in) up to distances in excess of 105 km (60 mi).
A successful pipeline project depends on proper economics. This will be the topic of
the next chapter.
11-14 REFERENCES
SLURRY PIPELINES
11.31
Abbot, J. 1965. Use of Hydrocyclones for Thickening and Recovery in the National Coal Board. Filtration
and Separation, 2, 3, 204–208, 234.
Abulnaga, B. E. 1990. An Internal Combustion Engine Featuring the Use of an Oscillating Liquid
Column and a Hydraulic Turbine to Convert the Energy of Fuels. Australian Patent AU-B-
20956/88.
Adams, W. I. 1986. Polyethylene Pipelines for Slurry Transportation. In 11th International Conference
on Coal Technology. Washington, D.C.: Coal and Slurry Technology Association.
Anand, S., S. K. Ghosh, S. Govindan, and D. B. Nayan. 1986. Maton Rock Phosphate Concentrate
Pipeline. Working paper, BHRA Group, Hydrotransport 10, Innsbruck.
Boggan, J. and R. Buckwalter. 1996. Slurry Pipeline Helps Remedy Corrosion at Record Height.
Pipeline and Gas Journal, 223.
Bomberger, D. R. 1965. Hexavalant Chromium Reduces Corrosion in a Coal-Slurry Pipeline. Materials
Protection, 4, 1, 41–48.
Brackebush, F. W. 1994a. Basics of Paste Backfill Systems. Mining Engineering, 46, 1175–1178.
Brackebush, F. W. 1994b. Basics of Paste Backfill Systems. Mining Engineering, 47, 1041–1042.
Brooks, D. A. and C. H. Dodwell. 1985. The Economic and Technical Evaluation of Slurry Pipeline
Transport Techniques in the International Economic Coal Trade. In 10th International Conference
on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology
Association.
Buckwalter, R. and A. Walters. 1989. Selection of coal slurry pipeline technologies for gasification
combined power cycle plants. In Proceedings of the 14th International Conference on Coal
and Slurry Technology. Washington, DC: The Coal and Slurry Association.
Burgess, K. E. 2000. Froth Pumping. Technical Bulletin No. 28. Sidney Australia: Warman International.
Coetzee, R. 1990. Wear and Corrosion of Mild Steel Tubes in Backfill and Backfill Feed Filtrate.
Report 332891. Physical Metallurgy Division. Council for Mineral Technology, South Africa.
Ercolani, D., E. Carniani, S. Meli, L. Pelligrini, and M. Primercio. 1988. Shear Degradation of Concentrated
Coal–Water Slurries in Pipeline Flows. In 13th International Conference on Coal
Technology, Denver, CO. Washington, D.C.: Coal and Slurry Technology Association.
Faddick, R. R. 1982. Ship loading Coarse-Coal Slurries. Working paper A-3, in 8th International
Conference on Solids in Pipes, Johannesburg, South Africa.
Gandhi, R. G. 1985. Fosferil phosphate slurry pipeline. In 10th International Conference on Coal
Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association.
Garbett, E. S. and S. M. Yiu. 1988. The Effect of Convective Heat on the Disintegration of a
Coal–Water Mixture in Pneumatic Atomization. In 13th International Conference on Coal
Technology, Denver, CO. Washington, D.C.: Coal and Slurry Technology Association.
Gillies, R. G., J. Schaan, R. J. Sumner, M. J. McKibben, and C. A. Shook. 2000. Deposition Veloci-

11.32 CHAPTER ELEVEN
ties for Newtonian Slurries in Turbulent Flows. Canadian Journal of Chemical Engineering,
78, 4, 704–708.
Hayashi, H. et al. 1980. Some Experimental Studies on Iron Concentrate Slurry Transport in Pilot
Plant. Working paper, BHRA Group, Hydrotransport 7, Sendai, Japan.
Hughes, C. V. 1986. Coal Slurry Pump Development Update. Mainline Pumps for the Belovo-
Novosibirsk Pipeline. In 11th International Conference on Coal Technology, Washington,
D.C.: Coal and Slurry Technology Association.
Klose, R. B. and H. W. Mahler. 1982. Investigations into the hydraulic transportation behaviour of
ore and coal suspensions with coarse particles. In Hydrotransport 8, Johannnesburg. Cranfield,
UK: BHRA Group.
Kreusing, H., and F. H. Franke. 1979. Investigations on the Flow and Pumping Behavior of Coal–Oil
Mixtures with Particular Reference to the Injection of Coal–Oil Slurry in the Blast Furnace.
Working paper C-2, BHRA Group, Hydrotransport 6, BHRA.
Landel, R. F., B. G. Mosen, and A. J. Bauman. 1963. In 4th International Conference on Rheology,
Brown University, Part 3, p. 663. New York: Interscience Publishers.
Leninger, D., W. Erdmann, and R. Kohling. 1978. Dewatering of Hydraulically Delivered Coal.
Working paper E-7, BHRA Group, Hydrotransport 5, Hanover.
Lokon, H. B., P. W. Johnson, and R. R. Horsley. 1982. A “Scale-up” Model for Predicting Head
Loss Gradients in Iron Ore Slurry Pipelines. Working paper B-2, BHRA Group, Hydrotransport
8.
Madsen, B. W., S. D. Cramer, and W. K. Collins. 1995. Corrosion in a Phosphate Pipeline. Materials
Performance, 34, 70–73.
McKibben, M., R. G. Gilles, and C. A. Shook. 2000. A Laboratory Investigation of Horizontal Well
Heavy Oil–Water Flows. Canadian Journal of Chemical Engineering, 78, 734–751.
Miller, J. W. and H. L. Hoyt. 1988. Evaluation of Polymers as Suspending Aids for Coal–Water
Slurries. In 13th International Conference on Coal Technology. Washington, D.C.: Coal and
Slurry Technology Association.
Morway, A. J. 1965. Stabilized Oiled Coal Slurry in Water. US Patent 31,201,168 assigned to Esso
Research & Engineering Co. N.J., USA.
Nordin, M. 1982. Slurry for Sale. Working paper F-2, BHRA Group, Hydrotransport 8.
Olofinsky, E. P. 1988. Belovo-Novosibirsk Coal Transportation Pipeline. In 13th International Conference
on Coal Technology, Denver, CO. Washington, D.C.: Coal and Slurry Technology Association.
Peterson, A. J. C. and K. Mackie. 1996. An Economic and Technical Assessment of the Hydraulic
Transport of Phosphate Ore. BHRA Group, Hydrotransport 13.
Pertuit, P. 1985. Gladstone Limestone Slurry Pipeline. In 10th International Conference on Coal
Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association.
Pertuit, P. 1985. Goiasferil Phosphate Pipeline. In 10th International Conference on Coal Technology,
Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association.
Pertuit, P. 1985. Hindustan Zinc Phosphate Pipeline. In 10th International Conference on Coal Technology,
Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association.
Pipelin, A. P., M. Weintraub, and A. A. Orning. 1966. Report of Investigation No. 6743, prepared for
the US Bureau of Mines.
Sanders, R. S., A. L. Ferre, W. B. Maciejewski, R. Giles, and C. Shook. 2000. Bitumen Effects on
Pipeline Hydraulics during Oil–Sand Hydrotransport. Canadian Journal of Chemical Engineering,
78, 4, 731–742.
Schaan, J., R. J. Sumner, R. G. Gillies, and C. A. Shook. 2000. The Effect of Particle Shape on
Pipeline Friction for Newtonian Slurries of Fine Particles. Canadian Journal of Chemical Engineering,
74, 4, 717–725.
Sellgrem, A., G. Addie, and S. Scott. 2000. The Effect of Sand–Clay Slurries on the Performance of
Centrifugal Pumps. Canadian Journal of Chemical Engineering, 78, 4, 764–769.
Shook, C. A., D. B. Haas, W. H. W. Husband, and M. Smail. 1979. Degradation of Coarse Coal Particles
during Hydraulic Transport. Working paper C-1, BHRA Group, Hydrotransport 6.
Steward, N. R. 1991. The Determination of Wear Relationships for FORSOC Fillset Binder Modified
Classified Tailings at High Relative Density. Report for Gold and Uranium Division of the
Anglo American Corporation of South Africa Ltd.

SLURRY PIPELINES
11.33
Swamanthan, S., A. Fair, and J. Wong. 1990. In Search of Mechanical Seals for Slurry Pumps. In
15th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.:
Coal and Slurry Technology Association.
Steward, N. B. 1996. An Empirical Evaluation of the Wear of Backfill Transport Pipelines. Working
paper, BHRA Group, Hydrotransport 13, Cranfield, England.
Thomas, A. D. 1976. Scale-up Methods for Pipeline Transport of Slurries. Int. Journal of Mineral
Processing, 3, 51–69.
Thompson, T. L. 1985. La Perla/Hercules Pipeline. In 10th International Conference on Coal Technology,
Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association.
Tian, H., G. Addie, and R. S. Hagler. 1996. Development of Corrosion Resistant White Irons for Use
in Phos-acid Service. Paper presented at the Annual Conference of Central Florida section of
the American Institute of Chemical Engineers.
Tillotson, I. S. 1953. Hydraulic Transportation of Solids. M.n. Congress Journal, 39, 1, 41–44.
Vanderpan, R. I. 1982. Proper Pump Selection for Coal Preparation Plants. In World Coal. San Francisco:
Miller Freeman Publications.
Venton, P. B. 1982. The Gladstone Pipeline. Working paper A-4, BHRA Group, Hydrotransport 8.
Venton, P. B. and T. J. Boss. 1996. An Analysis of Wear Mechanisms in the 155 km OK Tedi Copper
Concentrate Slurry Pipeline. Working paper, BHR Group, Hydrotransport, 533–548.
Walker, C. I. 1993. A New Alloy for Phosphoric Acid Slurries. Paper presented at the 1993 Clearwater
Convention, American Institute of Chemical Engineers.
Weston, M. D. 1985. SAMARCO Pipeline. In 10th International Conference on Coal Technology,
Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association.
Weston, M. D. and L. Worthen. 1987. Chevron Phosphate Slurry Pipeline commissioning and startup.
In Proceedings of the 12th International Conference on Coal and Slurry Technology.
Washington, DC: The Coal and Slurry Association.
Editorial Articles
Moving Mountains through a Slurry Pipeline. Engineering and Mining Journal, 195, 94–95, 1994.
A Conductance Based Solids Concentration Sensor for Large Diameter Slurry Pipelines. Journal of
Fluid Engineering, 122, 4
Variety of Slurry Pumps in Taconite Processing Plants. Skillings Mining Review, 70, 32, Aug 8,
1981.
Further Readings
Abulnaga, B. A. 2000. A Review of the Yichang Phophate Pipeline Feasibility Study. HATCH, unpublished.
USSR Plans Coal Slurry Pipelines. Oil and Gas Journal, 82, 58–59, 1984.
Braca, R. M. 1988. Use Needs Coal Slurry Pipeline. Pipeline and Gas Journal, 215, 32–36.
Catalano, L. 1983. Railroads Kill Eminent Domain for Coal-Slurry Pipelines. Power Journal, 127, 9.
Harvey, W. W. and Hossain, M. A. 1987. Co-recovery of Chromium from Domestic Nickel Laterites.
Journal of Metals, 39, 21–25.
Mahr, D. and B. Robert. 1986. Coal Slurry Pipelines Overland Belt Conveyors See Bright Future.
Power Engineering, 90, 24–28.
Maki, G. A. and D. M. Smith. 1983. Potash Mines and Mining/Saskatchewan/Thickeners/Design.
CIM Bulletin, 76, 57–62.
Maki, G. A., R. G. Roden, and P. J. Fullman. 1990. Stacking of Potash Mill Tailings. CIM Bulletin,
83, 96–98.
Marrey, D. T. 1985. Exporting Colorado Water in Coal Slurry Pipeline. Journal of Water Resources
Planning and Management, 111, 207–221.
Nalziger, R. H. 1988. Ferrochromium from Domestic Lateritic Chromites. Journal of Metals, 40,
34–37.
Nasr-El-Din, H., C. A. Shook, and M. N. Esmail. 1984. Isokinetic Probe Sampling from Slurry
Pipelines. Canadian Journal of Chemical Engineering, 62, 179–185.
Postlethwaite, J. 1987. The Control of Erosion–Corrosion in Slurry Pipelines. Materials Performance,
26, 41–45.
Postlethwaite, J., M. H. Dobbin, and K. Bergevin. 1986. The Role of Oxygen Mass Transfer in the
Erosion-Corrosion of Slurry Pipelines. Corrosion, 42, 514–521.

11.34 CHAPTER ELEVEN
Schaan, J., and C. A. Shook. 2000. Anomalous Friction in Slurry Flows. Canadian Journal of Chemical
Engineering, 78, 4, 726–730.
Shvartsburd, V. 1983. Pipelining and Burning Coal, Here are Important Criteria for Designing Coal
Slurry Pipelines. Oil and Gas Journal, 81, 91–95.
Wasp, E. J. 1983. Slurry Pipelines. Scientific American, 249, 48–55.

CHAPTER 9
POSITIVE DISPLACEMENT
PUMPS
9-0 INTRODUCTION
Positive displacement slurry pumps and mud transfer pumps play a major role in a number
of industries such as mining and metallurgical processes, chemicals, power generation,
porcelain and ceramics, and sugar refining. These pumps are versatile, efficient,
and suitable for pressures up to 17.3 MPa (2500 psi). Positive displacement pumps
have gained acceptance on long-distance mineral concentrate pipelines as their high
capital cost is recuperated through lower installation cost of electric systems, elimination
of booster stations, and high hydraulic efficiency, which is superior to centrifugal
pumps.
Plunger or diaphragm pumps do not handle large flow rates in excess of 100 m 3 /hr
(4400 US gpm), but they are suitable for a wide range of applications at higher volumetric
concentrations than centrifugal pumps. Positive displacement pumps can pump slurries
with a weight concentration of 70%.
9-1 SOLID PISTON PUMPS
Positive displacement slurry pumps are used in a number of industries (Table 9-1). Solid
piston pumps are reserved for the pumping of slurries of a low to medium abrasiveness
(Miller Number

9.2 CHAPTER NINE
TABLE 9-1 Applications of Positive Displacement Pumps
Industry Application
Mining Coal transportation (e.g., Novo Siberski pipeline, Black Mesa Pipeline)
Flotation material
Washery refuse
Deep mine dewatering of water with solid particles
Limestone, milk of lime
Potash rock salt, phosphate, iron ore, nickel ore concentrate
Bauxite, red mud, gold mud
Sand, pyrite, REA gypsum
Backfilling
Underground drainage
Filter press feed
Autoclave feed
Chemicals Salt slurry
Porcelain slurry
Pastes
Detergent slurry
Combustion furnace feed
Filter press feed
Power generation Coal and coal slurry
Flue
Pressurized fluidized bed combustion
Wet ash removal
Ship loading
Long-distance pipelines
Construction Bentonite, clay mash, cement
Porcelain Clay slurry
Filter press feed
Sugar Carbonation slurry
Sugar beet washing
Information provided by courtesy of Wirth-Maschinen and Bohrgerate, Germany.
TABLE 9-2 Comparison between Duplex and Triplex Pumps
Duplex Single Acting Duplex Double Acting Triplex Single Acting
2 cylinder liners 2 cylinder liners 3 cylinder liners
2 piston gaskets 4 piston gaskets 3 piston gaskets
2 cylinders 2 cylinders 3 cylinders
2 piston rods with packing
4 valves 8 valves 6 valves
Slurry does not come in Slurry does come in contact Slurry does not come in
contact with packing with packing contact with packing

POSITIVE DISPLACEMENT PUMPS
FIGURE 9-1 Concept of the double-acting duplex piston pump. (Courtesy of Wirth
Pumps.)
the piston, these pumps operate at lower speeds than single-acting duplex and triplex
pumps (Figure 9-2). The single-acting duplex pump seems to have disappeared from the
world of manufacturing.
For proper balancing, the pistons of duplex pumps are 180° out of phase (Figure 9-3),
but for triplex pumps they are 120° out of phase with each other (Figure 9-4).
Triplex pumps have a lower degree of oscillation than duplex units. The degree of
FIGURE 9-2 Concept of the triplex piston pump. (Courtesy of Wirth Pumps.)
9.3

9.4 CHAPTER NINE
FIGURE 9-3 Concept of gear mechanism for duplex piston pumps. (Courtesy of Wirth
Pumps.)
FIGURE 9-4 Concept of gear mechanism for triplex piston pumps. (Courtesy of Wirth
Pumps.)

POSITIVE DISPLACEMENT PUMPS
FIGURE 9-5 Pulsation diagram for triplex pump. (Courtesy of Wirth Pumps.)
variation of flow in the former is 23% (Figure 9-5) compared to 46% in the latter (Wallrafen,
1983; Figure 9-6).
Duplex and triplex slurry pumps are manufactured to a power frame of approximately
1500 kW (2000 bhp). The Black Mesa Pipeline featured 13 duplex pumps, each with a
driving power of 1250 kW (1675 bhp) to transport 4.8 million tons of coal over a distance
of 440 km (275 miles) (Wallrafen, 1983). Some of these pumps were manufactured by
Wilson-Snyder in the United States.
Piston slurry pumps are used extensively as mud transfer pumps. Gardner-Denver in
the United States offers a range of duplex pumps in the power range of 12–76 kW
(16–102 hp).
FIGURE 9-6 Pulsation diagram for duplex pump. (Courtesy of Wirth Pumps.)
9.5

9.6 CHAPTER NINE
FIGURE 9-7 Triplex pump TPK 7� × 12�/1600. Driving power 1200 kW (1600 hp). (Courtesy
of Wirth Pumps.)
9-2 PLUNGER PUMPS
For a long time, plunger pumps were not considered to be suitable for slurry transportation,
but in the late 1970s, manufacturers developed a suitable flushing system to minimize
wear of the plunger.
Plunger pumps are single acting. They use plungers instead of pistons (Figure 9-8)
with valves. They operate at 80–120 cycles per minute.
Plunger pumps are prone to wear. They are less expensive to purchase than diaphragm
pumps but have a higher maintenance cost. These pumps use three types of valves:
1. Free-floating valves
2. Spring-loaded spherical (Rollo) valves
3. Spring-loaded elastomer-seal (mud) valves
These valves are shown in Figure 9-9. It is important to minimize packing wear with
piston pumps. Smith (1985) proposed four methods:
1. Use of conventional packing of plungers at low speed with slurries of low abrasiveness
2. Provision of a clean, slurry-free environment for the packing rubbing surface (by synchronized
or continuous injection of water or cleaning fluid)
3. Separation of the slurry from the pumping element
4. Total isolation of the slurry from the packing (by providing a separate diaphragm
chamber)
The SAMARCO pipeline in Brazil used 14 plunger pumps with a driver power of 920
kW each to deliver 12 million metric tons of iron oxide ore concentrate over a distance of

POSITIVE DISPLACEMENT PUMPS
FIGURE 9-8 Schematic representation of a plunger pump.
400 km (250 mi) (Wallrafen, 1983). These triplex plunger pumps were manufactured by
Wilson-Snyder.
The Wilson-Snyder line of plunger slurry pumps features 21 different sizes from
45–1250 kW (60–1700 hp). They have also been used in Georgia, U.S.A. on a kaolin
pipeline. Kaolin is not very abrasive. These pumps have also been used for mine dewatering
from a depth of 1036 m (3400 ft). The water contained solids.
FIGURE 9-9 Categories of valves for slurry pumps. (a) Free floating; (b) spring-loaded
spherical (Rollo); (c) spring-loaded elastomer seal. (From Smith, 1985. Reprinted by permission
of McGraw-Hill.)
9.7

9.8 CHAPTER NINE
Some of the triplex plunger pumps are rated at 41.4 MPa (6000 psi) (Wilson-Snyder,
1977), such as the model 85-25. The volume capacity is rated from 363–2941 L/min
(96–777 US gpm).
9-3 PISTON DIAPHRAGM PUMPS
To handle abrasive slurries that piston pumps would find difficult, manufacturers such as
Geho Pumps (Netherlands), Wirth (Germany), and Gorman-Rupp (United States) have
developed pumps to use a diaphragm or a sort of flexible piston that comes in contact
with the slurry or sludge. Feluwa of Germany added a hose so that there is an isolating
bath of oil between the hose and the diaphragm. The pumps from Geho, Wirth, and
Feluwa feature a crankshaft mechanism to move the diaphragm but the Gorman-Rupp
pump uses an air cylinder to actuate the diaphragm.
Diaphragm piston pumps use a sort of oil chamber between a reciprocating piston (operated
by a crankshaft) and the diaphragm (Figure 9-10). Provided that no puncture occurs
in the diaphragm, slurry does not come in contact with the piston. A special control
system is installed to detect diaphragm rupture.
Wallrafen (1983) reported that Wirth manufactured its first piston diaphragm pump in
1969 to pump sand slurries. The first unit lasted 1000 hrs without having to replace worn
parts. By the early 1980s, wear life of 6000 hrs was achieved with rubber materials and
FIGURE 9-10 Concept of double-acting piston duplex diaphragm pump. (Courtesy of Wirth
Pumps.)

POSITIVE DISPLACEMENT PUMPS
proper design of the diaphragm. Diaphragm piston pumps are designed as duplex doubleacting
or as triplex single-acting pumps in a similar concept rather to solid piston pumps
(Figures 9-10 and 9-11).
Piston diaphragm pumps (Figure 9-12) are more expensive than plunger pumps. For
autoclave feed pumps, Geho developed a special design to handle slurries as hot as
200°C (392°F) at a high flow rate. Solids concentrations can be as high as75% and
pumps can operate at slurry temperatures up to 200°C. Typical uses in the mining industry
include:
� Long-distance slurry (mineral concentrate) pipelines (up to 300 km long)
� Clean and efficient tailings disposal
� High-pressure bauxite digester feed
� Autoclave and reactor feed
� Mine backfilling
� Mine dewatering (single stage)
� Hydraulic ore hoisting
With more than 400 piston diaphragm pumps installed on some of the world’s most demanding
long-distance pipeline applications, Geho Pumps has taken the opportunity,
through a significant research and development effort, to constantly improve piston diaphragm
pump design. This has resulted in numerous proprietary design improvements
and technical innovations relevant to severe slurry pumping.
FIGURE 9-11 Concept of single-acting piston triplex diaphragm pump. (Courtesy of Wirth
Pumps.)
9.9

9.10 CHAPTER NINE
FIGURE 9-12 Geho pump at Freeport. (Courtesy of Geho.)
Geho Piston diaphragm are used for tailings disposal and pumping at very high concentration
so that:
� Amount of free water is virtually eliminated, allowing slurry to be stacked or distributed
in layers
� Dry stacking requires less storage space
� Prevents contamination of the environment by leakage
� Rain and wind do not affect the solidified tailings
� Mechanical stability of tailings allows a high stack with rehabilitation possibilities after
use
Typical tailings applications of Geho pumps include:
� Bayswater Power Station—fly ash disposal (Australia)
� Nabalco, Gove Refinery—red mud disposal (Australia)
� Ledvice Power Station—fly ash disposal (Czech Republic)
� Pingguo Aluminium Company—red mud disposal (Peoples Republic of China)
� Khaperkheda Ash Handling Plant—fly ash disposal (India)
� National Aluminum Company—red mud disposal (India)
TABLE 9-3 Examples of Installation of Piston Diaphragm Pumps on Slurry Pipeline
and Tailings Applications
Location Manufacturer Installation application flow rate stated for each pump
Alsen Zementwerke Geho 3 piston diaphragm pumps to pump limestone slurry over
10 km (6.3 mi)
Antamina, Peru Wirth 4 piston pumps, 100 m3 /hr (440 gpm), 25.2 MPa (3650
psi), copper concentrate
Ashanti Goldfields Ghana Wirth 1 pump, 215 m3 /hr (947 gpm), 6 MPa (870 psi) backfill
slime (gold tailings)

TABLE 9-3 (continued)
POSITIVE DISPLACEMENT PUMPS
9.11
Location Manufacturer Installation application flow rate stated for each pump
Bajo Alumbrera, Argentina Geho 6 piston diaphragm pumps in 3 booster stations to
transport copper concentrate over a distance of 320 km
(225 mi) in a 150 mm (6 in) line, 91 m 3 /h at 217 bar
Cameco, Canada Wirth 2 pumps, 80 m 3 /hr (352 gpm), 12.5 MPa (1810 psi),
uranium ore
Cia Minera Disputada de Wirth 3 pumps 115 m 3 /hr (510 gpm), 2.5 MPa (360 psi) copper
Las Condes, Chile tailings
Codelco, Chile Wirth 3 pumps, 140 m 3 /hr (616 gpm), 3.5 MPa (507 psi), c
opper tailings
Course Nickel , Australia Wirth 2 pumps, 193 m 3 /hr (850 gpm), 5 MPa (725 psi), lateritic
nickel ore
Doña Ines Collahuasi, Chile Geho 2 piston diaphragm pumps for 203 km of copper
concentrate transport, 117 m 3 /h at 217 bar
ECPSA, Cuba Wirth 10 pumps, 200 m 3 /hr (800gpm), 6.4 MPa (920 psi),
iron–nickel slurry
Empresa Minera Yauliyacu, Wirth 2 pumps, 90 m 3 /hr (400 gpm), 14.4 MPa (2080 psi),
Peru tailings
Eskay Creek, Canada Wirth 1 pump, 25 m 3 /hr (110 gpm), 11.7 MPa (1700 psi) for a
6.5 km (4 mi) pipeline
Freeport, Indonesia Geho 2 piston diaphragm pumps to transport copper
concentrate over 120 km (75mi), 159 m 3 /h at 40 bar
Goldmine, South Africa Wirth 1 pump, 12 m 3 /hr (66 gpm), 12 MPa (1714 psi),
backfilling
ISCOR, Hillendake Mine, Wirth 2 pumps 450 m 3 /hr (1980 gpm), 7.4 MPa (1075 psi),
South Africa heavy mineral tailings
Jian Shan Geho 100 kms iron ore concentrate transport (PR of China), 2
piston diaphragm pumps, 216 m 3 /h at 153 bar
Nabalco, Australia Geho 3 piston diaphragm pumps for a highly concentrated red
mud slurry to a disposal area, 200 m 3 /h at 160 bar
Norilsk Nickel Combinat Geho 9 piston diaphragm pumps, 55 km of multimetallic ore
transport (North Siberia, Russia), 400 m 3 /h at 80 bar
Pasminco, Australia Wirth 3 pumps, 161 m 3 /hr (710 gpm), 12.5 MPa (1810 psi) zinc
and lead concentrate
Los Pelambres, Chile Geho 2 piston diaphragm pumps for 120 km pipeline
transportation of copper concentrate slurry, 165 m 3 /h
at 150 bar
Sicartsa, Mexico Geho 1 piston diaphram pump for iron ore concentrate slurry,
380 m 3 /h at 110 bar
J. R. Simplot, USA Geho 4 piston diaphragm pumps for 100 km transportation of
phosphate slurry, 97 m 3 /h at 228 bar
Batu Hijau, Indonesia Geho 2 piston diaphragm pumps for 120 km copper
concentrate slurry transportation, 123 m 3 /h at 228 bar
Cockburn Cement, Australia Geho 3 piston diaphragm pumps for shell and slurry transport,
206 m 3 /h at 65 bar
New Zealand Steel Geho 4 piston diaphragm pumps for ironsand concentrate
transportation, 194 m 3 /h at 100 bar
Rio Capim, Brazil Geho 2 piston diaphragm pumps for kaolin slurry
transportation, 293 m 3 /h at 57 bar
Minera Escondid, Chile Geho 1 piston diaphragm pum for copper concentrate slurry
transportation, 295 m 3 /h at 69 bar
OEMK, Ukraine Geho 4 piston diaphragm pumps for iron ore slurry
transportation, 540 m 3 /h at 74 bar

9.12 CHAPTER NINE
Geho piston diaphragm pumps are suitable for feeding autoclaves with ore slurry in different
mineral processes, such as in the aluminum, gold, and nickel industries. A special
design of the Geho piston diaphragm pump is developed for “hot” slurries. As the diaphragm
cannot be exposed to high temperatures, Geho Pumps developed a dropleg concept,
which allows transfer of hot slurry without having to cool the slurry down. Thus, the
proposed pump design excels in low maintenance cost, low energy cost, and high reliability.
Geho Pumps designed the dropleg concept in the early 1980s for feeding gold ore slurries
to autoclaves for pressure oxidation at installations in Nevada, U.S.A. Recently, Geho
Pumps developed an improved dropleg configuration for 200°C slurries. These developments
include, for example, a horizontal dropleg layout, a patented separator, improved
dropleg efficiency, patented slide mounting of the pump to compensate for thermal expansion,
etc. Typical examples of high-temperature autoclave feeding using Geho pumps
include:
� Bulong Nickel project—200°C laterite nickel slurry (Australia)
� Murrin Murrin—200°C laterite nickel slurry (Australia)
� American Barrick Phases I, II, and III—gold slurry (United States)
� Twin Creeks Phases I and II—gold slurry (United States)
Diaphragm piston pumps use two types of valves:
1. Ball valves
2. Conical valves
The ball valves are a kind of check valve on suction and discharge and move according to
outlet ball valve
air exhaust
Air
slurry
air piston
air inlet
diaphragm
FIGURE 9-13 Concept of air operated slurry diaphragm pump.
inlet ball valve

POSITIVE DISPLACEMENT PUMPS
fluid forces to close and open the inlet and the outlet. Their own weight plays a role in
opening and closing. They are more suitable for the lower end of pressures. When the
pressures are very high, the sealing of the ball valve cannot support the force. Conical
spring-actuated valves are then installed; the force of the spring helps to close the valves.
Conical valves allow operation at higher pressures due to metal-to-metal support.
Air driven diaphragm pump use a piston connected to the diaphragm. In addition, air
can be forced against the dry side of the diaphragm to help move the diaphragm. Air
leaves at the top piston assembly. The pump uses ball valves for the inlet and exit of the
slurry (Figure 9-13). The air-operated diaphragm pumps serve a niche of the market and
have a range of discharge from 15–100 mm (5/8–4 in) and flows up to 9 L/s (150 US
gpm). The maximum head from these pumps is of the order of 15 m (50 ft).
9-4 ACCESSORIES FOR PISTON AND
PLUNGER PUMPS
9.13
The pulsations of a positive displacement pump are transmitted to the slurry. To prevent
their propagation to the pipeline and its support, it is essential to install hydraulic dampeners
(Figure 9-14).
A hydraulic dampener is essentially a chamber with a diaphragm. On one side there is
slurry and on the other there is a gas such as nitrogen under compression to absorb the oscillations.
Dampeners are installed on the discharge of the pump, and in some cases on the
suction too.
The manufacturers of piston and diaphragm pumps provide a package of special tools
to install replacement parts.
FIGURE 9-14 Hydraulic dampener for diaphragm pumps. (Courtesy of Wirth Pumps.)

9.14 CHAPTER NINE
FIGURE 9-15 Peristaltic pump. (Courtesy of Gorman Pump Industries.)
Air operated diaphragm pumps are noisy and need a silencer on the exhaust of the air.
9-5 PERISTALTIC PUMPS
A peristaltic pump is essentially a hose that is pressed by a cam, an eccentric mechanism,
or three rollers on arms (Figure 9-15). The pressure is then transmitted to the fluid. They
are available in a range of flow from microliters/min up to 33 L/min (8.8 gpm) and pressures
up to 420 kPa (60 psi). They are popular for medical applications as they do not
cause damage to blood cells. Peristaltic pumps are used to transport highly concentrated
slurry at a small flow rate for a specific range of applications such as clay, gold, and platinum
slurries, and filter press feed. They are self-priming and develop a high vacuum, up
to 635 mm (25 in) on the mercury scale.
9-6 ROTARY LOBE SLURRY PUMPS
Rotary lobe pumps are a special form of positive displacement pump. They feature two
lobes (Figure 9-16) that rotate against each other like intermeshing gears. They are avail-

POSITIVE DISPLACEMENT PUMPS
able for flow rates in the range of 0–170 L/s (0270 US gpm) and for discharge pressures
up to 1.2 MPa (175 psi). They are self-priming up to a negative suction of 8 m (24 in) and
can handle viscous and abrasive slurries. They are capable of dry running up to 30 min.
The lobes for slurry handling are made from abrasion-resistant alloys or a steel core with
molded rubber surfaces. The casing is hardened.
Rotary lobe slurry pumps are used with certain soft slurries with mild abrasion characteristics
such as wastewater and sewage disposal, flotation slimes, digested scum, lime
slurry in waste treatment plants. They are also used in food processing to move potato and
starch pulp, mash, food paste, tomato paste, food wastes, and dairy waste and whey in milk
processing. They are used in the paper industry to pump lime slurry and adhesives. They
are used in the plastic recycling industry to pump plastic and Styrofoam cups. They are also
used in the construction industry for pumping bentonite slurries, clay slurries, and mud.
9-7 THE LOCKHOPPER PUMP
9.15
FIGURE 9-16 Rotary positive displacement pump. (Courtesy of Gorman Pump Industries.)
The lockhopper is not exactly a pump, but rather a system to pump slurry, including fairly
coarse material at extremely high pressure. It consists of two chambers that alternate in

9.16 CHAPTER NINE
water tank
Multi-stage
water pump
injecting water and slurry into the pipeline. Injection is provided by water pressure. The
water is separated from the slurry by a diaphragm in the form of a free-rolling rubber
spherical “piston,” a sort of double-acting piston with water on one side and slurry on the
other. On one side of the piston, slurry enters from feeding hoppers and is charged to the
pipeline. On the other side of the piston, water enters—pumped by high-pressure, multistage
water pumps—which propels the piston before being returned to water tanks. Figure
9-17 illustrates the lockhopper system. The lockhopper can be adapted to pump 2� coarse
coal, bauxite lumps, and other materials whose size would be beyond the range of diaphragm
and plunger pumps.
Although the lock hopper can be designed for very high pressures, the limitation is on
the practical size of pipeline pressure rating that can be selected. This is often of the order
of 21 MPa (3000 psi).
9-8 CONCLUSION
slurry hopper
Valve
Water Slurry Valve
Valve
Valve
FIGURE 9-17 Concept of the lockhopper system.
Slurry Pipeline
free rolling piston
Positive displacement pumps play a major role in the power industry and the pumping of
highly concentrated slurries and food and chemical pastes at high efficiency. Considerable
development in the last quarter of the 20th century by manufacturers of these pumps,
particularly diaphragm pumps, has led to a wide range of applications from long-distance
pipeline pumping stations, to mine dewatering, to pumping concrete with high degree of
reliability. Research continues in the field of high-temperature applications such as autoclave
feeds.

9-9 REFERENCES
POSITIVE DISPLACEMENT PUMPS
9.17
Wallrafen, G. 1983. Piston Pumps for the Hydraulic Transport of Solids. Bulk Solids Handling, 3, 1.
Wallrafen, G. 1985. Backfilling with Viscous Slurry Pumps. Bulk Solids Handling 5, no. 4: Wilson-
Snyder. 1977. Slurry Pumps. Texas: Wilson Snyder. Publication ADWS 28-77 (3M).
Smith W.1985. Construction of Solids-Handling Displacement Pumps. Chapter 9-17-3 in The Pump
Handbook, Karassik I. J., W. C. Krutzch, W. H. Fraser, and J. P. Messina (Eds.). New York:
McGraw-Hill.

CHAPTER 5
HOMOGENEOUS FLOWS
OF NONSETTLING
SLURRIES
5-0 INTRODUCTION
The rheology of non-Newtonian flows and homogeneous flows was examined in detail in
Chapter 3. With modern methods of grinding, the size of particles can be reduced to values
smaller than 70 �m (0.0028 in). As shown in Table 3-8, a wide range of metal concentrates
and tailings are pumped at a sufficiently high concentration with a small enough
particle size for the mixture to behave as a Bingham plastic. There are other clays and
slurries that may behave as pseudoplastics. In certain circuits of oil sands processing
plants with tar at the level of flotation, the slurry may behave as a thixotropic mixture.
With non-Newtonian flows, it is important to take into account the rheology, yield
stress, power law exponent, coefficient, and even the time response. Different models
have evolved over the years for Bingham and pseudoplastic slurries. Some of these models
put more emphasis on the laminar flow regime, in which roughness effects are negligible.
Some other models extend to the transition and turbulent regimes. The effect of pipe
roughness on friction loss factors in non-Newtonian flows remains a topic worth investigating
and researching.
For thixotropic slurries, methods are used to predict start-up pressure after a shutdown
of the pipeline, and the time required to clean the conduit of gelled material before resuming
pumping.
The equations for friction factors of non-Newtonian fluids are fairly complex and require
iteration and data on the rheology. Throughout the years, different authors have developed
equations for “modified” Reynolds numbers, Hedstrom numbers, etc. In this
chapter, equations developed by different authors will be reviewed. Through worked examples,
the reader will be shown methods of calculating the friction factor. It is the purpose
of this chapter to focus on the engineering side of the problem. The reader is strongly
advised to read through Chapter 3, to gain the fundamentals for this chapter.
For the practical engineer, who is more concerned with the actual design of a pipeline
or a pumping system, the numerous and different definitions of the so-called “modified
Reynolds number” can be very confusing. Every few years, an author develops a new definition
of the “modified Reynolds number” and claims to have found a relationship with
the friction factor. The cautious approach for an engineer is to assume that such a universal
relationship is illusive, and for every type of slurry there may be a model to use.
5.1

5.2 CHAPTER FIVE
Sometimes the use of two different methods can yield differences of 25–35% in the estimation
of the friction factor.
One important difference with the slurries described in Chapter 4 is that most non-
Newtonian slurries do not exhibit the stratification of solids, which is usual with coarse
particles.
5-1 FRICTION LOSSES FOR
BINGHAM PLASTICS
Bingham plastics were defined in Chapter 3. They are characterized by a yield stress that
must be overcome to start the flow. Examples of Bingham plastics are listed in Table 3-9
5-1-1 Start-up Pressure
The start-up pressure for pumping a Bingham plastic is expressed in terms of the true
yield stress:
4�0L Pst = � (5-1)
Di
The start-up pressure per unit length is obtained by dividing Equation 5-1 by the
length of the pipeline:
P st
� L
=
Examples for the starting pressure per unit length for slurries in 3�, 6�, 12�, and 18�
pipes are presented in Table 5-1.
The Reynolds Number for a Bingham slurry is expressed as:
The coefficient of rigidity � was defined in Equation 3-29 as
ReB = � (5-2)
�
� 0
4� 0
� Di
D iV� m
� = � + �� (3-29)
(d�/dt)
For Bingham slurries a nondimensional coefficient is defined as the plasticity number:
�0Di PL = � (5-3)
�V
The Hedstrom number is the product of the plasticity number and the Reynolds number
and is calculated as
2 D i �m�0 �2 �
He = (5-4)
Table 5-2 shows examples of Bingham slurry mixtures and the magnitude of the Hedstrom
number for flows in rubber-lined 6� (150 mm NB), 12� (300 mm NB) and 18� (450

5.3
TABLE 5-1 Starting Pressure per Unit Length for Certain Slurries in Pa/m
Starting pressure per unit length (Pa/m)*
3� Pipe 6� Pipe
Coefficient Sch 40, Sch 40, 12� Pipe S, 18� Pipe S,
Yield of rigidity, rubber lined, rubber lined, rubber lined, rubber lined,
Particle size, Density, stress, � mPa · s ID = 2.560� ID = 5.557� ID = 11.500� ID = 16.500�
Slurry d 50 kg/m 3 Pa (cP) (65 mm) (141.2 mm) (292 mm) (419 mm)
54.3% Aqueous suspension of 92% under 74 �m 1520 3.8 6.86 234 108 52 37
cement rock
Flocculated aqueous China clay 80% under 1 �m 1280 59 13.1 3631 1671 808 567
suspension No. 1
Flocculated aqueous China clay 80% under 1 �m 1207 25 6.7 1538 708 342 240
suspension No. 4
Flocculated aqueous China clay 80% under 1 �m 1149 7.8 4.0 480 221 107 75
suspension No. 6
Aqueous clay suspension I 1520 34.5 44.7 2123 977 473 332
Aqueous clay suspension III 1440 20 32.8 1231 567 274 192
Aqueous clay suspension V 1360 6.65 19.4 409 188 91 64
21.4% Bauxite

5.4
TABLE 5-2 Examples of Hedstrom Numbers for Bingham Slurries in 3�, 6�, 12�, and 18� Rubber-Lined Pipes
Hedstrom number*
3� Pipe 6� Pipe
Coefficient Sch 40, Sch 40, 12� Pipe S, 18� Pipe S,
Yield of rigidity, rubber lined, rubber lined, rubber lined, rubber lined,
Particle size, Density, stress, � mPa · s ID = 2.560� ID = 5.557� ID = 11.500� ID = 16.500�
Slurry d 50 kg/m 3 Pa (cP) (65 mm) (141.2 mm) (292 mm) (419 mm)
54.3% Aqueous suspension of 92% under 74 �m 1520 3.8 6.86 518,568 2,445,272 1.046 × 10 7 2.124 × 10 7
cement rock
Flocculated aqueous China clay 80% under 1 �m 1280 59 13.1 1,859,285 8,773,821 3.752 × 10 7 7.616 × 10 7
suspension No. 1
Flocculated aqueous China clay 80% under 1 �m 1207 25 6.7 2,840,039 1.472 × 10 7 6.078 × 10 7 1.234 × 10 8
suspension No. 4
Flocculated aqueous China clay 80% under 1 �m 1149 7.8 4.0 2,366,581 1.117 × 10 7 4.776 × 10 7 9.694 × 10 7
suspension No. 6
Aqueous clay suspension I 1520 34.5 44.7 110,885 523,259 2,237,759 4,541,865
Aqueous clay suspension III 1440 20 32.8 113,102 533,721 2,282,499 4,632,671
Aqueous clay suspension V 1360 6.65 19.4 101,528 479,100 2,048,910 4,158,568
21.4% Bauxite

mm) pipes. Laminar flows in large pipes are considered for certain slurries at relatively
high weight concentration (� 60%), or certain high-energy mixtures (e.g., crude oil with
fine and ultrafine coal).
Example 5-1
A slurry consists of a clay and water mixture. It is tested and classified as a Bingham mixture
with a yield stress of 17 Pa. The pipe inner diameter is 63 mm. The pipe length is
6500 m. Determine the start-up pressure, ignoring any static head.
Solution in SI Units
Using Equation 5-1:
Solution in US units
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
P st = 4� 0L/D i
P st = 4 × 17 × 6500/0.063 = 7,015,873 (1018 psi)
P st = 4� 0L/D i
� 0 = 17 Pa/6895 = 2.465 × 10 –3 psi
L = 6500 m/0.0254 = 255,905 in
Di = 63/25.4 = 2.48 in
4 × 2.465 × 10
Start-up pressure Pst = = 1017.6 psi
–3 × 255,905
���
2.48
5-1-2 Friction Factor in Laminar Regime
Buckingham (1921) was the first to develop an equation for a fully developed laminar
flow. This equation has since been modified by Hedstrom (1952) and others to express
the friction factor as a function of the Hedstrom and Reynolds numbers:
= – + (5-5)
or
fNL = �1 + – He
� (5-6)
This would occur below the critical Reynolds number or in transition between laminar
and turbulent flow. The last term between brackets in the equation is often considered
second order.
4
He
16 He
� � �3 7
ReB 6ReB 3 f NLReB 4
1 fNL He
� � � � 2 3 8
ReB 16 6ReB 3 f NLRe B
Example 5-2
A Bingham slurry with a concentration of 50% by weight is tested in a plastic-lined pipe
with an inner diameter of 2.5 in. The tests indicate a yield stress of 1.5 Pa, a slurry mixture
specific gravity of 1.54, and a coefficient of rigidity of 0.4 Pa · s. Assuming a flow
speed of 4 ft/s in a laminar regime, determine the friction factor by Buckingham’s equation.
5.5

5.6 CHAPTER FIVE
Solution in SI Units
Pipe ID = 2.5� or 63.5 mm
Speed = 4 ft/s or 1.219 m/s
Reynolds number = 0.0635 × 1.219 × 1540/0.4 = 298
Hedstrom number = (0.0635) 2 × 1540 × 1.5/(0.4 2 ) = 916.8
Using Equation 5-6 and ignoring the higher-order terms:
f NL � [16/Re B][1 + He/(6Re B)] = 0.0812
This a fanning factor, so the Darcy friction factor is 0.3248.
Figure 5-1 presents values of the friction factor versus the Reynolds number for a wide
range of Hedstrom numbers from 0 to 10 9 . The transition between laminar and turbulent
flows is shown by the dotted curve of the critical Reynolds number. From the point of
view of engineering, the most practical flows in pipes are in a range of Hedstrom numbers
between 10 5 and 10 8 , as shown in Table 5-2. The transition to turbulent flow will be examined
in more detail throughout this chapter. To appreciate the practical magnitude of
the laminar friction factor, Table 5-3 presents cases at a speed of 1 m/s (3.3 ft/sec) for rubber-lined
pipes in sizes of 3� (80 mm N.B.), 6� (150 mm N.B), 12� (300 mm N.B.), and
18� (450 mm N.B.). The Fanning friction factor based on the Buckingham equation is in
the range of 0.001 to 0.15.
FIGURE 5-1 Friction factor versus Reynolds number and Hedstrom number. (From Hill R.
A, P. E. Snoek, and R. L. Ghandi, Hydraulic transport of solids, in The Pump Handbook, 2nd
ed., I. J. Karassik et al. (Eds.), New York: McGraw-Hill. Reproduced by permission.)

5.7
TABLE 5-3 Friction Factor at a Speed of 1 m/s (3.3 ft/sec) for Bingham Mixtures in Rubber-Lined Pipes*
Fanning friction factor f N at a speed of 1 m/s (3.3 ft/sec)
3� Pipe 6� Pipe
Coefficient Sch 40, Sch 40, 12� Pipe S, 18� Pipe S,
Yield of rigidity, rubber lined, rubber lined, rubber lined, rubber lined,
Particle size, Density, stress, � mPa · s ID = 2.560� ID = 5.557� ID = 11.500� ID = 16.500�
Slurry d 50 kg/m 3 Pa (cP) (65 mm) (141.2 mm) (292 mm) (419 mm)
54.3% Aqueous suspension of 92% under 74 �m 1520 3.8 6.86 0.00778 0.00718 0.00691 0.00684
cement rock
Flocculated aqueous China clay 80% under 1 �m 1280 59 13.1 0.12541 0.12407 0.12348 0.12331
suspension No. 1
Flocculated aqueous China clay 80% under 1 �m 1207 25 6.7 0.00566 0.05586 0.05554 0.05545
suspension No. 4
Flocculated aqueous China 80% under 1 �m 1149 7.8 4.0 0.0186 0.0185 0.0183 0.0182
suspension No. 6
Aqueous clay suspension I 1520 34.5 44.7 0.0677 0.0639 0.0621 0.0616
Aqueous clay suspension III 1440 20 32.8 0.0426 0.0396 0.0382 0.0379
Aqueous clay suspension V 1360 6.65 19.4 0.01655 0.0146 0.01382 0.01359
21.4% Bauxite

5.8 CHAPTER FIVE
5-1-3 Transition to Turbulent Flow Regime
Hanks and Pratt (1967) analyzed extensive experimental data on critical Reynolds numbers
and proposed a relationship between the Reynolds and Hedstrom Numbers at transition as
ReBc = �1 – x He 4 1
� � 4
c + � x c� (5-7)
xc 3 3
where xc = �0/�wc = the ratio of the yield stress to the wall shear stress at the transition
from laminar to turbulent flow.
At the transition,
He = 16,800 � (5-8)
(1 – xc) 3
Figure 5-2 plots the magnitude of critical Reynolds number versus the Hedstrom number
for a number of Bingham slurries.
Example 5-3
Using Figure 5-2, determine the critical Reynolds number for a clay slurry at a Hedstrom
number of 10,000.
Solution
From Figure 5-2 Rec � 3700.
Wasp et al. (1977) defined the effective pipeline viscosity for laminar flow as
Critical Reynolds Number
10 5
10 4
10
3
3 4
10 10
x c
�w �e = � (5-9)
8V/Di
5
10
6
10
8
10
Hedstrom Number
FIGURE 5-2 The critical Reynolds number versus the Hedstrom number for flow in pipes.
(After Hanks, R. W., and D. R. Pratt. 1967.)

HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
In Chapter 3, in the description of the capillary tube test, the Buckingham equation was
derived. When ignoring the fourth-power term, the equation reduces to
�w � 8V�/Di + 4/3 �0 (3-62)
or
�e � ���1 + 8V�/Di + 4/3 �0 DI�0 �� �� (5-10)
8V/Di
6V�
In many laminar flow pipelines the term Di�0/(6V�) is much smaller than unity. In such
cases Equation 5-10 is then simplified to
�e � Di�0/(6V) (5-11)
At transition, Wasp et al. (1977) point out that this equation can be approximated to
�e � [Di�0/(6VTR)] using the effective pipeline viscosity, the critical Reynolds number is expressed as
ReBC = 6V 2 TR�/�0 At high shear rate for Bingham plastics, � e � � � � � (see Figure 3-10):
ReBC�0 VTR = � �� 6�
(5-12)
The Wasp method is based on numerous assumptions, and usually terminates by assuming
that the transition Reynolds number is in the range of 2000 to 3000 for numerous
Bingham slurries. In some respects, it is a useful tool for hand calculations. A more widely
accepted method since the mid 1980s is to compute the transition velocity that was proposed
by Wilson and Thomas, to be discussed in Section 5-4-3. It is then assumed that the
transition from laminar to turbulent flow occurs when the Wilson–Thomas and the Buckingham
equations intersect.
5-1-4 Friction Factor in the Turbulent Flow Regime
Hanks and Dadia (1971) developed a semiempirical equation for the turbulent flow of
Bingham slurries in closed conduits. These equations were modified by Darby (1981) and
Darby et al. (1992) to give a friction factor for the turbulent regime as
fNT = 10a b ReB (5-13)
where
a = –1.47[1 + 0.146 exp(–2.9 × 10 –5He)] b = –0.193
The values of the parameters a and b are based on empirical data for closed conduits.
Bingham slurries do not exhibit a sudden change from laminar to turbulent flow. Darby
et al. (1992) reviewed the work of previous authors and proposed to combine the laminar
and turbulent fanning friction factors into the following equation:
fN = ( f m NL + f m NT) (1/m) (5-14)
where
m = 1.7 + 40,000/ReB. (5-15)
5.9

5.10 CHAPTER FIVE
Equations 5-12 and 5-13 do not account for pipe roughness and are essentially for very
smooth pipes such as glass and high-density polyethylene pipes. Studies have been published
in the past on flow of Bingham plastics in the laminar regime, where roughness effects
are neglected. Thomas and Wilson (1987) have even argued that the non-Newtonian
fluids form a viscous sublayer in the boundary layer, that is usually thicker than with
Newtonian flows. This viscous sublayer is considered to suppress the contribution of
roughness and, in effect, Wilson and Thomas do support the assumptions made by Darby
(1981). Many concentrates are pumped at high volumetric concentrations but also at a relatively
moderate speed of 2–2.5 m/s (6.6–8.2 ft/s). We will, however, discuss the effects
of roughness in Section 5-7.
Example 5-4
In Figure 3-9, the relationship between the Bingham plastic apparent viscosity and the
shear rate was presented as
� = �(d�/d�) + �� at high shear rate � � ��. Considering an aqueous clay suspension III (Table 3-9) with a mixture density of 1440
kg/m3 (SG = 1.44), a yield stress of 20 Pa, and a coefficient of rigidity of 32.8 mPa · s as
reported by Caldwell and Babbitt (see references of Chapter 3), determine the friction factor
for flow at 2.5 m/s in a 63 mm ID pipe, using the Darby method. Determine also the
pressure drop per unit length.
Solution SI Units
Reynolds number:
Re = = = 6914.6
Hedstrom number:
(0.063)
He = = = 106,249
2 D × 1440 × 20
���
2 �VDI 1440 × 2.5 × 0.063
� ��
� 0.0328
��0 �2 �
(0.0328) 2
m = 1.7 + 40,000/Re
m = 1.7 + 40,000/6914.6 = 7.49
a = –1.47[1 + 0.146 exp(–2.9 × 10 –5He)] = –1.479
fT = 10aRe –0.193
fT = 10 –1.479 × 6914.6 –0.193 = 0.006
fL = �1 + – � � �1 + Re 16 Re
� �
Re 6He �
4
16 Re
� � �3 7
Re 6He 3f LHe fL = �1 + � = 0.00234
m m 1/m
fn = ( f L + f T )
Therefore,
fn = (0.002347.49 + 0.0067.49 ) 1/7.49 16 6914.6
� ��
6914.6 6 × 106,249
= 0.006
is the fanning friction factor

Darcy factor:
fD = 4fN = 0.006 × 4 = 0.024
Pressure drop per unit length:
dP/dz = �fDV 2 /(2Di) dP/dz = 1440 × 0.024 × 2.52 /(2 × 0.063) = 1,714 Pa/m (0.816 psi/ft)
5-2 FRICTION LOSSES FOR
PSEUDOPLASTICS
Pseudoplastic rheology was extensively discussed in Chapter 3, Section 3.4-2. Examples
of pseudoplastics are listed in Table 3-10.
5-2-1 Laminar Flow
A number of models have been developed for pseudoplastic flows. These treat the fluid as
a continuum.
5-2-1-1 The Rabinowitsch–Mooney Relations
Herzog and Weissenburg (1928) developed an equation for laminar time-independent,
viscous non-Newtonian flows. It was subject to further refinements by Rabinowitsch
(1929) and Mooney (1931).
For a circular pipe, a relationship is established between the shear stress � and the absolute
value of the rate of shear � = –du/dr
f(�) = –du/dr
Rabinowitsch and Mooney derived a general relationship for the shear rate at the wall:
du 8V 1 + 3�
–���w = ���� (5-16)
dr DI 4�
where
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
d[ln(Di �P/4L)]
� = ��
(5-17)
d[ln(8V/D)]
5-2-1-2 The Metzner and Reed Approach
Metzner and Reed (1955) developed an equation for the Reynolds number in laminar
flow as
D I � V 2–� �
where � is defined by Equation 5-16 and
� = K�8 (�–1) in SI units
ReMR = � (5-18)
�
� = g cK�8 (�–1) in USCS units
5.11

5.12 CHAPTER FIVE
K� = K� � n 1 + 3n
�
4n
In SI units (5-19a)
1 + 3n n
K� = K\gc�� 4n � In USCS units (5-19b)
The fanning friction factor is then expressed in the laminar flow regime in the conventional
manner but using the modified Reynolds number:
16
fNL = �
ReMR
(5-20)
Example 5-5
The pressure drop in a 80 mm ID pipe is to be determined for a slurry with S.G. = 1.37.
The power law exponent had been previously determined to be 0.4, and the power law
factor K as 16 dynes-spcn /cm2 . The speed of the flow is 1.35 m/s. Use the Metzner and
Reed approach to calculate the friction factor, assuming a shear rate of 600 s –1 .
Solution
From Equation 5-15:
8V 1 + 3�
–600 = ��� 4� �
Di
8 × 1.35 1 + 3�
–600 = ��� 0.08 4� �
–4.45 = (1 + 3�)/4�
–1.11 = 1/� + 3
Re MR =
K� = K� � n
� = 0.529
= 16� � n 1 + 1.2
� = 18.174
1.6
� = K�8 �–1 = 18.174 × 8 –0.47 = 6.84
Re MR =
1 + 3n
�
4n
D � V 2–� �
� �
0.08 0.529 × 1.35 1.471 × 1370
���
6.84
ReMR = 81.87
16
fn = � = 0.195
ReMR
The Metzner and Reed approach has become a classical method of dealing with
time-independent non-Newtonian fluids. It has been extended to Bingham slurries but
the opinion of the author is that this approach is fairly difficult to use for Bingham slur-

ies, and a more practical method would be the Darby approach, described in Section 5-
1-4.
The Metzner and Reed approach requires the engineer to assume a value of the shear
stress at the wall � w to calculate x. Such assumptions are very difficult to make for engineers
outside a research lab. It may be more practical to send samples of the slurry to a
rheology lab and to go from plots of yield stress versus weight concentration, as well as
from plots of viscosity versus weight concentration and shear rates to a more straightforward
computation of friction factors (Figure 5-3).
5.2.1.3 The Tomita Method
Tomita (1959) defined a fanning friction factor for power law fluids as
In the laminar flow regimes:
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
5.13
2Di�P 1 + 2n
fPL = ���� (5-21)
3L�mV 1 + 3n
2
n 2–n [1/n + 3] D i V �m
RePL = � ����� (5-22)
K
1–n
6
� ��
n 2 1/n + 2
FIGURE 5-3 Friction factor versus Reynolds number for power law factors. [From Hill R.
A, P. E. Snoek, and R. L. Ghandi, Hydraulic transport of solids, in The Pump Handbook, 2nd
ed., I. J. Karassik et al. (Eds.), New York: McGraw-Hill. Reproduced by permission.]

5.14 CHAPTER FIVE
16
fPL = � (5-23)
Remod
5-2-1-3 Heywood Method
Heywood (1991) proposed to define a modified Reynolds number for pseudoplastics as
Remod = � � n
� � n–1
(5-24)
where K and n are the consistency coefficient and flow behavior indexes for pseudoplastic
flows previously defined in Chapter 3.
In the laminar flow regime, Heywood (1991) used the conventional method of defining
the fanning friction factor in terms of the Reynolds number in the laminar flow regime
as previously discussed in Chapter 2, or
fNPL = 16/Remod (5-25)
The effective pipeline viscosity is expressed as
�e = K� � n
� � n–1
�mVDi 4n Di � � �
K 1 + 3n 8V
4n 8V
� � (5-26)
1 + 3n Di
5-2-2 Transition Flow Regime
Ryan and Johnson (1959) defined a critical Reynolds number for purely viscous pseudoplastics
as
Rec = (5-27)
The friction factor at the transition from laminar to turbulent, flow called the critical friction
factor is
(1 + 3n) 1
fNc = ��
(5-28)
(n+2)/(n+1)
(n + 2)
Table 5-4 tabulates the critical Reynolds number and fanning friction factor versus the
power factor “n.” The minimum friction factor is 0.0067 at n = 0.5. However, Heywood
(1991) deducted from various test data that the minimum value for fNc = 0.004, which is
even lower than the values indicated by Equation 5-28 (Figure 5.4).
2
6464n (n + 2)
�
404n
(n+2)/(n+1)
���
(1 + 3n) 2
5-2-3 Turbulent Flow
Various equations have been developed over the years for turbulent flow of pseudoplastics
in smooth pipes. These equations are based on empirical data and semitheoretical
models.
Using the modified Reynolds number as per Equation 5-17, Dodge and Metzner
(1959) developed the following semitheoretical equation for turbulent flow:

HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
TABLE 5-4 Critical Reynolds Number and Fanning Friction Factor versus the Flow
Behavior Index “n” According to the Ryan and Johnson Method
5.15
Flow Critical Critical Flow Critical Critical
behavior Reynolds fanning friction behavior Reynolds fanning friction
index “n” number factor, f NC index “n” number factor, f NC
0.1 1577 0.01015 0.9 2158 0.00741
0.2 2143 0.00747 1.0 2099 0.00762
0.3 2345 0.00682 1.1 2043 0.0783
0.4 2396 0.00668 1.2 1990 0.00804
0.5 2381 0.00672 1.3 1941 0.00824
0.6 2337 0.00685 1.4 1895 0.00844
0.7 2280 0.00702 1.5 1852 0.00864
0.8 2219 0.0072 1.6 1812 0.00883
1 4
(1–� /2) 0.4
� = � log10[Remod f NT ] – � (5-29)
0.75
1.2
�fN� T� � �
Although Equation 5-29 has been extensively used, it has its own limitations. Measuring
the power exponent “n” in laminar flow tests and then trying to apply it to turbulent
flows is asking for trouble, particularly for cases when n < 0.5. Heywood and Richardson
(1978) showed that pumping flocculated clays yielded higher experimental values of friction
coefficient than those predicted by Dodge and Metzner (1959), particularly when the
value of “n” had been obtained at low shear stress.
Note: Equation 5-20 does not incorporate the effects of roughness. Govier and Aziz
(1972) indicated that Equation 5-20 gives excellent agreement between calculated and experimental
data in the range of modified Reynolds numbers ReMR of 2900–36,000 and
Critical fanning friction factor
( from equation 5-28)
0.010
0.008
0.006
0.004
0.002
f NCR
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Flow index "n"
Re CR
2500
2300
2100
1900
1700
1500
FIGURE 5-4 Values of critical Reynolds number and critical fanning factor versus the flow
index “n” for pseudoplastic slurries based on Equation 5-27.
Modified nritical Reynolds number
(from Equation 5-27)

5.16 CHAPTER FIVE
modified power exponent s of 0.36–1.0. Equation 5-20 requires repeated iteration. The
use of a personal computer is recommended.
For power law fluids, Tomita (1959) extended his laminar flow model (discussed in
Section 5.2.1.3) to turbulent flows in smooth pipes by applying Prandtl’s mixing length
concept, and developed a different implicit equation:
1
� = 4 log10(RePL �fP� LT �) – 0.40 (5-30)
�fP� �LT
where RePL and fPLT were already expressed for the laminar flow in Equations 5-21 and 5-
22. Tomita’s equation was supported by 40 experimental data points on starch pastes and
lime slurries.
Irvine (1988) published the following equation:
F�(n)
fn = ��
(5-31)
where
F�(n) = � � 1/(3n+1)
2
� ��
n–1 7n n
8 7 (1 + 3n)
(5-32)
Example 5-6
At a volumetric concentration of 30%, a magnetite suspension has a power law coefficient
K of 12 dynes-secn /cm2 and a power law exponent of 0.2. If the slurry is homogeneous
and nonsettling at a speed of 1.5 m/s, determine the friction factor in a 101 mm ID
pipe, at a slurry density of 1600 kg/m3 .
Solution
From Equation 5-24, the modified Reynolds number is calculated as
Remod = � � 0.2
� � –0.8
1600 × 1.5 × 0.101 4 × 0.2 0.101
�� �� � = 202 × 0.5 × 45,697
12 × 0.1 1 + 3 × 0.2 8 × 1.5
Remod = 4615
It is necessary to check if the flow is turbulent.
From Equation 5-27:
Re c� =
[1/(3n+1)]
Remod
6464 n(n + 2) (n+2)/(n+1)
���
(1 + 3n) 2
Rec� = = 2143
Since Remod > Rec�, flow is therefore turbulent. Two different approaches will be used.
Irvine Method
From Equation 5-32:
F�(n) = � � 1/1.6
8 × 0.2
= 0.862
From Equation 5-31, the fanning friction factor is
0.2
6464 × 0.2(2.2)
2
� ��
–0.8 1.4 0.2
8 7 (1 + 0.6) (2.2/1.2)
���
(1 + 0.6) 2
8n n

0.862
fn = � = 0.00442
1/1.6 4615
Tomita Method
From Equation 5-22:
RePL = 4615 × 80.8 × 20.2 × 6(1.6/0.2) 0.8 /2.8 = 316,457
From Equation 5-30:
5.3 FRICTION LOSSES FOR
YIELD PSEUDOPLASTICS
Yield pseudoplastics were described extensively in Chapter 3. Examples were listed in
Table 3.11.
5-3-1 The Hanks and Ricks Method
In the laminar flow regime, Hanks and Ricks (1978), defined the fanning friction factor in
terms of the modified Reynolds number:
fNPL =
16
(5-33)
where
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
1
� = 4 log10(316,457 × �fP� LT �) – 0.40
�fP� �LT
Iteration 1 starts by assuming at fPLT � 0.004
correction fPLT = 0.0035
Iteration 2 starts at fPLT � 0.0042
fPLT = 0.00352
Iteration 3 starts at fPLT � 0.0045
fPLT = 0.003498
Iteration 4 starts at fPLT � 0.0035
fPLT = 0.00359
So by theTomita method we obtain a friction factor of 0.0035.
The Irvine method yields a friction factor 23% higher than the Tomita method. Both
methods do not account for roughness of the pipe wall.
� �Remod
� = (1 + 3n) n (1 – x) 1+n� + + � n x2 (1 – x) 2x(1 – x)
� �
1 + 2n 1 + n
2
�
1 + 3n
2 �yp x = = ��
fN · � · V
where �yp is the yield stress for pseudoplastic.
2
�yp �
�w
5.17
(5-34)

5.18 CHAPTER FIVE
For yield-pseudoplastics, Hanks and Ricks (1978), Heywood (1991) proposed to define
a modified Hedstrom Number as
Hemod = � � 2–n/n
2 Di �m �yp � � (5-35)
K K
The critical Reynolds number is established in terms of the modified Reynolds number
and Hedstrom number, as in Figure 5-5.
5-3-2 The Heywood Method
For the laminar flow regime fanning friction factor, Heywood (1991) modified the Buckingham
equation to
fNLY = �1 + 2Hemod ��1 – ���
(2n + 1) fNLY(Remod) 2
2Hemod ��
fNLY(Remod) 2
16
�
Remod
�1 + �1 + 2nHemod �����
(5-36)
fNLY(Remod) 2
4nHemod ���
(n + 1) fNLY(Remod) 2
5-3-3 The Torrance Method
Defining x = � y/� w, Torrance (1963) derived the following equation for turbulent friction
factor of a yield pseudoplastic:
Critical (Hanks & Ricks) Reynolds Number
3200
2800
2400
2000
1600
1200
800
400
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Flow Behavior Index "n"
4
He = 10
He = 0
He = 10 6
FIGURE 5-5 Values of the critical value of the “Hanks and Ricks Reynolds number” versus
the flow index “n” for yield pseudoplastic slurries.

� log10RePLC( f [1–n/2] 4.53
)�
+ (5n – 8) (5-37)
where RePLC = DnV 2–n�/K8n–1 0.68
� �
n
n
.
The work of Torrance was essentially an exercise in algebra, and has not been substantiated
by experimental data. Nevertheless it has been substantially quoted in the absence
of suitable confirming data.
5-4 GENERALIZED METHODS
Various models have been developed for complex non-Newtonian flows. The most important
ones are listed, but most were derived from empirical data.
5-4-1 The Herschel–Bulkley Model
Herschel and Bulkley (1928) developed a model that has been extensively applied to
sewage sludge, kaolin slurries, and mine tailings:
� = �y + j(�) n (5-38)
The apparent viscosity is
�a = = + j(�) n–1 � �y � � (5-39)
� �
where j is called the Herschel–Bulkley parameter.
5-4-2 The Chilton and Stainsby Method
Chilton and Stainsby (1998) indicated that the accuracy of the Herschel–Bulkley model
deteriorated at high shear rates. However, this may or may not be significant, depending
on the application. At high strain rates the model predicts that the viscosity tends to zero,
which is obviously incorrect.
In Chapter 3, Section 3-4-2-2 , the Sisko, Cross, Meter, and Bird rheological models
were presented. Chilton and Stainsby (1998) stressed the limitations of these models and
the shear rates at which they are valid.
In an effort to solve the Rabinowitsch and Mooney equations presented in Equations
5-15 and 5-16, Chilton and Stainsby (1998) proposed to express the pressure drop for a
Herschel–Bulkley fluid as:
where
�P
� L
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
= �� � – 2.95� + log 1 2.69
4.53
�� � � 10(1 – x)
�f� n
n
4j
�
D
8V
�
D
= � � n
� � n
� �� � n
3n + 1 1
1
��
� 4n
� 1 – x
x = = 4L� �0 0
� �Di�P �w
1 – ax – bx 2 – cx 3
5.19

5.20 CHAPTER FIVE
b =
1
a = �
(2n + 1)
2n
��
(n + 1)(2n + 1)
c =
For a Bingham slurry, j = � or the coefficient of rigidity. For a power law pseudoplastic
fluid, �0 = 0, j = �p or the power law viscosity. For a Newtonian, Hagen–Poiseuille
fluid, �0 = 0, � = 1, j = � or viscosity.
= � � =
Defining an effective viscosity as
� * = j� � n–1
� � n
� �� � n
2n
�P 4� 8V 32�V
� � � �2 L D D D
8V 3n + 1 1
1
� � � ��
(5-40)
2 3
Di 4n 1 – x 1 – ax – bx – cx
Chilton and Stainsby proposed their equation for a generalized Reynolds number as
�VDi ReMR = (5-41)
2
��
(n + 1)(2n + 1)
� �*
Using the value � (defined by Equation 5-16), the authors indicated that
3� + 1
� * = �L�� 4� �
and a modified Reynolds number is defined as
4��VD i
ReMR = ��
(5-42)
�L(3� + 1)
if the wall viscosity could be measured.
The friction factor in the laminar regime is then expressed as
16
fn = � (5-43)
ReMR
In the turbulent regime, for Herschel–Bulkley fluids Chilton and Stainsby derived
Re MR
fn = 0.079� � –0.25
��
2 4 n (1 – x)
Chilton and Stainsby then proposed another modified Reynolds number:
Re MR
(5-44)
R� =
This indicates that Equation 5-44 is a modified Blasius equation (see Chapter 2):
(5-45)
fn = 0.079(R�) –0.25 �� 2 4 n (1 – x)
(5-46)

HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
Without further proof, they proposed to follow the Prandtl equation of Newtonian fluids
as if it could be applied to non-Newtonian fluids:
f n –0.5 = 4.0 log10(R�f n 0.5 ) – 0.4 (5-47)
Deviations from experimental data were noticed at high Reynolds numbers due to the
limitations of the Herschel–Bulkley model on which basis this model was developed.
(See Figures 5-6 and 5-7.)
Example 5-7
Lab tests are conducted on sewage sludge in a 6� pipe. The density is 1018 kg/m 3 (S.G. =
1.018). The yield stress is measured as 1.28 Pa. The slurry is a Herschel–Bulkley fluid
mixture with a parameter j = 0.2. The power law coefficient is determined to be 0.74. Calculate
the friction factor for a flow of 350 L/s in a 18� pipe of 0.375� thickness, assuming
a wall shear stress of 1.6 Pa.
Pipe inner diameter = (18 – 2 × 0.375) × 0.0254 = 0.438 m
Pipe inner area = 0.25 × � × 0.438 2 = 0.151m 2
Pipe inner speed = 0.35/0.151 = 2.31 m/s
�0 1.28
x = � = � = 0.8
�w 1.6
5.21
FIGURE 5-6 The friction factor versus the Chilton–Stainsby Reynolds number for Bingham
mixtures. (From A. Chilton and R. Stainsby, 1998. Reproduced with permission of Journal of
Hydraulic Engineering, ASCE.)

5.22 CHAPTER FIVE
FIGURE 5-7 The friction factor versus the Chilton–Stainsby Reynolds number for pseudoplastic
mixtures. (From A. Chilton and R. Stainsby, 1998. Reproduced with permission of
Journal of Hydraulic Engineering, ASCE.)
� * = 0.2� � (0.74–1)
8 × 2.31 3 × 0.74 + 1
� ��
0.438
4 × 0.74
1
= 0.366���2 3
1 – ax – bx – cx �
� * = 0.2� � –0.26
8 × 2.31
�
0.438
� � 0.7 1
1
� ����� � � 0.75
(2.12 + 1)
��
2.96
1
5 ������
� 1 – 0.403 × 0.064 – 0.343 × 0.064 �
2 – 0.254 × 0.0643 � * = 0.3725
Re MR = 1018 × 2.31 × 0.438/0.3725 = 2765
This is transition Reynolds Number between laminar and turbulent flow. In laminar
regime fn = 16/ReMR = 0.0057.
�fnV Pressure drop = = 72 Pa/m
2 �P 2
� �
L Di
5-4-3 The Wilson–Thomas Method
� 1 – 0.8
1 – ax – bx 2 – cx 3
The Wilson–Thomas Method was developed in the 1980s for yield pseudoplastic and
power law slurries. Wilson (1985) and Thomas and Wilson (1987) assumed that the fluid

is a continuum, but that in non-Newtonian cases the viscous sublayer was thicker than
with the water. (See Chapter 2 for the definition of viscous sublayer.)
Defining a velocity VN for a Newtonian fluid at the same wall shear stress �w as for the
flow of a non-Newtonian fluid, a bulk velocity V is defined as
V = VN + Uf�2.5 ln� � + 11.6� �� (5-48)
with
= 2.5 loge� � (5-49)
Since Uf = (�w/�m) 1/2 , the Wilson–Thomas model requires the designer to assume a value
of wall shear stress.
The effective viscosity is defined as
�eff = � � n–1
� � n–1
(5-50)
which is slightly different than expressed by Equation 5-26.
In the laminar flow regime, the shear rate is expressed as
= � � 1/n
(5-51)
For Bingham fluids, the Wilson–Thomas equation is written as
V = VN + Uf�2.5 ln� � + x (14.1 + 1.25x)�
n + 1 1 – n
� �
2
n + 1
VN �Di Uf � �
Uf
�
4n 8V
� �
3n + 1 Di
8V 4n �w � � �
Di 3n + 1 K
1 – x
� (5-52)
1 + x
where
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
x = � 0/� w
ReMR = �mVDi(1 – x)/�� In the laminar flow regime
= �1 – x + x4 8V �w 4 1
� � � � � (5-53)
Di �� 3 3
(where �p = plastic viscosity), which is essentially the Buckingham equation. For Bingham
fluids in the laminar regime, if x � 0.5 then
8V�� 4
�w = � + � �0 Di 3
To obtain the transition velocity from laminar to turbulent flow, the following approach
based on the Wilson–Thomas method is recommended. In the turbulent regime, a
Reynolds number based on the friction velocity U f is defined as
Ref = �DIUf ��p
5.23

5.24 CHAPTER FIVE
The Wilson–Thomas velocity is defined by modifying Equation 5.52 to
1 – x
V = Uf�2.5 ln Re + 2.5 ln� � + x (14.1 + 1.25x)�� (5-54)
1 + x
The transition occurs at the intersection of Equations 5-53 and 5-54.
The Wilson–Thomas method was derived from first principles and does not use empirical
correction coefficients. It has proved to be correct for many slurries, but it sometimes
overpredicts losses for the Carson slurries (described in Equation 3-52).
Example 5-10 shows the method of calculation using the Wilson–Thomas equation.
Figure 5-8 compares the Wilson–Thomas method with the Hanks–Dadia friction factor.
Figure 5-9 compares it with experimental data.
5-4-4 The Darby Method: Taking into Account
Particle Distribution
Professor R. Darby (2000) from Texas A&M University recently published a new method
to predict the friction factor of power law non-Newtonian slurries. It is, however, much
closer to the domain of slurries and takes into account such concepts as the drag coefficient,
to which the reader was exposed in Chapter 3.
In a method reminiscent of the work of Wasp for compound heterogeneous flows
(see Chapter 3), Darby stated that the overall pressure drop for a non-Newtonian fluid
is essentially the pressure drop of the liquid phase plus the pressure drop due to the
solids:
�Pm = �Pf + �Ps (5-55)
for each fraction i of solids with an average diameter dpi, and with a volumetric concentration
Cv, the individual pressure drop must be computed. In a first iteration, the pressure
drop for the liquid phase is computed by treating it as homogeneous non-Newtonian liquid
by the various methods described in this chapter. A Froude number for the solid particles
is defined as
FIGURE 5-8 Comparison between the Wilson–Thomas and the Hanks–Dadia models for
friction factor of Bingham slurries. (From A. D. Thomas and K. C. Wilson, 1985. Reproduced
by permission of Canadian Journal of Engineering.)

Fr2 V
= = (5-56)
2
�� ��
(�s/�L – 1)d�g (�s/�L – 1)gDi where V t is the terminal velocity of falling particles.
At slip velocity V r between the solids and the liquid phases, with V as the carrier speed
of the suspension, a nondimensional pressure drop is defined as
If V r = V f – V s,
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
X = � � 2
��s �� �
Cv�L(�s/�L – 1)gL V
W r 2
(5-57)
X = � (5-58)
1 – Wr
where W r = V r/V. For small volume fraction 0 < C v < 0.25, X = X 0.
For other concentrations,
V t 2
V t
5.25
FIGURE 5-9 Comparison between the Wilson–Thomas and experimental data on pressure
losses for a limestone slurry. (From A. D. Thomas and K. C. Wilson, 1985. Reproduced by
permission of Canadian Journal of Engineering.)
X = X 0 + 0.1F R 2 (Cv – 0.25) (5.59)

5.26 CHAPTER FIVE
Procedures for Newtonian Mixtures:
1. Calculate the Froude number.
2. Determine Vt from the Archimedean number:
Ar =
d p 3 �f g(� s/� L – 1)
��
� L 2
with � f being the liquid viscosity.
If Ar < 15, the terminal velocity can be obtained from the Stokes equation
If Ar > 15, the particle Reynolds number is obtained after rearranging the
Dallavalle equation:
Re p = [(14.42 + 1.827�F�r�) 0.5 – 3.798] 2 = d pV t � L/� L
(5-60)
3. Calculate the Froude number from Equation 5-56.
4. The value of ratio Wr is then calculated from the Molerus diagram (Figure 5-10). The
values of X and �Ps/L are then calculated from Equations 5-59 and 5-60. This step
may be repeated for each range of particle size and summed up with other particle
sizes to get an overall pressure drop for solids. However for Non-Newtonian mixtures
additional procedures are needed.
FIGURE 5-10 Molerus diagram. (From R. Darby, 2000. Reproduced by permission of
Chemical Engineering.)

Terminal Velocity for Power Law Fluids
For a solid particle settling in a non-Newtonian fluid, defining
Z = (5.61)
as a nondimensional power law parameter and
C1 = [(1.88/n) 8 + 34] –1/8
1.33 + 0.37n
��3.7
1 + 0.7n
as another nondimensional power law parameter, Darby expressed the modified Reynolds
number as
with C d the drag coefficient:
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
4.8
�1/2 C d – C1
Re PL = Z� � 2
= � (5.62)
K
4g(�m/�L – 1)d� Cd = ��
(5.63)
Friction Factors
The transition from laminar to turbulent flow occurs at a critical Reynolds number:
RePLC = 2100 + 875(1 – n) (5.64)
Defining fL as the fanning factor in the laminar regime, and fT as the fanning factor in the
turbulent regime, Darby derived the following equations:
�
fn = (1 – �) fL + ��
(5.65)
–8 –8 1/8
( f T + f L )
where the laminar flow friction factor is
16
fL = � (5.66)
RePL
0.0682n –1/2
fT = ��
(5.67)
1/(1.87+2.39n)
RePL
At the transition from laminar to turbulent flow, the friction factor is expressed as:
fTR = 1.79 × 10 –4 (0.414+0.757n) exp(–5.24n)RePL (5.68)
and
1
� = � (5.69)
� 1 + 4
� = Re PL – Re PLC
Example 5-8
Slurry is required to flow in a 250 mm ID pipe and has the following characteristics:
�0 = 15 Pa
� = 43 mPa · s
� = 1450 kg/m3 3V t 2
d � n V t 2–n �f
5.27
(5.70)

5.28 CHAPTER FIVE
K = 0.4 Pa · sn n = 0.8
d85 = 65 �m
V = 1.76 m/s
Determine the friction factor. Check on critical Reynolds number:
RePLC = 2100 + 875(1 – 0.8) = 2275
Z = 1.13
C1 = 0.4236
Calculations will be done by a series of iterations.
Iteration 1. Assume the drag coefficient is Cd = 0.4, then
RePL = 1.13� � 2 4.8
�� = 597
1/2 0.4 – 0.4236
16 16
fL = � = � = 0.026
RePL 597
0.0682 × 0.8
fT = = 0.0141
–1/2
��
597 0.264
� = 597 – 2275 = –1678
� =
� � 0
�
fn = (1 – �)fL + ��
–8 –8 1/8
( f T + f L )
fn = fL = 0.026
By further iteration, the magnitude of the friction factor is refined.
5-5 TIME-DEPENDENT
NON-NEWTONIAN SLURRIES
1
� 1 + 4 –�
Time-dependent non-Newtonian flows are difficult to model. There is a certain lapse of
time to overcome before a stable friction factor can be obtained. Govier and Aziz (1972)
have published cases in which the lapsing time was as high as 1000 minutes with Prembina
crude oil. During this initial lapse of time, the pressure gradient for friction losses
dropped from an initial value of 72.5 Pa/m (0.0032 psi/ft) down to 18 Pa/m (0.0008 psi/ft)
by the time the flow had stabilized. Such a ratio of 4:1 is certainly not negligible.
Govier and Aziz (1972) indicated that once the initial period of stabilization is
reached, the general form of pressure loss equations are the same as for time-independent
non-Newtonian fluids. At the entry to a pipe, the flow may be laminar, but at a certain distance,
once the entrance effects are overcome, the flow can transit to turbulence.
The start-up pressures for thixotropic slurries may be quite high, particularly when
these slurries coagulate into gels inside the pipeline. During an initial of period of time,
gels must be expelled from the pipeline. Crude oils, fuel oils, bentonite clay, and drilling

muds are a concern to engineers. Water may be used as an expelling medium to clear up
the pipeline. Positive displacement pumps are preferred for thixotropic slurries to overcome
the high starting pressures.
Various models have been developed for thixotropic fluids. These slurries are sometimes
treated as Bingham plastics and sometimes as pseudoplastics, based on the experimental
data from test work.
The problem of pumping thixotropic froth in oil sand plants was a challenge to manufacturers
of pumps, and for a while the belief was that only positive displacement pumps
could be used. In 2000, the operators of oil-sand processing plants invited the manufacturers
of slurry pumps to develop new appropriate pumps. Research is being conducted at
the Saskatchewan Research Institute in Canada and is starting to yield new concepts of
pump design.
5-6 EMULSIONS
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
The concept of emulsions was introduced in Chapter 1. In an emulsion, one phase is suspended
as droplets rather than particles. Tar or bitumen, for example, can be suspended in
water up to a volumetric concentration of 70%. In some of the oil sand extraction processes
in Canada, the situation can be further complicated by the addition of flocculants. The
Venezuelan corporation, PDVSA, and its research branch, INTEVEP, conducted considerable
research on Orimulsion, a proprietary synthetic fluid composed of surfactants,
bitumen droplets, and water. The surfactants keep the bitumen droplets in suspension.
Nunez et al. (1996) published a comprehensive paper on the flow characteristics of
concentrated emulsions in water with a volumetric concentration in the range of 70–85%.
5-7 ROUGHNESS EFFECTS ON FRICTION
COEFFICIENTS
5.29
Szilas et al. (1981) published the following equation for pseudoplastic oil flow in rough
pipes at high Reynolds numbers:
= log10 (Remod(4fn) 1–n/2 ) + 1.511/n�4.24 + � – – 2.114 (5-71)
However, this equation does not include any terms for wall roughness.
Torrance (1963) developed a theoretical equation for yield pseudoplastics in fully turbulent
flow (at high Reynolds numbers) in rough pipes as
= 4.07 log10� � + 6.0 – (5-72)
where RePLC = �DnV 2–n /K8n–1 1 4
1.414 8.03
� � � �
�fn� n
n n
1
RePLC 2.65
� � �
�fn�
�
n
. This applies to the region where the friction factor is independent
of the Reynolds number, or at high Reynolds numbers.
Govier and Aziz (1972) suggest the following procedure:
� Compute the friction factor for the slurry in a smooth tube using one of the methods
described in Sections 5-2, 5-3, and 5-4.
� Using the Moody diagram determine the ratio of the friction factor for rough to smooth
pipe at the value of the Reynolds number (using Re B, Re PLC, or Re mod).

5.30 CHAPTER FIVE
Aral and Kaylon (1994) focused on highly concentrated suspensions and investigated the
effects of temperature as well as surface roughness.
To take into account the pipe roughness in laminar and turbulent flows, SRC (2000)
recommended the use of the equation of Churchill (1977):
where
8
�
Re
f n = 2�� � 12
+ (A + B) –1.5 � 1/12
A = � –2.457 ln�� � 0.9 7
16
� + 0.27 �/D
Re
i��
B = � � 16 37,530
�
Re
(5-73)
(5.74)
(5.75)
Example 5-9
Assuming turbulent flow and using the Wilson–Thomas model, calculate the bulk velocity
for a flow with the following characteristics:
�0 = 20 Pa
�w = 24 Pa
�� = 0.016 Pa · s
� = 3 × 10 –6 m
Di = 141 mm
� = 1350 kg/m3 x = �/�w = 20/24 = 0.833
Iteration 1. Assume VN = 2.5 m/s, then
Re = �VDi/� � = ��/(1 – x) = 0.016/(1 – 0.833) = 0.096
Re = �VDi/� Re = 1350 × 2.5 × 0.141/0.096 = 4967
Relative roughness �/Di = 3 × 10 –6 /0.141 = 0.000021
From Equations 5-74 and 5-75:
A = {–2.457 ln[(7/4967) ) 0.9 + 0.27 × 0.000021)] 16 = 3.86 × 1018 B = 1.1286 × 10 14
f n = 0.00949
Iteration 2.
Checking the value of V N. From Chapter 2:
�w = fn�V 2 /2
(2 × 24)/(0.00949 × 1350) = 3.75
V = 1.935 m/s
The friction velocity from Equation 2-5:
Uf = ��w�/�� = 0.133 m/s

HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
Using Equation 5-52:
1 – 0.833
V = 1.935 + 0.133�2.5 ln � + 0.833(14.1 + 1.25 × 0.833)�
1.833
2.815 m/s = 1.935 + 0.133[–5.989 + 12.61]
Therefore, the equivalent Newtonian fluid velocity is 1.935 m/s, and the mean velocity of
the Bingham slurry will effectively be 2.815 m/s.
Up until 1990, one author after another tried to treat non-Newtonian slurries as homogeneous
mixtures. The arbitrary assumption that the particle diameter was absent in many
equations when the diameter was smaller than 44 �m or 74 �m (depending on the author)
is now being challenged. The advent of new experimental methods, such as laser velocimeters,
is helping engineers understand the complexity of turbulent non-Newtonian
fluids. The work of Park et al. (1989) using laser velocimeters, and Pokryvalio and
Grozberg (1995) using electro-diffusion techniques on non-Newtonian slurries was analyzed
by Slatter et al. (1996), who confirmed the significance of high-intensity turbulence
at the wall. This encouraged them to postulate a theory of particle roughness. Slatter et al.
proposed that the d 85 particle size be used for particle roughness. (This may be an attempt
to correlate with the Nikrudase particle roughness of Newtonian slurries.) For large pipe,
when the actual roughness � exceeds the value d 85, � is used. For
� < d 85
� > d 85
d x = d 85
d x = �
For a yield pseudoplastic, Slatter et al. (1996) defined their Reynolds number, Re r, in
terms of the friction velocity, consistency factor K, and power coefficient n, as well as
roughness d x:
8�U f 2
5.31
Rer = (5.76)
If Rer > 3.32, then smooth wall turbulence occurs, and the mean bulk velocity V is expressed
as
= 2.5 ln� � – 2.5 ln Re ��
�0 + K(8Uf /dx) V Di � � r + 1.75 (5.77)
Uf 2dx
If Rer ( 3.32, then fully developed rough turbulent flow occurs, and the mean bulk velocity
V is expressed as
V Di � = 2.5 ln��� + 4.75 (5.78)
Uf 2dx
This correlation produces an abrupt transition from smooth turbulent to fully turbulent
flow at the wall of the pipe.
We have already explained that the Wilson–Thomas model was based on the assumption
that the viscous sublayer in non-Newtonian flows was thicker than with water, thus
suppressing the effect of roughness. The work of Slatter, Thorscalden, and Petersen
(1996) on mixtures of kaolin clay and sand indicates that this is not very achievable (Figure
5-11). The resultant pressure losses (Figure 5-12) are therefore higher. Figure 5-12
does indicate that the Torrance and the Wilson–Thomas models correlate well. Both models
were based on mathematical assumptions at 20 year intervals.
n

5.32 CHAPTER FIVE
FIGURE 5-11 A comparison between the Wilson–Thomas and Slatter, Thorscalden, and
Petersen models for the viscous sublayer. (From P. T. Slatter et al., 1996. Reproduced by permission
of BHR Group.)
FIGURE 5-12 A comparison of pressure drop per unit length between the Slatter,
Thorscalden, and Petersen and Wilson–Thomas models. (From P. T. Slatter et al., 1996. Reproduced
by permission of BHR Group.)

Example 5-10
A sand–kaolin mixture with a volumetric concentration of 19.5% is flowing in a 431 mm
ID pipe at an average speed of 1.8 m/s. The yield stress is 5.5 Pa, the coefficient of consistency
K = 0.124 Pa · sn , and the power coefficient n = 0.64. The specific gravity of the solids
are 2.65 and the d85 = 131 �m. Using the Slatter method, determine the friction factor.
Solution
Density of mixture:
�m = Cv(�s – �L) + �L �m = 0.19(1265) + 1000 = 1240 kg/m3 Iteration 1. Assume fully turbulent flow:
= 2.5 ln� � = 4.75 = 23.26
If V = 1.8 m/s, then Uf = 0.0774 m/s, since Uf = V �fD�/8�. fD = 0.0147
8 × 1240 × 0.0774
Rer = = 1.78
Iteration 2. Since Rer < 3.32, the equation to use is
2
�����
5.5 + 0.124(8 × 0.0774/0.131 × 10 –3 ) 0.64
V
0.5Di � ��–6
Uf 131 × 10
V
� Uf
5-8 WALL SLIPPAGE
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
0.5D i
= 2.5 ln� � + 2.5 ln Re � r + 1.75
d85
V
� = 18.51 + 1.44 + 1.75
Uf
V
� = 21.70
Uf
0.046 = (fD/8) fD = 0.01698
Uf = 0.0829 m/s
5.33
A phenomenon encountered with non-Newtonian mixtures is a tendency for the low-viscosity
constituent to migrate to regions of high shear and to lubricate the flow. One example
is the core annular flow of crude oil in water, where the more viscous material is lubricated
by the less viscous material. In the case of emulsions and certain non-Newtonian
slurries, lubrication occurs by a slip layer of water on the wall.
Mathematically, the concept of slip can be treated as a discontinuity. Heywood (1991)
proposed to represent slip by the slip velocity Vs. In the laminar flow regime, the total
flow in a pipe would be
Q = Vs + � � 3
� �w �
0
2 2 �D i � Di d�
� � � � d� (5-79)
4 8 �w dt

5.34 CHAPTER FIVE
Heywood (1991) noticed that there are no methods to evaluate slip in turbulent flows.
Evaluation of slip in laminar flow is conducted using the coaxial cylinder described in
Chapter 3.
Nunez et al. (1996) indicate that the migration of viscous droplets from the wall in an
emulsion exhibit the Segré–Silberberg effect. However, they pointed out that not all
emulsions experience slip and that the phenomena of slip appeared to be characteristic of
the crude oil or viscous component. In some respects, slip does reduce the friction factor
by lubrication from the least viscous phase. However, Aral and Kaylon (1994) found that
increasing the surface roughness tended to reduce or eliminate slip.
One particular problem with emulsions is the fracture of the droplets under high
shear rates. A form of comminution occurs as large droplets come in contact with other
ones. Degradation of non-Newtonian slurries under high shear rates is not well documented.
There is evidence that high shear rates occur in centrifugal pumps. Adequate clearance
may reduce the degradation of the emulsion or slurry but tends to reduce the efficiency of
the pump. Degradation can include a form of coalescence or formation of colloids and
larger droplets or particles. Clay ball formation is encountered in dredging operations after
passage through the pump.
5-9 PRESSURE LOSS THROUGH
PIPE FITTINGS
The method of the two K-factor was presented in Chapter 2 in Section 2.8. It was explained
that Hooper had established a general relationship
K1 1
K = � + K��1 + �
Re � DI-in
where
K1 is the value of K at a Reynolds number of 1
K� is the value of K at high Reynolds number
Di-in is the internal pipe diameter in inches
Johnson (1982) reviewed some of the problems of pumping non-Newtonian mixtures.
In his assessment of fittings for sewage, he indicated important discrepancies in the laminar
regime with losses 2–4 times as much as those for water flows. In the turbulent
regime, the losses were either of the order of those for water or higher. He recommended
that further studies be conducted for laminar flow, but for turbulent flows, the concept of
equivalent length be used.
In Chapter 2, concepts of pressure losses for Newtonian liquids were examined. Edwards
et al. (1985) reviewed the flow of non-Newtonian slurries in laminar regimes. They
recommended that the modified Reynolds number (using ReB or Remod) be used to correlate
with the loss factor Kf of Newtonian flows in laminar regimes.
In certain fittings such as globe valves, turbulence is enhanced by the geometry of the
fitting. Although the flow may be laminar in a straight pipe up to a transition to a
Reynolds number of 2000, turbulence in the globe valve may actually start at much lower
Reynolds number such as 900.
Much more work is needed on loss factors for non-Newtonian flows in transition and
turbulent regimes. Govier and Aziz (1972) had noticed the lack of any methodology to
compute loss from pipe fittings with non-Newtonian flows. They proposed that the

method of equivalent length be used, i.e., that the equivalent length of pipe fittings for
Newtonian liquids be added to computations of total length for pressure loss of the non-
Newtonian slurry.
5-10 SCALING UP FROM SMALL
TO LARGE PIPES
Non-Newtonian flows are complex. A lab test in a pumping loop can yield very useful
rheology data about the pressure drop. Scaling up to larger pipes is one method of predicting
pipeline flows. Heywood et al. (1992) proposed the following methods:
� For laminar flow, the plot of the shear rate (8V/Di) against the shear stress (Di�P/4L) is
independent of the pipe diameter and the pipe roughness, so that experimental data
could be converted directly into practical data for pipeline design.
� For turbulent flows, the Bowen method, which is essentially a modification of the Blasius
method, should be used. Bowen (1961) suggested the following modification to
the Blasius equation:
x w Di � = kV (5.80)
The shear stress is plotted against the flow rate Q to obtain the magnitude of the exponent
w. The result is then used to plot �/V w against the diameter Di to obtain the values of k and
x. The intersection of the turbulent and laminar flow curves gives the transition point.
Kenchington (1972) showed that this method showed great discrepancy when the ratio of
diameter between the large field pipe and the lab pipe exceeded 6 to 12 folds, so that large
pipe tests may be needed. The Bowen method assumed the same range of roughness between
a lab and a field pipe. It is therefore important to apply correction factors for roughness
when using this method.
5-11 PRACTICAL CASES OF
NON-NEWTONIAN SLURRIES
The equations presented in the previous sections of this chapter are fairly complex and are
based on so many assumptions that the practical engineer may feel lost. Some real examples
are needed to guide the designer of non-Newtonian systems.
5-11-1 Bauxite Residue
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
Want et al. (1982) reviewed the design of a bauxite residue pipeline for Alcoa Australia.
The plant disposed 4.75 Mtpy (million tonnes per year) of alumina. Tests conducted on
samples confirmed that the rheology of the slurry at concentrations in excess of 45% by
weight could be expressed by the Carson equation:
1/2 1/2 � w = � 0 + ���� �� 1/2 du
�
dr
where du/dr is expressed by Equation 5-15 and � Equation 5-16.
5.35

5.36 CHAPTER FIVE
In the case of Kwina red mud, tests indicated that the modified consistency coefficient
K� (Equation 5-18) was a complex function of the weight concentration:
K� = 3.88 × 10 –18 (100 Cw) 10.57 Pa/s� and that the value of � was
� = 2.55 × 106 (Cw × 100) –4.19 for Cw � 51%
� = 6.685 (Cw × 100) –0.926 for Cw > 51%
Applying the Dodge–Metzner model Want et al. (1982) applied Equation 5-20 to express
the consumed power under turbulent conditions as:
eT = 1.44 × 10 –4 150 – 100 Cw 2
fn ��� (100 Cw) � 1.5
(�Q) 3
�5 D i
in kW/km, and in laminar conditions as:
eL = 0.393 K�� � 1+�
� � 1+3�
64�mQ(150 – 100Cw) 1
��� � in kW/km
3 × 10 Di
Want et al (1982) discussed the thixotropic nature of red mud at high concentration, and
the importance of flocculants and dispersants on the rheology of this slurry. Referring to
Figure 5-13, it is clear that there is a change in the pressure drop per unit length as the
weight concentration is increased and the flow changes from turbulent to laminar. This
drop in pressure to a minimum at such a transition is often misunderstood, particularly because
it goes against the concepts examined in Chapter 4. The important parameters include
the diameter of the pipe, so that there is an optimum diameter, and an optimum
weight concentration for a given tonnage of fine solids to be transported.
Slurries may therefore be pumped at very high concentrations (Figure 5-14) using positive
displacement pumps (Figure 5-15) over long distances, provided that the correct
weight concentration is used near the turbulent to laminar transition region.
4� Cw Example 5-11
Using the data obtained by Want, examine pumping red mud bauxite residues at a speed
of 1.74 m/s in a 141 mm I.D. pipe at weight concentrations of 45% and 60%. Determine
the required power for a horizontal pipeline, 3 km long. Assume a density of 1350 at 45%
and 1800 at 60%.
Solution
At Cw = 45%:
K� = 3.88 × 10 –18 (100Cw) 10.57
K� = 1.157
� = 2.55 × 106 (100 Cw) –4.19
� = 0.302
Q = AV = � × 0.25 × 0.1412 × 1.74 = 0.0272 m3 /s
eT = 0.393 × 1.157� � 1.3
� � 1.906
64 × 1350 × 0.0272(150 – 45) 1
���� �
3 × 10 0.141
4� × 0.45
eT = 188 kW/km
For 3 km, this is equivalent to 565 kw or 758 hp.

At C w = 60%:
For 3 km, e L = 3123.9 kW or 4186 hp.
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
K� = 3.88 × 10 –18 (60) 10.57 = 24.2
� = 6.685(60) –0.926 = 0.151
e T = 1041.3 kW/km
� � 1.906 1
eL = 0.393 × 24.2� � 1.3
64 × 1800 × 0.0272 (150 – 60)
���� �
3 × 10 0.141
4� × 0.6
5.37
FIGURE 5-13 Pressure drop per unit length for red mud tailings. (From F. M. Want et al.,
1982. Reproduced by permission of BHR Group.)

5.38 CHAPTER FIVE
FIGURE 5-14 Slurry can be pumped in a non-Newtonian regime at high volumetric concentrations.
(Courtesy of Geho Pumps.)
5-11-2 Kaolin Slurries
Slatter et al. (1996) reported that Kemblowski and Kolodziejski (1973) found that the
Dodge and Metzner model did not well represent the flow of kaolin slurries. They derived
the following empirical equation:
and more generally:
4f n =
0.3164
�0.25 ReMR
4fn = �1/ReMR E
�m ReMR

where E, m, and � are empirical parameters and functions of the apparent flow behavior
index � (defined in Equation 5-16) and the modified Reynolds Number Re MR is defined
by Equation 5-17.
5-12 DRAG REDUCTION
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
5.39
FIGURE 5-15 The pumping of high concentration slurries and pastes may require positive
displacement pumps. (Courtesy of Geho Pumps.)
Ippolito and Sabatino (1984) showed that the addition of 3% salt to water tended to reduce
the friction factor of bentonite suspensions. Sauermann (1982) indicated that the addition
of 0.2 kg/ton of tripolyphsophate (Na 5P 3O 10) to gold slimes at a weight concentration
of 67.7% , and with particles smaller than 50 �m, reduced the pressure gradient in
laminar flow by as much as 30%. Some viscosity reducing agents were discussed in

5.40 CHAPTER FIVE
Chapter 3.There are, however, very little data published on other methods for reducing
friction losses of non-Newtonian slurries.
Schowalter (1977) discussed some aspects of drag reduction in non-Newtonian slurries
and reported certain cases of mixtures with a pressure drop actually lower than that of
water.
5-13 PULP AND PAPER
Pulp and paper pump slurries behave as non-Newtonian slurries. The following equations
have been reported by the Cameron Hydraulic Data book of IDP (1995), based on work
at the University of Maine. A modified Reynolds number is defined as
D I 0.205 Vs�g
ReMR = (5-81)
where
C = % consistency of the pulp, oven dry
g = 32.2 ft/s
� = density in slugs/ft3 �� 1.157 C
Vs = velocity, defined in ft/sec
A modified friction factor is defined as
3.97
f = � (5-82)
1.636
ReMR
A special equation for friction losses is therefore defined as
fV 2 LK 0
Hf = � (5-83)
DI
The correction factor K0 depends on the type of pulp. It is considered to be 1.00 for unbleached
sulfite softwood, 0.90 for bleached sulfite softwood, 0.90 for unbleached kraft
softwood, 0.90 for soda hardwood, 0.90 for reclaimed fiber, 1.0 for presteamed groundwood–softwood,
and 1.42 for stoned groundwood–softwood. (IDP 1995).
The following program calculates friction factors.
10 PRINT “program for stock , pulp and paper”
PRINT “based on the curves of the University of Maine”
PRINT “ which are correlations to the data of Brecht and Heller”
pi = 4 * ATN(1)
INPUT “name of project”; pr$
INPUT “name of client “; nc$
INPUT “date of calculations”; dat$
INPUT “ type of pulp”; pul$
INPUT “consistency of pulp in percent”; CS
C = CS/100
15 INPUT “ choose between (1) SI units (m) and (2) US units (feet) “; NB
IF ABS(NB) < 1 THEN GOTO 15
IF ABS(NB) > 2 THEN GOTO 15
IF (NB > 1) AND (NB < 2) THEN GOTO 15
GOSUB conversion
18 IF NB = 1 THEN INPUT “inner diameter (m)”, D1M
IF NB = 2 THEN INPUT “inner diameter (ft)”; D1US

IF NB = 1 THEN d1 = D1M * dl
IF NB = 1 THEN D1US = D1M/.3048
IF NB = 2 THEN d1 = D1US * dl
IF NB = 1 THEN INPUT “pulp flow rate (m3/hr)”, qm
IF NB = 2 THEN INPUT “pulp flow rate in USgpm”; qus
IF NB = 1 THEN q = qm/60000
IF NB = 2 THEN q = qus * 3.785/60000
a = .25 * pi * d1 ^ 2
v = q/a
PRINT USING “pulp speed = ###.### m/s”; v
vus = v/.3048
PRINT USING “PULP SPEED = ##.### ft/s”; vus
IF (C > .03) AND (vus > 8) THEN PRINT “speed exceeds 8 ft/s please use
larger pipe size”
IF C > .03 THEN GOTO 25
IF (C < .02) AND (vus > 10) THEN PRINT “speed exceeds 10 ft/s please use
larger”
IF C < .02 THEN GOTO 25
IF vus > 9 THEN PRINT “SPEED EXCEEDS 9 FT/S, PLEASE USE LARGER PIPE”
25 INPUT “DO YOU WANT TO USE A LARGER PIPE (Y/N)”; p$
IF p$ = “Y” OR p$ = “y” THEN GOTO 18
IF NB = 1 THEN INPUT “DENSITY OF STOCK IN KG/M3 (OFTEN ASSUMED TO BE 1000
KG/M3 “; DENSM
IF NB = 1 THEN DENSUS = DENSM * (62.4/1000)
IF NB = 2 THEN INPUT “DENSITY OF STOCK IN LBS/CU.FT (OFTEN ASSUMED TO BE
62.4 LBS/CU.FT”; DENSUS
REM0 = D1US ^ .205 * vus * DENSUS/CS ^ 1.157
FM0 = 3.97/REM0 ^ 1.636
PRINT USING “MODIFIED REYNOLDS NUMBER = #######”; REM0
PRINT USING “MODIFIED FRICTION FACTOR = #.####”; FM0
PRINT “ please choose between the following pulps”
PRINT “ 1- unbleached sulfite “
END
conversion:
IF NB = 1 THEN dl = 1
IF NB = 2 THEN dl = .3048
IF NB = 1 THEN ql = 3875/60
IF NB = 2 THEN ql = 60/3875
RETURN
5-14 CONCLUSION
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
5.41
The world of non-Newtonian flows is very complex and encompasses very different
flows, including pulp and paper, food, plastics, and clays. The use of various equations
developed by researchers do not yield the same values of the friction coefficient. The
problem is compounded by the fact that many rheological tests are conducted in tubes and
in laminar flows, yielding values of consistency factor K and exponent n outside the range
of turbulent flows. The practical engineer is often left to use his engineering judgment to
use the appropriate equation. Proper tests of the slurry flow at the correct range of shear
stresses are essential to avoid errors. Various reference books on the subject have equations
similar to Equations 5-20 to 5-25 without emphasizing the limitations to their use.
Because many of the equations for friction loss factors are implicit and require iteration
methods, the use of personal computers is encouraged.
The flow of non-Newtonian slurries is complex and requires a significant energy in-

5.42 CHAPTER FIVE
put. Methods have been developed over the years to reduce friction losses. These include
dilution, reduction of volumetric concentration of solids, removal of all flocculants, provisions
for air purging of the pipeline, addition of high-aspect-ratio fibers, addition of deflocculants
(soluble ionic compounds), reduction of the angularity of particles, and addition
of viscosity reducing agents. Not all these methods are always possible, and some
require capital investment.
It is hoped that the various worked examples in this book will help the practical engineer
to design slurry pipelines. It may be necessary to use more than one method and
compare results. There are many advantages to pumping slurries at high concentrations,
such as concentrates from process plants, food pastes, and ceramic slurry for the manufacture
of new materials.
5-15 NOMENCLATURE
a Nondimensional parameter and function of Hedstrom number
A Factor for friction in the Churchill equation
Ar Archimedean number
b Nondimensional parameter
B Factor for friction in the Churchill equation
C1 Nondimensional power law parameter
CD Drag coefficient
Cv Volume fraction of solid particles in the slurry mixture
Cw Weight concentration
d85 Particle diameter passing 85% (m)
dp Diameter of particle
dP/dz Pressure gradient per unit length
dx Equivalent roughness
Di Conduit inner diameter (m)
eT Consumed energy
E Empirical coefficient
fD Darcy friction factor
fL Laminar component of fanning friction factor for a Bingham plastic
fN Fanning friction factor
fNC Fanning friction at transition between laminar and turbulent flow
fNL Laminar component of fanning friction factor
fNLY Laminar component of fanning friction factor for a yield pseudoplastic
fNPL Fanning friction factor for a pseudoplastic in a laminar regime
Tomita laminar friction factor
fPLT fT fTR Turbulent component of fanning friction factor
Fr
Darby friction factor for transition from laminar to turbulent flows with power
law fluids
Froude number
g Acceleration due to gravity (9.8 m/s2 )
gc Conversion from slugs to pounds mass in USCS units
He Hedstrom number for Bingham plastic
He mod
Modified Hedstrom for yield pseudoplastic
j Herschel–Bulkley parameter
K Coefficient of consistency for power law fluids
K� Modified coefficient of consistency for power law fluids

L Length of conduit or pipe
m Exponent in Darby’s equation for fanning friction factor calculations
n Flow behavior index for pseudoplastic flows
P Pressure
Plasticity number
PL Pst Start-up pressure to pump a non-Newtonian slurry
Q Flow rate
R� Chilton and Stainsby proposed modified Reynolds number
Re Reynolds number
ReB Reynolds number for a Bingham plastic using the coefficient of rigidity for viscosity
ReBc Critical transition Reynolds number for a Bingham plastic using the coefficient
of rigidity for viscosity
Rec Reynolds number at transition
Remod Modified Hedstrom number for yield pseudoplastic
ReMR Modified Reynolds number for a power law fluid
RePL Tomita Reynolds number for a power law fluid
RePLC Tomita Reynolds number for a power law fluid at transition
Rer Slatter Reynolds number
Rep Particle Reynolds number
Uf Friction velocity
V Speed
VN Newtonian velocity
Vr Slip velocity
Vs Velocity of solids
Vt Terminal velocity of falling particles
VTR Transition velocity from laminar to turbulent flows
Ratio of slip velocity to slurry speed
W r
x Ratio of the yield stress to the wall shear stress
x c
Ratio of the yield stress to the wall shear stress at the transition from laminar to
turbulent flow
Z Settling factor for a non-Newtonian fluid
Subscripts
L Liquid
m Mixture
p Particle
Greek symbols
� Function for use of laminar and friction factors
� Increment
� Concentration by volume in decimal points
�P Pressure drop
��m Density change for the mixture
� Density
�L Density of liquid carrier in kg/m3 �m Density of slurry mixture in Kg/m3 � s
Density of solids
HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
� Modified flow behavior index for pseudoplastic flows
� Shear strain
d�/dt Wall shear rate or rate of shear strain with respect to time
5.43

5.44 CHAPTER FIVE
� Linear roughness (m)
� Coefficient of rigidity of a non-Newtonian fluid, also called Bingham plastic viscosity
�� Coefficient of rigidity at high shear rate
� Carrier liquid dynamic viscosity
�a Apparent viscosity of a pseudoplastic fluid
�e Effective pipeline viscosity
�e�� Wilson–Thomas effective viscosity
�L Viscosity of liquid carrier
�p Effective pipeline viscosity for pseudoplastic
� * Effective viscosity for Hagen-Poiseuille fluid
� �
Bingham plastic limiting viscosity of slurry mixtures (Poise)
� Empirical function of the power exponent n
� Pythagoras number (ratio of circumference of a circle to its diameter)
� Shear stress at a height y or at a radius r
�0 Yield stress for a Bingham plastic
�w Wall shear stress
�yp Yield stress for pseudoplastic
5-16 REFERENCES
Abulnaga, B. E. 1997. Slurcal—Computer Program for Non-Newtonian Flows. Fluor Daniel Wright
Engineers, Vancouver, BC, Canada (unpublished).
Aral, B. K., and D. M. Kaylon. 1994. Effects of temperature and surface roughness on time dependent
development of wall slip in steady torsional flow of concentrated Suspensions. Journal of
Rheology, 38, 957–972.
Bowen, R. L. 1961. Chemical Engineering, 143–150.
Buckingham, E. 1921. On plastic flow through capillary tubes. ASTM Proceedings, 21, 1154.
Chilton, R. A., and R. Stainsby. 1998. Pressure loss equations for laminar and turbulent non-
Newtonian pipe flow. Journal of Hydraulic Engineering, 124, 5, 522–529.
Clifton, R. A, and R. Stainsby. 1998. Pressure loss for laminar and turbulent non-Newtonian pipe
flow. Journal of Hydraulic Engineering, 124, 5, 522–529.
Churchill, S. W. 1977. Friction factor equation spans all fluid flow regimes. Chemical Engineering
84, 7, 91–92.
Darby, R. 2000. Pressure drop of non-Newtonian slurries, a wider path. Chemical Engineering, 107,
5, 64–67.
Darby, R. 1981. How to predict the friction factor for flow of Bingham plastics. Chemical Engineering,
88, 26, 59–61.
Darby, R., R. Mun, and V. Boger. 1992. Prediction friction loss in slurry pipes. Chemical Engineering,
September.
Dodge, D. W., and A. B. Metzner. 1959. Turbulent flow of non-Newtonian systems. Am. Inst. Chem.
Engr., 5, 2, 189–204.
Edwards, M. F., M. S. M. Jadallah, and R. Smith. 1985. Head losses in pipe fittings at Low Reynolds
Number. Chem. Engr, Res. Des., 63, 1, 43–50.
Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand
Reinhold.
Hanks, R. W. 1962. A Generalized Criterion for Laminar–Turbulent Transition in the Flow of Fluids.
Union Carbide report.
Hanks, R. W., and D. R. Pratt. 1967. On the flow of Bingham plastic slurries in pipes and between
parallel plates. Soc. Petr. Eng. Journal, 7, 342–346.
Hanks, R. W., and B. H. Dadia. 1971. Theoretical analysis of the turbulent flow of non-Newtonian
slurries in pipes. American Journal of Chemical Engineering, 17, 554–557.

HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES
5.45
Hanks, R. W., and B. L. Ricks. 1975. Transitional and turbulent pipeflow of pseudoplastic fluids.
Journal of Hydronautics, 9, 39–44.
Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Ind. Eng. Chem., 44, 651–656.
Herschel, W. H., and R. Bulkley. 1928. Measurements of consistency as applied to rubber benzene
solution. Proc. ASTM, 26, Part 2, 621–633.
Herzog, R. L., and K. Weissenberg. 1928. Kolloid Z, 46, 277.
Heywood, N. I. 1991. Pipeline design for non-settling slurries. In Slurry Handling, Brown, N. P., and
N. I. Heywood (Eds.). New York: Elsevier Applied Sciences.
Heywood, N. I., D. C. H. Cheng, and A. J. Carlton. 1992. Slurry systems. In Piping Design Handbook,
McKetta, J. J. (Ed.). New York: Marcel Dekker, pp. 585–622
Heywood, N. I., and J. F. Richardson. 1978. Rheological behavior of flocculated and dispersed
kaolin suspensions in pipe flow. Journal of Rheology, 22, 6, 559–613.
IDP (now called Flowserve). 1995. Cameron Hydraulic Data. NJ: IDP.
Ippolito, M., and C. Sabatino. 1984. Rheological behavior and friction resistance of colloidal aqueous
suspensions. In Proceedings of the IXth International Congress on Rheology, Mexico.
Mexico: Universidad Nacional Autonoma de Mexico.
Irvine. 1988. Experimental measurements of isobaric thermal expansion coefficients of Non-Newtonian
fluids. Heat Transfer, 1, 2, 155–163.
Johnson, M. 1982. Non-Newtonian fluid system design. Some problems and their solutions. In 8th
International Conference on the Hydraulic Transport of Solids in Pipe. Johanesburg, South
Africa. Cranfield, UK: BHR Group.
Kemblowski, Z., and J. Kolodziejski. 1973. Flow resistances of non-Newtonian fluids in transitional
and turbulent flow. Int. Chem. Eng., 13, 265–279.
Kenchington, J. M. 1972. In Proceedings of the 2nd International Conference on Hydraulic Transportation
of Solids.Cranfield, UK: BHR Group.
Metzner, A. B., and J. C. Reed. 1955. Flow of non-Newtonian laminar, transition and turbulent regions.
Am. Inst. Chem. Eng. Journal, 1, 4, 434.
Molerus, O. 1993. Principles of Flow in Disperse Systems. London: Chapman and Hall.
Mooney, M. J. 1931. Explicit formulas for slip and fluidity. Journal of Rheology, 2, 2, 210–222.
Nunez, G. A., M. Briceno, C. Mata, and H. Rivas. 1996. Flow characteristics of concentrated emulsions
of very viscous oil in water. The Journal of Rheology, 40, 3, 405–423.
Park, J. T., R. J. Munnheimer, T. A. Grimley, and T. B. Morrow. 1989. Pipe flow measurements of a
transparent Non-Newtonian slurry. Journal of Fluids Engineering, 111, 331–336.
Porkryvailo, N. A., and Y. G. Grozberg. 1995. Investigation of structure of turbulent wall flow of
clay suspensions in channel with electro diffusion method. In Proceedings of the 8th International
Conference on Transport and Sedimentation of Solid Particles, Prague, Czech Republic.
Rabinowitsch, B. 1929. Veber die viskositat und elastizitat von solen. Z. Phisik Chem. Ser. A., 145,
1–26.
Ryan, N. M., and M. M. Johnson. 1959. Transition from laminar to turbulent flows in pipes. Amer.
Inst. of Chem. Engr., 5, 433–435
Sauermann, H. B. 1982. The Influence of particle diameter on the pressure gradients of gold slimes
pumping In Proceedings of the 8th International Conference on the Hydraulic Transport of
Solids in Pipes. Jahannesburg, South Africa, August 1982, Paper E1, pp. 241–246. Cranfield,
UK: BRHA Group.
Schowalter, W. R. 1977. Mechanics of Non-Newtonian Fluids. New York: Pergamon Press.
Slatter, P. T., G. S. Thorvaldsen, and F. W. Petersen. 1996. Particle roughness turbulence. In Proceedings
of the 13th International Hydrotransport Symposium on Slurry Handling and Pipeline
Transport, Johannesburg, South Africa. Cranfield, UK: BRHA Group.
SRC. 2000. Slurry Pipeline Course, May 15–16, 2000, Saskatchewan Research Centre, Saskatoon,
Canada.
Szilas, A. P., E. Bobok, and L. Navratil. 1981. Determination of turbulent pressure loss Non-Newtonian
oil flow in rough tubes. Rheol Acta, 20, 487–496.
Thomas, A. D., and K. C. Wilson. 1987. New analysis of non-Newtonian turbulent flow, yield power
law fluids. Canadian Journal of Chemical Engineering, 65, 335–338.
Tomita, Y. 1959. On the fundamental formula of non-Newtonian flow. Bulletin of the Japan. Soc.
Mech.. Engr., 2, 7, 469–474.

5.46 CHAPTER FIVE
Torrance, B. McK. 1963. Friction factors for turbulent non-Newtonian fluid flow in circular pipes.
South African Mechanical Engineer, 13, 4, 89–91.
Want, F. M., P. M. Colombera, Q. D. Nguyen, and D. V. Boger. 1982. Pipeline design for the transport
of high-density bauxite residue slurries. In Proceedings of the 8th International Conference
on the Hydraulic Transport of Solids in Pipes, Johannesburg, South Africa. Cranfield,
UK: BRHA Group.
Wasp, E., J. Penny, and R. Ghandi. 1977. Solid-liquid flow slurry pipeline transportation. Clausthal,
Germany: Trans Tech Publications.
Wilson, K. C. 1985. A new analysis of turbulent flow of non-Newtonian fluids. Canadian Journal of
Chemical Engineering, 63, 539–546.
Further readings
Abulnaga, B. E. 1997. Channel 1.0 Computer Program For an Open Channel Slurry Flow. Fluor
Daniel Wright Engineers. Vancouver, BC, Canada. Internal report.
Al Fariss, T. and K. L. Pinder. 1987. Flow through porous media of a shear-thinning liquid with yield
stress. Canadian Journal of Chemical Engineering, 65, 391–405.
Bouzaiene, R., and D. Hassani-Ferri. 1992. A selection of pressure loss predictions based on slurry/backfill
characterization and flow conditions. C.I.M. Bulletin, 85, 959, 63–68.
Chlabra, R. P., J. F. Richardson, and R. Darby. 2000. Non-Newtonian flow in process industry, fundamentals
and engineering application. Chemical Engineering, 107, 4,
Draad, A. A., G. D. C. Kuiken, and F. T. M. Nieuwstadt. 1998. Laminar turbulent transition in pipe
flow for Newtonian and non-Newtonian fluids. Journal of Fluid Mechanics, 377, D25,
267–312.
Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Journal of Industrial Engineering
Chemistry, 44, 651–656.
Sandall, O. C., O. T. Hanna, and K. Amurath. 1986. Experiments on turbulent non-Newtonian mass
transfer in a circular tube. Am. Inst. Chem. Eng. Journal, 32, 2095–2098.
Steffe, J. F., and R. G. Morgan. 1986. Pipeline design and pump selection for non-Newtonian fluid
foods. Food Technology, 40, 78–85.
Wheeler, J. A., and E. H. Wissler. 1965. Friction factor: Reynolds number relation for the steady
flow of pseudoplastic fluids through rectangular ducts. Am. Inst. Chem. Eng. Journal, 11,
207–216.

PART TWO
EQUIPMENT AND
PIPELINES

CHAPTER 7
COMPONENTS OF
SLURRY PLANTS
7-0 INTRODUCTION
In Chapter 1, a typical circuit of a mineral process plant was presented. In Chapters 3
through 6, the theory of slurry flows was examined in detail for different rheology and
regimes. To achieve such complex flows, a number of important pieces of machinery,
such as mills, pumps, and valves, and drop boxes are needed. Together they form the slurry
preparation plant at the start of the pipeline and sometimes the slurry dewatering plant
when the concentrate or solids must be dried out for shipping, smelting, or burning as a
fuel. Their design is often complex and must account for wear and performance.
In simple layman’s terms, rocks that contain ores may be delivered in fairly large
pieces. These rocks may be obtained by blasting, special hydraulic jack hammers, excavators,
etc. (Figure 7-1). These large rocks need to be reduced to sufficiently small particles
to extract the ores—from as large as a few hundred millimeters (or dozens of
inches) down to a few millimeters or fractions of inches. This is done by a number of
steps, such as crushing, milling, grinding, screening, cycloning, vibrating, etc. Milled
rocks are then transported in slurry form and treated in different circuits such as flotation,
acid or cyanide leaching, and classification circuits. The concentrate may then be
thicked further for transportation to its final destination. The tailings are disposed of in
dedicated ponds.
The design of mineral processing plants has been the subject of numerous books, and
specialized books have been written for each piece of equipment. In this chapter, some of
the most important components of slurry systems will be introduced, with sufficient information
for the slurry engineer to appreciate the discharge from each type of equipment.
The next two chapters are devoted to pumps and valves and Chapter 10 is devoted to materials
for manufacturing. It would be beyond the scope of this book to dwell on the
chemistry of each process.
7-1 ROCK CRUSHING
Rock crushing is not part of the slurry circuit but is more of a preparatory step to the formation
of slurries. Crushing will therefore be reviewed briefly, as it is outside the scope
of this handbook.
7.3

7.4 CHAPTER SEVEN
FIGURE 7-1 Excavation is a primary source of materials for a mineral processing plant.
[Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]
Solid comminution is the process of reducing the size of particles. Two comminution
types are considered:
1. Dry comminution generally reduces rocks down to a diameter of 25 mm (1 in), by impact
and mechanical compression. This process involves jaw crushing, gyratory crushing,
cone crushing, and grinding using rod mills and ball mills.
2. Wet comminution generally reduces 25 mm (1 in) particles down to very fine sizes by
grinding and attrition in slurry form. This process involves semiautogenous mills, autogenous
mills, ball mills, hydrocyclones, columns, etc.
Comminution via a machine is measured by the reduction ratio, defined as 80% of the
particle size at the feed (F e80) to 80% of the particle size at the output (C r80).
The feed to a grinding mill must be crushed to a size appropriate to the grinding
process. Semiautogenous mills require little crushing; ball mills require a finer crushing.
A method of ore preparation that is now limited to narrow ore seams or veins in underground
mines is the so-called “run of the mine milling.” It consists of blasting the rocks
into lumps, usually of the order if 300 mm (12 inch) or larger. The most common approach,
however, is to crush the mined rock to an acceptable size.
7-1-1 Primary Crushers
Primary crushers absorb any size rocks (depending on the opening at the inlet) and reduce
their size down to 50–150 mm (2–6 in). Primary crushers are classified as:

� Jaw crushers
� Gyratory crushers
� Impact crushers
COMPONENTS OF SLURRY PLANTS
Some mines try to reduce the cost of crushing by blasting the rocks from mountains
and hills.
Crushing is essentially a process of reducing the size of a stone down to 25 mm (1 in)
(Figure 7-2). As this is difficult to achieve in a single stage, it is often encompassed in two
or three steps. The stones go through a cycle of primary crushing, secondary crushing,
and tertiary crushing. Special machines have been developed for each step of crushing
(Figure 7-3).
7-1-1-1 Jaw Crushers
These machines operate by compressing the rocks between a fixed plate and a moving
jaw (Figure 7-4). The rocks are fed from the top of the crusher. The fixed jaw or plate is
usually attached to the wall of a cavity. Through an eccentric mechanism or crankshaft, a
moving jaw presses the rocks against the walls of the crusher. Generally, the following
two types of machines are used:
1. In the overhead eccentric jaw crusher, also known as the single toggle crusher, the
moving plate is forced against the stationary plate by an eccentric mechanism driving
at its top, as well as by the rocking of a toggle connected to the bottom of the
moving plate.
FIGURE 7-2 Crushing is an essential step in handling hard rock, gravel, and mining ores as
well as for recycling. [Courtesy of Metso Minerals (formerly known as the companies Nordberg
and Svedala).]
7.5

7.6 CHAPTER SEVEN
fixed jaw
out
feed
(a) Jaw crusher
feed
bowl Head or mantle
(c) Impact crusher
Pivoting jaw
pitman
bowl
inclined bowl
FIGURE 7-3 Principles of crushing.
feed
(b) Gyratory crusher
feed
(b) Cone crusher
Head or mantle
FIGURE 7-4 Cross-sectional representation of a jaw crusher. [Courtesy of Metso Minerals
(formerly known as the companies Nordberg and Svedala).]
cone

COMPONENTS OF SLURRY PLANTS
2. The blake jaw crusher features a moving plate that pivots at the top but is oscillated at
the bottom.
The dimensions and shape of the plates affect the performance of the crusher. The
smaller the discharge gap, or required output size, the lower the tonnage from the crusher.
Jaw crushers work best on rocks that are not flat or slabs. With a feed opening of 1.67 ×
2.13 m (66 × 84 in) and a discharge gap of 200 mm (8 in), the crusher can handle a capacity
of 800 tph.
The walls and moving blade of the crusher are lined with a hard metal such as manganese
steel. The liners are removable for repairs once worn out. The liners may be flat,
plain, or ribbed.
The final output size of crushed particles depend on the setting of the plates (Figure 7-
5). Curves shown in Figure 7-5 indicate, for example, that for a closed setting of 100 mm
(4 in) the size particles will be at a maximum of 160 mm (6.375 in) with a significant portion
of particles smaller than 50 mm (2 in).
7-1-1-2 Gyratory Crushers
These machines operate on the principle of compressing the rocks in a cone (Figure 7-6)
The rocks fall into the cavity from the top. The moving part is an eccentric cone. The
FIGURE 7-5 The size of the output from jaw crushers depends on the plate setting. If the
closed side setting (c.s.s) is 100 mm (4�), the maximum product size is 160 mm (6 3 – 8�) and the
portion of fraction under 50 mm (2�) is approximately 35%. [Courtesy of Metso Minerals (formerly
known as the companies Nordberg and Svedala).]
7.7

7.8 CHAPTER SEVEN
Mainshaft sleeve
Spider bushing
Spider arm guard
Head nut
Spider
Concave fifth row
Concave fourth row
Concave third row
Concave second row
Concave first row
Inner deflector ring
Arm guard (inner)
Arm guard (outer)
Bottom shell
Tie rod nut
Gear housing shield
Positive air pressure
Eccentric
Seal ring
Eccentric support
Hydraulic cylinder
Cylinder sleeve
Cylinder shield
Piston cap
Cylinder head
Transmitter
Spider cap
Mainshaft
Retainer bar
Guide bushing
Seal retainer
Tie rod nut
Top shell
Upper mantle
Tie rod
Lower mantle
Floating ring bracket
Oil deflector ring
Dust seal bonnet
Floating ring
Floating ring retainer
Outer bushing
Pinion
Inner busing
Countershaft box
Countershaft
Balanced gear
Eccentric thrust washer
Eccentric thrust bearing
Swivel plate
Socket plate
Thrust plate
FIGURE 7-6 Cross-sectional drawing of a primary gyratory crusher. (Courtesy of Sandvik.)
rocks enter on the largest corner of the cavity but are compressed as the eccentric cone rotates.
The outside cone is sometimes called the bowl, and the rotating cone is called the
mantle. The bowl reduces in diameter toward the bottom, whereas the mantle increases in
diameter with depth in the opposite direction.
Gyratory crushers are preferred for slabs or flat-shaped rocks as they snap the rock
better. Gyratory crushers are manufactured to handle tonnage flows up to 3500 tph. Sandvik
purchased the line of Nordberg mobile primary gyratory crushers (Figure 7-7) that
can be moved from one site to another as the mine expands.
7-1-1-3 Impact Crushers
These machines operate on the principle of a set of rotating hammers hitting against the
rocks. The hammers are fixed to a cylinder. The feed is from the top and as the rocks feed
in, they fall between a breaker plate and the rotating cylinder. The hammers produce the
required impact to chip the rocks. Impact crushers work best on rocks that are neither
abrasive nor silica-rich, as these cause rapid wear of the hammers. Metso Minerals manufactures
impact crushers (Figure 7-8) for primary and secondary crushing. Figure 7-9
shows typical gradation curves.

7-2 SECONDARY AND
TERTIARY CRUSHERS
Crushing the rocks is often achieved in two or three stages. The secondary and tertiary
crushing machines resemble the machines used during primary crushing. They consist of
vertical cone crushers or horizontal cylinder crushers. The former type is the most widespread.
7-2-1 Cone Crushers
COMPONENTS OF SLURRY PLANTS
FIGURE 7-7 Large mobile gyratory crushers are designed with a special frame and wheels
to permit relocation from one area of the mine to another. (Courtesy of Sandvik.)
Cone crushers operate on the same principle as gyratory crushers. This allows a gradual
reduction of the area between the two cones. The rotating cone or mantle is inclined, thus
providing a combination of impact loads and compression loads. By comparison with the
gyratory crusher, the outer bowl is inverted, and the mantle rotates at much higher speeds.
There are two types of cone crushers:
1. The standard type (for secondary crushing)
2. The short head type (for tertiary crushing)
7.9

7.10 CHAPTER SEVEN
FIGURE 7-8 Cross-sectional cut through an impact crusher. (Courtesy of Sandvik.)
The two types of cone crushers have different bowl shapes. The standard has a wider feed
and is used for larger stones. The short head has a more shallow feed and tighter space
surrounding the mantle. The short head is therefore used for finer crushing.
Because of the continuous wear of the surfaces, adjustment of the cone crusher is essential.
By measuring power on a continuous basis, a feedback loop readjusts the mantle.
Screens on the output of the crusher facilitate the separation of coarse and fine stones. In a
closed circuit, the coarser stones are returned to the crusher. The fine stones could clog
the crusher and must be removed.
The diameter of cone crushers may be as low as 0.91 m (36 in) for a capacity of 50–80
tph, or as high 2.13 m (84 in) for a capacity of 500–1100 tph. The finer the output, the
smaller is the tonnage.

Figure 7-10 presents a cross-sectional drawing of the Metso Minerals cone crusher
and Figure 7-11 shows gradation curves of the output from HP cone crushers. Metso Minerals
manufactures complete portable cone/screen plants (Figure 7-12) that are relocated
from one area of the mine to another.
7-2-2 Roll Crushers
Roll crushers consist of two counterrotating cylinders. The gap between the cylinders is
adjusted by threaded bolts. Roll crushers can use springs to hold the cylinders in place.
Each cylinder is then driven by its own belt drive.
Roll crushers are used for less abrasive stones than cone crushers. They are most effective
on soft and friable stones, or when a close-sized product is required.
7-3 GRINDING CIRCUITS
COMPONENTS OF SLURRY PLANTS
FIGURE 7-9 Performance curves of an impact crusher. (Courtesy of Sandvik.)
The dry ore from crushers is stored in a stockpile (see Figure 1-10). The stockpile then
feeds the milling circuit (Figure 7-13). It is claimed that grinding accounts for 60% of the
power consumption of a mineral process plant. Elliott (1991) indicates that for a typical
copper or zinc concentrator, grinding consumes 12 kWh/t, crushing 2–3 kWh/t, and the
rest of the plant 2–3 kWh/t. Obviously, the finer the grinding, the higher the energy consumption.
There are two main forms of grinding:
1. Dry grinding when the water content is 34% water by volume
7.11

7.12 CHAPTER SEVEN
FIGURE 7-10 Cross-sectional cut through a cone crusher.(Courtesy of Sandvik.)
Between 1% and 34%, the slurry is very difficult to handle and grinding is inefficient. In
some plants, an initial grinding process may be followed by some form of classification
such as flotation or magnetic separation, which in turn is followed by a second grinding
process. This approach tends to eliminate at an early stage a good portion of the gangue
(see Chapter 1).
It is not possible to achieve the particle size needed through a single grinding phase
unless coarse output is required. When a coarse product is required, crushed materials are
transported to a rod mill via a conveyor belt and the output is delivered from the rod mill.
This is essentially an open circuit.
Closed circuits (Figures 7-14–7-16) may include SAG and ball mills, hydrocyclones,
and centrifuges. Grinding mills are designed with different approaches to feed and discharge
(Figure 7-17). The energy required to reduce the size of a particle is usually a
function of its diameter raised to an exponent. Holmes (1957) indicated that this exponent

FIGURE 7-11 Gradation curves of cone crushers. (Courtesy of Sandvik.)
FIGURE 7-12 Mobile cone and screen plants. (Courtesy of Sandvik.)
7.13

7.14 CHAPTER SEVEN
reclaim water
Crushers
SAG Mill
SAG Mill discharge
Cyclone
Feed
Pumps
conveyor
Stockpile
Monorail
Water
Sprays
cyclone overflow
Ball Mill
Mill Feed
Conveyor
coarse
Water
Sprays
Auto
Sampler
reclaim
water
Reclaim
water
Mill Feed Stockpile
Belt
Feeders
To Rougher Flotation
FIGURE 7-13 Flow chart of a grinding circuit. The stockpile of ore feeds the SAG mill, and
the ore is processed even further by ball mills.
is not a constant but a variable. His method of iteration is fairly complex and would require
a computer program.
For wet grinding, which is where the slurry circuit starts, the resistance to comminution
is measured by a grindability work index. It is established by test work. Bond (1952)
defined the grindability work index � from the power W (in kWh per ton) required to reduce
the feed size F (mm) to the final product size Cr (mm):
–1/2 –1/2 W = 10�(Cr80 – Fe80 ) (7-1)
Equation 7-1 is based on reduction of the rock size in a 2.44 m (96 in) ball mill. This
equation applies in the case of wet grinding, which is often the first step in a slurry circuit.
Typical examples of the grindability work index � are presented in Table 7-1.
The feed, its shape, and mechanical properties ultimately influence the performance of
the grinding circuit and the degree of efficiency of ore extraction. The performance of the
grinding process is dependent on a successful grinding operation.
In an autogenous mill, the feed itself is used as a grinding medium. The larger the particles,
the more energy they release on impact with each other. A coarse feed (larger than

7.15
Conveyor from stock pile
mill feed box
feed
rods
primary grinding mill
separation of
grinding medium
gear
mill discharge pump box
cyclone overflow (fines)
separation of
grinding balls
cyclone feed pump
or mill discharge pump
FIGURE 7-14 Two-stage closed circuit for grinding and classification of ore.
hydrocyclone
coarse cyclone underflow
recirculated to ball mill
ball mill
feed
mill feed box

FIGURE 7-15 View of a closed circuit grinding copper ore. In the back of the photo is the
large 12.2 m (40 ft) diameter SAG mill that receives the ore from the stockpile. In the front, the
ball mill grinds the underflow from the hydrocyclone.
FIGURE 7-16 View of the hydrocyclones set at a height of 30 m above the base of the SAG
mill. The overflow is diverted to centrifuges to separate the gold ore from the lighter copper
ore. The copper ore is then diverted to the ball mill (on the left-hand side of the photo) for secondary
grinding.
7.16

feed
feed
balls
COMPONENTS OF SLURRY PLANTS
out
FIGURE 7-17 Schematic representation of different types of grinding mills.
feed
7.17
slurry grate
(a) Overflow mills (wet grinding only)
(b) Diaphragm or grate mills
- Used for rod mills in open circuits and ball mills
- Not suitable for rod mills, and mostly used for
in closed circuit
closed circuit
Grinding with maximum specific area and suitable
- Used for Autogeneous and Semi-Autogeneous Grinding
for very fine output
for very fine output
Simple and robust - Coarser output than overflow mills
rods
(c) peripheral central port discharge (d) peripheral discharge at the end
Peripheral discharge mills are essentially reserved for rod mill grinding, wet or dry
Used for coarse grind where close control of final feed size is required , either coarse or fine
suitable for open or closed circuits
TABLE 7-1 Typical Examples of Grindability Work Indices (For Wet Grinding in a
Ball Mill)
Grindability
Material work index Reference
Barite 5 Elliott (1991)
Bauxite 9 Elliott (1991)
Clay 7 Elliott (1991)
Coal 11 Elliott (1991)
Dolomite 11 Elliott (1991)
Feldspar 12 Elliott (1991)
Fluorspar 9 Elliott (1991)
Granite 15 Elliott (1991)
Limestone 12 Elliott (1991)
Magnetite 10 Elliott (1991)
Quartz 13 Elliott (1991)
Quartzite 10 Elliott (1991)
Sandstone 7 Elliott (1991)
Shale 16 Elliott (1991)
Taconite 23 Elliott (1991)
feed
rods
out
feed

7.18 CHAPTER SEVEN
150 mm or 6 in) is important for a fully autogenous mill. Typically, the feed has an 80%
passing size of 200 mm (8 in).
In a semiautogenous (SAG) mill, steel or high chrome white iron balls are added to the
circuit as a grinding medium. As they rotate and are carried away by centrifugal forces,
they fall by gravity and impact against the feed or crushed rocks. Due to the difference in
density between the steel balls (typically 7610 kg/m3 or a specific gravity of 7.61) and
rocks (with a range of specific gravity of 1.3 to 4.0), smaller steel balls in a SAG mill
have the effect of large rocks in fully autogenous mills. The d80 of the feed, called F80 in
SAG mills, is typically 110 mm (4.5 in).
In a mineral process plant, the process of comminution is one of the least efficient and
highest consumers of power. A number of equations are used to define the process of dry
grinding. These are described by Elliott (1991).
Equation 7.1 is often called Bond equation. In practice it is modified by multiplying
the right hand side of the equation by so-called “inefficiency factors,” E1 to E9. Dry grinding correction factor E1. For dry grinding circuits, without the addition of
water, an inefficiency factor, E1 = 1.3, is applied.
Product size correction factor E2. Another efficiency factor in terms of the final
product size is defined as E2. If the final product is classified at 80% of the passage diameter,
then E2 = 1.2. If the final product is classified at 95%, then E2 = 1.57 (see Table 7-2).
Diameter correction factor E3. For a mill with the diameter Dm (in meters), a coefficient
E3 is defined as
E3 = (2.44/Dm) 0.2 (7-2a)
If the diameter of the mill is expressed in inches then
E3 = (96/Dmus) 0.2 (7-2b)
where Dmus is the diameter of the mill in inches.
Oversize correction factor E4. The optimum rock size fed into a rod mill is given as
Feop = 16,000 (13/�) 1/2 expressed in �m (7-3)
and for a ball mill:
Feop = 4000 (13/�) 1/2 expressed in �m (7-4)
TABLE 7-2 Inefficiency Factor E 2 for Grinding
Circuits
Product size
control reference % passing E 2
50% 1.035
60% 1.05
70% 1.10
80% 1.20
90% 1.40
92% 1.46
95% 1.47
98% 1.70
Source: “The Science of Communition,” Brochure No.
0647-05-98-N-English, Nordberg, Helsinki, Finland, 1998.

COMPONENTS OF SLURRY PLANTS
If the size of the feed is larger than the optimum size Feop, (i.e., if Fe80 � Feop), then E4 =
1 if Fe80 > Feop (the case of oversized feed); then
Fe80 – Feopt Cr80 E4 = 1 + (� – 7)���� (7-5)
Feopt*
When Equation 7-5 yields a result smaller than 1.0, the result should be corrected to E4 =1.0. This equation should not be used in the case of a rod mill used to feed a ball mill, in
which case, E4 = 1.0.
Fineness correction factor E5. If the crushed output diameter Cr80 is less than 75 �m,
then it is necessary to calculate a fineness correction factor E5, defined as
Cr80 + 10.3
E5 = ��
(7-6)
1.145Cr80
Otherwise E5 = 1.
Correction factor for high/low ratio of reduction rod milling E6. For a rod mill,
defining the length of the mill as Lm and the diameter as Dm, a ratio Rr0 is defined as
Rr0 = 8 + (5Lm/Dm) (7-7)
The material reduction ratio is defined as
Rr = Fe80/Cr80 (7-8)
If Rr > (Rr0 ± 2), then
(Rr – Rr0) E6 = 1 + � � (7-9)
Otherwise a correction factor E6 = 1 is assumed.
Correction factor for the low reduction ratio for ball mills. If Rr < 6, or when the
ratio of the ball mill feed to the product output sizes is smaller than 6.0, a correction factor
E7 is defined as
2(Rr – 1.35) + 0.26
E7 = ��
(7-10)
2(Rr – 1.35)
2
��
150
If the computation of Equation 7-10 exceeds the magnitude of 2.0, it is highly recommended
to conduct lab tests and to contact the manufacturer of the mills.
Correction factor for rod mills E8. The rod milling feed factor is where the material
is fed into a rod mill from an open circuit crusher. Elliott (1991) suggested 1.4 as the magnitude
of E8. However, if the source is a closed circuit with rod milling followed by ball
milling, then E8 is 1.2.
Correction factor for rubber-lined mills E9. When grinding balls are smaller than
80 mm or 3.25 in, rubber liners are used to line the inside walls of the mill. When grinding
balls are larger than 80 mm or 3.25 in, metal liners are used.
Rubber liners (Figure 7-18) are thicker than metal liners, use more space, and absorb
more impact energy than their metal counterparts. It is customary to apply a correction
factor E9 = 1.07 for rubber liners.
The final power required to mill the feed is then obtained after multiplying all the correction
factors by Bond’s equation (7-1).
Iteration to consumed energy:
Wf = W(E1 × E2 × E3 × E4 × E5 × E6 × E7 × E8 × E9) (7-11)
� Fe80
7.19

7.20 CHAPTER SEVEN
FIGURE 7-18 Rubber lining of SAG mills supplied to the Murin–Murin project in Australia
to treat nickel-rich laterites. [Courtesy of Metso Minerals (formerly known as the companies
Nordberg and Svedala).]
Equation 7-11 is useful to determine the power to grind down rocks. It must be corrected
for worn-out liners, ball charges, and slurry density. It is therefore recommended that in
the initial phase of the design of a mineral process plant, lab tests be conducted.
Some of the empirical coefficients and equations for E 1 to E 9 were developed assuming
a recirculation load of 250%. This means that the charge load of coarse material that
is returned to the mill is about 250% of the fresh feed in a closed circuit. This is not always
the case. The author was once involved in the design of a copper concentrate plant
for a Peruvian mine in which the presence of soft high clay in the ore increased viscosity
tremendously at a weight concentration of 50% to 60%. It became necessary to add water,
dilute the slurry, and cut down the recirculation load.
When the rocks in the feed are large, and milling is dominated by impact loads, Equation
7.1 should not be used to compute the work index load.
Some of the empirical coefficients and equations for E 1 to E 9 were developed for a final
output size with 80% passing 100 �m. (mesh 140). When C r80 < 100 �m, Equation
7.11 does not give correct results.
Example 7-1
An ore with a grindability index � = 13 is to be ground in a rod mill with feed from a
closed-circuit crusher. The feed has a diameter F e80 of 26 mm (1 in). The final product is
required at 80% to be C r80 of 10 mm (0.4 in) at a mass throughput of 350 tons/hour
(770,000 lbs/hour). Estimate the power consumed by the rod mill.

Solution
Using Equation 7-1, the work input to the rod mill is
W = 10 × 13(10 –1/2 – 26 –1/2 ) = 130(0.3162 – 0.1961) = 15.61 kWh/ton
For wet grinding, E1 = 1. For closed-circuit grinding E2 = 1; E3 will be calculated after
other factors.
The oversize feed factor E4 is obtained from Equation 7.3.
Feop = 16,000(13/13) 1/2 = 16,000 �m or 16 mm
Since Feop < Fe80, then
E4 = {[(26/10) + (13 – 7)(26 – 16)]/16}/(26/10) = 0.3846(2.6 + 3.75) = 2.442
Since Cr80 > 75 �m, then E5 = 1.
From Equation 7-8, the reduction ratio of the material Rr = 26/10 = 2.6. Rr0 will be calculated
after selecting the rod mill. Since Rr < 6 then
E7 = [2(2.6 – 1.35) + 0.26]/2(2.6 – 1.35) = 1.104
E8 = 1.2 since it is a closed circuit crusher.
Iteration to consumed energy
Wf = W(E1 × E2 × E3 × E4 × E5 × E6 × E7 × E8) Wf = 15.61 × 1 × 1 × E3 × 2.442 × 1 × E6 × 1.104 × 1.2 = 50.5 × E3 × E6 kWh/ton
Since the feed is 350 tons per hour, the total energy consumption would be
350 ton/h × 50.5 kWh/ton E3 × E6 = 17,675 kW × E3 × E6 This would require a number of mills in parallel. From Equation 7-2, if the mill diameter
of 6 m (19.7 ft) is selected, then
E3 = (2.44/6) 0.2 = 0.833
Rod mills with a length to diameter ratio of 2 are selected:
Rr0 = 18
and since Rr < (Rr0 ± 2),
E6 = 1
Final power consumption is 42.067 kWh/ton or total of 14,723 kW (19,736 hp).
With modern technology, a SAG mill should be considered as an alternative to the rod
mill (see Tables 7-3 and 7-4).
7-3-1 Single-Stage Circuits
COMPONENTS OF SLURRY PLANTS
7.21
When finer material is required, a ball mill is used in a closed circuit. The feed is ground
and then classified to separate coarse from fine solids. The coarse solids, also called oversized
particles, are returned back to the mill for further grinding. This is called the “recirculation
load” and the circuit is considered a closed circuit. In a dry circuit, the classifier
may be a set of vibrating screens.
In a typical copper or zinc circuit, the recirculation load can be as high as 250–350%
of the new feed. The mill and mill discharge pumps must then be sized for the combination
of recirculation load and new feed.

7.22 CHAPTER SEVEN
TABLE 7-3 Estimates of Bond Energy Consumption per Mass for Grinding
Rocks (W i)
Mineral Specific gravity (kWh/sh.ton) (kWh/tonne)
Andesite 2.84 18.25 20.08
Barite 4.50 4.73 5.20
Basalt 2.91 17.10 18.81
Bauxite 2.20 8.78 9.66
Cement clinker 3.15 13.45 14.80
Clay 2.51 6.30 6.93
Coal 1.4 13 14.3
Coke 1.31 15.13 16.84
Copper ore 3.02 12.72 13.99
Diorite 2.82 20.90 22.99
Dolomite 2.74 11.27 12.40
Emery 3.48 56.70 62.45
Feldspar 2.59 10.80 11.88
Ferro-chrome 6.66 7.64 8.40
Ferro-manganese 6.32 8.30 9.13
Ferro-silicon 4.41 10.01 11
Flint 2.65 26.16 28.78
Fluospar 3.01 8.91 9.8
Gabbro 2.83 18.45 20.3
Glass 2.58 12.31 13.54
Gneiss 2.71 20.13 22.14
Gold ore 2.81 14.93 16.42
Granite 2.66 15.13 16.64
Graphite 1.75 43.56 47.92
Gravel 2.66 16.06 17.67
Gypsum rock 2.69 6.73 7.40
Iron ore, hematite 3.53 12.84 14.12
Iron ore, hematite—specular 3.28 13.84 15.22
Iron ore, magnetite 3.88 9.97 10.97
Iron ore, oolitic 3.52 11.33 12.46
Iron ore, taconite 3.54 14.61 16.07
Lead ore 3.35 11.90 13.09
Lead–zinc ore 3.36 10.93 12.02
Limestone 2.66 12.74 14
Manganese ore 3.53 12.20 13.42
Magnesite 3.06 11.13 12.24
Molybdenum 2.70 12.80 14.08
Nickel ore 3.28 13.65 15.02
Oil shale 1.84 15.84 17.43
Phosphate rock 2.74 9.92 10.91
Potash ore 2.40 8.05 8.86
Pyrite ore 4.06 8.93 9.83
Pyrhotite ore 4.04 9.57 10.53
Quartzite 2.68 9.58 10.54
Quartz 2.65 13.57 14.93
Rutile ore 2.80 12.68 13.95
W i
W i

TABLE 7-3 Continued
7-3-2 Double-Stage Circuits
A rod mill in an open circuit may be followed by a ball mill in a closed circuit. This is
called a double-stage circuit and is often a wet process. The output from the rod mills is a
slurry that contains a high proportion of coarse stones. The slurry is pumped via “mill discharge
pumps” to a hydrocyclone. The underflow from the cyclone is then fed to a ball
mill. From there, the output from the ball mill is fed once again to the hydrocyclone via
the pump.
In some circuits, the rod mill discharge is fed first to the ball mill before reaching the
hydrocyclone. The hydrocyclones then feed the ball mills by gravity. A set of ball mill
discharge pumps may then pump the output to a second classification circuit. The ball
mill discharge has its own sets of slurry pumps.
7-4 HORIZONTAL TUMBLING MILLS
In a horizontal tumbling mill, the actual body of the mill rotates and imparts energy to the
grinding medium (balls or rods) and to the slurry. The combination of centrifugal forces
and gravity forces from falling media act to create energy transmission by impact against
the mineral. There are three categories of horizontal tumbling mills:
1. rod mills
2. ball mills
3. autogenous and semi-autogenous mills
COMPONENTS OF SLURRY PLANTS
Mineral Specific gravity (kWh/sh.ton) (kWh/tonne)
Shale 2.63 15.87 17.46
Silica sand 2.67 14.10 15.51
Silicon carbide 2.75 25.87 28.46
Slag 2.74 10.24 11.26
Slate 2.57 14.30 15.73
Sodium silicate 2.10 13.40 14.74
Spodumene ore 2.79 10.37 11.41
Syenite 2.73 13.13 14.44
Tin ore 3.95 10.90 11.99
Titanium ore 4.01 12.33 13.56
Trap rock 2.87 19.32 21.25
Zinc ore 3.64 11.56 12.72
From Denver Sala Basic. Selection Guide for Process Equipment. Reproduced by permission of
Metso Minerals (formerly known as the companies Nordberg and Svedala).
7.23
Basically a horizontal tumbling mill is a cylinder lined on the inside with wear-resistant
alloy liners. The liners are fixed to the shell by T-bolts and nuts on the outside. The
cylinder is carried by hollow trunnions running side bearings at each end.
W i
W i

7.24
TABLE 7-4 Selection Guide for Grinding Mills
Rod
__________________
Autogenous
Ball Vertical
_________________ Peripheral ______________________ _____________
Mineral Primary Secondary Overflow Discharge Overflow Grate Pebble Spindle Tower Vibrating Hammer
Ores (ferrous and nonferrous) � � � � � � � � � �
Preponderance of fine aggregates � � � �
Talc and ceramic materials �
Cement raw materials � � �
Cement clinker �
Coal and petrol, coke � � � �
Silica ceramics, etc. (must be free of iron) � �
Production to a specific particle diameter or mesh � � � � � � � � � � �
Production to a specific surface area � �
Wet grinding � � � � � � � � � �
Dry grinding � � � � � � � �
Damp feed (1%–15% moisture) �
Large feed (

COMPONENTS OF SLURRY PLANTS
7.25
A special chamber on the tip is installed to feed the material and the grinding medium.
The liners on the inside are designed to be ribbed either along the length of the cylinder or
in spiral shaped ribs. Ni-hard is the most commonly used alloy for liners (Figure 7-19),
but manganese steel and chrome steel are also used. In some designs, rubber is used as the
liner material (Figure 7-18)
It is important to cut down maintenance costs and periods of outage to replace
liners. Many mines open their mills once a month for maintenance and to replace the
worn liners. The grinding medium is steel rods in the case of rod mills and steel balls
in the case of ball mills. As the cylinder or tumbler rotates, the heavy rods and balls
are lifted by the ribs of the liners. The rods and balls fall by gravity after a certain angle
of rotation is reached. The impact in turn fractures and grinds the rocks into smaller
stones.
The spacing between the ribs of the liner is critical. Too narrowly spaced ribs may
jam the coarser rocks and delay their fracture. Speed of rotation is extremely important.
At a certain speed, the material, which is lifted by friction against the liner, starts to fall
down. The cascading effects of stone against stone causes grinding by attrition. The material
output is fine but wear is high. As the speed of the mill is increased, grinding
takes place by impact of the rods or balls against the rocks. As the speed increases even
further, centrifugal forces become sufficient for the material to centrifuge. This speed is
called “the critical speed of the mill.” Mills are designed to operate at 75% of their critical
speed.
The diameter of the rods is often 50 mm (2 in) but can be set by the designer of the
mill. It is, however, important to separate the rods or balls from the slurry at the discharge
of the mill before they enter the slurry pump. The successful separation of steel balls from
the slurry involves proper design of trommels, a mechanism to catch the balls, and
screens on top of the pump box. Ideally, the balls should be recycled back to the feed of
the milling unit.
FIGURE 7-19 Worn-out metal liners removed during monthly maintenance of a SAG mill.

feed
7.26 CHAPTER SEVEN
7-4-1 Rod Mills
Rod mills are a type of fine crusher and can reduce the size of rocks down to 1 mm (0.04
in). They perform better than a fine crusher in less than optimum conditions when the
feed is damp or contains clay.
Typically, the length to diameter ratio of the rod mills is 1.5 to 2.5. Milling occurs by
impact of rod against rod. The stones are trapped between the rods and disintegrate. The
coarser stones are the first to break. The finer escape milling. Rod mills are not used on
closed circuits.
In the last 20 years, the mining industry has tended to replace rod mills with large autogenous
and semiautogenous mills
7-4-2 Ball Mills
In ball mills, metal balls are used as the grinding media. The balls are made of a variety of
materials. Steel balls are forged. High chrome balls are cast with 28% chrome and are
available from special foundries. About 1 kg of balls is used per ton of stone. Small balls
with a diameter of about 25 mm (1 in) are preferred to larger ones in order to maximize
the area of contact between balls and stones.
The slurry weight concentration in a ball mill is 65–80%. Excessive concentration will
cause the particles to stick to the balls and will decrease the effectiveness of grinding. The
ball mill may then “freeze” and spill out its contents, causing costly downtime to empty
the mill. For this reason, the weight concentration should not be allowed to exceed 80%.
A trunnion at the discharge of the ball mill separates balls from slurry. The balls are
then conveyed back to the feed. Balls gradually wear out through repeated feeding to the
mill and must be replaced.
Ball mills are built in different diameters up to a maximum of 6.5 m (21 ft), and in
power drives up to 9650 kW (13,000 hp). Their shape is determined by the type of output
(Figure 7-20).
7-4-3 Autogenous and Semiautogenous Mills
Autogenous and semiautogenous (AG and SAG) mills are extremely large mills with a
maximum diameter of 12.2 m (40 ft). In the last few years, Siemens and ABB have devel-
balls
slurry
discharge
Cascade mills (wet and dry grinding)
- Used for autogneous and semiautogeneous milling
in closed circuit
- Primary Grinding with minimum retention time
for very fine output
- Diameter to length ratio 2:1
balls
feed
slurry
(b) Conical shape mill
- Suitable for fine discharge
discharge
FIGURE 7-20 The shape of ball grinding mills is determined by the type of discharge and ore.

oped and installed “wraparound” motors. In this design, the outside diameter of the tumble
on one side is part of the rotor of the motor (Figure 7-21). These motors are manufactured
up to a power size of 7000 kW (9400 hp).
The large diameter of these mills maximizes the impact forces. Although the feed is
typically 150–180 mm (6–7 in) in diameter, the output can be as fine as 0.3 mm (0.012
in). Particles tend to cleave along their natural grain boundaries.
Six to ten percent of steel balls are added on a continuous basis to the feed to assist
grinding through a separate entry. Wet milling and grinding is less dusty and less noisy
than dry grinding. The feed and output trunnions are on opposite sides. The trommel on
one side catches the steel or high chrome balls to prevent them from falling into the pump
box.
7-5 AGITATED GRINDING
COMPONENTS OF SLURRY PLANTS
7.27
FIGURE 7-21 SAG mill with wrap-around or ring motor. [Courtesy of Metso Minerals (formerly
known as the companies Nordberg and Svedala).]
Agitation is another method of grinding. The whole body of the mill may sit on springs
and be agitated by crankshafts or an eccentric mechanism driven by a motor. Another ap-

7.28 CHAPTER SEVEN
proach is to have a rotor, an agitator, and a rotating hammer inside the mill to impart energy.
7-5-1 Vertical Tower Mills
A vertical mill was developed in Japan for grinding fine minerals. The tower mill is a
combination of vertical tanks, a screw mixer, a mill, and a classifier. From a chute on the
side of the mill, the rocks, steel balls, and liquid are introduced. The vertical screw rotates
around a vertical shaft and creates an upward vertical counterflow. The finer materials
float to the top and are led to a side chamber for classification, while the heavier and
coarser solids sink with the steel balls (Figure 7-22).
The diameter of the balls is 6–32 mm (0.25–1.25 in). Size reduction of the ore is limited
to 5 mm (0.197 in) due to limited grinding. Vertical tower mills are manufactured for a
maximum output of 100 tph. They require limited floor space and have a low consumption
of power.
7-5-2 Vertical Spindle Mills
The vertical spindle mill (also known as SAM) uses a central vertical multistage mixer
(Figure 7-23). Each stage consists of a number of wolfram carbide pins fixed on a hollow
shaft. They provide horizontal stirring. This machine operates with feed smaller than 1
mm (16 mesh) for fine and ultrafine wet or dry grinding. The units are small and compact
and can be relocated within the plant. Maximum power is 75 kW per unit.
7-5-3 Roller Mills
Roller mills are used for soft grinding of industrial minerals in a dry state. The mill consists
of a rotating table on a vertical axis. Two rollers rotate around their own shafts at an
angle with respect to each other. The rollers are spring loaded. The output is diverted to
dry cyclones and the oversized material is fed back to the roller mill.
A new generation of high-pressure roller mills has appeared on the market since the
1980s. A very high level of torque is transmitted to the rollers to maximizing the crushing
loads. High-pressure rollers are mainly used in cement plants, diamond processing (when
the extraction is from rocks, as it is in Canada), and to a certain extent in the field of metalliferrous
minerals.
7-5-4 Vibrating Ball Mills
The body of the mill consists of a central feed chamber and two side chambers. The feed is
from the top and the discharge from the central chamber is at the opposite end. The whole
body of the mill sits on four strong springs. Two electric motors synchronized by V-belts
rotate an eccentric mechanism linked to each of the side chambers (Figure 7-24). This machines
uses fine feed smaller than 5 mm (mesh 4) and is particularly suited for difficult material
with an energy index W i > 30 kWh/sh.ton. These are essentially small machines with
maximum motor size rated at 55 kW or 75 hp. However, they are often chosen over tumbling
mills for lower installation cost, lower operating cost, less floor space, increased
grinding flexibility, and improved product control within the limitation of their size. Rods
or balls may be used as grinding media within open or closed circuits with these machines.

7.29
�������� ���� ��� ����
���������� ��� ���� �� ��������
����� ��� ������� �������
����� ������ ��� ��������
������ ����
����� ����
���������� ���
������� ��� ����� ��������
������������� �� ������ ��������
������ ����
FIGURE 7-22 Slurry circuit of vertical grinding tower mill for solids with a maximum diameter of 6.4 mm (1/4�).

7.30
vertical bars for wall
protection and help to grinding
helix for upwards pumping
while mixing and grinding
solids feed classifier box
FIGURE 7-23 Vertical spindle mill slurry circuit.
launder for fines (output)
recirculation of coarse material
M
A
A
Z
D
P U -
A
K
2 5
slurry pump

two motors in parallel at
each end
7-5-5 Hammer Mills
In the hammer mill, a central rotor arm is fitted with rings of arms that crush and mill the
feed against the wall of the mill. It is essentially used for dry milling of low-abrasive and
friable minerals such as cement, coal, gypsum, and limestone. It is considered by some
engineers a crusher rather than a mill.
7-6 SCREENING DEVICES
COMPONENTS OF SLURRY PLANTS
feed
one side chamber at each end
discharge
FIGURE 7-24 Vibrating mill.
7.31
In Chapters 1 and 3, the concept of d 50 was introduced as the particle size diameter below
and at which 50% of the particles can pass through the opening of a sieve. The
same concept applies to the definition of screen size. The screen size aperture is equal
to d 50.
An ideal screen would let all particles equal to or smaller than d 50 pass through. This is
not always the case, as the performance of the screen depends on a variety of factors:
� Screen deck size. In order for all particles to use the screen effectively, the layer of
solids above the screen needs to be very thin. This means a large deck size for a given
mass of solids. For economical reasons, this is not possible and a thick layer of solids
forms on the smaller screens.
� Vibration. To move away the coarse particles that block the passage of the finer ones,
it is essential to oscillate the screen. The amplitude of the oscillation must match the
specifics of the solids. Too much vibration could cause the solids to float as a cloud
without passing through the screens.
� Presentation angle. Ideally, the solids should be fed as normal as possible to the screen.
This means that the solids should come in at a 90° angle. Unfortunately, this is not always
possible.
� Screen material. Screens are manufactured of metal, rubber, and even fiberglass. Metal
screens have a wider aperture than rubber, which is more flexible and less prone to particle
binding.
� Moisture content. Sprays are sometimes added to screens to improve their efficiency
and flush the solids. Sprays suppress clouds of fine particles.

7.32 CHAPTER SEVEN
7-6-1 Trommel Screens
Trommel screens are essentially rotating cylindrical mesh. These trommels rotate at a
slight angle of inclination to facilitate the removal of material. Trommel screens can be
supplied as a set of concentric screens of different aperture. The finest screens are the
quickest to show wear.
7-6-2 Shaking Screens
Shaking screens move in a horizontal reciprocal motion along the length of the screen.
Solids are fed in a horizontal circular movement. The discharge moves in the direction of
the horizontal movement of the screen. The motion of the solids changes from circular at
the feed to eccentric, and finally to horizontal shaking.
7-6-3 Vibrating Screens
These screens are set at an angle with respect to the horizontal. The vibration occurs at a
right angle to the screen by the rotation of unbalanced counterweights on a shaft above
the screen. Vibration can also be induced by electromagnets and oscillating currents. Vibration
levels are high and the screens must be mounted on vibration isolation rubber
pads. These screens are extremely noisy and exceed 100 dBA levels of noise, nevertheless,
they are the most widely used.
7-6-4 Banana Screens
Banana screens are essentially stationary screens. The sieve is bent around a curved
screen. The top of the screen is vertical and solids are fed from the top. Particles pass successive
wedge bars and solids are removed between them based on the trigonometric
opening normal to the fall. To avoid clogging, the bars are pneumatically tilted at regular
intervals. These screens can be designed to sieve particles as fine as 50 �m.
7-7 SLURRY CLASSIFIERS
Classification is the process that separates coarse from fine. Various methods use the effect
of size, density, and magnetic and electrostatic properties of the solids. When the
weight concentration is smaller than 15%, particles settle in a “free settling” mode. When
the weight concentration increases, turbulence promotes settling of the heavier particles
faster than the lighter particles. Two families of density classifiers are available:
1. Classifiers that use the principle of free settling to achieve size separation
2. Classifiers that use the particles’ hindered settling speed for density separation and for
concentration of a particular mineral
7-7-1 Hydraulic Classifiers
In a hydraulic classifier, solids are fed at the top through a chamber that leads into a column.
Water is pumped from the bottom of the column. The counterflow moves into a

number of successive columns or stages. Products settle in accordance with the principles
described in Chapter 3. Solids are removed at the bottom through a restriction such as a
spigot. The spigot valve opens and closes in accordance with the applied pressure from
the accumulation of solids. This is the principle of operation of the hydrosizer, which is
often connected to other gravity and magnetic separators.
7-7-2 Mechanical Classifiers
A mechanical classifier is a combination of an open channel flow, a weir, and a mechanical
device to remove solids. The trough to which the slurry is directed is inclined
with respect to the horizontal. On one side, a weir is constructed opposite to the direction
of the flow. The heavier solids deposit upstream of the weir, whereas the finer
solids pass over it. This system operates on the principle of the sliding bed described in
Chapters 4 and 6.
The weir is designed to minimize turbulence. A mechanical rake or rotating
Archimedean screw removes the coarse solids. Water drains away as the solids are removed.
The speed of the rake or rotating shaft is critical to the efficiency of separation.
The height of the weir must be adjustable to change the depth of the pool, the rising velocity,
and the cut point between coarse and fine.
The addition of water controls the density of the slurry. Dilution may be required for
effective separation, but excessive water dilution may have to be followed by thickening
after classification.
Mechanical classifiers are expensive to install but in some applications they are selected
for high-density valuable minerals as they assist in their immediate recovery without
requiring further complicated flotation circuits. In some respects, flotation circuits use
some of the principles of mechanical classifiers by using a circular internal weir, a mixer,
an underflow pump, and a separate froth pump.
7-7-3 Hydrocyclones
COMPONENTS OF SLURRY PLANTS
Hydrocyclones (Figure 7-25) are the most common classifiers in the mining industry.
They require little space and operate on the pressure from the mill discharge pumps (typically
104–152 kPa, 15–22 psi). They are typically used to classify solids from a size of 40
to 400 �m (mesh 325 to 35).
The principle of the cyclone relies on creation of a vortex; sometimes primary and secondary
vortexes are created by feeding the material tangentially. In a vortex, a certain
pressure field is created to counterbalance centrifugal forces. In the cyclone, the applied
pressure is converted into a swirling motion. The intensity of the swirl is measured as the
swirl number:
S =
angular momentum
���
axial momentum
7.33
If the swirl flow number exceeds 0.5, the swirl is classified as a strong swirl. Strong swirl
is associated with a low-pressure zone at the core. A strong swirl is associated with an important
pressure drop. The cyclone feed gauge pressure at the inlet flange is often of the
order of 70–100 kPa (10 to 15 psi).
The swirling chamber of the cyclone is where the separation starts. The inlet area to
the cyclone is often of the order of 5–7% of the chamber area, or the inlet diameter is be

7.34 CHAPTER SEVEN
FIGURE 7-25 A number of hydrocyclones may be used in parallel to classify slurry flow in
a grinding circuit.
tween 20% and 25% of the swirling chamber diameter. The inlet nozzle is rectangular in
shape in some sheet metal fabricated cyclones, or circular in fiberglass cyclones.
Inside the swirling chamber, a pipe section protrudes from the top of the cyclone. It is
called the vortex finder. It must extend below the feed entrance to avoid shortcuts of unclassified
slurry to the top discharge or overflow. The diameter of the vortex finder is typically
32% to 36% of the swirling chamber diameter. The finer and lighter particles flow
out of the hydrocyclone through the vortex finder (Figure 7-26).
In some of the earlier metal fabricated designs, the swirling chamber consisted of a
single cylinder. In fiberglass designs, it is split into two halves, which are individually
lined with removable rubber liners. The cyclone chamber is followed by a cylindrical
chamber with a depth approximately equal to its diameter. This chamber provides some
retention time.
The cylindrical transition chamber is followed by a conical chamber, often designed
with an included angle between 10 and 20 degrees. It provides further retention time.
At the bottom of the cyclone, the apex is installed. It acts as a sort of nozzle or orifice.
For different applications, different orifice diameters may be used, and for different apex
diameters, different pressures are required. The apex is therefore a sort of controlling element
to the cyclone. The minimum apex orifice diameter is on the order of 10% of the
swirling chamber diameter, and the largest orifice diameter is on the order of 35%. In either
case, the apex must allow the flow of the coarse materials. At the bottom of the apex,
the discharge is called the cyclone underflow. At the top of the vortex finder, the discharge,
which consists of fines, is called the cyclone overflow.
For primary grinding circuits, the underflow typically contains 50 to 53% by volume

Involute design
inlet pipe
COMPONENTS OF SLURRY PLANTS
vortex finder
fiberglass body
discharge pipe
for coarse solids
(cyclone underflow)
discharge of finer
particles (cyclone overflow)
cylindrical feed
chamber (top half)
cylindrical chamber
(bottom half)
7.35
Conical section
angle from 10 to 20 degrees
Rubber liner
Apex
FIGURE 7-26 A cross-sectional representation of a rubber-lined cyclone. (Courtesy of Mazdak
International Inc.)
of solids, whereas for regrinding circuit, the underflow typically delivers 40 to 45% solids
by volume.
Hydrocyclones can be manufactured from dough-molded compound fiberglass, cast
iron, or sheet metal lined with polyurethane. Metal and fiberglass cyclones are lined with
rubber or with hard metal (Ni-hard or 28% chrome white iron). Burgess and Abulnaga
(1991) presented a finite element analysis of fiberglass cyclones.
The performance of the hydrocyclone is calculated by using a partition curve similar
to a screen curve. This gives the d 50 size, or 50% probability at the cut point. This cut
point is defined as the condition for which 50% of the feed will be discharged as coarse
particles in the cyclone underflow and 50% as fines or cyclone overflow. For every cyclone
design, there is a base d 50C or cut-off for the recovery (Figure 7-27).
Cyclones are usually operated in a steady mode with constant pressure. Surges can
lead to unfavorable air entrainment. To maintain constant pressure from the pumps, the
pump box must have a constant level of slurry. To adjust the slurry level, the sump must
be provided with a water addition mechanism.
Normal feed to cyclones consists of a slurry at 30% solids concentration by weight.
Some mines operate with slurry weight concentrations as high as 35%. Higher concentration
by weight imposes higher pressures of operation, which can cause a reduction in
efficiency of operation of the hydrocyclone while coarsening the cut point. Vortex finders
are changed in accordance with the required cut. A larger diameter vortex finder
tends to coarsen the overflow while increasing its discharge flow rate at a constant pressure.

7.36 CHAPTER SEVEN
100
50
Recovery to underflow, %
0
ACTUAL RECOVERY
d
Diameter of particles in micrometers
The size of the discharge spigot is selected in order to maximize the flow of solids
at high density and to reduce the flow of water in the underflow. Too small a spigot
tends to dewater the underflow and to break the air core while reducing the overall efficiency.
Because of the wide range of slurries with different particle sizes, the cutoff d50C is
used to normalize the particle size (Arterburn, 1982). The actual particle diameter from a
recovery is divided by the d50C size and a parameter X is defined as:
X = particle diameter/d50C particle diameter
The recovery to the underflow (Arterburn, 19xx) on a corrected basis is defined as:
Rr = (7-15)
Using the base d50C, Arterburn (1982) proposed to use three correction factors for an application,
C1, C2, C3, or
d50C(application) = d50C(base) × C1 × C2 × C3 (7-16)
The base d50C is defined as a polynomial function of the cyclone swirling chamber diameter.
Arterburn proposed
d50C = 2.84(D/100) 0.66 e
(7-17)
where D is the cyclone chamber diameter in meters.
The correction factor C1 is based on the volumetric concentration of solids fed to the
cyclone:
4X – 1
��
4X 4 e + e – 2
50c
CORRECTED RECOVERY
FIGURE 7-27 Typical particle recovery curve for the underflow from a hydrocyclone.

COMPONENTS OF SLURRY PLANTS
7.37
C1 = � � –1.43
(7-18)
Equation 7-18 applies in a range CV < 0.4.
The pressure drop between the feed nozzle and the cyclone overflow �P is used to
compute the second correction factor C2: C2 = 3.27(�P) –0.28 53 – 100CV ��
53
(7-19)
where �P is expressed in kPa. To minimize energy losses, �P should be in the range of
40 to 70 kPa, Arterburn (1982), particularly for coarse separation in grinding circuits.
The final correction factor C3 is based on the density of the solids with respect to the
liquid density:
1650
C3 = � (7-20)
�� �S – �L Tables 7-5 and 7-6 show the typical size ranges for cyclones.
Example 7-2
A hydrocyclone with a diameter of 250 mm is selected for a flow rate of 15 L/s at a pressure
of 140 kPa. The specific gravity of the solids is 4.8 and the volumetric concentration
is 0.3. If the pressure drop between the feed and the overflow is maintained at 50 kPa, determine
the corrected d50C. Assuming a discharge coefficient of 0.5 and a remaining pressure
of 20 kPa at the apex, determine the underflow capacity for an apex diameter of 80
mm if the underflow density is 2000 kg/m3 .
From Equation 7-17 the base d50C = 2.84 (D/100) 0.66 = 23.77 �m. From Equation 7-18
the correction factor C1 is
C1 = � � –1.43
= 3.3
From Equation 7-19 the correction factor C2 is
C2 = 3.27(�P) –0.28 = 3.27 (50) –0.28 53 – 30
�
53
= 1.0935
TABLE 7-5 Typical Range of Sizes for Cyclones Operating at Pressures from 20 to 500
kPa (3–72 psi)
Diameter Diameter
(of swirling (of swirling
chamber) chamber)
in mm Capacity in L/s in inches Capacity in USgpm
100 1 L/s @ 20 kPa–6 L/s @500 kPa 4 16 gpm@ 3psi–96 gpm @ 72 psi
150 3 L/s @ 20 kPa–15 L/s @500 kPa 6 48 gpm @ 3psi–240 gpm @ 72 psi
250 7 L/s @ 20 kPa–35 L/s @500 kPa 10 110 gpm@ 3psi–555 gpm @ 72 psi
380 12 L/s @ 20 kPa–60 L/s @500 kPa 15 190 gpm@ 3psi–950 gpm @ 72 psi
510 26 L/s @ 20 kPa–140 L/s @500 kPa 20 410 gpm@ 3psi–2200 gpm @ 72 psi
660 50 L/s @ 20 kPa–250 L/s @500 kPa 26 793 gpm@ 3psi–3963 gpm @ 72 psi
760 85 L/s @ 20 kPa–450 L/s @500 kPa 30 1350 gpm@ 3psi–7100 gpm @ 72 psi

7.38 CHAPTER SEVEN
TABLE 7-6 Typical Cyclone Size versus Particle Cut
Diameter of hydrocyclone Discharge cut size
The final correction factor C3 is based on the density of the solids with respect to the
liquid density:
1650
C3 = �� = 0.66
�� 4800 – 1000
d50C application = 23.77 × 3.3 × 1.0935 × 0.66 = 57 �m
The flow rate in the underflow is determined from nozzle equations:
Q = C D · A�2���P�/�� = 0.5 · 0.00503 �2�0� = 0.01124 m 3 /s = 11.24 L/s = 178 USgpm
7-7-4 Magnetic Separators
In beach and mineral sand plants as well as in taconite processing plants, minerals have
magnetic properties. The presence of a magnet would attract the ferrous ores and separate
them from other solids. This is the principle of magnetic separation. Magnetic separators
work on two principles.
1. An electromagnetic drum set in a stream
2. A belt driven by an electromagnetic drum on which solids in a dry state or slurry form
are allowed to pass to separate the ferrous ores
7-8 FLOTATION CIRCUITS
25 mm (1�) 5 �m (mesh 2500)
100 mm (4�) 40 �m (mesh 380)
250 mm (10�) 75 �m (mesh 200)
500 mm (20�) 150 �m (mesh 100)
Flotation is a method of separating solids from streams by creating a froth to which they
are attracted. Thus in a slurry circuit, flocculants are added to create a froth rich with the
metal concentrate. The trick is to make mineral particles hydrophobic, or water repellant.
Flotation involves the selected “adsorption” of hydrocarbons (e.g., ethyl xanthate) on liberated
minerals (e.g., chalcopyrite), which can then be attached to and transported by air
bubbles in the slurry to a so-called froth layer and then separated from the hydrophilic
(wetted) particles.
For flotation to be efficient, it must be repeated a few times in a circuit that includes a
rougher, a scavenger, and a cleaner as shown in Figure 7-28.
The collector in a flotation circuit consists of a hydrophobic hydrocarbon chain of
melecules (grease or wax) that repels the mineral-laden water and causes it to attach itself
to the passing air bubble. The surface chemistry is divided into three categories:

feed
circulating
tails
1. Physical adsorption with a free energy of the collector smaller than 5 kcal/mol
2. Chemisorption with a free energy of the collector larger than 30 kcal/mol
3. Intermediary stages between adsorption and chemisorption
Sulfide minerals are relatively easier to separate by chemisorption because they can
use the major collectors such as xanthates and dithiosphophates. Certain special additives
with high surface energy capabilities can also be added to separate different grades of sulfides
(e.g., to sink pyrite while floating chalcopyrite).
Oxide minerals (e.g., hematite, apatite. etc.) and silicate minerals are more difficult to
separate by flotation than sulfide minerals. For oxides and silicate minerals, flotation is
difficult because it is done by adsorption with minimal free surface energy using anionic
fatty acids and cationic amines, which operate essentially by electrostatic forces.
When various ores are present, flotation may be done in stages using tanks in series. In
each tank, a different pH level may be set or different collectors may be added, with the
output from each tank going to a different circuit for further treatment.
Depressants are chemicals that make the particle surface hydrophilic and nonfloatable.
Typical depressants include bichromate, cyanide, zinc sulphate, and lime.
Activators are chemicals that make the surface of nonfloating particles active for collector
attachment. Typical activators are copper sulphate and sodium sulphide.
The pH value is a determining factor in many flotation circuits. It is adjusted by using
various chemicals such as lime, caustic soda, sulfuric acid, etc.
Frothers are chemicals that are used to decrease the surface tension of water in order to
� Develop improved stability in the pulp
� Achieve smaller and better bubble size
COMPONENTS OF SLURRY PLANTS
rougher
cleaner
Air bubble Water
particles
circulating concentrate
concentrate
concentrate
tails scavenger tails
FIGURE 7-28 Flow chart for flotation circuit with rougher, scavenger, and cleaner.
Fig 7-28
7.39

7.40 CHAPTER SEVEN
� Create a suitable froth layer
� Help destroy froth, after which it is removed
Typical frothers include alcohols and pine oil.
The size of the flotation tank is based on the required flow rate as well as the required
retention time, an aeration factor, and a scale factor:
Q·tr · sc volume = � (7-21)
a
where
s c = scale factor = 0.85 for plants
= 1.0 for pilot plants
= 1.7 for lab batch
Q = flow rate
t r = retention time
a = aeration factor = 0.85
Example 7-3
A flow rate of 500 m3 /hr requires a retention time of 20 minutes. Size the tank assuming
four tanks.
volume = (500/60) · 20 · 1/0.85 = 196 m3 196/4 = 49 m3 per tank
The number of cells in a flotation circuit is determined by the degree of metallurgical
control and the concern for short-circuiting. It used to be believed that the correct approach
would consist of small cells and longer banks. However, with the advent of special
mixers and good aeration techniques, it is now possible to use larger tanks (Figure 7-29).
Flotation circuit can be very simple or very complex. A simple circuit such as used
with coal, achieves floatation in a single step and does not involve cleaning of the froth.
In a more complex circuit, an initial stage, called the rougher, is added; it acts as a preconcentrator.
The flocculated output goes then to a second stage, called cleaning, that is
done at higher dilution and is sometimes associated with regrinding at various stages.
When the ore grade is fairly low but the mineral is of high value, a scavenger is used
for additional preconcentration.
Froth is a real challenge in the design of pumps. This will be reviewed in Chapter 8.
7-9 MIXERS AND AGITATORS
Mixers or agitators (Figure 7-30) are very important components of mineral and chemical
process plants. They are used in various stages such as flotation circuits, leaching circuits,
gold adsorption on carbon, preparation of special chemicals such as milk of lime, preparation
of feed for pipelines, and final storage where sedimentation is likely to occur.
Mixers are used in the gold leaching processes. Special tanks for carbon in pulp (CIP)
or carbon in leach (CIL) are built with mixers. The largest diameter of these tanks is approximately
17 m (56 ft).
For large plants, the process of flotation or leaching in a single tank is not very efficient.
To increase productivity, tanks are installed in series (up to five stages or five tanks
in a series), thus eliminating possible short-circuiting.

feed
air
agitator impeller
COMPONENTS OF SLURRY PLANTS
concentrate
FIGURE 7-29 Simplified flotation circuit.
gangue
FIGURE 7-30 Top-entry agitator. (Courtesy of Hayward Gordon, Canada.)
7.41

T
H = D
7.42 CHAPTER SEVEN
There are processes that require a single mixing tank with one or two agitator mixers;
an example is the preparation milk of lime, where solids are simply mixed with water and
the mixers are used to prevent sedimentation.
Special processes in solvent extraction plants require gentle mixing while pumping the
organic solution over a weir. For example, in a copper SX-EW plant the organic solution
is kerosene-based; it absorbs the copper sulfates and separates them from other solids. For
these applications, manufacturers have developed special pump mixers, which are essentially
large, open shrouded pump impellers with a large number of vanes.
Most tanks used in mining are built with a height to diameter ratio around unity and use
a single-stage mixer. The mixer is of a vertical shaft design (Figure 7-30). The impeller diameter
is in the range of 30% to 45% of the tank diameter. The impeller is usually situated
at about 30% of the depth of the tank. Baffles at the wall of the tank break the vortex that is
formed by the agitator. These baffles have a width of 8% of the tank diameter.
Certain processes use tall, concentric tanks (sometimes called Pechuka tanks) with
two agitators in a series on a single shaft. These are more common in South Africa than in
North America.
Horizontal agitators are installed on the periphery of very large tanks, particularly in
the pulp and paper industry. They have not been popular in slurry mixing tanks in mineral
processes as they are difficult to maintain.
A vertical mixing tank (Figure 7-31) is fit with baffles at the walls to break the vortex
generated by the agitator. A certain gap is left between the baffles and the wall of the
tank.
In some respects, the propeller-type mixer causes continuous mass flow against a
stationary flat bottom. If the speed of rotation is not sufficient, a stagnation area develops.
The levels of turbulence in the tank, as well as the shape of the bottom of the tank,
feed
D = 0.3 D
A T
to 0.45 D
T
D T
b = D T/12 to D T/10
gap =
(1/72)
tank
diameter
FIGURE 7-31 Typical dimensions for the design of mixing tanks.
C = DA
= DT
/3

COMPONENTS OF SLURRY PLANTS
7.43
are important parameters. Sometimes a conical deflector is installed at the bottom of the
tank.
Mixing can be done by using an outside pump. The tank must then have a conical bottom.
The discharge of the tank at the bottom flows to a pump. The discharge of the pump
is returned to the tank and passes through a jet mixer.
For tanks with a flat bottom, the discharge may be from the bottom or through a pipe
on the side (Figure 7-32).
For very viscous mixtures, anchor agitators are recommended. The blades are vertical
and rotate fairly close to the wall surface of the tank. For some difficult and frothy pulps
in biological slurry treatment, helicoidal mixers are installed (Figure 7-33).
For some complex mixtures, the agitator may incorporate a hollow shaft to sparge
oxygen, an impeller to break up the froth at a high level, and one at the bottom of the shaft
to mix the slurry (Figure 7-34).
The propeller-type agitator is the most common in the mining industry. Its design can
be examined from various angles: mechanical strength, speed of operation, hydrofoil
shape of the blades, etc. The shaft is designed for the “jamming” condition or “start-up”
in a settled tank. The main force is taken at 75% of the maximum radius blade span meas-
feed feed
bottom and side discharge bottom and central discharge
feed feed
Side discharge
Top discharge
FIGURE 7-32 Various patterns of discharge from the mixing tank in accordance with the required
degree of agitation.

Anchor mixer for very viscous mixtures Helicoidal mixer for very Helicoidal viscous mixer mixtures for v
FIGURE 7-33 Anchor and helicoidal mixers are used for particularly viscous mixtures.
So Sole lepl plate ate to suppor support t
Top of vessel vessel structure
mixer
Top column
(flanged)
foam breaker
Bottom Bo ttom column (flanged)
oxygen discharge
7.44
hollow shaft vertical motor
Ho Hollow llow shaft for for
central oxygen oxygen flow
threaded coupling
Op Open en housing for for
foam breaker
Gre Grease ase lubricated sleeve
bearing & packing packing
Mixer
FIGURE 7-34 Complex mixer for biological slurries with central injection of oxygen and
with a baffle to skim the froth.

COMPONENTS OF SLURRY PLANTS
7.45
ured from the center of the shaft. For severe duty application, the shaft is designed for 2.5
times the rated motor torque without exceeding the yield stress (or 0.2% proof stress) of
the shaft material. For light-duty mixers, the shaft is designed for 1.5 times the rated motor
torque without exceeding the yield stress of the shaft material.
The shaft must not operate at speeds higher then 70% of the first critical speed if the
impeller is not dynamically balanced, or higher than 85% of the first critical speed if the
impeller is dynamically balanced. The deflection of the shaft should be limited, particularly
in the case of anchor agitators, so that the blades do not hit the walls or baffles.
The shaft is supported between two bearings in a cantilever arrangement. The bearings
may be part of a vertical gearbox or an independent bearing assembly. For large agitators,
the gearbox and bearing assembly are integral.
In Figure 7-35, the flow between the two levels z1 and z2 contracts across the propeller,
which induces a velocity Vi. Since the flow at the free surface as well as at the bottom
of the tank is negligible, this flow resembles the ground effect of a hovering helicopter.
From airscrew theory, the thrust across the propeller is:
2 T = 2A�V i (7-22)
where
Vi = the induced velocity
A = area of flow across the propeller
Vi = �T�/2�A��� (7-23)
where
T = thrust
3 power = TVi = 2�AVi (7-24)
Another important theory used to calculate thrust is the blade theory. In Chapter 3,
the concept of lift and drag around an aerofoil or an aircraft wing was introduced. The
blade of a propeller mixer is essentially a rotating wing exposed to a flow velocity V
and a rotating speed in rpm. Due to the contraction of the stream across the propeller,
the flow velocity is half the induced velocity. Since the blades are set at a certain pitch
angle � (quite often 40–45°), they are at an angle of attack with respect to the relative
speed. The relative speed is the vectorial addition of these two perpendicular speeds
(Figure 7-35).
2 2 2 W= �¼�V� i�+� r� �� � (7-25)
where the angular speed � = 2�N/60 and N is rotations per minute.
When the flow approaches a blade at a relative speed W and an angle of incidence �
with respect to the chord of the blade, a certain pressure distribution develops around the
blade. The result is a lift force L perpendicular to W and a resistance drag force D tangential
to it. A good designer keeps the angle of attack at a value that corresponds to the maximum
lift-to-drag ratio. For every airfoil, a plot of lift-to-drag curve (Figure 7-36) is obtained.
If the blade is pitched at an angle �, then the vertical force is
Y = lift cos � – drag sin � = L cos � – D sin � (7-26)
and the horizontal force
X = L sin � + D cos � (7-27)
The vertical force Y is opposite to thrust, whereas the horizontal force X multiplied by
the radius gives the resistant torque.
Near the tips of the propeller, the flow degrades due to the presence of tip vortices.

7.46
z
1
z 2
feed
V /2
i
V
i
b = D T/12 to D T/10
gap
angle of
incidence
W
W
V /2
i
Y = L cos � – D sin �
Lift
U = r
pitch
angle of
blade
Drag
FIGURE 7-35 Induced velocity and hydrodynamic forces for a propeller-type mixer.
W
X = L sin � + D cos �

Lift coefficient
C L
1.3
angle of
incidence
10 o
COMPONENTS OF SLURRY PLANTS
W
stall
Angle of attack
(or incidence )
Lift
7.47
The load distribution for many propellers is a maximum at 75% of the radius (Figure 7-
37) so that the effective torque is measured at this region.
In order to predict the performance of a full-scale mixer, a test may be conducted on a
reduced scale model under laboratory conditions. Performance is scaled up to large units
using nondimensional factors such as the power factor from the theory of rotating equipment.
In basic terms, it means that two mixers of the same geometrical design (but differ-
Drag
Drag coefficient
C
D
0.1
stall
10
Angle of attack
(or incidence )
o
FIGURE 7-36 Lift and drag forces as a function of the angle of incidence with respect to the
flow.
radial distribution of load mixer propeller
blade
FIGURE 7-37 Distribution of the total force as a function of the span of the blade.

7.48 CHAPTER SEVEN
ent sizes) will behave in similar ways with the same fluid and tank conditions. They
would have the same power number. The power number is defined as
P
Cp = � (7-28)
3 5 N D �
where
D = tip diameter of the mixer
N = rotational speed (rpm)
� = density of the slurry
P = power
The mixer Reynolds number is defined as
ND
Rem = (7-29)
2� �
2�
The relationship between the power number and the Reynolds number is shown in
Figure 7-38. Examples of the power number are presented in table Table 7-7.
The ability of the mixer impeller to pump or induce flow is measured and defined by a
nondimensional flow factor:
Q
CQ = � (7-30)
3 ND
It is also a function of the Reynolds number, as shown in Figure 7-39.
Gates et al. (1976) examined the use of mixerss to maintain solids in suspension. An
equivalent volume Vol eq is defined is defined as
FIGURE 7-38 Power coefficient versus Reynolds number. The top curve is typical of flatblade
mixers with wide blades. The middle curve is typical of flat-blade mixers with narrow
blades. The bottom curve is typical of pitched-blade mixers. (Reproduced by permission of
Hayward Gordon.)

COMPONENTS OF SLURRY PLANTS
TABLE 7-7 Typical Power Number for Mixers in Turbulent Flows
Type 3 Blades 4 Blades
45° flat-pitched blades 1.6 1.7
Propeller (marine) 0.7 0.8
Hydrofoil 0.2 0.5
7.49
Voleq = Sm Vol (7-31)
where Sm is the specific gravity of the slurry mixture.
The terminal velocity for spheres was discussed in Section 3-1-3-1. A design settling
velocity Vd is correlated to Vt by a correction factor fw: Vd = fwVt (7-32)
where
Vd = design settling velocity
Vt = the terminal (or free settling) speed
The correction factor fw is presented in Table 7-8 as a function of weight concentration.
This empirical coefficient was developed by Chemineer Inc., based on experimental
work. It is often difficult to predict the nature of the flow or the drag coefficient near an
impeller blade. At weight concentrations in excess of 15%, the solids start to interact, hindering
settling so that the settling velocity must be adjusted.
The level of agitation is very important to the mechanics of suspension. Chemineer
Inc. developed a scale of agitation from 1 to 10, summarized by Gates and al. (1976) as in
Table 7-9. Figure 7-40 shows the level of suspension of solids in correlation with the
Chemineer scale. Often, manufactures define mixing as simple, mild, medium, vigorous,
or violent (Figure 7-40).
The science of mixing and keeping solids in suspension is highly empirical. The engineer
should take into account existing similar installations as well as lab work results.
Because of wear associated with slurries, a simple flat blade system is used to design
C =
Q
Flow Coefficient
Q
N D 3
Laminar Transition
Reynolds Number
Turbulent
2
Re = ND /(2 )
FIGURE 7-39 Flow coefficient versus Reynolds number.
D/H = 0.40
D/H = 0.45
D/H = 0.50
Ratio of Impeller
diameter to tank height

7.50 CHAPTER SEVEN
TABLE 7-8 The Correction Factor f w Presented as a Function of
Weight Concentration
Solids weight concentration (%) Factor f w
2 0.8
5 0.84
10 0.91
15 1.0
20 1.1
25 1.2
30 1.3
35 1.42
40 1.55
45 1.70
50 1.85
From Gates et al., 1976, reprinted by permission from Chemical Engineering.
TABLE 7-9 Chemineer Scale for Agitation of Solids in Suspension
Scale of agitation Description
1–2 At levels 1–2, agitation is required for minimal suspension of solids.
Agitators capable of working at an agitation level of 1–2 will:
� Produce motion of all of the solids of the design-settling velocity in the
vessel
� Permit moving fillets of solids on the bottom, which are periodically
suspended
3–5 Agitation levels 3–5 characterize most chemical process industries solids
suspension applications. This scale range is typically used for dissolving
solids.
Agitators capable of working at an agitation level of 3–5 will:
� Suspend all of the solids of design velocity completely off the vessel
bottom
� Provide slurry uniformity to at least one-third of the fluid batch height
� Be suitable for slurry draw-off at low exit-nozzle elevations
6–8 Agitation levels 6–8 characterize applications where the solids suspension
level approaches uniformity.
Agitators capable of scale level 6 will:
� Provide concentration uniformity of solids to 95% of the fluid batch
height
� Be suitable for slurry draw-off up to 80% of the fluid batch height
9–10 Agitation levels 9–10 characterize applications where the solids suspension
uniformity is the maximum practical.
Agitators capable of scale 9 will:
� Provide concentration uniformity of solids to 98% of the fluid batch
height
� Be suitable for slurry draw-off by means of overflow
From Gates et al., 1976. Reprinted by permission from Chemical Engineering.

a. Unstable particles are on
vessel bottom (Scale of
agitation = 1)
the mixer. The blades are often pitched at an angle of 45° with respect to the horizontal
plane. To size such a flat-bladed propeller in a mixing tank, with baffles at 90° to each
other, baffle width of 1/12th of the tank diameter, and offset from the wall with a gap of
1/72nd of the tank diameter, Gates et al. (1976) proposed the following empirical equation
in USCS units:
Din = 394� � 0.2 HP
� (7-33a)
SmN 3n where
Din = diameter in inches
HP = power in hp
n = number of impellers
N = speed in rev/min
Sm = specific gravity of the slurry mixture
To express Equation 7-33a in SI units:
COMPONENTS OF SLURRY PLANTS
b. Particles swept off vessel
bottom (Scale of agitation = 3)
Dimp = 37.57� � 0.2 P
�
SmN3n 7.51
c. Solids are homogeneously
distributed (Scale of
agitation = 9)
FIGURE 7-40 Intensity of agitation yields different patterns of solid suspension. (From
Gates et al., 1976. Reproduced by permission from Chemical Engineering.)
(7-33b)
where
Dimp = diameter in meters
P = power in Watts
To correlate between the levels of agitation in a tank, the speed of rotation, and diameter
of the impeller, Gates et al. (1976) plotted the scale of agitation versus �, a factor defined
in U.S. units as:
� = N3.75 2.81 Dimp /Vd
with Vd expressed in ft/min, D in inches, and N in rev/min (Figure 7-41).

7.52 CHAPTER SEVEN
FIGURE 7-41 Chart to determine speed and diameter of a mixer versus the solids suspension
scale. [From Gates et al. (1976). Reprinted by permission of Chemical Engineering.]
Example 7-4
A tank contains 600 ft3 of a slurry mixture. The specific gravity of the solids is 4.1 and the
average particle size d50 is 0.01 ft. It is required to be designed for overflow output at an agitation
scale of 9. The weight concentration of the mixture is 20%. Size the mixer, assuming
a single impeller with an impeller-to-tank diameter of 0.4, baffles 1/12th the tank diameter
and baffle gap of 1/72nd the tank diameter; use Figure 7-34. Assume viscosity of 1
cP. Compare the power consumption with a mixer operating at a Chemineer scale of 5.
Solution in USCS Units
Since most tank have a diameter equal to the height, and assuming a volume of 90% the
tank volume, the effective volume occupied by the slurry is
3 3
Vol = 0.9 × 0.25 × �DT = 600 ft
Hence,
DT = 9.47 ft
T = 9.47/0.9 = 10.52
Dimp = 0.4 × 10.52� = 4.21� or approximately 50.5 inch
For a scale of 9 and D/T = 0.4, � = 30 × 10 10 , so
� = N 3.75 D 2.81 /V d
We must determine V d. The particles are coarse enough to assume a drag coefficient C D =
0.44. Substituting into Equation 3-7
4(�S – � 4(3.1) × 32.2 × 0.01
CD =
L)gdg �� = ��� 2 3 × V t
3�LV t 2
Vt = 1.5 ft/s = 90 ft/min
From Table 7-8, the correction factor fw = 1.0 and Vd = VT, or
� =30 × 10 10 = N3.75D2.81 /Vd = N3.7550.52.81 /90 = 679.5 N3.75 N = 202 rev/min
To determine the required horsepower, Equation 7-33a is used:
Din = 394� � 0.2 HP
�
SmN 3n

The specific gravity of the mixture is obtained by using Equation 1-4:
or S m = 1.128.
COMPONENTS OF SLURRY PLANTS
100
���
15/4100 + (100 – 15)/1000
� m = = 1128 kg/m 3
[50.5/394] × (1.128 × 202 3 ) 0.2 = HP 0.2
or HP = 322 hp. For a scale of 5, and D/T = 0.4, � = 5 × 10 10 , so
� = 5 × 10 10 = N 3.75 D 2.81 /V d = N 3.75 50.5 2.81 /90 = 679.5 N 3.75
N = 125 rev/min
To determine the required horsepower, Equation 7-33a is used:
[50.5/394] × (1.128 × 1253 ) 0.2 = HP0.2 7.53
or HP = 76 hp. Going from a mild level of agitation at a 5 on the Chemineer scale to level
9 increases the power consumption 4.24 fold.
The shear rate was previously defined in Chapter 2 as the rate of change of the velocity
with respect to height above the wall. More generally for a mixer, the shear rate is the
change of velocity with respect to depth. The induced velocity is essentially created by
the impeller, and the maximum shear rate is experienced at the tip of the blade. There is,
however, an average shear rate estimated for the impeller zone, and an average in bulk of
the tank.
For a radial impeller, all the flow is discharged at the tip of the impeller, and all the
solids are subject to the same speed and head. In the case of the propeller or axial turbine,
the velocity distribution is proportional to the radius. All solids pass through a higher
shear zone in the case of the radial machine.
The closer the impeller is to the bottom of the tank, the more the induced velocity is
suppressed. A proximity factor h c/D is defined as the ratio of the gap of the impeller to the
diameter. This principle is similar to those in the world of aeronautics. The ground effect
is essentially the pressure exerted by a helicopter or airplane. The closer it is to the
ground, the more pronounced is the effect and, eventually, the induced drag is reduced
when the machine flies very close to the ground. In the case of the mixers, the flow is restricted
as the impeller gets closer to the bottom of the tank. This affects power consumption.
A radial mixer tends to induce flow tangentially, whereas an axial machine tends to
induce it vertically. Propeller and radial mixers tend to create different patterns of recirculation
around their blades.
Mounting more than one impeller on the same shaft gives different patterns of performance
based on the mutual interference that the different inpellers exert on each other.
The power from two axial turbines is not twice the power from a single propeller, as the
induced velocity of the second propeller is not twice the induced velocity of the first propeller.
In fact, in some applications, the first propeller is smaller in diameter than the bottom
propeller. However, two radial impellers closely spaced consume more than twice
the power of a single unit.
In sewage treatment plant processes as well as in chemical and mining processes, gas
is sparged in the liquid. The impeller of the mixer is then used to provide dispersion of the
gas and circulation of the tank contents. For radial machines, a small radial impeller is installed
at the bottom to mix the gas. In the case of hydrofoils, different approaches are
used. If the shaft and blades are hollow, gas may be pumped through the shaft and blades.
In other applications, the impeller is contained within a cylinder that is within the tank.
This prevents flooding the impeller with gas.

7.54 CHAPTER SEVEN
A hydrofoil type of mixer would have the highest pumping capacity but would develop
the lowest head or shear at constant horsepower or torque. Hydrofoil mixers are often
chosen for high-flow applications.
For heat transfer, solid suspension, blending, or solid dissolving, bulk pumping is critical.
A hydrofoil mixer would be the preferred machine. However, for gas–liquid contacting,
molecular mixing, solid dispersion (reduction of agglomerates), an axial flow, flatblade
machine or radial mixer are preferred options.
The mixing of non-Newtonian slurries is fairly complex and must rely on experimental
data. Wasp et al. (1977) proposed that the power consumption for mixing non-Newtonian
slurries is a function of the Reynolds number, the Hedstrom number, and the
Froude number:
N
= fn� , , � (7-34)
In other words, the power consumption is based on the Reynolds number, Hedstrom number,
and Froude number based on the impeller diameter.
The mixing of non-Newtonian fluids is common in the manufacture of polymers and
considerable data is available. It is, however, not very wise to extrapolate to non-Newtonian
slurry mixtures.
In the last 25 years of the 20th century, larger and larger mixers have been built. Certain
shaft failures and blade failures have occurred on some large agitators. Some were
due to corrosion of the bolts holding the blades and others due to jamming of the shaft. It
is important to understand that starting an agitator from fully settled conditions can be
very stressful to the machine.
The reader should consult appropriate reference books on machine design and gearbox
design. A service factor of 1.5 is a minimum for sizing the gearbox. It would be beyond
the scope of this book to explore the selection of gearboxes for agitators. However, the
designer of slurry mixers should be aware of certain important mechanical criteria, such
as critical speed.
To examine the distribution of loads on the blade, the program “Agitblade” may be
used.
2 P
2� �0 Dimp � � � � 3 5 2 2 2 �N D imp �NDimp �N Dimp g
Program “Agitblade” for Propeller Blade Loads
CLS
REM propeller design for AGITATOR
PI = 3.1415
DIM R(20), PITCH(20), V(20), W(20), ALPHA(20), BETA(20), BLPHA(20)
DIM C(20), CL(20), CD(20), CT(20), CU(20), T(20), INCIA(20)
DIM P(20), TK(20), VV(20), VH(20), VS(20), PK(20)
INPUT “FLUID DENSITY”; DENS
INPUT “radius at hub “; RHUB
INPUT “ RADIUS AT TIP”; RTIP
AP = PI * RTIP ^ 2
100 PRINT “OPTIONS FOR COMPUTATION”
PRINT “1-CALCULATIONS ASSUMING KNOWLEDGE OF INDUCED VELOCITY”
PRINT “ USING MOMENTUM THEORY”
PRINT “2- BLADE THEORY CALCULATIONS FOR BLADEWISE DISTRIBUTION OF
FORCES”
PRINT “ AND FLOW CHARACTERISTICS”
PRINT “3- VORTEX FLOW CALCULATIONS”

COMPONENTS OF SLURRY PLANTS
INPUT “YOUR OPTION “; OPT
IF OPT = 1 THEN 300
IF OPT = 2 THEN 2000
IF OPT = 3 THEN 6000
300 INPUT “NUMBER OF PROPELLERS ON THE SAME SHAFT”; NU
IF NU > 1 THEN PRINT “CALCULATIONS FOR FIRST PROPELLER”
INPUT “VELOCITY UPSTREAM THE PROPELLER “; VS
IF VS = 0 THEN 305
INPUT “IS THIS VELOCITY PARALLEL TO SHAFT (Y/N)”; V$
IF V$ = “Y” OR V$ = “y” THEN 305
INPUT “INCLINATION OF U/STREAM VELOCITY WRT SHAFT AXIS IN DEGREES”;
INC
INCI = INC * PI/180
305 FOR M = 1 TO NU
VS(M) = VS
PRINT
PRINT
PRINT “CALCULATION FOR PROPELLER “; M
INCIA(M) = INCI * 180/PI
VV(M) = VS * COS(INCI)
VH(M) = VS * SIN(INCI)
GOTO 320
310 VV(M) = VS
320 IF VV(M) = 0 THEN PRINT “COMPONENT OF U/S VEL PARALLEL TO PROP
IS NIL”
321 INPUT “IS THE HYDROSTATIC PRESSURE EQUAL ON BOTH SIDES OF THE
PROP (Y/N)”; PH$
IF PH$ = “Y” OR PH$ = “y” THEN 340
INPUT “HYDROSTATIC PRES UPSTREAM PROP “; PHUS
INPUT “HYDROSTATIC PRES DOWNSTREAM PROP “; PHDS
340 INPUT “DO YOU KNOW THE INDUCED VELOCITY (Y/N)”; IV$
IF IV$ = “N” OR IV$ = “n” THEN 600
INPUT “INDUCED VELOCITY “; VU(M)
VU = VU(M)
T(M) = DENS * AP * SQR((VV(M) + VU) ^ 2 + VH(M) ^ 2) * VU ^ 2 +
(PHUS – PHDS) * AP
P(M) = T(M) * VV(M) + .5 * T(M) * VU(M)
TK(M) = T(M)/1000
PRINT USING “ THRUST = ###########.#### kN”; TK(M)
PK(M) = P(M)/1000
PRINT USING “POWER = #########.### kW”; PK(M)
VH = VH(M)
VS = SQR((VV(M) + VU) ^ 2 + VH(M) ^ 2)
INCI = ATN(VH/(VV(M) + VU(M)))
NEXT M
LPRINT “CALCULATIONS BASED ON MOMENTUM THEORY”
FOR M = 1 TO NU
7.55

7.56 CHAPTER SEVEN
LPRINT
LPRINT
LPRINT “CALCULATION FOR PROPELLER “; M
LPRINT USING “VELOCITY UPSTREAM SHAFT = ###.## m/s”; VS(M)
LPRINT USING “ITS COMPONENT PARRALLEL TO SHAFT= ###.## m/s “; VV(M)
LPRINT USING “ITS COMPONENT PERPANDICULAR TO SHAFT = ###.## m/s “;
VH(M)
LPRINT USING “ITS INCLINATION WRT TO SHAFT = ###.# deg “; INCIA(M)
LPRINT USING “INDUCED VELOCITY = ###.## m/s “; VU(M)
LPRINT USING “ RESULTANT THRUST = #####.## KN”; TK(M)
LPRINT USING “ INDUCED POWER CONSUMPTION = #####.## KW”; PK(M)
NEXT M
GOTO 8000
600 PRINT “YOU MAY HAVE TO CALCULATE THE INDUCED VELOCITY BY
ASSUMING A CERTAIN THRUST MAGNITUDE”
2000 INPUT “VERTICAL SPEED”; V
INPUT “PITCH AT HUB”; PITCHR
INPUT “PITCH AT TIP”; PITCT
INPUT “REQUIRED TIP SPEED “; VTIP
RPS = VTIP/RTIP
PRINT USING “THE ROTATIONAL SPEED IN ######.## RAD/S “; RPS
RDIV = (RTIP - RHUB)/10
PITDIV = (PITCHR - PITCT)/10
INPUT “chord at the root”; CR
INPUT “TIP CHORD “; CT
REM CALCULATE THE ADVANCE RATIO OF THE PROPELLER
J = V/(2 * PI * RTIP)
PRINT “ADVANCE RATIO OF PROP “; J
REM CALCULATE BLADE AREA AND HENCE ASPECT RATIO
SB = (RTIP - RHUB) * .5 * (CT + CR)
AR = (RTIP - RHUB) ^ 2/SB
PRINT “BLADE ASPECT RATIO”; AR
PRINT “BLADE AREA “; SB
CD0 = .015
K = .1/(1 + 2/AR)
KD = K ^ 2/3 * AR
REM K IS THE LIFT COEFFICIENT SLOPE DCL/DALPHA IN THE LINEAR RANGE
PRINT “LIFT SLOPE PER DEGREE IN THE LINEAR RANGE “; K
ACR = (CR - CT)/(RTIP - RHUB)
CDIV = (CR - CT)/10
DTIP = 2 * RTIP
INPUT “POWER NUMBER “; CP
P = CP * DENS * RPS ^ 3 * DTIP ^ 5 * CP/2 * PI
PK = P/1000
TOR = PK/RPS
PRINT USING “REQUIRED TORQUE = #####.## KNm”; TOR
PRINT USING “REQUIRED POWER #####.## KW”; PK
FOR N = 1 TO 11
R(N) = RHUB + (N - 1) * RDIV
PITCH(N) = PITCHR - (N - 1) * PITDIV

COMPONENTS OF SLURRY PLANTS
V(N) = R(N) * RPS
W(N) = SQR(V(N) ^ 2 + V ^ 2)
BLPHA(N) = ATN(V/V(N))
ALPHA(N) = BLPHA(N) * 180/(2 * PI)
BETA(N) = PITCH(N) - ALPHA(N)
C(N) = CR - ACR * (R(N) - RHUB)
CL(N) = K * BETA(N)
CD(N) = CD0 + KD * CL(N)
CT(N) = CL(N) * COS(BLPHA(N)) - CD(N) * SIN(BLPHA(N))
CU(N) = CL(N) * SIN(BLPHA(N)) + CD(N) * COS(BLPHA(N))
T(N) = CT(N) * .5 * W(N) ^ 2 * DENS * C(N) * RDIV
NEXT N
RPM = RPS * 60/(2 * PI)
LPRINT “AGITATOR PROPELLER”
LPRINT USING “VERTICAL SPEED = #####.### m/s”; V
LPRINT USING “HUB RADIUS = ###.#### m TIP Radius = ####.#### m “;
RHUB, RTIP
LPRINT USING “TIP SPEED = ####.### M/S”; VTIP
LPRINT USING “ANGULAR SPEED = ######.## RPM”; RPM
LPRINT USING “VERTICAL SPEED = ####.## m/s “; V
LPRINT USING “PITCH AT TIP= ####.## ; PITCH AT HUB = ####.##”;
PITCT, PITCHR
LPRINT “BLADE AREA”; SB
LPRINT “BLADE ASPECT RATIO”; AR
LPRINT “APPROX LIFT SLOPE COEF IN LINEAR RANGE”; K
LPRINT “APPROX LIFT/DRAG POLAR RATIO “; KD
LPRINT USING “PROPELLER AREA ######.## m^2”; AP
LPRINT “LOCAL RADIUS PITCH ANGLE ANGLE OF INCIDENCE TOTAL VELOCITY
ALPHA”
FOR N = 1 TO 11
LPRINT R(N), PITCH(N), BETA(N), W(N), ALPHA(N)
NEXT N
LPRINT “ LOCAL RADIUS,CHORD, LIFT COEF, DRAG COEF, THRUST COEF,
TANG FORCE COEF”
7.57
FOR N = 1 TO 11
LPRINT R(N), C(N), CL(N), CD(N), CT(N), CU(N)
LPRINT
NEXT N
6000
8000
INPUT “DO YOU WANT TO PROCEED WITH OTHER THEORIES OF DESIGN (Y/N) “;
D$
IF D$ = “Y” OR D$ = “y” THEN 100
END
REM CT=THRUST COEFFICIENT PARALLEL TO PROPELLER SHAFT
REM CU= FORCE COEFFICIENT PERPANDICULAR TO SHAFT AND CAUSING
RESISTANCE

7.58 CHAPTER SEVEN
The designer of mixers with hydrofoils should be aware that the center of pressure
does not always correspond with the center of gravity of the blade. The center of pressure
could be as far as 25% of the blade chord on high-aspect-ratio blades or those with a high
ratio of blade length to blade chord. A pitching–bending moment results; it must be considered
when sizing the bolts of the blade.
It is important to check the stress on the shaft of an agitator. The following program,
written in QuickBasic, allows one to check for the case of a shaft with two impellers in
series.
Program “Dblagit.Bas” for Two Agitators Mounted on a Shaft
PRINT “double agit”
pi = 4 * ATN(1)
‘INPUT “are you using SI units (Y/N)”; l$
l$ = “n”
IF l$ = “n” OR l$ = “N” THEN conv = .0254
IF l$ = “y” OR l$ = “Y” THEN conv = 1!
‘INPUT “distance from bottom bearing to first coupling”; ab
ab = 9.41
‘INPUT “shaft O.D. for first section of shaft ab “; dab0
‘INPUT “shaft I.D. for first section of shaft ab “; dabi
dab0 = 5
jab = (pi/32) * (dab0 ^ 4 - dabi ^ 4) * conv ^ 4
‘INPUT “distance from first coupling to second coupling”; bc
bc = 135.64
‘INPUT “shaft O.D. for second section of shaft ab “; dbc0
‘INPUT “shaft I.D. for second section of shaft ab “; dbci
dbc0 = 8.625
dbci = 7.625
‘sch 80
jbc = (pi/32) * (dbc0 ^ 4 - dbci ^ 4) * conv ^ 4
‘INPUT “distance from second coupling to first impeller”; cd
cd = 48
‘INPUT “shaft O.D. for third section of shaft cd “; dcd0
‘INPUT “shaft I.D. for first section of shaft cd “; dcdi
‘dcd0 = 6.625
‘dcdi = 6.065
dcd0 = dbc0
dcdi = dbci
jcd = (pi/32) * (dcd0 ^ 4 – dcdi ^ 4) * conv ^ 4
‘INPUT “distance from first to second impeller”; de
de = 150
dde0 = dcd0
ddei = dcdi
jed = jcd
‘INPUT “ power on top impeller”; p1
p1 = 21 * 746
‘INPUT “ power on bottom impeller”; p2
p2 = 35 * 746
INPUT “rpm”; rpm

w = rpm * 2 * pi/60
‘assume start up torque factor 250%
ts = 2.5
t1 = p1/w
t2 = p2/w
PRINT “torque from top impeller”; t1
‘INPUT “diameter of first impeller”; dimp1
dimp1 = 84
‘INPUT “diameter of second impeller”; dimp2
dimp2 = 66
r1 = dimp1 * conv/2
r2 = dimp2 * conv/2
f1 = t1/(.75 * r1)
PRINT “ tangential force on top impeller”; f1
f2 = t2/(.75 * r2)
PRINT “tangential force on bottom impeller”; f2
ma = (f1 * conv * (ab + bc + cd) + f2 * conv * (ab + bc + cd +
de))/10
‘assume unsymmetry of forces at 10% radius
may = SQR(ma ^ 2 + .75 * (t1 + t2) ^ 2)
mayn = ma/1000000!
PRINT USING “total bending moment at a = ####.## MNm”; mayn
syab = mayn/jab
PRINT USING “shaft stress syab = ###.## MPa”; syab
IF syab > 100 THEN PRINT “warning the required stress limit is 100
MPa”
mb = (f1 * conv * (bc + cd) + f2 * conv * (bc + cd + de))/10
mby = SQR(mb ^ 2 + .75 * (t1 + t2) ^ 2)
mbyn = mby/1000000!
PRINT USING “total bending moment at b = ####.## MPa”; mbyn
sybc = mbyn/jbc
PRINT USING “shaft stress syab = ###.## MPa”; sybc
IF sybc > 100 THEN PRINT “warning the required stress limit is 100
MPa”
mcd = (f1 * conv * cd + f2 * conv * (cd + de))/10
mcd = SQR(mcd ^ 2 + .75 * t2 ^ 2)
mcdn = mcd/1000000!
PRINT USING “total bending moment at c = ####.## MPa”; mcdn
sycd = mcdn/jab
PRINT USING “shaft stress sycd = ###.## MPa”; sycd
IF sycd > 100 THEN PRINT “warning the required stress limit is 100
MPa”
7-10 SEDIMENTATION
COMPONENTS OF SLURRY PLANTS
7.59
Sedimentation is a form of separation of solids from liquids by using gravity forces rather
than electrostatic, chemical (flotation), or magnetic forces. Sedimentation may be
achieved by gravity forces, using thickeners and clarifiers. On the other hand, it may be
accomplished by centrifugal forces, as in centrifuges. In gold extraction circuits, an inter-

7.60 CHAPTER SEVEN
mediary centrifuge is sometimes installed between the hydrocyclones and the ball mill
feed box. Centrifuges are sometimes called concentrators because they permit the extraction
of some of the heavy metals by applying a very high centrifugal force such as 60
times the acceleration due to gravity (60 g).
7-10-1 Gravity Sedimentation
Gravity sedimentation is classified as thickening or increasing the concentration of the
feed stream, or clarification or the removal of solids from relatively dilute streams. The
former is used to prepare the feed for tailings and concentrate pipeline flow, or for the removal
of tailings on trucks. The latter is more frequently used in sewage and waste treatment
plants, where the volume of solids is considerably smaller than in tailings and concentrate
flows.
Considerable research on the use of flocculants in the last quarter of the twentieth century
has lead to more concentrated sedimentation with less thickener. It would be beyond
the scope of this book to discuss all these new flocculants.
In simple terms, a clarifier or a thickener is essentially a sedimentation tank. To make
the sedimentation uniform, a rake or arm rotates slowly but continuously. A relatively
clear layer of liquid forms at the top and is withdrawn through an overflow box feeding a
launder. The slurry in the thickener is denser at lower and lower layers. The bottom of the
thickener forms a shallow cone with the center feeding into an underflow pipe to a separate
launder or pump.
The actual feed to the thickener is through a launder to the center. A feed box leads the
slurry to a depth lower than the relatively clear water. Some special processes use intermediary
mixing chambers where flocculants are added to accelerate the precipitation.
The tank itself may be shallow and called a shallow thickener, or deep and called a
deep thickener. The decision to choose either is often based on various parameters such as
the final weight concentration, the rate of sedimentation, the viscosity, the design of the
rake, as well as other parameters. This is at the basis of the design of the thickener (Figure
7-42).
The actual process of sedimentation in a tube is based on the settling (or terminal)
speed that was discussed at great length in Chapter 3. It is also depicted in Figure 7-43.
Initially, the slurry is uniformly mixed. Gradually, the solids sink, forming three layers of
liquid: free of solids, a dilute mixture, and a relatively dense layer. Eventually, all the
solids in the dilute layer sediment out, leaving only two layers, one of water and one of a
dense mixture with solids at minimum void ratio. The use of certain chemicals can accelerate
the sedimentation of solids.
The correlation between the terminal velocity of a sphere V t and the sedimentation
speed V s is correlated to the void fraction � (Cheremisinoff, 1984) by the following equation:
Vs = Vt� 2X(�) (7-35)
Where X(�) is a function of the void ratio that must be determined by tests. The void ratio
is
Vol f
� = ��
(7-36)
Volf + Volp where
Volp = volume filled by the particles
Volf = volume of liquid filling the space between the particles

7.61
FIGURE 7-42 Schematics of a thickener used for sedimentation of solids.

7.62 CHAPTER SEVEN
height of dense phase
fig 743
For thickened sludges with a void ratio smaller than 0.7, Cheremisinoff (1984) proposed
the following correlation:
�
Vs = 0.123 Vt� � (7-37)
3
Spheres can actually compact in a very dense pattern to a minimum void ratio of
0.215, but Cheremisinoff (1984) indicated that the average void ratio from thickeners was
0.6. For nonspherical and coarse particles, the situation becomes more complex because
of the shape factor (discussed in Chapter 3), and it is the norm to conduct sedimentation
tests on samples of the slurry before designing the thickener.
7-10-2 Centrifuges
clear water boundary
dense phase boundary
FIGURE 7-43 Response of gravity sedimentation with time.
� 1 – �
time (minutes)
Centrifuges use centrifugal force as a means to separate solids from liquids. Liquid is fed
into the inlet and a rotating bowl is used to apply the centrifugal force, similar to a clothes
drier that separates liquid from clothes by continuously rotating the clothes. Obviously,
with slurry, it is more complex (Figure 7-44).
The centrifugal force is defined as
F = mR�2 (7-38)
where
� = 2�N/60
R = radius of rotation
The ratio of the centrifugal force to the weight is called the centrifugal number Nc: Nc = mR�2 /mg = R�2 /g (7-39)
For liquid-to-liquid separation, the centrifugal number may be as high as 60,000 for
certain tubular sedimentation designs. The mining industry is concerned with wear, so
slurries are separated at centrifugal numbers smaller than 100.
Cheremisinoff (1984) stated that the settling velocity of a particle in turbulent motion
(Re > 500) in a centrifuge is Ks times as much as the free settling velocity, where

R
Ks = 2�N � �� g
The Reynolds number for the particle is calculated using the radial velocity:
Re =
For very fine particles with Re < 2, the migration is in laminar flow:
For transition flow with 2 < Re < 500
COMPONENTS OF SLURRY PLANTS
FIGURE 7-44 Centrifugal separator. (Courtesy of Knelson Concentrators.)
Ks = 4� 2N2 R
� �
4� 2N2R �
g
K s = � � 0.71
(7-40)
(7-41)
(7-42)
Consider a simple vertical centrifuge as in Figure 7-36. The solids in the slurry move
toward the wall at a speed u s toward the radius R w, while the liquid moves toward the axial
feed tube at a speed u L toward the radius R a. If the solids are at a volumetric concentration
C V with a flow rate Q, the solids move at a speed u s as
Q s = 2�R 0Hu s = C vQ
Separation will occur when u s > C vQ/2�R 0H.
�2�RNd p
��
60�
� g
7.63

7.64 CHAPTER SEVEN
Example 7-5
A small centrifuge with a diameter of 150 mm is designed to handle 1.5 tons/hr of solids
at a volumetric concentration of 40%. The density of the solids is 3000 kg/m3 . The height
of the cone is 125 mm. Determine the minimum speed of solids for separation from liquid.
Solution
Since the density is 3000 kg/m3 and the centrifuge handles 1500 kg/hr, the volume flow
rate of solids is 0.5 m3 /hr, or 0.139 kg/s.
For separation, us > 0.139/(2 × � × 0.15 × 0.125) and
us > 1.18 m/s
Considering the settling velocity of many particles, it is obvious that this centrifuge
can handle the coarse particles found in certain mining systems.
7-11 CONCLUSION
To achieve many of the tasks described in this chapter, slurry must be transported from
one point to another. This may be done by gravity flow, by open channel flow, or by
pumping. The pump is the workhorse of slurry transportation and will be analyzed in the
next two chapters.
A lot of different equipment is used in the processing of mineral ores. These were reviewed
in this chapter more in terms of their place in the slurry circuit. The performance
of the equipment depends on many factors such as proper sizing and the characteristics of
rocks and soils that too often cause extensive wear. The materials selected for processing
by such equipment will be examined in Chapter 10, as they are also used as criteria in the
manufacture of pumps.
7-12 NOMENCLATURE
A Area of flow across the propeller
c Blade chord
C1, C2, C3 Coefficients of a hydrocyclone
CD Drag coefficient
CL Lift coefficient
CQ Flow coefficient
Cp Power coefficient
Cr80 CVL d50 D
d80 of the output wet ground rocks
Volume fraction of liquid phase in a slurry tank
d50 cut point of a hydrocyclone
Drag force
Di Conduit diameter (m)
Din Diameter of mixer in inches
Dimp Mixer impeller diameter
Dm Mill diameter in meters
Dmus Mill diameter in inches
DT Mixer tank diameter
e Natural number
E1 Dry grinding factor

E2 Factor for open circuit grinding to be expressed in terms of the final classification
of solids
E3 Mill diameter factor
E4 Oversize feed factor for grinding
E5 Fineness factor for ground or crushed particles
E6 Reduction ratio factor for ball or rod mills
E7 Low reduction ratio factor for ball or rod mills
E8 Correction factor for rod mills
E9 Correction factor for rubber-lined mills
Fe80 Feop fw d80 of the feed rocks
Optimum size of feed to a ball or rod mill
Correlation factor for a mixer between design settling velocity and terminal
velocity of solids
g Acceleration due to gravity (9.8 m/s2 )
H Height of mixer above bottom of tank
HP Horsepower
L Lift force
n Number of impellers
N Rotational speed in rev/min
P Power
Q Flow rate (m3 /s)
Rc Recovery of underflow from a cyclone
Re Reynolds number
ReB Reynolds number for a Bingham plastic, using the coefficient of rigidity for
viscosity
Rr material reduction ratio in a grinding circuit
S Swirling number
T Thrust force
Uslip Slip speed between liquid and solids in a mixer
V Average velocity of flow (m/s)
Vt Terminal velocity of solids
W Consumed power for wet grinding
Greek letters
� Angle of incidence
� Void fraction
� Wet grinding factor
� m
Density of slurry mixture (kg/m 3 or dlugs/ft 3 )
�s Density of solids in mixture (kg/m3 or dlugs/ft3 )
� Factor of energy dissipation before the hydraulic jump in a free fall
� Concentration by volume in decimal points
� Shear strain
� Pythagoras number (ratio of circumference of a circle to its diameter)
� Duration of the shear for a time-dependent fluid
� Density
� 0
Yield stress for a Bingham plastic
� Kinematic viscosity (usually expressed in Pascal-seconds or poise)
� Angular velocity of particle
Subscripts
L Liquid
m Mixture
COMPONENTS OF SLURRY PLANTS
7.65

7.66 CHAPTER SEVEN
p Particle
s Solids
7-13 REFERENCES
Arterburn, R. A. 1982. The sizing and selection of hydrocyclones. In Design and Installation of
Communution Circuits, A. L. Mular and G. V. Jergensen (Eds.). New York: Society of Mining
Engineers.
Bond, F. C. 1952. Third theory of comminution. Trans. AIME, 193, 484.
Burgess, K. E. and B. Abulnaga. 1991. The application of finite element analysis of Warman pumps
and process equipment. Paper presented at the Fifth International Conference on Finite Element
Analysis, University of Sydney, Sydney, Australia.
Cheremisinoff, N. P. 1984. Pocket Handbook for Solid–Liquid Separations. Houston: Gulf Publishing.
Dickey, D. S. and J. G. Fenic. 1976. Dimensional analysis for fluid agitation systems. Chemical Engineering
Elliott, A. J. 1991. Solids, communition, and grading. In Slurry Handling, edited by N. P. Brown and
N. I. Heywood. New York: Elsevier Applied Sciences.
Gates, L. E., J. R. Morton, and P. L. Fondy. 1976. Selecting agitator systems to suspend solids in liquids.
Chemical Engineering, May 24.
Holmes, J. A. 1957. A contribution to the study of comminution, a modified form of Kick’s law.
Trans. Inst. Chem. Engrs., 35, 125–156.
Mular, A. L. and N. A. Jull. 1978. The selection of cyclone classifiers, pumps, and pump boxes for
grinding circuits. In Mineral Processing Plant Design, A. L. Mular and R. B. Bhappu (Eds.).
New York: Society of Mining Engineers.
Oldshue, J. Y. 1983. Fluid Mixing Technology. New York: Chemical Engineering.
Stephiewski, W. Z. and C. N. Keys. 1984. Rotary-Wing Aerodynamics. New York: Dover Publications.
Stone, R. 1971. Types and costs of grinding equipment for solid waste water carriage. Paper 19 in
Advances in Solid–Liquid Flow in Pipes and Its Applications, edited by I. Zandi. New York:
Pergamon Press, pp. 261–269:
DENVER-SALA. 1995. Selection Guide for Process Equipment. Colorado Springs: Svedala Industries.
Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow Slurry Pipeline Transportation.
Aedermannsdorf, Switzerland: Trans Tech Publications.
Weisman, J., and I. E. Efferding. 1960. Suspension of slurries by mechanical mixers. Am. Inst.
Chem. Eng. Journal, 6, 419–426.
Further readings
Su, Y. S., and F. A. Holland. 1968. Agitation and mixing of non-Newtonian fluids. Chem. &
Process. Eng., 49, 77–79.
Turner, H. E., and H. E. McCarthy. 1965. Fundamental analysis of slurry grinding. AIChE, 15,
581–584.

CHAPTER 4
HETEROGENEOUS FLOWS
OF SETTLING SLURRIES
4-0 INTRODUCTION
A practical engineer sifting through the literature on slurry flows would be astonished by
the number of different equations. Since the work of the French scientists Durand and
Condolios in 1952, and British scientists Newitt et al. in 1955, engineers and scientists
have continued to develop new equations for deposition velocity and friction losses.
This chapter reviews the evolution of models of Newtonian slurry flows from wellgraded
and uniform particle sizes to complex mixtures of coarse and fine particles. For
this purpose, some equations are listed in a historical context, to demonstrate their evolution
to the reader. The sheer number of equations demonstrates how complicated heterogeneous
flows are.
A number of factors interact in a horizontal pipe. The flow takes the form of different
regimes, and includes everything from a simple stationary bed at low speed to a pseudohomogeneous
flow at high speeds. For each regime, equations have been developed over
the years to account for the mean particle diameter, the diameter of the conduit, the density
of the particles, their drag coefficient, the speed of flow, etc. As a result, there are many
angles from which a heterogeneous flow of settling solids can be examined. The followers
of Durand and Condolios put great emphasis on the drag coefficient of the solid particles,
whereas the followers of Newitt prefer to focus on the terminal velocity. As we have
clearly demonstrated in Chapter 3, the drag coefficient and the terminal velocity are interrelated.
Examining a flow of slurry is often an exercise of playing with a murky liquid in
which very little can be seen. Sand does not behave like coal and there is not a single universal
law that may apply to the transportation of solids by liquids. It is therefore important
to rely on database, historical information, and empirical data.
In the past, engineers have tried to simplify the complexity of slurry flows by defining
certain transition velocities. With the use of modern research tools, there is an emerging
approach of rejecting the concept of an abrupt change from one state of flow to another,
and a tendency to consider such a change over a band of the speed. Different approaches
have been developed to examine the mixture of coarse and fine particles from superimposed
layers to two-layer models.
This book is intended for engineers, and various examples are included in the text. The
purpose of such examples is to simplify the use of complex equations. With modern personal
computers, which use simple languages such as quick basic, an engineer can efficiently
calculate friction losses for a heterogeneous slurry flow.
4.1

4.2 CHAPTER FOUR
4-1 REGIMES OF FLOW OF A
HETEROGENEOUS MIXTURE IN
HORIZONTAL PIPE
The history of slurry pipelines was briefly presented in Section 1-10 of Chapter 1. Two
schools are credited for laying the foundations of modern hydro-transport engineering;
SOGREAH in France, and the British Hydro-mechanic Research Association of the United
Kingdom. Starting in 1952, Durand and Condolios of SOGREAH published a number
of studies on the flow of sand and gravel in pipes up to 900 mm (35.5 in) in diameter.
Based on the specific gravity of particles with a magnitude of 2.65, they proposed to divide
the flows of nonsettling slurries in horizontal pipes into four categories based on average
particle size as follows:
� Homogeneous suspensions for particles smaller than 40 �m (mesh 325)
� Suspensions maintained by turbulence for particle sizes from 40 �m (mesh 325) to
0.15 mm (mesh 100)
� Suspension with saltation for particle sizes between 0.15 mm (mesh 100) and 1.5 mm
(mesh 11)
� Saltation for particles greater than 1.5 mm (mesh 11)
This initial classification was refined over the next 18 years by Newitt et al. (1955),
Ellis and Round (1963), Thomas (1964), Shen (1970), and Wicks (1971). Due to the interrelation
between particle sizes and terminal and deposition velocities, the original classification
proposed by Durand has been modified to four flow regimes based on the actual
flow of particles and their size. Referring to Figures 4-1 and 4-2, there are four main
regimes of flow in a horizontal pipe
1. Flow with a stationary bed
2. Flow with a moving bed and saltation (with or without suspension)
3. Heterogeneous mixture with all solids in suspension
4. Pseudohomogeneous or homogeneous mixtures with all solids in suspension
velocity
Fully suspended
lenticular
deposits
stationary
deposits
with ripples
suspended with moving bed
suspended with saltation
blocked
pipe
FIGURE 4-1 Flow regimes of heterogeneous flows in terms of speed versus volumetric concentration
(after Newitt et al., 1955).

particle size
Flow with a stationary bed
Two special cases shown in Figure 4-1 are not considered to be at the limits of these
regimes of flows. They are lenticular deposits at very low speeds but low solid concentration,
and blocked pipes at high solid concentration.
Slurry flows that have some form of segregation or separation of solids in layers are
called “heterogeneous” flows, whereas the slurries themselves are called “settling” slurries.
4-1-1 Flow With a Stationary Bed
When the slurry flow speed is low, the bed thickens. As the fluid above the bed tries to
move the solids by entrainment, they tend to roll and tumble. The particles with the lowest
settling speed move as an asymmetric suspension, whereas the coarser particles build
up the bed. As the speed drops even further, the pressure to maintain the flow becomes
quite high and eventually the pipe blocks up.
Flow with saltation and asymmetric suspension occurs above the speed of blockage.
This means that the coarser particles “sand up,” whereas the finer particles continue to
move. Certain tailing lines have exhibited this phenomenon. In fact, when a process plant
is built with a tailing line too large to handle the initial flow, the operator may choose to let
the bottom of the pipe sand up to reduce the effective cross-sectional area of the pipe. This
principle has been successfully applied to pipelines in a variety of countries. Saltation can
eventually lead to blockage of a pipe. It may result in a number of problems, such as water
hammer, wear, and freezing in cold climates. Most engineering specifications require that
the pipeline be designed to operate at speeds higher than those associated with saltation.
4-1-2 Flow With a Moving Bed
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
Flow with a moving bed
with or without suspensions
Heteregeneous flow with all
solids in suspension
Flow as a homogeneous or
pseudohomogeneous suspension
Mean velocity
FIGURE 4-2 Flow regimes of heterogeneous flows in terms of particle size versus mean velocity
(after Shen, 1970).
When the speed of the flow is low and there are a large number of coarse particles, the
bed moves like desert sand dunes. The top particles are entrained in the moving fluid
4.3

4.4 CHAPTER FOUR
above the bed. Consequently, the upper layers of the bed move faster than the lower layers
in a horizontal pipe. If the mixture were composed of a wide range of particles with
different sizes and settling velocities, the bed would be composed of the particles with the
highest settling speed. Particles with a moderate settling speed are maintained in an asymmetric
suspension, with most particles concentrated in the lower half of the pipe, whereas
the particles with the lowest settling speed move as a symmetric suspension
4-1-3 Suspension Maintained by Turbulence
As the flow speed increases, turbulence is sufficient to lift more solids. All particles move
in an asymmetric pattern with the coarsest at the bottom of a horizontal pipe covered with
superimposed layers of medium- and fine-sized particles. Many particles may strike the
bottom of the pipe and rebound. Wear on the bottom of the pipe must be taken into account
in maintenance schedules and the pipes must be rotated at intervals suggested by
the slurry engineer in order to maintain an even wear pattern of the internal wall of the
pipe. Although the flow is not symmetric, from the point of view of power consumption,
this regime may be the most economical for transporting a certain mass of solids.
Wilson (1991) calls all flows below V 3 fully stratified flows and all flows above V 3
fully suspended flows. The transition from fully stratified to fully suspended flows is considered
by this author to be fairly complex and should be represented by sigmoid or ogee
curves. It is a transition over a range of the speed and not an abrupt transition at a single
value of the speed. The work of Wilson and colleagues will be examined in Section 4-4-5.
4-1-4 Symmetric Flow at High Speed
At speeds in excess of 3.3 m/s (10 ft/s), all solids may move in a symmetric pattern (but
not necessarily uniformly). Sometimes this flow is called pseudohomogeneous because of
its symmetry around the pipe axis. Power consumption is a linear relationship of the static
head multiplied by the velocity, but is proportional to the cube of velocity needed to
overcome friction losses. Power consumption in pseudohomogeneous mixtures of coarse
and fine particles may be excessive for long pipelines. Pseudohomogeneous mixtures of
fine or ultrafine particles may occur at speeds as low as 1.52 m/s (5 ft/s). One definition of
fine and coarse particles was explained Govier and Aziz (1972), who proposed the following:
� Ultrafine particles: d p < 10 �m (mesh 1250), where gravitational forces are negligible.
� Fine particles: 10 �m < d p < 100 �m (mesh 1250 < d p < mesh 140), usually carried fully
suspended but subject to concentration gradients and gravitational forces.
� Medium sized particles: 100 �m < d p < 1000 �m (mesh 240 < d p < mesh 15), will
move with a deposit at the bottom of the pipe and with a concentration gradient.
� Coarse particles: 1000 �m < d p < 10,000 �m (0.039 in < d p < 0.394 in). These are seldom
fully suspended and form deposits on the bottom of the pipe.
� Ultracoarse particles are larger than 10 mm (0.4 in). These particles are transported as
a moving bed on the bottom of the pipe.
Considering particle sizes while ignoring their density is meaningless. Practical engineers
do shift the boundaries between different sizes based on the density of the particles.
There is no question that beads of high-density polyethylene will behave differently than

sand particles with the same average diameter because the former is lighter than water
while the latter is 2.65 times heavier than water.
4-2 HOLD-UP
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
The previous section describes how different layers of solids move with different speeds,
from the bottom, coarser particles, to the finer particles at the top of the horizontal pipe.
The theory of hold-ups complicates this process, however. Hold-ups are due to velocity
slip of layers of particles of larger sizes, particularly in the moving bed flow pattern.
Newitt et al. (1962) conducted speed measurements of a slurry mixture in a horizontal
pipe. In the case of light Plexiglas pipe, zircon or fine sand did not result in local slip; particles
and water moved at the same speed. However, for coarse sand and gravel, they observed
asymmetric suspension and a sliding bed. They also observed that in the upper layers
of the horizontal pipe, the concentrations of larger particles were the same as for finer
solids, but were marked by differences in the magnitude of the discharge rate of the lower
layers.
4.3 TRANSITIONAL VELOCITIES
The four regimes of flow described in Section 4-1 can be represented by a plot of the
pressure gradient versus the average speed of the mixture (Figure 4-5). The transitional
velocities are defined as
� V 1: velocity at or above which the bed in the lower half of the pipe is stationary. In the
upper half of the pipe, some solids may move by saltation or suspension.
Ratio distanc e from bottom of pipe
to the inner diameter (y/D ) I
1.0
0.8
0.6
0.4
0.2
0
0
0.03
C
v
0.07
0.10
0.14
0.05 0.1 0.15 0.20
Discharge solids concentration Cy
FIGURE 4-3 Distribution of concentration of solids in a pipe versus average volumetric
concentration.
4.5

4.6 CHAPTER FOUR
� V 2: velocity at or above which the mixture flows as an asymmetric mixture with the
coarser particles forming a moving bed.
� V 3 or V D: velocity at or above which all particles move as an asymmetric suspension
and below which the solids start to settle and form a moving bed.
� V 4: velocity at or above which all solids move as a symmetric suspension.
Volumetric Concentration (%)
30
20
10
0
0
1 2 3 4 5
Velocity (m/s)
FIGURE 4-4 Simplified concept of particle distribution in a pipe as a function of volumetric
concentration and speed.
Pressure drop per unit of length
Velocity (ft/sec)
0 5 10 15 20
1
2
slurry
3
V V V V 1 2 3
4
stationary bed
moving bed
asymmetric flow
direction of flow
4
water
symmetric flow
6
Speed of flow
FIGURE 4-5 Velocity regimes for heterogeneous slurry flows.

V 3 is effectively the deposition velocity, often called in the past the Durand velocity
for uniformly sized coarse particles. It is no longer recommended that it be called the Durand
velocity, as tests in the last 20 years have led to new equations that include the effects
of particle size and composition of the slurry. The magnitude of the velocity depends
on the volumetric concentration (Figure 4-7).
4-3-1 Transitional Velocities V 1 and V 2
The transitional velocity V 1 is obviously not used for the operation of slurry lines. It may
be of interest in lab research, instrumentation, and monitor of start-up.
The transitional velocity V 2 is determined individually from pressure measurements of
the pressure gradient. The main focus of the tests is to determine the height of the bed and
to derive a stratification ratio.
Wilson (1970) developed a model for the incipient motion of granular solids at V 2. He
assumed a hydrostatic pressure exerted by the solids on the wall and proposed the following
equation:
1
� �L
� �� + �sin � – �P � – sin � cos � Rw ��s �� ��
� L
4
� tan �r
�s(S – 1) Cvb(sin � – cos �)g
= ���
(4-1)
2
where
(�P/L) 2 = pressure gradient at 2
� = half the angle subtended at the pipe center due to the upper surface of the bed,
in radians
�s = coefficient of static friction of the solid particles against the wall of the pipe
Rw = cross-sectional area of the bed divided by the bed width
�r = angle of repose of the solid particles
Cumulative passing
(%)
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
100
10
FIGURE 4-6 Concept of d 50 by cumulative passing percentage versus particle size.
� Di
0
0 10 100 1000
d 50
Particle mesh diameter ( m)
4.7

4.8 CHAPTER FOUR
S = ratio of density of solids to density of liquid
Cvb = volume fraction solids in the bed
(When USCS units are used, express density in slugs/ft3 rather than lbm/ft3 ).
For 0.7 mm (mesh 24) sand with water in a 90 mm (3.5 in) pipe, Wilson measured � =
0.35 and concluded that the assumptions of hydrostatic distribution of the granular pressure
were correct.
4-3-2 The Transitional Velocity V 3 or Speed for Minimum
Pressure Gradient
The transitional velocity V3 is extremely important because it is the speed at which the
pressure gradient is at a minimum. Although there is evidence that solids start to settle at
slower speeds in complex mixtures, operators and engineers often referred to transitional
velocity as the speed of deposition.
Durand and Condolios (1952) derived the following equation for uniformly sized sand
and gravel:
VD = V3 = FL{2 · g · Di[(�s – �L)/�L]} 1/2 (4-2)
where
FL = is the Durand factor based on grain size and volume concentration
V3 = the critical transition velocity between flow with a stationary bed and a heterogeneous
flow
Di = pipe inner diameter (in m)
g = acceleration due to gravity (9.81 m/s)
�s = density of solids in a mixture (kg/m3 )
�L = density of liquid carrier
The Durand factor FL is typically represented in a graph for single or narrow graded
particles, as in Figure 4-7 after the work of Durand (1953). However, since most slurries
Durand Velocity Factor
F
D
2.0
1.0
0
C = 5%
V
C = 2%
V
C
V
= 15%
C
V
= 10%
0 1.0 2.0 3.0
Particle diameter (mm)
C
V
= 15%
C
V
= 5%
For single or narrow
graded slurries
Based on Schiller
equation using d
50
FIGURE 4-7 Durand velocity factor versus particle size—comparison between the conventional
values for single graded slurries and Schiller’s equation using d 50 for wide graded slurry.

are mixtures of particles of different sizes, this plot is considered too conservative. The
Durand velocity factor has been refined by a number of authors.
In an effort to represent more diluted concentrations, Wasp et al. (1970) proposed including
a ratio between the particle diameter and the pipeline diameter. Wasp proposed
the use of a modified factor F� L so that
1/2
VD = V3 = F� L�2gDi � � � 1/6
�S – �L dp (4-3)
Schiller and Herbich (1991) proposed the following equation for the Durand velocity
factor based on the d 50 of the particles:
F L = {(1.3 × C v 0.125 )[1 – exp (–6.9 d50)]} (4-4)
where d 50 is expressed in mm.
Some reference books define a Froude number as Fr = F L · �2�. The particle size d 50
is the statistically determined particle size below which half (or 50%) would be equal or
smaller to that set size. The following example illustrates the concept of d 50.
Example 4-1
A sample of slurry is sieved for particle size. The data is collected in the laboratory (see
Table 4-1). Plot the data on a logarithmic graph and determine the d50. Solution
The data is plotted in Figure 4-6; the d50 is determined to be 145 �m.
Example 4-2
A slurry mixture has a d 50 of 300 �m. The slurry is pumped in a 30 in pipe with an
ID of 28.28�. The volumetric concentration is 0.27. Using Equations 4-4 and 4-2, determine
the speed of deposition for a sand–water mixture if the specific gravity of sand is
2.65.
Solution in SI Units
From Equation 4-4:
F L = (1.3 × 0.27 0.125 )(1-exp (–6.9 × 0.3))
F L = 1.1 × 0.8738
F L = 0.964
From Equation 4-2 the deposition velocity is
TABLE 4-1 Data for Example 4-1
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
� �L
V 3 = 0.964 (2 × 9.81 × 28.25 × 0.0254 × 1.65) 1/2
V 3 = 4.64 m/s
Particle size (�m) 425 300 212 150 106 75 53 45 38 –38
Cumulative 97.2 87.1 68.3 51.3 35.9 20.5 14.5 11.8 10.8 —
passing (%)
� Di
4.9

4.10 CHAPTER FOUR
Solution in USCS Units
From Equation 4-4:
FL = (1.3 × 0.270.125 )(1 – exp (–6.9 × 0.3))
FL = 1.1 × 0.8738
FL = 0.964
From Equation 4-2 the deposition velocity is
V3 = 0.964 (2 × 32.2 × 28.25/12 × 1.65) 1/2
V 3 = 15.25 ft/sec
Various curves have been published for the magnitude of F L. They are often plotted
for a single graded size and use difficult to read logarithmic scales. For the sake of accuracy,
Table 4-2 tabulates the magnitude of F L between 0.08 mm < d 50 < 5mm on the basis
TABLE 4-2 The Coefficient F L Based on Schiller’s Equation Using the d 50 of the
Particles for Particles Between 0.080 and 5 mm for Volumetric Concentration up to
30%. F L = {(1.3 × C v 0.125 )[1 – exp(–6.9 d50)]}
d 50 (mm) C V = 0.05 C V = 0.10 C V = 0.15 C V = 0.20 C V = 0.25 C V = 0.30
0.08 0.379 0.414 0.435 0.451 0.464 0.474
0.10 0.446 0.486 0.511 0.530 0.545 0.557
0.12 0.503 0.549 0.577 0.599 0.616 0.630
0.14 0.554 0.604 0.635 0.658 0.677 0.693
0.16 0.598 0.652 0.686 0.711 0.731 0.748
0.18 0.636 0.693 0.729 0.756 0.777 0.795
0.20 0.669 0.730 0.768 0.796 0.818 0.837
0.25 0.735 0.801 0.843 0.874 0.898 0.919
0.30 0.781 0.852 0.896 0.929 0.955 0.977
0.35 0.814 0.888 0.934 0.968 0.995 1.018
0.40 0.837 0.913 0.961 0.996 1.024 1.048
0.45 0.854 0.931 0.980 1.015 1.044 1.068
0.50 0.866 0.944 0.993 1.029 1.058 1.083
0.55 0.874 0.953 1.002 1.039 1.069 1.093
0.60 0.880 0.959 1.009 1.046 1.076 1.101
0.65 0.884 0.964 1.014 1.051 1.081 1.106
0.70 0.887 0.967 1.017 1.055 1.084 1.109
0.75 0.889 0.969 1.020 1.057 1.087 1.112
0.80 0.890 0.971 1.021 1.059 1.089 1.114
0.85 0.891 0.972 1.023 1.060 1.090 1.115
0.90 0.892 0.973 1.023 1.061 1.091 1.116
1.00 0.893 0.974 1.0245 1.062 1.092 1.1172
1.5 0.8939 0.9748 1.0255 1.063 1.0931 1.1183
2 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
2.5 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
3.0 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
3.5 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
4.0 0.8940 0.9749 1.0255 1.063 1.0932 1.1184
5.0 0.8940 0.9749 1.0255 1.063 1.0932 1.1184

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
of Schiller’s equation. The magnitude of FL based on d50 is smaller than values published
in the literature for single graded slurry mixtures (lab mixtures using a uniform size of
particles). A number of authors have confirmed that this is the case (Warman International
Inc., 1990).
In order to compare the conventional magnitude of FL based on single and narrow
graded particles to the Schiller equation, both ranges of FL are plotted in Figure 4-7.
With a more complex approach that takes into account the actual viscosity of the slurry
mixture and the density of the particles, Gillies et al. (1999) developed an equation for
the Froude number F in terms of the Archimedean number (which we will discuss in Section
4-4-5 for stratified coarse flows):
4
3 Ar = d p �L(�s – �L)g (4-5)
To estimate the deposition velocity V3, Gilles et al. (1999) developed an equation for
the Froude number based on the Archimedean number:
Fr = aArb (4-6)
where
Fr = FL · �2�
For Ar > 540, a = 1.78, b = –0.019
For 160 < Ar < 540, a = 1.19, b = 0.045
For 80 < Ar < 60, a = 0.197, b = 0.4
For Ar < 80, the Wilson and Judge (1976) equation can be used, which expressed the
Froude number as
Fr = (�2�)�2.0 + 0.30 log dp 10� �� (4-7)
This correlation is useful in the range of 10 –5 < (dp/DiCD) < 10 –3 �
DiCD .
To determine the drag coefficient, the actual density of the liquid should be used,
whereas the viscosity should be corrected for the presence of fines.
Example 4-3
Water at a viscosity of 0.0015 Pa · s (0.0000313 slugs/ft-sec) is used to transport sand
with an average particle diameter of 300 �m (0.0118 inch). The volumetric concentration
is 0.27. The pipe’s inner diameter is 717 mm (28.35�). Using the Gilles equation (Equation
4-6), determine the deposition velocity if the specific gravity of sand is 2.65. Assume
CD = 0.45.
Solution in SI Units
d 50
� Di
� 3�L 2
= = 0.4 × 10 –3
0.3
�
717
Iteration 1
Assuming C D > 10 –3 , by the Wilson and Judge correlation (Equation 4-7):
Fr = (�2�)�2.0 + 0.30 log 0.003
10����� 0.717 × 0.45
Fr = 1.54
FL = Fr/�2� = 1.54/�2� = 1.09
4.11

4.12 CHAPTER FOUR
The specific gravity of the mixture is determined as:
Sm = Cv(Ss – SL) + SL = 0.27 (2.65 – 1) + 1 = 1.446
VD = FL�[2�g�D� i(��� s/��� L�–� 1�]� = 4.82 m/s
Iteration 2
Ar = = 258.98
for 160 < Ar < 540, a = 1.19, b = 0.045.
From equation 4.6:
Fr = aArb = 1.19 · 258.980.045 = 1.53
FL = F/�2� = 1.53/�2� = 1.082
VD = FL[2gDi(�s/�L – 1)] 0.5
4 × 9.81 (3 × 10 –4 ) 3 × 1000 (1650)
����
3(1.5 × 10 –3 ) 2
Solution in USCS Units
V D = 1.082[2 · 9.81 · 0.717 · (1.65)] 0.5 = 5.21m/s
d 50
� Di
= = 0.4 × 10 –3
0.00118
�
28.23
Iteration 1
Assuming C D > 10 –3 , by the Wilson and Judge correlation (Equation 4-7):
Fr = (�2�)�2.0 + 0.30 log 0.00118
10����� Fr = 1.54
FL = 1.54/2 = 1.09
The specific gravity of the mixture is determined as:
Sm = Cv(Ss – SL) +SL = 0.27(2.65 – 1) + 1 = 1.446
VD = 1.09[2 · 32.2 · (28.23/12) (2.65 – 1)] 0.5
VD = 17.23 ft/sec
Iteration 2
The particles’ diameter is 0.984 · 10 –3 ft
The density of water is 1.93 slugs/ft3 The density of sand is 5.114 slugs/ft3 Water dynamic viscosity is 0.0000313 slugs/ft-sec
4(0.984 · 10
Ar = = 259
–3 ) 3 28.23 × 0.45
× 1.93(5.114 – 1.93) · 32.2
������
3(0.0000313) 2
for 160 < Ar < 540, a = 1.19, b = 0.045.
From equation 4.6,
Fr = aArb = 1.19 · 258.980.045 = 1.53
FL = 1.53/�2� = 1.082
VD = FL[2gDi(�s/�L – 1)] 0.5
V D = 1.082[2 · 32.2 · 2.35 · (1.65)] 0.5 = 17.1 ft/s

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.13
The solution by the Gilles equation is within the limits set by Schiller in Example 4-2.
In these two different examples, we applied two different formulae but obtained consistent
results. This demonstrates the sensitivity of approaches to equations derived from
empirical equations. It may be necessary sometimes try to solve a problem using two different
equations, and to use common sense when similar results are obtained.
Table 4-3 presents values of the Archimedean number, the resultant magnitude of the
factor F L for particles d 50 in the range of 0.08 to 50 mm for a specific gravity of 1.5,
which is typical of coal-based mixtures. Most coals may be pumped with different sizes
of particles as discussed in Chapter 11. The viscosity may be due to the presence of cer-
TABLE 4-3 The Coefficient F L Based on Gilles Equation for Particles Between 0.080
and 50 mm of Specific Gravity of 1.500 (e.g., Coal) as a Function of Viscosity
� = 1 cP � = 5 cP � = 10 cP
_____________________ _______________________ _______________________
Archimedean Archimedean Archimedean
d 50 (mm) number Ar F L number Ar F L number Ar F L
0.08 3.35 Eqn 4-7 0.13 Eqn 4-7 0.033 Eqn 4-7
0.10 6.54 Eqn 4-7 0.26 Eqn 4-7 0.065 Eqn 4-7
0.12 11.3 Eqn 4-7 0.45 Eqn 4-7 0.113 Eqn 4-7
0.14 17.9 Eqn 4-7 0.72 Eqn 4-7 0.18 Eqn 4-7
0.16 26.8 Eqn 4-7 1.07 Eqn 4-7 0.27 Eqn 4-7
0.18 38.1 Eqn 4-7 1.53 Eqn 4-7 0.38 Eqn 4-7
0.20 52.3 Eqn 4-7 2.1 Eqn 4-7 0.52 Eqn 4-7
0.25 102 0.89 4.1 Eqn 4-7 1.02 Eqn 4-7
0.30 177 1.062 7.1 Eqn 4-7 1.77 Eqn 4-7
0.35 280 1.084 11.2 Eqn 4-7 2.80 Eqn 4-7
0.40 419 1.104 16.75 Eqn 4-7 4.19 Eqn 4-7
0.45 596 1.420 23.8 Eqn 4-7 5.96 Eqn 4-7
0.50 818 1.43 32.7 Eqn 4-7 8.18 Eqn 4-7
0.55 1088 1.437 43.5 Eqn 4-7 10.9 Eqn 4-7
0.60 1413 1.445 56.51 Eqn 4-7 14.1 Eqn 4-7
0.65 1796 1.451 72 Eqn 4-7 18 Eqn 4-7
0.70 2243 2.457 89.7 0.84 22.4 Eqn 4-7
0.75 2579 1.463 110.4 0.914 27.6 Eqn 4-7
0.80 3348 1.469 134 0.99 33.5 Eqn 4-7
0.85 4016 1.474 161 1.058 40 Eqn 4-7
0.90 4768 1.478 191 1.066 48 Eqn 4-7
1.00 6540 1.487 262 1.081 65 Eqn 4-17
2.00 52320 1.547 2093 1.455 523 1.12
3.00 176580 1.583 7063 1.489 1765 1.45
4.00 418560 1.610 16742 1.514 4185 1.475
5.00 817500 1.63 32700 1.533 8175 1.494
6.00 1415640 1.647 56505 1.55 14126 1.51
8.00 3348480 1.674 133939 1.575 33485 1.534
10.00 6540000 1.696 261600 1.595 65400 1.554
20.00 5.23 × 10 7 1.764 2092800 1.66 523200 1.616
30.00 17.7 × 10 8 1.805 7063202 1.698 1765800 1.654
40.00 41.86 × 10 8 1.835 16742404 1.726 4185601 1.682
50.00 81.75 × 10 8 1.859 32700008 1.749 81750020 1.703

4.14 CHAPTER FOUR
tain fines, as with peat coals or degradation of the coal during pumping over long distances,
or the use of a heavy medium such as magnetite at high concentration as a carrier
for coal in a water mixture.
Table 4-4 presents values of the factor F L for particles d 50 in the range of 0.08 to 50
mm for a specific gravity of 2.65, which is typical of sand and tar-sand-based mixtures.
The largest particles are often found in tar sand applications, with some contribution of
the tar or oil to viscosity. In this table, there was no need to present the Archimedean
number, as this was demonstrated in the previous table.
Newitt et al. (1955) preferred to express the speed of transition between “saltation”
flow and heterogeneous flow in terms of the terminal velocity of particles (previously discussed
in Chapter 3):
V 3 = 17 V t
(4.8)
The reader should refer to Equation 3-18, which corrects the terminal velocity of a single
particle to a mass of particles at higher volumetric concentration. Although Equation
4-8 has served as the basis of many models, we will later discuss the recent corrections
proposed by Wilson et al. (1992).
The approach to obtain the magnitude of V 3 is basically to conduct a test and measure
pressure drop per unit length of pipe. V 3 is considered to occur at the minima, or the point
of minimum pressure drop. W. E. Wilson (1942) expressed the pressure gradient of noncolloidal
solids by referring to clean water and by proposing a correction to the
Darcy–Weisbach equation (discussed in Chapter 2). He expressed the consumed power
due to friction by the following equation:
FIGURE 4-8 These taconite tailings must be pumped above a deposit velocity of 13 ft/s in
14� pipe due to the size of the particles.

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
TABLE 4-4 The Coefficient F L Based on Gilles Equation for Particles Between 0.080
and 50 mm of Specific Gravity of 2.65 (e.g., Sands and Oil Sands) as a Function of
Viscosity
d50 (mm) � = 1 cP, FL � = 5 cP, FL � = 10 cP, FL 0.08 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.10 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.12 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.14 Eqn 4-7 Eqn 4-7 Eqn 4-7
0.16 0.837 Eqn 4-7 Eqn 4-7
0.18 0.964 Eqn 4-7 Eqn 4-7
0.20 1.061 Eqn 4-7 Eqn 4-7
0.25 1.093 Eqn 4-7 Eqn 4-7
0.30 1.421 Eqn 4-7 Eqn 4-7
0.35 1.433 Eqn 4-7 Eqn 4-7
0.40 1.444 Eqn 4-7 Eqn 4-7
0.45 1.454 0.8 Eqn 4-7
0.50 1.462 0.906 Eqn 4-7
0.55 1.470 1.016 Eqn 4-7
0.60 1.478 1.065 Eqn 4-7
0.65 1.485 1.076 Eqn 4-7
0.70 1.491 1.087 Eqn 4-17
0.75 1.497 1.097 0.847
0.80 1.502 1.107 0.915
0.85 1.507 1.116 0.984
0.90 1.512 1.423 1.054
1.00 1.521 1.431 1.072
2.00 1.583 1.489 1.450
3.00 1.620 1.524 1.484
4.00 1.647 1.549 1.509
5.00 1.668 1.569 1.528
6.00 1.685 1.585 1.544
8.00 1.713 1.611 1.569
10.00 1.735 1.632 1.589
20.00 1.805 1.698 1.654
30.00 1.847 1.737 1.692
40.00 1.877 1.766 1.720
50.00 1.901 1.789 1.742
fDV CwVt � �
2gDi V
where
�Hf = head loss due to friction (in units of length)
fD = Darcy–Weisbach friction factor
C1 = constant
Equation 4-9 may also be reexpressed as
�H f = L� + C 1 � (4-9)
�H f g
� L
4.15
fDV C1CwVt g
= + � (4-10)
V
2
�
2Di

4.16 CHAPTER FOUR
By differentiating this equation with respect to V, we obtain for the minimal value
or
2 f DV
� 2Di
f DV
� Di
=
=
–C 1C wV t g
��
V 2
C 1C wV t g
� V 2
V 3 =
at constant friction factor fD, or
[C1CwVt gDi] Vmin = (4-11)
1/3
C1CwVt gDi ��
fD
��
f D 1/3
The magnitude of the Darcy friction factor for water flow in rubber lined and HDPE
pipe was computed for pipes from 2� to 18� and results presented in Chapter 2.
Wilson (1942) defined a factor C3 to determine whether the particles will settle to
form a bed:
C3 = (4-12)
If C3 > 1 most particles with a terminal velocity Vt will stay in suspension. If C3 � 1 most
particles with a terminal velocity Vt will settle out.
Whereas the equations of Newitt et al. (1955) and Wilson (1942) focused on the terminal
velocity, the work of Durand and Condolios (1952) focused on the drag coefficient for
sand and gravel. Zandi and Govatos (1967) and Zandi (1971) extended the work of Durand
to other solids and to different mixtures. They defined an index number as
V
Ne = (4-13)
At the critical value when Ne = 40, the flow transition between saltation and heterogeneous
regimes occurs. This statement infers that when Ne < 40 saltation occurs, and when
Ne � 40 heterogeneous flow develops. These results, based on a mixture of different particle
sizes, did not apply to the work of Blatch (1906), who concentrated on particles of a
uniform size (sand 20–30 mesh in water). Babcock (1967) reinterpreted this work and
demonstrated that for finely graded particles the transition occurred at an index number of
10. It is obvious that a complex mixture of particles of different sizes can increase the
magnitude of the transition index number.
2 2Vt ��
(�Hf fDgDi/L) 1/2 CD ��
CvDig(�s/�w – 1)
1/2
Example 4-4
Tailings from a mine consist of solids at a volumetric concentration of 20%. The specific
weight of the solids is 4.2. The pipe diameter is 8� with a wall thickness of 0.375� and
rubber lining of 0.5�. The particle Albertson shape factor is 0.7. The dynamic viscosity is
3 cP. The average d50 = 0.4 mm. Determine the speed of transition from saltation using
the Zandi approach as expressed by Equation 4-13.
Solution in SI Units
Pipe inner diameter Di = 8� – 2 · (0.5 + 0.375) = 6.25� = 158.75 mm

Iteration 1
Let us first assume a transition from saltation at 3 m/s and let us determine the drag coefficient
of the particles in water at the stated dynamic viscosity:
Rep = 1000 · 0.0004 · 3/0.003 = 400
From Table 3.7, CD = 1.09.
The transition from saltation occurs when Ne = 40. From Equation 4-13, using SI units:
Ne =
Ne = 9.43.
Iteration 2
Let us first assume a transition from saltation at 6 m/s and let us determine the drag coefficient
of the particles in water at the stated dynamic viscosity:
Rep = 1000 · 0.0004 · 6/0.003 = 800
From Table 3.7, CD = 1.15.
Ne =
Ne = 39.
The transition from saltation therefore occurs at a speed of 6.1 m/s.
Solution in USCS Units
Iteration 1
Pipe diameter = 8� – 2 · (0.375 + 0.5) = 6.25� = 0.521 ft
Let us first assume a transition from saltation at 10 ft/s and let us determine the drag coefficient
of the particles in water at the stated dynamic viscosity.
Particle size = 0.4 mm/304.7 mm = 1.3128 × 10 –3 ft
� = 0.003/47.88 = 6.265 × 10 –5 lbf-sec/ft2 Density of water = 62.3 lbm/ft3 /32.2 ft/sec = 1.935 slugs/ft3 1.935 slugs/ft
Re =
From Table 3.7, CD = 1.09.
3 × 1.3128 × 10 –3 ft × 10 ft/sec
�����
6.265 × 10 –5 lbf-sec/ft2 9 · �1�.0�9�
����
0.2 · 0.15875 · 9.81 · (4.2/1 – 1)
36 · �1�.1�5�
����
0.2 · 0.15875 · 9.81 · (4.2/1 – 1)
= 406
N e = 9.73.
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
N e =
100 · �1�.0�9�
����
0.2 · 0.5208 ft · 32.2 ft/sec · (4.2/1 – 1)
4.17
Iteration 2
Let us first assume a transition from saltation at 20 ft/s and let us determine the drag coefficient
of the particles in water at the stated dynamic viscosity:
Rep = 406 · (20/10) = 804

4.18 CHAPTER FOUR
From Table 3.7, CD = 1.15.
20
Ne =
Ne = 39.97.
The transition from saltation therefore occurs at a speed of 20 ft/sec.
2 · �1�.1�5�
����
0.2 · 0.5208 ft · 32.2 ft/sec · (4.2/1 – 1)
4-3-3 V4: Transition Speed Between Heterogeneous and
Pseudohomogeneous Flow
For the transition to pseudohomogeneous flows, Newitt et al. (1955) expressed the speed
in terms of the terminal velocity of particles as
V4 = (1800 gDiVt) 1/3 (4-14)
Refer to Chapter 3 and Equation 3-18 to calculate terminal velocity.
Govier and Aziz (1972) applied Newton’s law (i.e., CD = 0.44) for particles immersed
in a fluid to Equation 4-14 to yield
1/3 V4 = 38.7D i (S – 1) 1/6 4gdp � (4-15)
3CD
Govier and Aziz (1972) analyzed the work of Spells (1955) on solid particles with a
diameter 80 �m < dp < 800 �m (mesh 180 < dp < 20) and derived the following equation:
0.816 0.633 1.63 V4 = 134CD D i V t (4-16)
This equation was derived in USCS units with the diameter expressed in feet and the velocity
in feet per seconds.
Example 4-5
An ore with a specific gravity of 4.1 is to be pumped in a pseudohomogeneous regime in
a 24 in pipe with an ID of 22.23 in. The drag coefficient of the particles is assumed to be
0.44. The estimated flow rate is 12,000 US gpm. The particles have a sphericity of 0.72
and a diameter of 250 �m. Solve for V4. Solution in SI Units
Q = = 0.757 m3/s
Pipe ID = 22.25 × 0.0254 = 0.565 m
Cross-sectional area = 0.251 m2 Average speed of flow = 3.02 m/s
Sphericity = Asp/Ap = 0.72
dsp = �0�.7�2� � 2�5�0� = 218 �m
4 × 0.218 × 10
Vt = ��������
Vt = 0.142 m/s
–3 12,000 × 3.785
��
60,000
× 9.81 (4100 – 1000)
����
3 × 0.44 × 1000

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
By Newitt’s equation (Equation 4.14):
V4 = (1800 × 9.81 × 0.565 × 0.142) 1/3
V4 = 11.22 m/s
Alternatively using Equation 4.16:
Di = 1.854 ft
Vt = 0.466 ft/sec
0.816 0.633 1.63
V4 = 134C D D i V t
V4 = 134 × 0.440.816 × 1.8540.633 × 0.4661.63 = 29.19 ft/sec or 8.9 m/s
Solution in USCS Units
Q = 12,000 · 0.002228 = 26.736 ft3 /sec
Pipe ID = 22.25/12 = 1.854 ft
Cross-sectional area = 2.7 ft2 Average speed of flow = 9.9 ft/sec
Sphericity = Asp/Ap = 0.72
dsp = �0�.7�2� � 2�5�0� = 218 �m = 0.000715 ft
The density of water is 1.93 slugs/ft3 The density of solids is 7.913 slugs/ft3 Vt = �������
Vt = 0.465 ft/s
By Newitt’s equation (Equation 4.14):
V4 = (1800 × 32.2 × 1.854 × 0.465) 1/3
4 × 0.000715 × 32.2 (7.913 – 1.93)
����
3 × 0.44 × 1.93
V4 = 36.83 ft/sec
Alternatively, using Equation 4.16:
0.816 0.633 1.63
V4 = 134C D Di V t
V 4 = 134 × 0.44 0.816 × 1.854 0.633 × 0.466 1.63 = 29.19 ft/sec
4-4 HYDRAULIC FRICTION GRADIENT OF
HORIZONTAL HETEROGENEOUS FLOWS
4.19
Having been able to determine the speed for transition from one regime to another, the
slurry engineer must determine the loss of head per unit length due to friction, called the
hydraulic friction gradient (Equation 2-24). The hydraulic friction gradient for the slurry
(i m) is higher than the hydraulic friction gradient for an equivalent volume of water. Since
the first slurry pipelines were built, engineers and scientists have tried to correlate the
losses with slurry to those of an equivalent volume of water.
It was initially assumed that the friction losses would increase in proportion to the vol-

4.20 CHAPTER FOUR
umetric concentration of solids. A term im was then defined as the friction head of the mixture
in equivalent meters (or feet) of the carrier fluid (e.g., water) per unit of pipe length.
In Chapter 2, the friction hydraulic gradient was introduced by Equation 2-24 and is
defined as:
fDV i =
There are a number of models to predict friction losses and they are essentially based
on the interaction forces between solids and liquid carrier. Some use the drag coefficient,
others use the terminal velocity of the solids, and some consider the solids to be moving
as a bed with a layer of liquid and suspended fines above it.
To reflect the increase in friction head due to the volumetric concentration of solids,
Durand and Condolios (1952) proposed a nondimensional ratio
im – iL Z = � (4-17)
CviL where
Cv = the volumetric concentration of solids
im = pressure gradient for the slurry mixture in meters of water
iL = pressure gradient for an equivalent volume of water or carrier fluid in meters of water
The reader should not confuse head of slurry in meters or feet of slurry with meters or
feet of water. This is not a barometer or some instrument measuring pressure; for this reason
everything is kept consistent by using meters or feet of water. By itself, the term i relates
only to clear water having the same velocity as the slurry flow. It is convenient to
use water as a reference benchmark. (See Figure 4-9.)
2
�
2gDi
Head loss per pipe length
in equivalent (m/m) or (ft/ft)
C V2
water
C V1
C
V3
Average velocity of flow
i m
i L
FIGURE 4-9 Concepts of the hydraulic friction gradients i m and i L for slurry mixture and for
water.

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.4.1 Methods Based on the Drag Coefficient of Particles
Based on their analysis of test data from 11 references for sand in particle sizes ranging
up to 1 inch (25.4 mm), in pipes with a diameter range from 1.5 inch to 22 inch, and in
volumetric concentration up to 22%, Zandi and Govatos (1967) derived an equation for
the index number Ne (equation 4-13) in terms of the volumetric concentration, and some
empirical parameters:
� = (4-18)
Or from equation 4-13:
Ne = (4-19)
or � = CvNe. They plotted this function against a parameter � to express head loss as
� = = K(�) m (4-20)
where
iL = hydraulic gradient in terms of water density for a flow of clean water with a mean
velocity V
im = hydraulic gradient in terms of water density for a slurry flow with a mean velocity
V
K, m = constants
On a logarithmic scale they obtained:
For � > 10, K = 6.3 and m = –0.354
For � < 10, K = 280 and m = –1.93
The data is presented in Figure 4-10.
The dramatic change in values of K and m at � = 10 has encouraged researchers to develop
more sophisticated models that we shall review in the rest of this chapter.
Substituting for the value of 40 of the index coefficient, V3 may be expressed as
[40 CvDi g(�s – �w)/�w] V3 = (4-21)
1/2
V
�
�
Cv
im – iL �
CviL ���
2 1/2 C D
��
Dig(�s/�w – 1)
C D 1/4
4.21
Equation 4-21 is therefore a modified version of Equation 4-2. Equation 4-4 is a different
approach, as it accounts for particle size, which is often easier to measure than the
drag coefficient. Example 4-4 has shown that some iteration is necessary to obtain the velocity
at which the transition from saltation to asymmetric flow occurs.
Despite its simplicity, this method continues to be used by dredging engineers who
usually deal with sand and gravel mixtures of less than 20% concentration by volume.
The personal experience of the author is that often mines and dredging systems have
to be designed in very remote areas where there are no slurry labs to conduct tests. This is
an unfortunate fact, and sometimes an “overconservative” approach based on Durand,
Zandi, and other authors is the only alternative. However, the author does encourage engineers
of slurry systems to plan well ahead and test data to avoid very expensive field corrections.

4.22 CHAPTER FOUR
i L
� Cv i L
� =
10000
1000
100
10
1
.1
Durand
&
Condolios
FIGURE 4-10 The Zandi–Govatos factors for heterogeneous slurry flows. (From Zandi and
Govatos, 1967, reprinted with permission from ASCE.)
Shook et al. (1981) modified Zandi’s equation by proposing “in-situ concentration of
particles” C t rather than volumetric concentration:
� t = K� m
� t =
i m – i L
� iLC t
Zandi &
Govtes
RANGE OF
1 NUMBER
� 0–40
� 40–310
� 310–1550
� 1550–3100
.01
.01 0.1 1.0 10 100 1000
V
� =
2�C� D�
��
Di g(�s/�L – 1)
They measured a magnitude of m = –1 for one single type of coal in different pipe sizes.
They measured different values of K for different coals. The in-situ concentration C t remained
constant with speed, but the volumetric concentration of solids C v that could be
moved increased with V. This concept will be reexamined in Section 4.10 as part of the
two-layer models.
Example 4-6
Using Equations 4-19 to 4-20, consider the pumping of solids in a 305 mm (12 in) ID pipe
at a speed of 3.045 m/s (10 ft/s) and a volumetric concentration of 18%. Assume a drag
coefficient of 0.45 for the solid particles and a specific gravity of 2.65. Determine the increase
in the pressure gradient for flow in the pipe due to the presence of solids.

Solution in SI Units
V = 3.045 m/s
pipe Di = 0.305 m (12 in)
3.045
Ne = = 68.67
Ne 68.67
� = � = � = 381.5
Cv 0.18
� > 10 then K = 6.3 and m = –0.345
2 × �0�.4�5�
���
0.18 × 0.305 × (2.65 – 1)
� = = K � –0.345 = 0.81
= 0.81 × 0.18 = 0.145
im � = 1.145
iL
The slurry causes an increase of pressure gradient of 14.5% by comparison with water
at the same velocity.
Using the approach developed by Durand and Condolios, the fanning friction factor
for the slurry is correlated with the friction factor for an equivalent volume of water by
the following equation:
gDi(�s – �L) 3/2
fDm = fDL�1 + Kf Cv���� �
(4-22)
2 V �L�C� D�
Wasp et al. (1977) deducted that the coefficient Kf is between 80 and 150, depending
on the slurry. The most common value is actually 81 for most sands according to Govier
and Aziz (1972) (see Table 4-5).
Example 4-7
Using Equation 4-22, determine the correction for the friction factor for the portion of
solids in a slurry mixture of uniform size distribution. The slurry is pumped at the rate of
16,000 gpm in a rubber-lined 22.75� ID pipe. The volumetric concentration is 22%. Assume
K f = 85 and C D = 0.45. Use the Swain–Jaime equation to determine f L. The specific
gravity of the solids is 2.65. The dynamic viscosity of water is 2.7 × 10 –5 lbf-sec/ft 2 .
Solution in SI Units
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
i m – i L
� iL
Q = = 1.009 m3 /s
Pipe ID = 22.75 (0.0254) = 0.5778 m
Area of pipe = 0.262 m2 16,000 (3.785)
��
60,000
Velocity = 3.85 m/s
Dynamic viscosity = 0.00129 mPa · s
4.23

4.24 CHAPTER FOUR
TABLE 4-5 Correction of Friction Factor Due to Volumetric Concentration of Solids
Based on Equation 4-22 Assuming K = 81
gD i (� s – � L)
��
V 2 �L�C� D�
For the water:
1,000 (3.85) 0.5778
Re = ��� = 1,723,292
0.00129
Absolute roughness of rubber = 0.00015 m.
Relative roughness
� 0.00015
� = � = 0.0002596
DI 0.5778
0.25
fD = ����� = 0.0151
[log10{(0.0002596/3.7) + (5.74/1,723,2920.9 )} 2 ]
Solution in USCS Units
f Dm – f DL
� CV f DL
0.01 0.081 0.45 24.451
0.02 0.229 0.50 28.638
0.03 0.421 0.55 33.039
0.04 0.648 0.60 37.645
0.05 0.906 0.65 42.448
0.06 1.190 0.70 48.024
0.07 1.500 0.75 52.611
0.08 1.833 0.80 57.959
0.09 2.187 0.85 63.477
0.10 2.561 0.90 69.159
0.15 4.706 0.95 75.002
0.20 7.245 1.00 81.000
0.25 10.125
0.30 13.31
0.35 16.77
0.40 20.49
fm = fL�1 + 85 · 0.22� � 1.5
9.81 · 0.578 · 1.65
�� �
f m = f L · 18.067 = 0.273
Q = 35.63 ft3 /sec
22.75
Pipe ID = � = 1.896 ft
12
Area = 2.823 ft 2
gD i (� s – � L)
��
V 2 �L�C� D�
3.85 2 �0�.4�4�5�
f Dm – f DL
� CV f DL

Velocity = 12.62 ft/s
Dynamic viscosity = 1.29 cP = 0.0129 · 0.002089 lbf-sec/ft2 = 0.00002695 lbf-sec/ft2 Re = � � = 1.7 × 106 Absolute roughness of rubber = 0.00049 ft
Relative roughness of rubber = 0.0002596
fD = 0.0151
fm = 0.0151�1 + 85 × 0.22� � 1/5
32.2 × 1.896 × 1.65
���� � = 0.27
12.622 62.3 12.62 (1.896)
� ��–4
32.2 2.695 × 10
�0�.4�5�
An increase of the friction factor by 18-fold appears to be very high. The engineer in
charge of such a problem should seriously consider redesigning the system. At this stage,
the reader is encouraged to become familiar with the basic equations before applying
them to compound systems.
Equation 4-17 can be expressed in terms of the drag coefficients of the solid particles,
the pipe inner diameter, the density of the solid and liquid phases, the speed, and an experimental
factor Ke: im – iL Dig(�s/�L – 1) 1 3/2
Z = � = Ke������� (4-23)
2
CviL V �C�D�
Babcock (1968) was very critical of all equations using pressure gradients based on
the work of Durand and Condolios or their followers. Geller and Gray (1986) did not
agree with Babock’s criticisms and spelled out some of the misgivings. Govier and Aziz
(1972) did confirm that errors of the order on 40% have occurred in predicted values of Z,
but for all intents and purposes, these equations were the best available till the early
1970s. Herbich (1991) agreed with the value of 81 for most dredged sands and gravel.
Sand and gravel are typically dredged, then pumped at a volumetric concentration smaller
than 20%.
4.4.2 Effect of Lift Forces
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.25
It may be considered that the magnitude of the constant m is based on a very large magnitude
of data. In an innovative study at the Canada Center for Mineral and Energy Technology
(CANMET), Geller and Gray (1986) conducted an extended analysis that demonstrated
that lift forces had an effect on the pressure gradient. This study, rather than
dismissing the ideas of Durand, supported the previous work and gave it more importance.
Reviewing the work of Babock (1971), Geller and Gray (1986) indicated that for fine
to intermediate sizes (80/100 quartz sand with d � = 0.16 mm) the value of m was –0.25. In
addition, they concluded that lift forces are at a maximum when the volumetric concentration
C v is less than 0.23. For intermediate sands at higher volumetric concentration, the
lift forces seem to be minimal. This is an important factor to consider (for an understanding
of lift forces review Chapter 3, Section 3.1).
Furthermore, there is an important coefficient of mechanical friction � p, which results
from the sliding displacement between solids in contact, which is distinct from the viscous
friction.

4.26 CHAPTER FOUR
4-4-3 Russian Work on Coarse Coal
There are no universally accepted models for coarse coal. Work in the former Soviet
Union on coarse coal was reported by Traynis (1970) and reviewed by Faddick (1982).
From Russian data, the following two equations were reported. For deposition velocity:
V3 = [Dig] 1/2 [(�c – �hm)/�c] (4-24)
For the hydraulic gradient for coal:
1/3
���
[ fDLkCD] 1/3
�s – �L Cvc(�s – �hm) � � ��
�L k CdV�L where
Cv = total volumetric concentration of solids
Cvc = volumetric concentration of coarse solids
K = constant for coarse coal = 1.9
CD = drag coefficient considered to be 0.75 for the coarse coal fraction
�hm = density of heavy medium produced by the fines
i m = i L� 1 + C v� � + � · �� (4-25)
For the other terms, see Section 4-14.
�g�D� i�
Example 4-8
Coarse coal is to be pumped in a rubber-lined 18 in pipe steel with an inner diameter of 17
in. A screen analysis of the coal indicates that it has a distribution of 20% passing 200 microns.
The velocity of pumping is 4.5 m/s and the total weight concentration is 52%. The
specific gravity of the coal is 1.35. Determine the hydraulic gradient due to wall friction
in the horizontal pipeline. Assume a water dynamic viscosity of 1.2 cP, but correct for
viscosity due to solids using Einstein’s equation. Assume a drag coefficient of 0.75 for
the coarse coal.
Solution
Since the weight concentration is 52%, the specific gravity of the mixture is
Sm = SL/(1 – (CW(Ss– SL)/Ss) = 1/(1 – 0.52(1.35 – 1)/1.35) = 1.156
The volumetric concentration is
Cv = CwSm/Ss = 0.52 · 1.156/1.35 = 0.445
The weight concentration of the fines is 20%.
Density of the heavy medium carrying the fines is
Smf = SL/(1 – (CWf(Ss– SL)/Ss) = 1/(1 – 0.104(1.35 – 1)/1.35) = 1.028
Volumetric concentration of the fines = 0.2 · 0.445 = 0.089.
Calculations in SI Units
Pipe ID = 17 (0.0254) = 0.432 m
Area of pipe = 0.146 m2 Velocity = 4.5 m/s
The dynamic viscosity is corrected to take in account the presence of fines at a volumetric
concentration of 0.089. The dynamic viscosity of water is 1.2 cP, the Einstein–Thomas
equation is applied:

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
� = �L(1 + (2.5 · 0.089) + (10.05 · 0.0892 ) + 0.00273 exp (16.6 · 0.089)] = 1.314
�L = 1.577cP
1,000(4.5) 0.432
Re = �� = 1,232,720
0.001577
Absolute roughness of rubber = 0.00015 m.
Relative roughness:
� 0.00015
� = � = 0.000368
DI 0.432
fDL = = 0.0162
iL = fDV 2 /(2gDi) = 0.0162 · 4.52 0.25
�����
[log10{(0.000368/3.7) + (5.74/1,232,720
/(2 · 9.81 · 0.432) = 0.0387 m/m
Using Equation 4.25:
0.9 )} 2 ]
im = �1 + 0.445� � + 1350 – 1000 �(9�.8�1� ·� 0�.4�3�2�)� 0.8 · 0.445 · (1350 – 1028)
�� � �� · �����
im = 0.0387[1 + 0.445(0.35)] + [0.0368)] = 0.0815m/m
The presence of coal effectively doubles the head losses. The deposition velocity is
expressed from Equation 4-24:
V3 = [0.432 · 9.81] 1/2
[(1350 – 1028)/1350] 1/3
���
[0.0162 · 1.9 · 0.75] 1/3
1000 1.9 · 0.75 · 4.5
1000
V3 = 4.48 m/s
Calculations in USCS Units
Pipe ID = 17� = 1.417 ft
Area of pipe = 1.576 ft2 Velocity = 4.5 m/s = 14.76 ft/sec
The dynamic viscosity is corrected to take in account the presence of fines at a volumetric
concentration of 0.089. For the water, dynamic viscosity = 1.2 cP = 0.012 · 0.002089 lbfsec/ft2
= 2.507 × 10 –5 lbf-sec/ft2 . The Einstein–Thomas equation is applied:
� = � L(1 + (2.5 · 0.089) + (10.05 · 0.089 2 ) + 0.00273 exp (16.6 · 0.089)] = 1.314 � L
= 3.294 × 10 –5 lbf-sec/ft 2 .
For the water, the density is 1.934 slugs/ft3 .
Re = = 1.23 × 106 1.934 · 14.76 · 1.417
���
3.294 × 10 –5
Absolute roughness of rubber = 0.000492 ft.
Relative roughness:
�
� DI
0.000492
= � = 0.000368
1.417
4.27

4.28 CHAPTER FOUR
fDL = = 0.0162
iL = fDV 2 /(2gDi) = 0.0162 · 14.762 0.25
�����
[log10{(0.000368/3.7) + (5.74/(1.23 × 10
/(2 · 32.2 · 1.417) = 0.0387 ft/ft
Using Equation 4.25, and substituting density with specific gravity
6 ) 0.9 )} 2 ]
im = �1 + 0.44� � + 1.350 – 1 �(3�2�.2� �· 1�.4�1�7�)� 0.8 · 0.445 · (1.350 – 1.028)
� �� · �����
�
1 1.9 · 0.75 · 14.76
1.0
im = 0.0387[1 + 0.445(0.35)] + [0.0368)] = 0.0815ft/ft
The presence of coal effectively doubles the head losses. The deposition velocity is
expressed from Equation 4-24:
V 3 = [1.417 · 32.2] 1/2
V3 = 14.71 ft/sec
The coal slurry is therefore being pumped just above the deposition speed, and therefore
at the minimum pressure gradient for horizontal pipelines.
4-4-4 Equations for Nickel–Water Suspensions
Ellis and Round (1963) conducted tests on a mixture of nickel particles and water and derived
the following equation:
� = = K(�) m = 385 � –1.5 im – iL � (4-26)
CviL The constants K and m are therefore different from those reported by Zandi and Govatos
(1967) for sand particles, as expressed by Equation 4-20.
4-4-5 Models Based on Terminal Velocity
Newitt et al. (1955) conducted tests in pipes smaller than 150 mm (6 in) and proposed to
express Z in terms of the terminal velocity (instead of the drag coefficient).
Z = = K2� � (4-27)
where
K2 = an experimentally determined constant. For small pipes, K2 = 1100.
Vm = mean velocity of mixture
For solids of different sizes, Newitt suggested a weighted mean diameter as
dpm = � n
im – i �s – �L gDiVt � � �3 Cvi �L V m
dpimi/mt (4.28)
where
m i = the mass of solids with particle diameter of d p
m t = total mass of solids
[(1.350-1.028)/1.350] 1/3
���
[0.0162 · 1.9 · 0.75] 1/3
i=1

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.29
Hayden and Stelson (1968) proposed a modification of the Durand–Condolios equation
using the terminal velocity instead of the drag coefficient:
= 100� � 1.3
gDi[(�m – �L)/�L]Vt ���
(4-29)
V
Geller and Gray (1986) pointed out that the equations of Durand, Newitt, and Babcock
converged when m = –1.
Newitt et al. (1955) minimized the importance of lift forces when a bed cannot form
because of lift forces on particles. However, the work of Bagnold (1954, 1955, 1957) indicated
that the submerged weight of particles separated from the bed was transmitted to
the bed or the pipe wall under the same conditions. Thus, mechanical friction can contribute
to head loss.
It may be argued that sometimes it is easier to measure the terminal velocity rather
than the drag coefficient, particularly with oddly shaped particles. As Chapter 3 clearly
demonstrated, both parameters are interrelated.
2 im – iL �
CviL �g�d� ��� p( m�)/ ��� L�–� 1�)�
Example 4-9
The tailings from a small mine are pumped at a weight concentration of 40%. They consist
of crushed rock at a specific gravity of 3.2. The d85 of the particles is 1mm. For a flow rate
of 280 m3 /hr, a smooth high-density polyethylene pipe with an internal diameter of 138 mm
is selected. Using Newitt’s method as expressed By equations 4.27 and 4.29, determine the
head loss due to the presence of solids, assuming a dynamic viscosity of 1.8 cP.
Solution in SI Units
Pipe flow area = 0.25 · � · 0.1382 = 0.01496 m2 Average velocity of flow = Q/A = (280/3600)/0.01496 = 5.2 m/s
Particle Reynolds number using the density of water = Rep = 0.001 · 3.71 · 1000/0.0018
= 2063
Since Rep > 800, the flow is turbulent and Newton’s law is used to calculate the terminal
velocity:
Vt = 1.74(dp · g · (�p – �L)/�L) 1/2 = 1.74(0.001 · 9.81 · 2.1) 1/2 = 0.25 m/s
By Newitt’s method, the transition between saltation and motion occurs at 17Vt or
V3 = 17 · 0.25 = 4.25 m/s
Since the weight concentration is 40%, the specific gravity of the mixture is
Sm = SL/(1 – (CW(Ss– SL)/Ss) = 1/(1 – 0.4(3.1 – 1)/3.1) = 1.372
The volumetric concentration is Cv = (1.372 – 1)/2.1 = 0.177
Using equation 4.27, and assuming K2 = 1100,
Z = 1100 · (2.1) · (9.81 · 0.138 · 0.25/5.23 ) = 5.563
im/i = 1 + 0.177 · 5.563 = 1.985
Using equation 4.29:
= 100� � 1.3
9.81 · 0.138 · 2.1 · 0.25
��� = 11
5.2
im/iL = 1 + 0.177 · 11 = 2.95
2 [9.81 · 0.001 · 2.2) 1/2
im – iL �
CviL

4.30 CHAPTER FOUR
This example and the use of these two equations indicates that the empirical coefficients
of 1100 in the Newitt method for fine coal and sand, or the empirical coefficient of 100
for sand from the Hayden and Stelson equation, do not converge for similar results. Testing
would be recommended to confirm the magnitude of these coefficients.
4.5 DISTRIBUTION OF PARTICLE
CONCENTRATION IN COMPOUND SYSTEMS
The reader may be familiar with the concepts developed in the 1950s and 1960s on uniformly
graded solid particles. In reality, slurries often consist of a wide distribution of
particles. The coarser ones tend to move at the bottom of the horizontal pipe, and the finer
ones move above these bottom layers. Understanding the distribution of these particles
in layers above layers is essential for a correct estimation of the friction losses.
Initially, the work was done in the 1930s and 1940s on open channel flows and is
discussed in Chapter 6, Section 6-2-3. The distribution of volumetric concentration is
shown to be a function of depth of the liquid in an open channel flow, raised to a exponent.
The exponent is a function of the relation of the terminal velocity to the friction
velocity.
Ismail (1952) was the first to extend the approach of Vanoni to closed conduits. He focused
initially on rectangular closed conduits. This test work demonstrated that the concentration
was an exponential function:
C Vt Log10� � = (y – a) (4-30)
� �
CA Es
where
Es = the mass transfer coefficient
a = height of layer A above bottom of the conduit
y = distance from the lower boundary
C = volumetric concentration of the particle diameter under consideration
CA = volumetric concentration of height “A”
For many pipes, C/CA is considered by Wasp et al. (1977) to be 0.08 DI from the top of
the pipe. Wasp et al. (1977) examined the distribution of concentration of The Consolidation
Coal Company’s Ohio coal pipeline at a height of 8% from the bottom of the conduit
and at 8% from the top of the conduit; they reinterpreted the work of Ismail (1951) and
devised the following equation:
C 1.8 Vt log10 � = –��� (4-31)
CA �KxUf where
Uf is the friction velocity (discussed in Chapter 2)
Kx is the Von Karman constant
� = constant of proportionality
Hsu et al. (1971) reexamined the work of Ismail by proposing a polar coordinate system
(r, �) for the analysis of the distribution of concentration in a pipe:
C(r, �)
� C(0, 0)
V t
� Uf
r cos � cos �
� ��
RI me
= exp� � �� (4-32)

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
where
� = the angle from the horizontal
� = angle from the vertical starting at the lowest quadrant point
RI = inner diameter of pipe
r = local radius for a point in the flow
Equation 4-30 can be reduced to
C Vt log10 = � �
� � (constant) (4-33)
CA Uf
The extent by which the Von Karman constant Kx is suppressed by turbulence is difficult
to assess.
Ippen (1971) conducted an analysis of turbulent suspensions in open channel flows.
This work showed that the concentration close to the lower boundary was the most important
factor suppressing the Von Karman constant. This may not be astonishing when we
consider that beds of coarse particles form in this region at low speeds.
Hunt (1969) developed an equation for diffusion of heterogeneous flows:
d(Cv) ES � + (1 – Cv)CvVt = 0 (4-34)
d(y)
where Cv is the volumetric concentration of solids.
This equation shows that when coarse and fine particles are pumped together under
certain conditions, the flows may exhibit an increase in concentration of fine particles
with increasing height.
Example 4-10
Using Hunt’s equation, prove that the ratio of concentration at 0.08 DI from the top is the
concentration at pipe center expressed by
VR
log10��� = –1.8 Z
VRa
where
VR = Cv/1 – Cv a = the reference plane at 0.08 DI It has already been shown in Equation (4-31) that
C Vt log10��� = –1.8�
CA �KxUf Let us confirm that Hunt’s approach applies:
E s
dC v
� dy
+ (1 – C v)C vV t = 0
VR =
DC v
� dVR
C v
� 1 – Cv
= (1 – C v) 2
4.31

4.32 CHAPTER FOUR
But some of Hunt’s equation shows that
Then
Or
–V t
� Es
=
(1 – Cv) Cv = · = (1 – Cv) 2
dC dVR
dV
� � �
dVR dy
dy
E s
dC
� dy
–V t
� Es
This is the same as the Equation 4-34.
= � � (1 – Cv) 2
dC dC dVR dVR
� � � �
dy dVR dy
dy
–V t(1 – C v)C v
��
Es
dVR
Cv = � (1 – C
dy
v)
dVR
� (1 – Cv) + VtCv = 0
dy
dVR Cv Es � + Vt � = 0
dy (1 – Cv)
The approach discussed in the previous paragraph is sometimes classified as the distributed
concentration approach. The analysis is based on establishing the plane for reference
CA, usually at 0.08 diameter.
It has been demonstrated that
C
Vt log10��� = – 1.8�
CA �KxUf If � is assumed to be unity and there is no suppression for the Von Karman constant,
i.e., Kx = 0.4, then
C Vt log10��� = –4.5��� (4-35)
CA Uf
Thomas (1962) commented that the Durand–Condolios approach was limited to sand
and similar solids and proposed a more general criterion of evaluating flow of slurries in
terms of the ratio Vt/Uf or ratio of free-fall velocity to friction velocity. He indicated that
when
Vt � > 0.2 (4-36)
Uf
the solids would be transported as a heterogeneous slurry.
Charles and Stevens (1972) suggested that Equation 4.32 should be modified to correspond
to C/CA < 0.13, whereas Charles and Stevens’ criterion corresponds to C/CA < 0.27.
The Thomas criterion as expressed by Equation 4-31, corresponds to C/CA < 0.13,
whereas the Charles and Stevens’ criterion corresponds to C/CA < 0.27.

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
Thomas (1962) indicated that the minimum transport condition for particles depends
on a number of factors, and derived the following equation for glass beads:
= 4.90� �� � 0.60
� � 0.23
Vt dpUf 0 � �S – �L � � � � (4-37)
Uf � DiUf 0 �L
where
� = kinematic viscosity of water
Uf 0 = friction velocity at deposition for limiting case of infinite dilution
Thomas (1962) defined a critical friction velocity at which the slurry starts to deposit
for a given concentration as
UfC= Uf 0�1 + 2.8 (�C� V�)� � 1/3 Vt �
(4-38)
The approach of Thomas is implicit. It means that to predict U f, it is important to
measure friction loss as a function of velocity. It is then necessary to establish the deposition
velocity using Equations 4-34, 4-35, and 4-36.
4-6 FRICTION LOSSES FOR COMPOUND
MIXTURES IN HORIZONTAL
HETEROGENEOUS FLOWS
Many slurries resulting from dredging, cyclone underflow, and tailings disposal are not
pumped with single-sized particles. Some authors such as Newitt et al. (1955) proposed
the use of a weighted average particle diameter but Hill et al. (1986) proposed that the
particles should be divided. The finer particles would move as a heterogeneous flow,
while the coarser particles would move as a bed by saltation. The equations of friction
loss for each fraction or size of solids should be calculated as in Sections 4-4-1 and 4-4-3.
Hill et al. (1986), Wasp et al. (1977), and Gaesler (1967) demonstrated that this approach
worked well when applied to pumping water–coal mixtures.
The compound or heterogeneous–homogeneous system is the most important and
most common in slurry transportation. It involves coarse and fine particles. The fines
move as a homogeneous mixture while the remainder move as a heterogeneous mixture.
To conduct this analysis, the rheological and physical properties of the solids must be
known.
This method was pioneered by Wasp et al. (1977) and in some respects was further developed
by the “stratification model” described later on. The heterogeneous mixture or
bed motion is based on the method of concentration in relation to a reference layer, as described
by Equation 4-30.
The method proposed by Wasp et al. (1977) can be summarized as follows:
1. Divide the total size fraction into a homogeneous fraction using Durand’s equation.
2. Calculate the friction losses of the homogeneous fraction based on the rheology of the
slurry, assuming Newtonian flow.
3. Calculate the friction losses of the heterogeneous fraction using Durand’s equation.
4. Define a ratio C/C A for the size fraction of solids based on friction losses estimated in
steps 2 and 3.
� Uf 0
4.33

4.34 CHAPTER FOUR
5. Based on the value of C/C A, determine the fraction size of solids in homogeneous and
heterogeneous flows.
Re-iterate steps 2 to 5 until convergence of the friction loss.
Example 4-11
A nickel ore slurry needs to flow by gravity at a weight concentration of 28%. The design
flow rate is 1631 m3 /hr. The slurry was tested in a 159 mm pipeline with a roughness coefficient
of 0.016 mm at a weight concentration of 26.3%. The results of the pressure drop
versus speed are presented in Table 4-2. No data was made available on the drag coefficients
or terminal velocity of the solids.
The particle size distribution of the originally milled ore is presented in Table 4-3.
Special screens would be installed to screen away the coarsest particles (larger than 0.850
mm). Conducting a friction loss for a rubber lined steel pipe would be a better option.
(See Tables 4-6 and 4-7.)
The solids density was measured as 4074 kg/m3 . At a weight concentration of 26.3%,
this corresponds to a slurry density of 1244 kg/m3 . Volumetric concentration is
�m CV = CW = 0.08%
� �s
Using the Thomas–Einstein equation for dynamic viscosity correction:
� = �L(1 + (2.5 · 0.08) + (10.05 · 0.082 ) + 0.00273 exp(16.6 · 0.08)] = 1.274 · �L Analysis of Test Results
Water at a temperature of 20° Celsius has a dynamic viscosity of 1 mPa · s. Slurry viscosity
is therefore 1.274 mPa · s, and the Reynolds number is
1244(V)DI Re = �� = 155,256(V) = 294,986
–3
1.274 × 10
where V = 1.9 m/s
The slurry was tested in a pumping test loop. The lab tests indicated a pressure drop of
270 Pa/m at this velocity. The +0.850 mm solids were screened away prior to pump tests.
TABLE 4-6 Pressure Drop versus Speed in a 159 mm ID Steel Pipe at a Weight
Concentration of 26.3% (Example 4-11)
Temperature 20°C Temperature 35°C
______________________________________ ______________________________________
Velocity (m/s) Pressure drop (kPa) Velocity (m/s) Pressure drop (kPa)
0.61 0.063
1.00 0.085 1.00 0.079
1.5 0.175 1.51 0.169
1.9 0.270 1.91 0.259
2.3 0.360 2.30 0.358
2.7 0.525 2.70 0.487
3.1 0.688 3.11 0.628
3.5 0.847 3.50 0.793
4.0 1.046 4.00 0.988

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
TABLE 4-7 Particle Size Distribution Prior to Screening the
Coarsest Solids (Example 4-11)
Size (mm) Volumetric concentration
+ 0.850 14.3%
–0.850 to +0.400 1.61%
–0.400 to +0.200 1.91%
–0.200 to +0.105 1.41%
–1.05 to +0.044 1%
–0.044 79.8%
4.35
Table 4-8 indicates the new volumetric concentration of the solids in the slurry after
screening the +0.850 mm solids.
The method developed by Wasp et al. (1977) has been used very successfully over the
last 25 years for Newtonian slurries and will be used in the present calculations. The
roughness of a steel pipe is 0.046 mm. Assuming that the –0.044 mm particles were transported
by turbulence above the moving bed of coarser particles, the Swain–Jain equation
may be used in the range of 5000 < Re < 100,000,000 to determine the friction coefficient
of the homogeneous part of the mixture:
fD = = 0.017
where fD = the Darcy friction factor
For the density of 1244 kg/m3 , the pressure losses of the carrier fluid (including the
–0.044 mm) at a first iteration is therefore
0.017(1.9
Loss = = 240 Pa/m
The lab test measured 270 Pa/m; the losses due to the moving bed are therefore 31 Pa/m.
Using Table 4-8, apply the Wasp method for calculating the pressure losses of the moving
bed. It will be assumed initially that the –0.044 mm particles are part of the homogeneous
liquid layer above the bed.
It is essential first to determine the drag coefficient and the particle Reynolds number.
2 0.25
����
{log10 [(�/Di)/3.7 + 5.74/Re
) 1244
��
(2) 0.159
0.9 ]} 2
TABLE 4-8 Particle Size versus Volume Concentration in the Slurry (Example 4-11)
New volumetric Volumetric concentration
Original volumetric concentration C V in the slurry
concentration C V in the solids (at overall solids C V
Particle size (mm) in the solids (after screening) of mixture at 8%)
+0.850 14.3% — —
–0.850 to + 0.400 1.61% 1.88% 0.15%
–0.400 to + 0.200 1.91% 2.23% 0.178%
–0.200 to +0.105 1.41% 1.65% 0.132%
–1.05 to +0.044 1% 1.17% 0.093%
–0.044 79.8% 93.1% 7.45%

4.36 CHAPTER FOUR
Two cases will be considered: spheres and particles with an Albertson shape factor of 1.0
for the sake of simplicity.
To calculate the particle Reynolds number, the density of 1244 kg/m3 , viscosity of 1.3
mPas, and the speed of 1.9 m/s of the carrier fluid are used:
Rep = 1,818,154 dp where dp = the average particle size.
To calculate the drag coefficient of a sphere, the Turton equation (Equation 3.8a) is
used. Results are summarized in Table 4-9.
Wasp et al. (1977) recommend using Durand’s equation for each fraction of solids to
determine the increase in pressure losses due to the moving bed:
gDi(�s – �L)/�L 1.5
�Pbed = 82 �PLCvbed���� V 2 �C�D�
After determining the Darcy friction factor at the pipe diameter of 0.159 m and the
speed of 1.9 m/s at a liquid loss of 219 Pa/m, the loss due to each fraction becomes
1.5
�Pbed = 17,490 Cvbed� �
Results of calculations are presented in Table 4-10.
The total friction loss is therefore 240 Pa/m + 151.4 = 391.4. By comparison with the
measured 270Pa/m, the calculations for the bed are higher and can be refined by the
method of concentration using Equation 4-30:
log10� � = –1.8
At 391.4 Pa/m, the equivalent fanning factor is
391.4 = 2 ffV 2
1
�
�C�D�
C Vt � �
CA �KxUf �
�
Di
391.4(0.159)
fN = �� = 0.0069
2 2(1.9 )1,244
To calculate Uf, use Equation 2-25 from Chapter 2:
U = Um�( �f N/ �2�)� = 1.9�(0�.0�0�6�/2�)� = 0.1116 m/s
Assuming Kx = 0.4 and � = 1, we can iterate the results.
TABLE 4-9 Drag Coefficient for Particles in Example 4-11, Assuming
Spherical Shape
Drag coefficient
Particle size Average particle Particle Reynolds Drag coefficient for a particle with
distribution (mm) size (mm) number for a sphere shape factor of 1
–0.850 to + 0.400 0.63 1145 0.395 0.474
–0.400 to + 0.200 0.3 545 0.545 0.572
–0.200 to +0.105 0.15 272 0.706 0.7413
–1.05 to +0.044 0.07 127 1.02 1.07

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
TABLE 4-10 Calculated Losses for Each Fraction of Solids in the Moving Bed in the
Lab Test (Example 4-11)
To determine the terminal velocity, we turn to Chapter 3, Equation 3-7:
V t 2 =
C D =
4(� S – � L) gd g
��
3�LV t 2
4 (4.074 – 1.244) 9.81 d g
���
3 (1.244) C D
29.76 d
2 g
V t = �
CD
The iterated pressure loss is 349.7 Pa/m, which is still higher than the measured 270
Pa/m. For further iteration, the fanning factor must be recalculated:
349.7 = 2f fV 2
349.7 (0.159)
ff = �� = 0.00616
2 2 (1.9 ) 1244
Uf = 0.106 m/s
With this new iteration we are converging toward 107 + 240 = 347, which is above the
measured 270Pa/m.
Ellis and Round (1963) indicated that Durand’s equation coefficient of 82 was too
high for nickel suspensions. We may therefore divide 270/347 = 0.778 to obtain the new
value of 63.8 for K.
Pipeline Sizing for the Design Flow Rate of 1631 m 3 /hr at a Weight Concentration of 28%
The weight concentration of 28% corresponds to a volumetric concentration of 8.7% and
a mixture density of 1267 kg/m 3 using the solids density of 4074 kg/m 3 . The concentration
of solids in the bed is tabulated in Table 4-11. The flow of 1631 m 3 /hr corresponds to
0.453 m 3 /s.
Consider a 20� OD pipe with a wall thickness of 0.375�, rubber lined with a rubber
thickness of ¼�. The internal diameter of the pipe would be D I = [20 – 2(0.375+0.25)]
= 18.75� or 477 mm. The cross-sectional area of the pipe would be 0.178 m 2 and the average
flow speed of the slurry would be calculated as V = 0.453/0.178 = 2.55 m/s. Applying
the Thomas–Einstein equation to the volumetric concentration of 8.7% gives an
�
� Di
4.37
Calculated losses Calculated losses for
Particle size Average particle for spherical particles particles with Albertson
distribution (mm) size (mm) (Pa/m) shape factor of 1.0 (Pa/m)
–0.850 to + 0.400 0.63 58.31 50.87
–0.400 to + 0.200 0.3 53.85 51.93
–0.200 to +0.105 0.15 32.9 31.73
–1.05 to +0.044 0.07 17.56 16.87
Total for bed 162.62 151.4

4.38 CHAPTER FOUR
TABLE 4-11 Iteration for Calculated Losses for Each Fraction of Solids in the
Moving Bed, Based on the Distribution of Concentration—for Lab Tests (Example 4-11)
Average Drag Iterated
particle coefficient Terminal Iterated pressure
Particle size size for a velocity concentration loss
distribution (mm) (mm) sphere (mm/s) –1.8 V t�·K x·U f C/C A (Pa/m)
–0.850 to + 0.400 0.63 0.395 6.89 –0.27 0.537 31.31
–0.400 to + 0.200 0.3 0.545 4.047 –0.163 0.687 36.97
–0.200 to +0.105 0.15 0.706 2.515 –0.1015 0.79 26
–1.05 to +0.044 0.07 1.02 1.43 –0.057 0.877 15.4
Total for bed 109.68
effective viscosity of the mixture of 1.305 mPa · s at 20° C. The pipeline Reynolds number
is therefore
1267(2.55) 0.477
Re = �� = 1,180,931
–3
1.305 × 10
For commercially available rubber-lined pipes, the roughness is 0.00015 m.
Considering a 477 mm ID pipe, rubber lined, the relative roughness is therefore
0.000315. Applying the Swamee–Jain equation, the Darcy friction factor is calculated as
f D = 0.01578. Loss of carrier fluid is calculated as
0.01578 (2.55 2 ) 1,267
���
2 (0.477)
= 136.3 Pa/m
Using the Wasp method, and applying the Durand’s equation, the calculations yield
�Pbed = 63.8 (136.3)� � 1.5 9.81
��
1 1.5
�Pbed = (18,216) Cvbed�� �C�D� �
The drag coefficient is calculated at the particle Reynolds number using the speed of
2.55 m/s, viscosity of 1.305 mPa · s, and density of 1267 kg/m 3 . Re p = 2,475,747 (d p). Results
are presented in Table 4-12.
The Durand equation may then be applied to each fraction of solids. The results are
shown in Table 4-13.
Total losses for slurry mixture are therefore calculated as 136.3 + 165.9 = 302 Pa/m.
At 302 Pa/m, the equivalent fanning factor is
302 = 2f f V 2
2.55 2 �C�D�
�
�
Di
302 (0.477)
ff = �� = 0.0089
2 2 (2.55 ) 1244

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
To calculate Uf, we use Equation 2-15 from Chapter 2:
ff 0.0089
Uf = U�� � = 2.55���
2
2
4.39
Uf = 0.170 m/s
Assuming Kx = 0.4 and � = 1, we can iterate the results based on the distribution of concentration,
as per Table 4-14. Total friction losses = 136 + 129 = 265 Pa/m or 0.0217
m/m.
TABLE 4-12 Second Iteration for Calculated Losses for Each Fraction of Solids in the
Moving Bed, Based on the Distribution of Concentration—Lab Tests (Example 4-11)
Average Drag Iterated
particle coefficient Terminal Iterated pressure
Particle size size for a velocity concentration loss
distribution (mm) (mm) sphere (mm/s) –1.8 Vt�·K x·Uf C/CA (Pa/m)
–0.850 to +0.400 0.63 0.395 6.89 –0.287 0.516 30.1
–0.400 to +0.200 0.3 0.545 4.047 –0.173 0.671 36.13
–0.200 to +0.105 0.15 0.706 2.515 –0.108 0.78 25.66
–1.05 to +0.044 0.07 1.02 1.43 –0.061 0.868 15.24
Total for bed 107
TABLE 4-13 Drag Coefficient of the Solids in the Pipeline (Example 4-11)
Particle size Average particle Particle Reynolds Drag coefficient
Drag coefficient
for a particle with
distribution (mm) size (mm) number for a sphere shape factor of 1
–0.850 to + 0.400 0.63 1547 0.414 0.497
–0.400 to + 0.200 0.3 743 0.493 0.52
–0.200 to +0.105 0.15 384 0.602 0.632
–1.05 to +0.044 0.07 186 0.827 0.861
TABLE 4-14 Calculated Loss for Each Fraction of Solids in the Moving Bed in the
20� Pipeline (Example 4-11)
Volumetric Calculated losses
Drag coefficient concentration in for particles
Average for a particle the slurry (at (with the Albertson
Particle size particle size with shape overall solids C V shape factor of
distribution (mm) (mm) factor of 1 of mixture at 8.7%) 1.0 (Pa/m)
–0.850 to + 0.400 0.63 0.497 0.164% 50.47
–0.400 to + 0.200 0.3 0.52 0.194% 57.71
–0.200 to +0.105 0.15 0.632 0.144% 37
–1.05 to +0.044 0.07 0.861 0.102% 20.79
Total for bed 165.97

4.40 CHAPTER FOUR
TABLE 4-15 Iteration for Calculated Losses for Each Fraction of Solids in the
Moving Bed, Based on the Distribution of Concentration—for 20� Pipeline
(Example 4-11)
Average Drag Iterated
particle coefficient Terminal Iterated pressure
Particle size size for a velocity concentration loss
distribution (mm) (mm) sphere (mm/s) –1.8 V t�·K x·U f C/C A (Pa/m)
–0.850 to +0.400 0.63 0.497 6.14 –0.163 0.687 34.7
–0.400 to +0.200 0.3 0.52 4.14 –0.1093 0.777 44.85
–0.200 to 0.105 0.15 0.632 2.65 –0.07 0.851 31.5
–1.05 to +0.044 0.07 0.861 2.42 –0.063 0.86 17.88
Total for bed 128.93
The purpose of Example 4-11 was to demonstrate the method developed by Wasp. A
number of pipelines have been constructed around the world using this technique and the
practical engineer needs to be familiar with this method as well as with the two-layer
model and stratified flow models that we will explore later.
The following computer program is based on this methodology.
CLS
DIM dp(50), cvdp(50), rep(50), vt(50), cvn(50), dpbed(50), cd(50)
DIM cvind(50), dpav(50), z(50), cca(50), dpnew(50), dfbed(50)
pi = 4 * ATN(1)
DEF fnlog10 (X) = LOG(X) * .4342944
INPUT “name of ore and project”; ore$, proj$
INPUT “date “; dat$
INPUT “your name please “; name$
PRINT “ please choose between the following system of units”
PRINT “ 1- SI units”
PRINT “ 2- US Units”
PRINT
INPUT “ 1 or 2”; ch
10 PRINT
IF rt$ = “Y” OR rt$ = “y” THEN PRINT “

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.41
IF cwe = 1 THEN INPUT “weight concentration in percent”; cwin
IF cwe = 2 THEN INPUT “volume concentration in percent”; cvin
IF cwe = 0 OR cwe > 2 THEN GOTO 12
PRINT
IF cwe = 1 THEN cw = cwin/100
IF cwe = 1 THEN sgm = sgl/(1 - cw * (1 - sgl/sgs))
IF cwe = 1 THEN cv = (sgm - sgl)/(sgs - sgl)
IF cwe = 1 THEN PRINT USING “specific gravity of mixture = ##.##, cv #.###”;
sgm; cv
IF cwe = 2 THEN cv = cvin/100
IF cwe = 2 THEN sgm = cv * (sgs - sgl) + sgl
IF cwe = 2 THEN cw = cv * sgs/sgm
IF cwe = 2 THEN PRINT USING “specific gravity of mixture = ##.##, cw =
#.###”; sgm; cw
INPUT “hit any key to continue”; jk$
CLS
PRINT
22 INPUT “pipe outside diameter in inches”; d0
IF d0 = 0 THEN GOTO 22
INPUT “wall thickness in inches”; tw
INPUT “liner thickness in inches”; tl
D1 = d0 - 2 * (tw + tl)
PRINT “inside pipe diameter in inches”; D1
d1m = D1 * .0254
a1 = pi * d1m ^ 2/4
14 v1 = q1m/a1
v1us = v1/.3048
IF rt$ = “Y” OR rt$ = “y” THEN GOTO 18
INPUT “viscosity of slurry in cPoise or mPa-s”; viscp
visc = viscp/1000
18 ReL = sgl * 1000 * d1m * v1/visc
PRINT “Reynolds Number of carrier liquid”; ReL
IF rt$ = “Y” OR rt$ = “y” THEN GOTO 20
IF ch = 1 THEN INPUT “pipe roughness in meters”; em
IF ch = 2 THEN INPUT “pipe roughness in feet”; ef
IF ch = 2 THEN em = ef * .3048
edi = em/d1m
PRINT “relative roughness”; edi
20 a = (edi/3.7 + 5.7/ReL ^ .9)
b = fnlog10(a)
fd = .25/b ^ 2
PRINT “darcy factor for carrier liquid”; fd
fan = fd/4
dpl = fd * v1 ^ 2 * sgm * 1000/(2 * d1m)
slopliq = 2 * fan * v1 ^ 2/(9.81 * d1m)
PRINT USING “head loss per length = ####.##### = “; slopliq
PRINT USING “press drop due to carrier liquid = #####.## Pa/m”; dpl
PRINT
Ub = 9.81 * d1m * (sgm/sgl - 1)/v1 ^ 2
PRINT “Ub”; Ub
INPUT “hit any key to continue”; jk$
INPUT “hit any key to continue”; jk$
PRINT “ starting from the top size you are asked to input particle size for “
PRINT “ each fraction and its volumetric concentration as part of the
solids”
FOR i = 1 TO 50
IF i = 1 THEN PRINT “top fraction”
PRINT “size”; i

4.42 CHAPTER FOUR
IF rt$ = “Y” OR rt$ = “y” THEN GOTO 140
95 INPUT “particle size (in microns) and cumulative volume conc.(%)”; dp(i),
cvdp(i)
140 IF cvdp(i) > 100 THEN GOTO 95
IF i = 1 THEN GOTO 85
cvind(i) = –cvdo + cvdp(i)
dpav(i) = (dpo + dp(i))/2
GOTO 90
85 cvind(i) = cvdp(i)
dpav(i) = dp(i)
90 PRINT USING “average particle size = ###### micron,av.volume conc =
###.#### %”; dpav(i); cvind(i)
cvdo = cvdp(i)
dpo = dp(i)
rep(i) = dpav(i) * 10 ^ -3 * v1 * sgm/visc
PRINT “particle reynolds number”; rep(i)
IF (rep(i) > 2) AND (rep(i) < 500) THEN cd(i) = 18.5 * rep(i) ^ -.6
IF rep(i) < 2 THEN cd(i) = 27 * rep(i) ^ (–.84)
IF (rep(i) > = 500) AND (rep(i) < 200000) THEN cd(i) = .44
vt(i) = (4 * 9.81 * dpav(i) * .000001 * (sgs – sgl)/(3 * cd(i))) ^ .5
IF vt(i) >(v1/2) THEN PRINT “warning deposit velocity is higher than half
the average flow velocity”
PRINT USING “drag coef for av.diam = #.###;terminal velocity = #.#### m/s”;
cd(i); vt(i)
dfbed(i) = 82 * cvind(i) * cv * (Ub ^ 1.5) * (cd(i) ^ -.75)/100
PRINT “ correction to friction factor”; dfbed(i)
‘PRINT “pressure drop due this fraction”; dpbed(i)
hm = hm + dfbed(i)
PRINT USING “ TOTAL correction to friction factor = ###.###”; hm
IF dp(i) = 0 THEN GOTO 130
IF cvdp(i) = 100 THEN GOTO 130
‘INPUT “DO YOU WANT TO CONTINUE (Y/N)”; LKJ$
‘IF LKJ$ = “n” OR LKJ$ = “N” THEN np = i
‘IF LKJ$ = “n” OR LKJ$ = “N” THEN GOTO 130
120 np = i
NEXT i
130 fannew = fan * (1 + hm)
PRINT “total friction factor for slurry”; fannew
pressure = fannew * 2 * sgm * 1000 * v1 ^ 2/d1m
slope = pressure/(9810 * sgm)
PRINT USING “pressure = #####.## Pa/m”; pressure
PRINT “slope of slurry or head per unit length”; slope
150 INPUT “DO YOU WANT TO DO ITERATION BASED ON CONCENTRATION C/CA (y/n)”; HT$
IF HT$ = “N” OR HT$ = “n” THEN GOTO 325
DFANNEW = fannew
uf = v1 * SQR(DFANNEW/2)
PRINT USING “ FRICTION VELOCITY = ##.### m/s”; uf
FOR i = 1 TO np
IF i = 1 THEN dhm = 0
z(i) = –1.8 * vt(i)/(.38 * uf)
cca(i) = 10 ^ z(i)
PRINT “size,av diam and c/ca “; i, dpav(i), cca(i)
dpnew(i) = dfbed(i) * cca(i)
dhm = dpnew(i) + dhm
NEXT i
DFANNEW = fan * (dhm + 1)
PRINT “REVISED FANNING FACTOR = “; DFANNEW
pressureit = DFANNEW * 2 * sgm * 1000 * v1 ^ 2/d1m

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
PRINT USING “iterated pressure = #######.## Pa/m = “; pressureit
slopeit = pressureit/(9810 * sgm)
PRINT “revised slope “; slopeit
325 INPUT “do you want to repeat the calculation for another flow rate
(Y/N)”; rt$
IF rt$ = “Y” OR rt$ = “y” THEN GOTO 10
LPRINT “REVISED FANNING FACTOR = “; DFANNEW
LPRINT USING “iterated pressure = #######.## Pa/m = “; pressureit
LPRINT “revised slope “; slopeit
525 RETURN
When it was developed in the early 1970s, the Wasp method was considered state of
the art. It does, however, ignore or minimize one important parameter, namely the shear
stress between the different superimposed layers. This is a topic that the two-layer method
attempts to tackle, as we shall see in Section 4-10. It does, therefore, tend to predict pressure
losses higher than those from stratified flows in certain circumstances of bimodal
(fine and coarse) distribution. Nevertheless, the Wasp method remains a very useful
method to this day for the design of pipelines, particularly when it is supported by lab
tests, as we showed in Example 4-11.
4-7 SALTATION AND BLOCKAGE
Most modern engineering specifications for the design of slurry pipelines categorically
forbid flow at speeds below V 3. However, the instrumentation engineer needs to know the
pressure rise in saltation or at blockage. Herbich (1991) argued that motors should be
sized to handle the flow in saltation, and there are incidences where it may be economical
to reduce the cross-sectional area of the pipe by allowing flow over a stationary bed.
4.7.1 Pressure Drop Due to Saltation Flows
4.43
In saltation, there is a bed at the bottom part of the horizontal pipe (Figure 4-11) and different
approaches are use to evaluate the pressure losses.
FIGURE 4-11 Concept of Saltation.
Fines in suspension
Saltation bed

4.44 CHAPTER FOUR
Newitt et al. (1955) derived the following equation for saltation flow with d50 > 0.025
mm (0.001 in) and for pipes smaller than 25.4 mm (1 in):
2 Z = = 66{[�s/�L – 1]gDi/Vm } (4-39)
Babcock (1968) conducted tests on water suspensions of coarse sand and steel shot.
He expressed a nondimensional ratio of the square of the speed to the product of the pipe
inner diameter and the acceleration of gravity. His tests indicated that in the range of 5 <
(V 2 im – i
�
Cvi
/gDi) < 40 the friction losses could be expressed as
�s gDi Z = 60.6� – 1� (4-40)
But for a water suspension of taconite, in the range of 1 < (V2 /gDi) < 12:
�s Z = 66�
gDi – 1� (4-41)
Newitt’s equation (Equation 4-39) is the most commonly used for saltation flows.
Example 4-12
The slurry described in Example 4-7 is in saltation at 1.5 m/s. Determine the resultant
friction factor.
Solution in SI Units
Re = = 671,860
0.25
fd1 = ����� = 0.01572
[log10{(0.0002596/3.7) + (5.74/671,860
Using the Newitt equation (4-39), which was derived for small pipes, would have given
im – iL 1.65 × 9.81 × 0.5778
Z = � = 66����� = 274
2
CviL 1.5
im � – 1 = 60.35
iL
im = 61.35i
Calculating the density of the mixture as:
�m = Cv(�s – �L) + �L = 0.22 (1,650) + 1,000
0.9 )}] 2
1.5 × 1,000 × 0.5778
���
0.00129
�m = 1363 kg/m3 or S.G. = 1.363
Using the Newitt approach with fL = 0.01572
0.01572 × 1.5
im = � � 2
��
�P
� �z
� �L
� �L
2 × 0.5778 × 9.81
� Vm 2
� Vm 2
61.35 = 0.1914 m/m
= � mgi m = 1363 · 9.81 · 0.914 = 2560 Pa/m

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
A more advanced but less known method to estimate the friction gradient for saltation
flows is the Graf–Acaroglu method that is presented in Chapter 6, which we will also discuss
in the next section with a worked example.
4.7.2 Restarting Pipelines after Shut-Down or Blockage
4.45
The loss of power, a water hammer situation, or the freezing of a pipeline are occurrences
that require adequate understanding of the power and pressure needed to restart a
pipeline. The concept of the hydraulic radius is defined in Chapter 6 and is essential to
understand blockage.
Vallentine (1955) studied the blockage of a mixture of 0.53 mm (No. 1) and 1.05 mm
(No. 2) sand in 50 mm (2 inch) and 150 mm (6 inch) pipe. He proposed to write the Darcy
equation in terms of a function for blockage:
fD · Q
H = = = �(B) (4-42)
2 fD · Q L
�5 8gD I 2L ��
(8 · g · RH) A2 fD · V 2L ��
(4 · RH) 2 · g
where B is the blocked area of pipe and �(B) is a function of blockage.
Vallentine proposed that the blocked area B is a function of the total flow, pipe diameter,
flow rate of solids, and flow rate of mixture:
1/2 1/3 � L Q m
B = fn� ���
(4-43)
[�s – �L) 1/2 Q
�2.5 1/3
D I Q s
Herbich (1991) plotted these functions. They are presented in Figures 4-12 and 4-13.
In Chapter 6, the work of Graf and Acaroglu is examined in Section 6.5.4. Schiller
(1991) proposed to apply their equation (Equation 6.66b) to the problem of flow over a
BLOCKAGE B
1.0
0.9
0.8
No. 1 Nonuniform (Vallentine)
0.7
No. 2 '' ( “ )
0.6
0.5
0.4
0.3
0.2
0.1
Uniform Sands (Craven)
0.0
0 1 2 3 4 5 6 7 8
Q
� d 2.5
�
Q s
��� � –1/3
� �
�s – �w Q
FIGURE 4-12 Blockage factor B. (From Herbich, 1991, reprinted with permission from
McGraw-Hill, Inc.)

4.46 CHAPTER FOUR
FIGURE 4-13 Blockage chart. (From Herbich, 1991, reprinted with permission from
McGraw-Hill, Inc.)
stationary bed. Schiller established the hydraulic radius of the bed in terms of the ratio of
the mean velocity of flow to the deposition velocity V 3 as
or
BLOCKAGE Q (cfs)
10
9
8
7
6
5
4
3
2
1
0
Q s
V/�(4� ·� g� ·� R� H�)� = V3/�(D� I �· g�)�
R H = 0.25 [(V/V 3) 2 ]D I
Combining these two equations with the Graf–Acaroglu equation (6.64), Schiller proposed
to replace the slope by head losses per unit length. Schiller proposed to replace d p
by d 50 for a mixture of particles of different sizes:
{0.25 [(V/V3) 2 ]DI} �[( ���s/��� L�–� 1�)g� d� 3 50�]�
CV · V · = 10.39� � –2.52
�� ���
(4-44)
Example 4-13
Considering that the slurry of Example 4.1 is partially blocked at a speed of 12 ft/sec. Determine
the resultant head per unit length to maintain flow:
V3 = 15.25 ft/sec
Solution in SI Units
V = 3.66 m/s
D I = 0.718 m
(�s – �L)d50] �L(h/L) [0.25 [(V/V3) 2 ]DI] The hydraulic radius is therefore [0.25 [(.787) 2 ]0.718] = 0.111 m.
2.0%
� Q 60%
1.0%
0.75%
0.5%
0.25%
0.10%
0.05%
Qs � = Relative Rate of Sediment
Q
Transport
B = Average Pipe Area Blocked
50%
40%
30%
20%
10%
5%
2%
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.50
PIPE DIAMETER d (ft)

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
0.27 · 3.66 · 0.111
���
�[(
= 10.39� � –2.52
��
�1�.6�5�)9�.8�1� (�3� � 1�0� (h/L) 0.111
–4 �) 3 �]�
5247 = 10.39[(h/L) 2.52 [839090.5]
h/L = 0.0527
Solution in USCS Units
V = 12 ft/sec
DI = 2.357 ft
The hydraulic radius is therefore [0.25 [(12/15.25) 2 ]2.357] = 0.365 ft.
d50 = 0.000984 ft
0.27 · 12 · 0.365
���
�[(
= 10.39� � –2.52
��
�1�.6�5�)3�2�.2�(0�.0�0�0�9�8�4�) (h/L) 0.365
3 �]�
5256 = 10.39[(h/L) 2.52 [844444]
h/L = 0.0526
Wood 1979 attempted to simplify this complex problem by proposing that the pressure
gradient between the incipient motion velocity V2 (when particles start to leave the
bed) and the actual deposition velocity V3 (when the particles start to move as a heterogeneous
flow) is essentially composed of three components:
1. A component required to overcome the boundary shear stress and turbulence
2. A component required to accelerate the slurry due to changes in mean velocity of the
slurry
3. A component required to accelerate the particles that leave the bed
The third component is the principle source of increase in the pressure gradient. The
particles are assumed to be eroded from the bed at a rate based on flow conditions. If this
rate exceeds the capability of the flow to suspend the solids, they tend to fall back into the
bed, until the process is restarted.
The analysis of Wood is based on open channel flow, which is the topic of Chapter 6.
Wood treats the stationary bed by considering its hydraulic diameter, and proceeds to apply
a modified Zandi and Govatos approach.
4-8 PSEUDOHOMOGENEOUS
OR SYMMETRIC FLOWS
Above V 4, the flow is pseudohomogeneous or symmetric (but not necessarily uniform).
With negligible hold-up, Govier and Aziz (1972) proposed to express the pressure loss in
terms of the ratio of the friction fanning factor for the slurry mixture and the liquid mixture
or as
i m – i
� Cvi L
fNm�m – fNL�L = ��
(4-45)
fNL� L
1.65(3 × 10 –4 )
1.65(0.000984)
4.47

4.48 CHAPTER FOUR
where
fNm = fanning factor of mixtures
fNL = fanning factor of liquid at equivalent volume
In Chapter 1, the increase in dynamic viscosity due to the volumetric concentration of
solids was discussed; it can be expressed as the Einstein–Thomas equation:
� m = � L[1 + AC v + BC v 2 + C exp (DCv)]
where A, B, C, and D are constants. Since the fanning friction factor is a function of
the Reynolds number, the ratio of the Reynolds numbers of the mixture and liquid
would be
Re m =
ReL =
The ratio of the Reynolds number helps to establish a ratio of friction factors:
2 = = {1 + ACv + BCv + C exp(DCv)}� + 1� (4-46)
The next step consists of using the Blasius equation for transition flows:
= {ReL/Rem) 0.25 �LVmDi �
�m
Rem �L Cv(�s – �L) � � ��
ReL �m
�L
fm � (4-47)
fL
Although Govier and Aziz (1972) did not discuss turbulent flow, their analysis can be
extended. For turbulent flows up to Reynolds numbers from 5000 to 100,000,000, which
is well outside the range used for mining, the Swamee–Jain equation (Equation 2.19) can
be used to obtain the ratio of friction factors:
4-9 STRATIFIED FLOWS
� mV mD i
� �m
0.9 2
{log10(0.27 �/Di + 5.74/ReL }
0.9 2
{log10(0.27 �/Di + 5.74/Rem }
fm/fL = ����
(4-49)
In Section 4.6, it was clearly indicated that Wasp et al. (1977) extended the Durand and
Zandi approach to a concept of multiple and superimposed layers of particles of different
sizes and volumetric concentration, with a logarithmic concentration distribution. This
method uses the drag coefficient of the particles as an important parameter.
Another school of researchers, particularly lead by Shook, Wilson, and Gilles in Canada,
focused on refining the original models of Newitt, which were based on the terminal
velocity. These authors proposed that the compound mixture of coarse and fine particles
can be simplified to what they called “stratified flows,” in which the fines slide over a
moving bed of coarse solids.
Newitt, as expressed by Equation 4-5, had proposed that the deposition velocity was a
mere factor of 17 times the terminal velocity, Wilson (1991) indicated that this approach
was not confirmed by tests for large pipes. The concept he proposed was that the flow
was going through a gradual transition from a fully stratified to a fully suspended flow,
with a gradual change in the pressure gradient.

Defining a parameter � in terms of the pressure gradient:
� = (4-50)
Moreover, plotting � versus the speed, as in Figure 4-14, allows one to find a speed
V50 halfway between a fully stratified flow and fully suspended flow. The slope on this
plot is defined as M.
In the region of partially stratified flow, the logarithmic plot confirms that the pressure
gradient � is a function of the velocity raised to the power (–M). M is calculated
from the slope at V50 considered to be the point at which half-stratified flow occurs, as in
Figure 4-14.
Using V50 as a reference velocity, and considering that the value of � at V50 is effectively
half the mechanical sliding coefficient �p, Wilson (1992) proposed that � = fn(V,
V50, �p, M)
Determining V50 is no easy task. In this partially stratified model, it is argued that the
lifting of particles by turbulence is strongly influenced by their diameter. Thus, the fines
tend to be transported better by the fluid than the coarse solids at the bottom of the pipe.
These particles are transported by eddies, and the largest eddy would be equal to the pipe
diameter. The pipe diameter was therefore proposed as a reference. The resistance to motion
of the solid particles is proposed to be the result of a contact load associated with mechanical
sliding friction in the coarser bed and fluid friction associated with the friction
velocity of the carrier fluid. In this complex environment, Wilson et al. (1992) defined a
new settling velocity as
[{�s – �L}g�L] Vs = 0.9 Vt + 2.7� � (4-51)
1/3
im – iL ��
{�m/�L – 1}
��
log (i - i )/( - 1 )
m L m L
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
p
fully stratified flow
V
50
�L 2/3
partially
stratified
gradient M
Speed
FIGURE 4-14 Concept of V 50 for stratified flows.
4.49
Fully suspended
flow

4.50 CHAPTER FOUR
The two constants 0.9 and 2.7 were obtained from test data on sand. The velocity V 50 is
then expressed in terms of this settling velocity:
V50 = Vs�(8�/f �dL�)� cosh (60 dp/DI) (4-52)
where
cosh = the hyperbolic trigonometric function
fdL = Darcy friction factor for an equal volume of water
In tests conducted for sand in water at 20° C, � is 0.1 m/s at a particle size of 0.40 mm, indicated
that the magnitude of �p, the mechanical friction coefficient, was 0.44. From Figure
4-14:
� = 0.5 � p(V/V 50) –M = 0.22(V/V 50) –M (4-53)
For a mixture of particles of different diameters, a settling velocity V s85 at d 85 and V s50
at d 50 are used to calculate a standard deviation � s:
Vs85 cosh (60 d85/DI) �s = log����� (4-54)
Vs50 cosh (60 d50/DI) 2 –1/2 The coefficient M is expressed as M = (0.25 + 13�s ) .
The magnitude of Vm/V50 or ratio of mean velocity of the mixture to V50 is defined as the
stratification ratio. The friction losses may be expressed in terms of the stratification ratio.
When d85/d50 < 2, the slurry is considered to be narrowly graded, and M is set at 1.7.
For 2 < d85/d50 < 5, 1.7 > M > 0.4.
There is no question that the approach of Wilson et al. (1992) is extremely interesting,
but it is more complex than the methods proposed by Equations 4-3 to 4-7. It requires the
support of testing, a database, a computer software, and a personal computer.
4.10 TWO-LAYER MODELS
Khan and Richardson (1996) explained that Shook and Roco (1991) developed a two-layer
model for stratified slurry flows, which may be summarized as follows:
� The slurry flow of heterogeneous mixtures is considered to consist of two layers, each
with its own velocity of motion and volumetric concentration; but it is assumed that
there is no slip between liquid and solid phases.
� The solids in the upper layer are fully suspended, are at a volumetric concentration
C Vu, and move at a velocityV U.
� The coarser solids in the lower layer are considered to be packed. However, because of
their irregular shape, there is a certain void fraction between the particles. For sand, the
lower layer is considered to be at a maximum volumetric concentration C VU of 60%
and move at a velocityV B.
The total area of flow for the upper layer of fines and the lower layer of coarser particles
is A = A B + A U
For the mixture, the mass balance VA = V B A B + V U A U
For the liquid phase, (1 – C V)VA = (1 – C VU) V U A U + (1 – C VB)V B A B
For the solid phase, C VVA = C VBV B A B + C VUV U A U = C VUAV + (C VB – C VU)V B A B

Referring to Figure 4-15:
2 AB = 0.25 D I(� – sin � cos �)
2 AU = 0.25 D I(� – � + sin � cos �)
The upper wet perimeter is
WPU = DI(� – �)
WPB = DI� At the interface WPi = DI sin �
The momentum and force balance is expressed as
dP
–AU � = �UWPU + �iWPi for the upper layer
dx
dP
–AB � = �LWPB – �iWPi + fc�F for the lower layer
dx
dP
–A � = �UWPU + �LWPB + fc�F for the entire pipe
dx
fc is the friction factor due to the Coulombic friction between the particles and the pipe
and �F is the total force per unit length exerted normal to the pipe (e.g., weight of the bed
per unit length, etc.)
2 At low to moderate concentrations �B = 0.5�LL fNBV B. Certain solids are at a concentration
CVB – CVU. They are considered to be of a buoyant weight that is supported at the
wall as a result of interparticle contacts. Averaged over the entire cross-sectional area of
the pipe, these particles define the “contact load” CC, which is calculated as follows:
CC = [CVB – CVU] � (4-55)
A
where
AB = cross-sectional area of the lower layer
A = cross-sectional area of the entire pipe
The upper layer volumetric concentration CVU is obtained from the mean in-situ concentration
CX and the total contact load CC as CVU = Cr – CC. A parameter CX is defined as the in-situ concentration in the x-direction, as
CX = CVB + CVU (4-56)
The relationship between CC and CX is established as an experimental correlation:
where C VB = 0.60.
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
C C
� CX
= e –� (4-57a)
� = 0.124 Ar –0.061 [(V 2 /(gd p)) 0.28 ][(d p/D i) –0.431 ](� S/� L– 1) –0.272 (4-57b)
And for sand slurries, according to SAC (2000):
� = –0.122 Ar –0.12 [(V/V D) 0.30 ][(d p/D i) –0.51 ](� S/� L– 1) –0.255 (4-57c)
The Archimedean number Ar is defined in Equation 4-5.
A B
4.51

4.52 CHAPTER FOUR
The density of the suspended concentration CVU and the density of the carrier liquid �L are then used to compute the shear stresses in the two layers (Figure 4-15).
A fanning friction factor is then obtained for each layer based on some average velocity
of the flow in the pipe, density of the carrier liquid, and dynamic viscosity of the carrier
liquid, and using the relative roughness of the pipe �/Di. The Churchill (1977) equation is then used to calculate the friction factor as
fn = 2[8Re –12 + (A + B) –1.5 ) 1/12 (4-58)
where
A = [–2.457 ln[(7/Re) 0.9 + 0.27 �/Di)] 16
B = (37530/Re) 16
The Reynolds number is calculated on the basis of the average pipe flow speed, since
the speed of flow in each layer is not known at this point.
An equivalent “sand roughness” is then defined as the ratio of particle size to pipe diameter
(d p/D I). At the interface between the bottom and top layer, a friction factor is derived
by modifying the Colebrook equation (Equation 2-17):
Vertical distance (y)
above bottom quadrant
D I
WP
B
V B V U
Speed
A U
2
Vertical distance (y)
above bottom quadrant
A
B
WP
U
WP UB
typical real distribution
CVU
Solids volumetric concentration
FIGURE 4-15 Two-layer modeling of coarse and fine particle mixtures.
C VB

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
fNi = (4-59)
where
Y = 0 for dp/Di < 0.0015
Y = 4 + 1.42 log10 (dp/Di) for 0.0015 < dp/Di < 0.15
and Ar (the Archimedean number) < 3 × 105 2(1 + Y)
���
[4 log10(dp/Di) + 3.36] 2
For bimodal mixtures (mixtures of fine and coarse particles), the Saskatchewan Research
Council (SRC) (2000) suggested that the effects of the fines on the viscosity of the
carrier liquid should be used to calculate the apparent viscosity �f. However, the actual
density of the liquid should be used without corrections for volumetric concentration of
fines.
Khan and Richardson (1996) pointed out that the concept of an equivalent sand roughness
is very speculative as the transition between one layer and the other. By subsequent
iterations, the friction factor and velocity for each of the two layers are obtained from the
shear stress as the product of the friction factor and the dynamic pressure:
2
�U = 0.5fNU�UVU � B = 0.5f NB� BV B 2
2 2 fNB�BV BWPB D i gfc(�s – �L)(CVL – CVU)(1 – CVB)(sin � – � cos �)
�BWPB = �� + ������ (4-60)
2
2(1 + CVU – CVB) where fc is the coefficient of kinematic friction between the particles and the pipe.
The total normal force per unit length was defined by Shook as
2 D i g(�s – �L)(sin � – � cos �) (CVB – CVU) (1 – CVB) �F = ��������
(4-61)
2
1 – (CVB – CVU) where � is the half angle formed by the bottom layer with respect to the center of the pipe.
To appreciate the complexity of this approach, SRC (2000) indicated that this approach
yielded six unknowns (VB, VU, �, Cr, Cc, and –dP/dx). The numbers of iterations
that are required are better dealt with on a computer.
The two-layer models have gained wide acceptance in the oil–sand industry. The following
example is an illustration at the first level of iteration.
Example 4-14
Sand slurry in a pipe is flowing at 6.5 m/s (21.3 ft/sec). The pipe diameter is 717 mm
(28.35�) pipe and the sand particle diameter dp = 360 �m (0.0142�).The volumetric concentration
was presented to be 0.27. Upon review of the composition of the sand, it was
noticed that 15% of the solids were fines smaller than 74 �m. If the lower bed is packed
at 60%, the contact load Cr = 0.30, and the Columbian friction factor fc is 0.50, determine
the pressure gradient (assume water dynamic viscosity 1 cP, and sand S.G. 2.65).
Volumetric concentration in the upper layer consists essentially of fines:
CVU = 0.15 · 0.27 = 0.0405
CVB = 0.85 · 0.27 = 0.23
By the Einstein–Thomas equation, the dynamic viscosity of the carrier liquid needs to
be corrected for a concentration of 0.0405 in the upper layer:
� = � L(1 + (2.5 · 0.0405) + (10.05 · 0.0405 2 ) + 0.00273 exp (16.6 · 0.0405)] = 1.123
� L = 1.123cP
4.53

4.54 CHAPTER FOUR
4 × 9.81 (3.6 × 10
Ar = = 666
–4 ) 3 × 1000 (1650)
����
3 (1.123 × 10 –3 ) 2
� = 0.124 Ar –0.061 [(V 2 /(gd p)) 0.28 ][(d p/D i) –0.431 ](� S/� L – 1) –0.272
= 0.124 (666 –0.061 ) (6.5 2 /(9.81 · 3.6 × 10 –4 ) 0.28 ][0.0005 –0.431 ]1.65 –0.272
= 395.52
= e –395.52 = 0
Density of the liquid and fines in the upper layer is �U = .0401 · 2650 + (1 – .0401) ·
1000 = 1066 kg/m3 CC �
CX
If it is assumed that, due to the void fractions, the lower layer is full at 60%, and since
the volumetric concentration is 0.23, then the area used is 0.23/0.6 = 0.383.
2 2
area = 2[0.25D I (� – 0.5 sin � cos �)] = 0.383 × 0.25�DI
� � 80 degrees � 0.444 �
A B = 0.25 D I 2 (� – sin � cos �) = 0.25 × 0.717 2 x(1.396 – 0.171) = 0.1574 m 2
A U = 0.25 D I 2 (� – � + sin � cos �) = 0.25 × 0.717 2 x(0.556� + 0.171) = 0.246 m 2
WPU = DI(� – �) = 1.252 m
WPB = DI� = 1 m
The total normal force per unit length was defined by Shook as
0.717 (0.23 – 0.0405) (1 – 0.23)
�F = ����� 2 9.81 (2650 – 1066)(0.985 – 1.39 × 0.1736)
������
= 632 N/m
Since WP B = 1 m,
2
2 fNB�BV BWPB �BWPB = �� + fc �F
2
fNL 1066V
2 �B(1) = + 0.5 · 632 = 533 fNBV B + 316
dp/DI = 0.00035/0.717 = 0.000488, so Y = 0
2 LL(1)
��
2
f NI = 2/[4 log 10(0.717/0.00035) +3.36] 2 = 0.00725
To determine the friction factors for the upper and lower layers it is required to determine
the speed in each layer. To obtain the difference between the velocity in the upper
and lower layers, it is essential to obtain the shear stress �i between the two layers, or to
make some assumptions and to proceed with further iterations. In a first iteration, it shall
be assumed that VU = 1.1 VLL. AV = ABVB + AUVU 0.25 · � · 0.7172 · 6.5 = 0.1574 m2 (VL)+ 0.246 m2 (1.1VL) VB = 6.14 m/s
VU = 6.75 m/s
1 – (0.23 – 0.0405)

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
The following calculations require the reader to review some of the concepts of Chapter
6.
The hydraulic diameter (Equation 6-2) for the upper layer is 4AU/WPU = 4 ×
0.246/1.252 = 0.786 m.
The Reynolds Number for the upper layer is ReU = 1066 · 6.75 · 0.786/(1.123 · 10 –3 ) =
5,036,209.
The pipe is rubber coated to a roughness of 0.00015 m, �/Di = 0.0002.
Applying Churchill’s equation to the upper layer:
A = {–2.457 ln[(7/Re) 0.9 + 0.27 �/Di)} 16 = 1.192 · 1022 B = (37530/Re) 16 = 9.045 · 10 –35
fnu = 2[8Re –12 + (A + B) –1.5 ) 1/12 = 2[3 · 10 –80 + 7.684 · 10 –34 ] 1/12 = 0.0035
2 2 �U = 0.5fNU�UV U = 0.5 · 0.0035 · 1066 · 6.75 = 85 Pa
For the lower layer the hydraulic diameter is 4ABWPB = 0.6296 m.
The Reynolds number for the lower layer is ReU = 1066 · 6.14 · 0.6296/(1.123 · 10 –3 )
= 3,669,531.
The pipe is rubber coated to a roughness of 0.00015 m, �/Di = 0.000238.
Applying Churchill’s equation to the upper layer:
A = {–2.457 ln[(7/Re) 0.9 + 0.27 �/Di)]} 16 = 1.006 · 1021 B = (37530/Re) 16 = 1.433 · 10 –32
fnu = 2[8Re –12 + (A + B) –1.5 ) 1/12 = 2[1.34 · 10 –78 + 3.134 · 10 –32 ] 1/12 = 0.0047
2 2 �B = 0.5fNB�BVB = 0.5 · 0.0047 · 1066 · 6.14 = 94 Pa
2 2 2 2 At the interface �i, 0.5fNi�U(V U – V B) = 0.5 · 0.0035 · 1066 · (6.74 – 6.14 ) = 14.68 Pa.
dP
–AU � = �UWPU + �iWPi = 85 · 1.252 + 14.68 · 14.68 · 0.717 · 0.985 = 117 Pa/m
dx
Since A U = 0.246 m 2 , then dP/dx = 475.6 Pa/m.
Equation 4-60 is not applicable at high concentration of fines, as the slurry starts to behave
as a non-Newtonian mixture. For particles with d 50 finer than 74 �m, the method
does not give very reliable results.
For flows above a deposited bed (flows with saltation), SRC (2000) proposed to treat
the upper layer as a noncircular flow. This means that the hydraulic diameter must be determined
from the wetted area. The difference in roughness between these two surfaces
(upper surface of the bed) and pipe roughness is not well discussed. The hydraulic diameter
is calculated as D HB = 4A B/(WP B).
It is considered that for pipe flows, the deposit velocity is a function of the pipe diameter
raised to the power of 0.4 for noncircular channels and the friction loss gradient is a
function of the ratio V 2 /D H.
By defining V U as the velocity of the upper layer and V 3 the critical velocity at which
the bed deposits, the following equation is established:
V U
� V3
0.4 DHB 4.55
= � (4-62)
0.4 D

4.56 CHAPTER FOUR
Friction loss per unit length
i 1
The ratio of friction gradients is
i m
� i3
A U
deposited bed upper layer
2 V U D
� � 2 V 3 DHB
= = � � 0.2
(4-63)
where i 1 is pressure gradient at V 1.
At deposition, the flow rate in the upper layer Q U is lower than the flow rate through
the deposited bed Q B, and from the ratio of velocities as per Equation 4-57:
Q U
� QB
Q U = A UV U
Q 3 = AV 3
D HB
� D
= � � 0.2
Flow rate
FIGURE 4-16 Flow above a deposited bed—two-layer model.
D
�
DHB
A U
� A
WP
U
WP UL
Deposited solids
(4-64)
d

4.11 VERTICAL FLOW OF COARSE
PARTICLES
Newitt et al. (1961) conducted tests for flows of solids in vertical pipes. For fine solids
they derived the following empirical equation:
= 0.0037� � 1/2 gDi � �� � 2 Di �S (4-65)
In a vertical flow, it would not be possible to develop dunes, a bed, or saltation. There
is no concentration gradient and the flow may be treated as pseudohomogeneous for friction
loss calculations, as discussed in Section 4-4-3.
Since the flow in a vertical pipe is pseudohomogeneous, a simple instrument to measure
flow rate of slurry is the inverted U column, which consists of 4 elbows and 2 vertical
pipe spools (Figure 4-17). The first vertical branch must be sufficiently high to eliminate
entrance effects (previously discussed in Chapter 2). Toward the top of the pipe, a pressure
tap measures the static pressure. Pressure loss occurs through the two elbows. Another
pressure tap is added on the downward section of the pipe. The inverted U column is
calibrated on water and the pressure loss is a function of the flow rate as well as the density
of the slurry:
where C d is the discharge coefficient, or
�V
�P = K
2
�
2
2 VmDi Q = � Cd 4�
Q = C dD i�(2� ��P�/�� m�)�
The density of the mixture may be calculated from the input data or measured using a
nuclear radiation density gage.
flow
z A
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
i m – i L
� iLC v
2
1
� V 2
FIGURE 4-17 Inverted U tube piping for measuring flow of a slurry mixture.
� dp
� �L
3
4
z B
4.57

4.58 CHAPTER FOUR
In a vertical flow, in addition to the friction losses as discussed for a pseudohomogeneous
flow, there is a hydrostatic pressure gradient, so that the total pressure
drop is:
2 �P = �mg[�z + fmV m �z/(2gDi)]
�P = pressure loss between two points
�z = height difference between two points
For further reading, see the work of Einstein and Graf (1966) who described such a
flow and a concentration meter for water–sand mixtures.
4-12 INCLINED HETEROGENEOUS FLOWS
Between the two types of horizontal and vertical flows that we discussed in this chapter,
there is a range of inclined flows that are important but are not the subject of extensive
studies. In fact, it may be argued that in many plants, most flows are either horizontal or
vertical, with elbows and fittings between them. An understanding of inclined heterogeneous
flows is essential for certain long overland pipelines, thickener feed systems,
pumpbox feed systems, and, in some respects, to shed light on open channel flows. Until
very recently, attention was focused on uniformly graded slurries.
Worster and Denny (1955) indicated that if the pipe is inclined from the horizontal, the
fraction loss is
where
i � = pressure gradient at �
i m = pressure gradient of the mixture in the horizontal pipe
i � = i + (i m – i) cos � (4.65)
This equation suggests that the pressure loss is lower in an inclined pipe than in a
horizontal pipe. It also suggests that the pressure gradient is the same upward or downward.
The experimental work of Kao and Hawang (1979) indicates that this is not correct. In
fact, they noticed that the friction losses for upflows increased up to a certain magnitude
of the angle of inclination and then decreased. In the case of downflows, they measured a
reduction of friction losses from the values of the horizontal pipe.
Wilson et al. (1992) have discussed the effect of pipe inclination on their V 50,
and suggest that Worster and Denny’s equation be modified by using (cos �) 1.85 instead
of cos �. They also published data on certain particles with a diameter between 1 mm
and 6 mm. The test data indicated that the Durand factor for deposition velocity F L (see
Equation 4-2) increased up to an angle of inclination of 30 degrees and by as much as
38%. They also noticed that the deposition velocity V 3 increased by 50% at 30 degrees
pipe inclination, but then they noticed a drop at an angle of 40 degrees. They did not
conduct further tests. Interestingly, they noticed a reduction of the Durand factor F L by
0.3 at a negative inclination of 20 degrees. There is a dearth of information on flow in
inclined pipes, and as overconservative as it may be, Worster and Denny’s equation
continues to be used. This approach should change, particularly when the angle of inclination
is less than 30 degrees or up to –20 degrees.

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.12.1 Critical Slope of Inclined Pipes
4.59
Pipeline have to go through dunes and hills. Certain slurry pipelines may follow the topography
(Figure 4-18) but others require pipe bridges to avoid sedimentation of the slurry
after a shutdown or power failure (Figure 4-19). A question often raised in designing a
pipeline is determining the critical slope of the pipe to avoid areas of blockage once the
flow is shut down. To avoid blockage of the line, it is necessary to eliminate areas of steep
slopes. A commonly used design restriction of 10–16% (5.7–9°) is often adopted when
there is no knowledge of the critical slope. The idea is that the slurry would settle without
segregation to a “soft” consistency and not migrate down to the steepest slope in the
pipeline during shutdown.
Kao and Hwang (1979) criticized this approach as being extremely stringent because
it adds to construction costs and capital expenses. They implied that it would be better to
properly understand the critical slope rather than to use a rule of thumb.
Shook et al. (1974) measured the maximum inclination of a 50 mm (2 in) ID Perspex
pipe on a sand–water mixture. They reported that:
� The maximum rising angle is 14°, or slope of 24%, before the solids start to slide back.
� The sliding bed was at the interface of the solid bed and the pipe wall rather than within
the settling bed.
FIGURE 4-18 Tailings pipelines follow the slope of the hills and use soil friction produced
by partial burial as an anchor.

4.60 CHAPTER FOUR
FIGURE 4-19 Pipeline carrying taconite tailings with an important portion of coarse particles
required pipe bridges to avoid blockage after shutdown or power failure.
� The critical angles for the sand bed increases from 22–26°(slope of 40–48.7%) with
the decrease of the size of particles. The critical angle is neither affected by the ratio
d p/D I nor by the concentration of the solids.
Durand and Gilbert (1960) derived the following equation for inclined pipe:
i m� = i L + i B(cos �)
where
im = energy gradient for mixture
iL = energy gradient for liquid
iB = energy gradient for solid bed
Kao and Hwang (1979) observed that the critical slope for glass beads and for sand occurred
at 23°(42% slope) from the horizontal. For other substances such as coal and walnut
shells, the initial motion appeared to occur at the interface between the particle bed
and the pipe wall. This suggested that the internal friction between irregularly shaped
coarse particles was higher than the friction at the wall of the pipe.
The critical slope Kao and Hwang (1979) defined was the value for initial particle motion.
For sand or glass beads it was 27° ± 2°, and for coal and walnut shells it was 37° ±
2°.
Craven and Ambrose (1953) investigated the effect of tube inclination on the head loss
for a pipe partially blocked with sediments and for a pipe flowing full. They found that at

HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.61
a given average speed, an adverse slope in excess of 10% increased the pressure losses by
25% or more by comparison with a horizontal pipe.
4-12-2 Two-Layer Model for Inclined Flows
Matsouk (1996) developed a two-layer model for inclined pipes. His tests were conducted
in a 150 mm (6�) pipe at angles of inclination between –35 and +35 degrees. The fundamental
equations for this model for the upper layer are
d(P – �Ugh) �UWPU + �UBWPUB �� = ��
(4-67)
dx
AU
For the lower layer they are
d(P – �Bgh) �BWPB – �iWPi + fc�FN cos � + FW sin �
�� = ����� (4-68)
dx
AL where FW is the submerged weight of the sediments in the lower layer. The force balance
for the whole pipe is then
d(P – �Ugh) �BWPB + �UWPU + fc�FN cos � + FW sin �
�� = ����� (4-69)
dx
A
Equations 4-67 to 4-69 are then solved in a similar manner as presented in Section 4-10.
Matsouk pointed out that his approach was different than the models of Shook and Roco
(1991) (the SRC model), Lazarus (1989), and Lazarus and Cook (1993), which did not include
the pipe axis component of the submerged weight (due to buoyancy) FW sin �. The
tests of Matsouk did not confirm the Worster and Denny equation. At pipe inclinations
close to the angle of internal friction of the transported solids, the behavior of the solids
was different for inclining and descending pipes under the same speed and volumetric
concentration. The difference was larger with coarse than with fine solids. Matsouk con-
z
P
2
L
P + g z
x
P
1
submerged lower layer
of coarse solids
FIGURE 4-20 Concept of the two-layer bipolar flow of slurry at an angle of inclination.

4.62 CHAPTER FOUR
cluded that the deformation of the lower layer was essentially due to the submerged
weight of the solids at the bottom of the pipe.
4-13 CONCLUSION
In this chapter, we have shown that two principal schools of heterogeneous slurry flows
have developed over the last 50 years—one around the Durand–Condolios approach and
the other around the Newitt approach. The former evolved gradually until Wasp modified
it for multilayer compound systems. The latter gradually evolved to yield the two-layer
model.
There is no consensus as to which model to use. Some have argued that the Wasp
method was more suitable for coal, whereas the two-layer model was better for sand. This
is based on the number of papers published on two-layer models for sand slurry mixtures,
emanating from the great interest in Canadian oil sands.
The science of slurry flows is continuously evolving and needs to be understood in the
context of its evolution. It would be a mistake to favor one school of thought over another.
Many successful pipelines have been designed using either model. The slurry engineer
should appreciate that these models are effective tools to be used with correct empirical
coefficients obtained by experimental testing of samples in pumping loops.
After reading this chapter, some readers may get the feeling that designing a slurry
flow is a combination of science and art. Slurry dynamics may appear to be an exercise of
examining each mixture for its properties, much as the physician must examine each patient
before administering the cure. The flow of coarse particles in water or mixtures of
coarse and fine solids in a liquid is complex. When data is not well accumulated, it is recommended
to conduct slurry tests in a pump test loop.
The mixture of coarse and fine particles can lead to a concentration gradient of solids
in horizontal pipe. The coarse particles tend to flow in the lower layers, whereas the fines
flow in the upper layers. Certain authors recommend determining the pressure losses of
fine solids separately from coarse solids at the corresponding volumetric concentration
and particle diameter of each size. Methods based on concentration ratio for each layer
have been developed. A process of iteration is needed to achieve a final estimation of the
pressure loss due to the bed.
New models for inclined flows are appearing, such as the work of Matsouk (1996) on
inclined two-layer models The correct design of inclined flow must be based on empirical
data on the critical slope. The principles reviewed in this chapter apply to dredging and
transporting sand, gravel, coal, steel shot, and rocks from SAG, rod, or ball mills, cyclone
underflows, and tailings, etc., which often have particle sizes larger than 70 �m (mesh
200).
The practical engineer needs to appreciate the limitations of each method and that
models have often been developed for a certain range of particle sizes based on experimental
data. It is wise to check the original data.
Because the new methods of stratified flows or two-layer models use the actual hydraulic
diameter of the bed, whereas the Wasp and Durand methods use the actual pipe diameter,
it is easy to get confused. In fact, some of the proponents of the two-layer models
leave the impression that Wasp and Durand are using the “wrong pipe diameter.” This is
not the approach to take. It is wiser to recognize that the Wasp and Durand methods are
useful tools for the range of slurries for which they were developed. This includes concentrations
of coarse particles up to a volumetric fraction of 20%. This covers, in fact,
most dredged gravels and sands, coal in a certain range of sizes, as well as crushed rocks.
It is also important to appreciate that the work of Zandi and Govatos (1967) was based on

sands up to a volumetric concentration up to 22%, and it would be erroneous to push the
envelope of application of their equations beyond such a range.
In the last thirty years, considerable progress has been made with stratified flows. The
new two-layer models push the envelope of understanding beyond the limits of the models
of Durand, Zandi, and Wasp into the range of volumetric concentrations of 30%. But
these models have their limitations too. Certainly it would be unwise to use the two-layer
model when the d 50 is smaller than 74 micrometers. There is still a considerable amount
of experimental data associated with these stratified models needed to obtain the Conlombic
friction factor and to determine the difference in velocity between the upper and lower
layers. When the particles are not too coarse, such a difference is not too large, and approximations
such as those proposed by Richardson and Khan are justified, but when the
d 50 is around 400 micrometers or higher, the difference in velocity and shear between the
upper and lower layers become important.
In Chapter 6, the analysis presented in this chapter will be extended to include open
channel flows. The Acaroglu–Graf equation presented in Chapter 6 was applied here to
flows with saltation.
4-14 NOMENCLATURE
a Height of layer A above bottom of conduit
A Cross-sectional area of the entire pipe
AB Cross-sectional area of the lower layer
Ap surface area of particle
Ar The Archimedean number
AU Area of upper layer of flow in the two-layer model
b Factor used to calculate the Archimedean number
B blocked area of pipe
C volumetric concentration of the particle diameter under consideration
CA CC CD CE Ct Cv Cvb Concentration of solid particles at a reference plane A (usually at 0.08 DI) Contact load in the Shook–Roco two-layer model
Drag coefficient
Coefficient of discharge
In-situ concentration
Concentration of solids by volume
Volume fraction of solids in the bed
C v bed Concentration of solids in the moving bed
C vi
CVL CVU Cw CX C1 C3 dg dp d85 d50 DH Di dsp HETEROGENEOUS FLOWS OF SETTLING SLURRIES
4.63
Concentration of solids in the moving bed of fraction i
Concentration of solids by volume in the lower layer in the Shook–Rocco model
Concentration of solids by volume in the upper layer in the Shook–Rocco model
Weight concentration
In-situ concentration in the two-layer model
Constant
Constant
Diameter of spherical particle
Average diameter of the particle
Sieve passage diameter for 85% of the particles
Sieve passage diameter for 50% of the particles
Hydraulic diameter of a noncircular flow
Inner diameter of pipe
diameter of equivalent sphere using the sphericity factor

4.64 CHAPTER FOUR
Es fc fD fDL FL fN fNL fNm fNB fNU Fr
The mass transfer coefficient
coefficient of kinematic friction between particles and pipe
Darcy–Weisbach friction factor
Darcy friction factor for equivalent volume of water
Durand–Condolios coefficient to determine the deposition velocity
Fanning friction factor
Fanning friction factor for equivalent volume of water
Fanning friction factor for mixture
Fanning friction factor for the bottom layer in the two-layer model
Fanning friction for the top layer in the two-layer model
Froude number
g Acceleration of falling objects due to gravity (9.78–9.82 m/s2 )
i Equivalent pressure gradient of water at the same volume as the slurry
im Energy gradient of slurry mixture in equivalent m of water per m of pipe length
i� Pressure gradient for pipe at inclination �
i1 i3 K
Pressure gradient at V1 Pressure gradient at V3 Coefficient or constant
Ke Experimental factor for the pressure gradient
Kf Coefficient in the Durand equation for pressure drop
Kx Von Karman constant
K2 An experimentally determined constant
K3 Coefficient proportional to the mechanical friction factor �
L Length of conduit
Ne Index number
m Power coefficient in Zandi’s models
mi mt M
The mass fraction of solids with particle diameter of dp Total mass of particles
Slope of the log scale of pressure gradient versus velocity of a stratified flow
P Pressure loss
PW Wetted perimeter
Q Flow rate
QB Flow rate in the lower layer of the two-layer model
QU Flow rate in the upper layer of the two-layer model
Ri Inner diameter of a pipe
r Local radius for a point in the flow
Re Reynolds number
Rem Reynolds number of the mixture
ReL Reynolds number of the liquid carrier
RH Hydraulic radius
R1 Cross-sectional area of the bed divided by the bed width
s Specific gravity of the solids
Sf Specific gravity of liquid
Sm Specific gravity of slurry mixture
U Average velocity
Uf Friction velocity
Ufc Critical friction velocity at which the solids start depositing
Uf 0 Friction velocity at deposition for limiting case of infinite dilution
V Velocity
VB Velocity in the bottom layer
Deposition velocity (also called V3) V D

Vf VL Vm Vmin VR
Velocity of carrier fluid
Velocity of the lower layer in the two-layer model of Shook and Roco
Mean velocity of a mixture
Velocity at minimum pressure drop
Ratio of solid volume concentration of solids to liquid concentration
Vs Settling velocity in the stratified flow model
Vt Terminal velocity of the solid particle
VU Velocity of the upper layer in the two-layer model of Shook and Roco
V1 Velocity for which flow with a stationary bed is started
V2 Velocity for which flow with a moving bed is started
V3 Deposition velocity or velocity above which solids start to move
V4 Velocity above which all solids move as a pseudohomogeneous mixture
V50 Velocity at 50% stratification
WP Wetted perimeter
y Distance from the lower boundary (pg 33)
Z Nondimensional parameter to express difference between friction losses due to
the slurry and an equivalent volume of water
Subscripts
B Bottom layer
bed Due to the moving bed
i Fraction i
m Mixture
U Upper layer
Greek letters
� Constant of proportionality
� Roughness
� Angle from the vertical starting at the lowest quadrant point
� The angle from the horizontal
� Pressure loss factor
�r The angle of repose of solid particles
�L Density of liquid carrier
�m Density of the slurry mixture
�s Density of solid sediments
�U Density of suspended fines and carrier liquid in the upper layer of the two-layer
model
�W Density of water
�B Shear stress for the lower layer in the two-layer model
�U Shear stress for the upper layer in the two-layer model
� factor used to compute ratio of concentrations in the two-layer model
� Factor to determine speed at 50% stratification in the Wilson model
�L Dynamic viscosity of the carrier liquid
�m Dynamic viscosity of the slurry mixture
�p Mechanical friction coefficient
�s Granular stress of the solid particles
� Kinematic viscosity of water
�L Viscosity of carrier liquid at equal volume
� M
HETEROGENEOUS FLOWS OF SETTLING SLURRIES
Viscosity of slurry mixture
� Product of the index number and the volumetric concentration
�s Coefficient of static friction of the solid particles against the wall of the pipe
4.65

4.66 CHAPTER FOUR
4-15 REFERENCES
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Acaroglu, E. R., and W. Graf. 1968. Designing conveyance systems for solid–liquid flows. Paper
presented at the International Symposium on Solid–Liquid Flow in Pipes, March 3–7, University
of Pennsylvania, Philadelphia.
Babcock, H. A. 1967. Head losses in pipeline transportation of solids. Paper presented at First World
Dredging Conference, WODCON I, the Netherlands.
Babcock, H. A. 1968. Heterogeneous flow of heterogeneous solids. Paper presented at International
Symposium on Solid Liquid Flow in Pipes, March 3–7, University of Pennsylvania, Philadelphia.
Babcock, H. A. 1971. Heterogeneous flow of heterogeneous solids. In I. Zandi (Ed.), Advances in
Solid–Liquid Flows in Pipes and its Applications. pp. 125–148. Oxford: Pergamon Press.
Bagnold, R. A. 1954. Gravity-free dispersion of large spheres in a Newtonian fluid under shear.
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Bagnold, R. A. 1955. Some flume experiments on large grains but little denser than the transporting
fluid, and their implications. Proc. Inst. Civ. Eng., 4, 3, 174–205.
Bagnold, R. A. 1957. The flow of cohesionless grains in fluids. Phil. Trans. Roy. Soc., 249, 235–297.
Blatch, N. S. 1906. Water filtration at Washington, DC, discussion trans. Amer. Soc. Civ. Eng. 57,
400–408.
Charles, M. E., and G. S. Stevens. 1972. The pipeline flow of slurries—transitional velocities. Paper
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84, no. 7: 91–92.
Craven, J. P., and H. H. Ambrose. 1953. The transportation of sand in pipes. Engineering Bulletin
(University of Iowa), 34, 67–88.
Durand R. 1953. Basic Relationship of the transportation of solids in experimental research. Proc. of
the International Association for Hydraulic Research—University of Minnesota, September
1953.
Durand, R., and E. Condolios. 1952. Experimental investigation of the transport of solids in pipes.
Paper presented at Deuxieme Journée de l’hydraulique, Societé Hydrotechnique de France.
Durand, R., and R. Gilbert. 1960. Transport hydraulique et refoulement des mixtures en conduites.
Transactions École des Ponts et Chaussees, 130, 3–4.
Einstein, H. A., and W. H. Graf. 1966. Loop systems for measuring sand–water mixtures. Journal of
Hydraulic Division, Am. Soc. Civ. Eng. 92, HY1, paper 4608, 1–12.
Ellis, H. S., and G. F. Round. 1963. Laboratory studies on the flow of nickel–water suspensions.
Canadian Mining and Metallurgical Bulletin, 56, 773–781.
Faddick R. R. 1982. Ship Loading Coarse Coal Slurries. In The 8th International Conference on Hydraulic
Transport of Solids in Pipes, Johannesburg, South Africa. Cranfield, UK: BHRA
Group.
Gaessler, H. 1967. Experimentelle und Theoretische Untersuchungen uber die Stromungsvorgange
Beim Transport von Festoffen in Flassigkeiten durch Horizontale Rohrleitungen. Doctoral thesis.
Technische Hochschule, Karlsruhe, Germany. Quoted in G. W. Govier and K. Aziz, The
Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold Co., 1972, pp. 668
–670.
Geller, L. B, and W. M. Gray. 1986. Selected Theoretical Studies Made in Conjunction with the Joint
Canada/FRG Research Project on Coarse Slurry, Short Distance Pipeline. CANMET SP 86-
16 E–Government of Canada Publications
Gillies, R. G. J. Schaan, R. J. Sumner, M. J. McKibben, and C. A. Shook. 1999. Deposition velocities
for Newtonian slurries in turbulent flows. Paper presented at the Engineering Foundation Conference,
Oahu, HI. Submitted for publication in the Canadian J. Chem. Eng. Reference cited by
Saskatchewan Research Council (2000). Slurry pipeline course handout.
Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand
Reinhold.

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Hayden, J. W., and T. E. Stelson. 1968. Hydraulic conveyance of solids in pipes. Paper presented at
the International Symposium on Solid–Liquid Flow in Pipes, March 3–7, University of Pennsylvania,
Philadelphia.
Herbich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill.
Hill, R. A., P. E. Snock, and R. L. Gandhi. 1986. Hydraulic transport of solids. In The Pump Handbook,
2nd ed. Edited by I. J. Karassik et al. New York: McGraw-Hill.
Hsu, S. T., A. V. Beken, L. Landweber, and J. F. Kennedy. 1971. The distribution of suspended sediment
in turbulent flows in circular pipes. Paper presented at the American Institute of Chemical
Engeering Conference on Solids Transport in Slurries, Atlantic City, NJ.
Hunt, I. N. 1969. Turbulent transport of heterogeneous sediment. Quarterly Journal Mechanics and
App. Maths., 22, 234–246.
Ippen, A. T. 1971. A new look at sedimentation in turbulent streams. Journal of Boston Soc. Civil
Eng., 58, 3, 131–163.
Ismail, H. M. 1952. Turbulent transfer mechanism and suspended sediment in closed channels.
Trans. Amer. Soc. Chem. Eng., 117, 2500, 409–447.
Kao, D. T. Y., and L. Y. Hwang. 1979. Critical slope for slurry pipelines. Paper presented at the Hydrotransport
6 Conference of the British Hydromechanical Research Association, Cranfield,
England.
Khan, A. R., and J. F. Richardson. 1996. Comparison of coarse slurry pipeline models. In Proceedings
of Hydrotransport 13, pp. 259–281. Cranfield, UK: BHR Group.
Lazarus, J. H. 1989. Mixed-regime slurries in pipelines. I. Mechanistic model. Journal of Hydraulic
Engineering, ASCE, 115, 11, 1496–1509.
Lazarus, J. H. & Cooke, R. 1993. Generalised mechanistic model for heterogeneous flow in a non-
Newtonian vehicle. In Proceedings of Hydrotransport 12, pp. 671–690. Cranfield, UK: BHR
Group.
Matsouk V. 1996. Internal structure of slurry flow in inclined pipe—Experiments and mechanistic
modeling. In Proceedings of Hydrotransport 13, pp. 187–210. Cranfield, UK: BHR Group.
Newitt, D. M., J. F. Richardson, M. Abbott, and R. B. Turtle. 1955. Hydraulic conveying of solids in
horizontal pipes. Trans Inst. of Chem. Eng., 33, 93–113.
Newitt, D. M., J. R. Richardson, and J. B. Glibbon. 1961. Hydraulic conveying of solids in vertical
pipes. Trans. Inst. of Chem. Eng., 39, 93–100.
Newitt, D. M., J. R. Richardson, and C. A. Shook (Eds.). 1962. Symposium on Interaction between
Fluids and Particles, London. London: Institution of Chemical Engineers.
Raj, R. S. 1972. Pressure loss in hydraulic transport of solids in inclined pipes. Paper presented at
Hydrotransport 2, Coventry, England.
Saskatchewan Research Council. 2000. Slurry Pipeline Course—SRC Pipe Flow Technology Center,
Saskatoon, Canada, May 15–16.
Schiller, R. E., and P. E. Herbich. 1991. Sediment transport in pipes. In Handbook of Dredging, Edited
by P. E. Herbich. New York: McGraw-Hill.
Shen, H. W. 1970. Sediment transportation mechanism—Transportation of sediments in pipes. Journal
Hydraulics Division Am. Soc. Civ. Eng., 96, 1503–1538.
Shook, C. A. 1981. Lead Agency Report for MTCM Cooperative Research Project. Report E-725-6-
C-81. Report prepared for the Saskatchewan Research Council, Saskatchewan, Canada.
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Report IX. Report prepared for the Saskatchewan Research Council, Saskatchewan,
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Translated and Edited by W. C. Cooley and R. R. Faddick. Terraspace Inc. report, April 1970,
pp 17–19.
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3, 232.
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April, 349–355.
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Wasp, E. J. et al. 1970. Deposition velocities, transition velocities, and spatial distribution of solids
in slurry pipelines. Paper read at 1st International Conference on Hydraulic Transportation of
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and Its applications, pp. 1–38, I. Zandi (Ed.). Oxford: Pergamon Press.

CHAPTER 3
MECHANICS OF
SUSPENSION OF
SOLIDS IN LIQUIDS
3-0 INTRODUCTION
The physical principles of flow of complex mixtures are based on the interaction between
the different phases, which may mix well or move in superimposed layers. In this chapter,
the basic concepts of motion of particles in a carrying fluid will be presented, as well as the
effect of their concentrations and boundaries. In the previous two chapters, we reviewed the
physical properties of solids, single-phase flows, and some aspects of mixtures of both.
Concepts of non-Newtonian mixtures are reviewed so the reader can understand the
principles used to analyze complex homogeneous flows of very fine particles at high volumetric
concentration.
The physics of solid–liquid mixtures have been the subject of many publications, particularly
by chemical and nuclear engineers. In this chapter, an effort is made to focus on
the practical equations that a slurry engineer may use to accomplish his/her tasks. The engineer
may have to use more than one equation when assessing a mixture to make an engineering
judgment.
3-1 DRAG COEFFICIENT AND TERMINAL
VELOCITY OF SUSPENDED SPHERES
IN A FLUID
One fundamental aspect to the transportation of solids by a liquid is the resistance, called
the drag force, that such solids will exert, and the ability of the liquid to lift such solids,
called the lift force. Both are complex functions of the speed of the flow, the shape of the
solid particles, the degree of turbulence, and the interaction between particles and the
pipe. One approach is to look at a vehicle that we have all come to know—the airplane.
This distraction from the complex world of slurry flows is justifiable.
3-1-1 The Airplane Analogy
When an airplane flies in a horizontal plane, it is subject to the forces of downward gravity,
upward lift, and drag opposite to its flight path. To maintain steady flight, its engines
3.1

3.2 CHAPTER THREE
must develop sufficient thrust to overcome drag. The airplane must also fly above its
stalling speed.
The lift and drag are aerodynamic forces (Figure 3-1). They are proportional to the
surface area, the density of air, the inclination of the airplane body with respect to speed,
and the square of the speed. For the airplane wing, these forces are expressed as
L = 0.5 CL�V 2Sw (3-1)
D = 0.5 CD�V 2Sw (3-2)
where
� = density of the fluid
V = cruising speed of airplane
CL = lift coefficient of wing airfoil
CD = drag coefficient of wing airfoil
The aerodynamic drag consists of two components: the profile drag and induced drag.
The induced drag is proportional to the square of the lift. Airfoils are designed to maximize
the lift-to-drag ratio, or to develop the most lift at the least drag penalty:
2 CD = CD0 + kwC L (3-3)
where
CD0 = the profile drag
kw = a function of the shape of the wing (minimum for an elliptical wing and for a wing
flying in ground effect)
The value of the drag and lift coefficients are determined by the shape of the flying ob-
Thrust
Wing lift
Weight
Forces on an aircraft in
steady horizontal flight
Drag
Stabilizer lift
Weight
Thrust
Drag
Forces on a rocket in
vertical flight
FIGURE 3-1 Lift and drag forces on moving objects.
Buoyancy
Drag
Weight
Forces on a free-falling
particle immersed in a fluid

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
ject, but also by the physical properties of a fluid, particularly the density, viscosity, and
speed of motion. Nondimensional analysis, an important branch of fluid dynamics, allows
the expression of these relationships by characteristic numbers. The Reynolds Number
has already introduced in Chapter 2.
For an airplane in a steady horizontal linear flight, the lift must overcome weight and
the thrust drag. A rocket flying in a vertical plane must develop sufficient thrust to overcome
drag forces as well as weight:
L = W and T = D For an Airplane
T = W + D For a rocket in vertical flight
3-1-2 Buoyancy of Floating Objects
The principle of Archimedes is well known. It states that the buoyancy force developed
by an object static in a fluid is equal to the weight of liquid of equivalent volume occupied
by the object. When the density of the object is less than the density of the liquid, the object
floats, and in the inverse situation, the object sinks.
For a sphere immersed in a fluid of density �L, the buoyancy force is calculated from
the weight of fluid the particle displaces:
3 FBF = (�/6)d g�Lg (3-4)
where
FBF = buoyancy force
dg = sphere diameter
g = acceleration due to gravity (9.78–9.81 m/s2 )
3-1-3 Terminal Velocity of Spherical Particles
Although most solids are not spherical in shape, the sphere is the point of reference for
the analysis of irregularly shaped solids.
3-1-3-1 Terminal Velocity of a Sphere Falling in a Vertical Tube
When a sphere is allowed to fall freely in a tube, the buoyancy and the drag forces act vertically
upward, whereas the weight force acts downward. At the terminal or free settling
velocity, in the absence of any centrifugal, electrostatic, or magnetic forces
W = D + FBF (3-5)
� �dg 3�Sg = � �dg 3 2
� �
�d
2 g
�Lg + 0.5 CD�LV t� � (3-6)
�
6
�
6
�
4
The drag coefficient corresponding to free fall of the particle is calculated as
4(�S – �L)gdg CD = ��
(3-7)
3�LV t 2
where
d g = sphere diameter
g = acceleration due to gravity, typically 9.8 m/s 2 or 32.2 ft/sec 2
3.3

3.4 CHAPTER THREE
Vt = the terminal (or free settling) speed
�s = the density of the solid sphere in kg/m3 or slugs/ft3 �L = the density of the liquid
The terminal (or sinking) velocity is measured using a visual accumulation tube with a
recording drum. Various mathematical models have been derived for the drag coefficient.
Turton and Levenspiel (1986) proposed the following equation:
0.413
0.657 CD = (1 + 0.173Re p ) ���
(3-8)
1 + 1.163 × 104 24
� –1.09
Rep
Re p
Example 3-1
Using the Turton and Levenspiel equation, write a small computer program in quickbasic
to tabulate the drag coefficient of a sphere.
LPRINT “ Drag coefficient vs. Reynolds Number based on
Turton, R., and O. Levenspiel”
RE0= 1
15 FOR I=1 TO 10
RE=I*RE0
CD= (24/RE) * (1+0.173*RE^0.657)*(0.413/(1+11630*RE^-1.09)
PRINT USING “RE= ###### ; Cd = ##.#### “; RE,CD
NEXT I
IF RE>1E6 THEN GOTO 30
RE0=RE
TABLE 3-1 Particle Reynolds Number and Corresponding Drag Coefficient for a
Sphere Based on the Equation of Turton and Levenspiel (1986) as per Example 3-1
Particle Drag Particle Drag Particle Drag
Reynolds coefficient, Reynolds coefficient, Reynolds coefficient,
number, Rep CD number, Rep CD number, Rep CD 1 28.1520 80 1.2266 6000 0.3983
2 15.2735 90 1.1571 7000 0.4042
3 10.8485 100 1.0994 8,000 0.4151
4 8.5809 200 0.5025 9,000 0.4151
5 7.1908 300 0.6793 10,000 0.4200
6 6.2459 400 0.6085 20,000 0.4497
7 5.5588 500 0.5617 30,000 0.4617
8 5.0349 600 0.5281 40,000 0.4671
9 4.6211 700 0.5029 50,000 0.4697
10 4.2851 800 0.4832 60,000 0.4709
20 2.6866 900 0.4675 70,000 0.4713
30 2.0940 1,000 0.4547 80,000 0.4713
40 1.7729 2,000 0.3990 90,000 0.4711
50 1.5670 3,000 0.3878 100,000 0.4707
60 1.4216 4,000 0.3883 200,000 0.4653
70 1.3124 5,000 0.3927 300,000 0.4609

Drag Coefficient C D
25
20
15
10
5
GOTO 15
30 END
0
0 2 4 6 8 10
Rep
MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
30 0
C D
C D
0
6
4
2
1.2
1.0
0.8
0.6
0.4
0.2
20
200
40
400
Results are tabulated in Table 3-1 and presented in Figure 3-2 in a linear scale rather
than a logarithmic scale. Linear scales are sometimes more useful to the mine operator
who is in a remote area and has little time to waste on difficult logarithmic graphs
3-1-3-2 Very Fine Spheres
For small particles in the range of a diameter d50 < 0.15 mm (0.0059 in), the most common
equation was created by Stokes and reported by Herbich (1991) and Wasp et al. (1977),
who indicate that the main forces are due to the viscosity effect in the laminar flow regime:
D = 3��dg (3-9)
In the laminar regime, the drag coefficient is inversely proportional to the Reynolds number,
i.e., CD = 24/Rep. The terminal velocity is expressed by Stoke’s equation:
2 (�S – �L)d gg Vt = ��
(3-10)
18�L�
Stokes’s equation is limited to particle Reynolds numbers smaller than 0.1, but has often
been used for particle Reynolds Numbers as large as 1 (based on sphere diameter d g).
60
600
80
800
100
Rep
3
10
Rep
CD CD D C
FIGURE 3-2 Drag coefficient of a sphere for Reynolds number smaller than 300,000.
0.6
0.4
0.2
0
0.6
0.4
0.2
0
0.6
0.4
0.2
1X10 5
2000
4
2X10
4000
4
4X10
Rep
5
3X10
6000
4
6X10
8000
4
8X10
3.5
10 4
Rep
5
1X10
Rep

3.6 CHAPTER THREE
From Equation 3-10, Herbich (1968) pointed out that the radius of particles for which the
validity of the equation is in doubt is expressed as
4.5� 2 � L
� (�S – � L)
R = � � 3/2
This equation is not set in stone for all situations. Rubey (1933) demonstrated one example
by showing that Stoke’s law does not apply to spherical quartz suspended in water
when the particle diameter exceeds 0.014 mm (0.00055 in, mesh 105).
3-1-3-3 Intermediate Spheres
For the range of particle Reynolds numbers between 1 and 1000, i.e., when
dpV0 �
1 < � < 1000
�
Govier and Aziz (1972) reported that Allen (1900) derived the following equation:
(� � – � L)g
Vt = 0.2� � 0.72
��
�L
(3-11)
Example 3-2
A slurry mixture consists of fine rocks at an average particle diameter of 140 �m, with a
particle density of 2800 kg/m3 . The carrier liquid is water with a dynamic viscosity of 1.5
× 10 –3 Pa · s. The volumetric concentration of the solids is 12%. Determine the terminal
velocity of the particles.
Solution
Using Equation 1-9, the dynamic viscosity of the mixture is
�m = �L[1 + 2.5C� + 10.05C� 2 + 0.00273 exp(16.6C�)]
= 1.5 × 10 –3 [1 + 2.5 × 0.12 + 10.05(0.12) 2 + 0.00273 exp (16.6 × 0.12)]
�m = 2.197 × 10 –3 Pa · s.
Let us check the magnitude of the Reynolds number:
= = 4.468
The Allen law applies in a transition regime:
Vt = 0.2 [9.81 × 1.8] 0.72
(140 × 10 –6 ) 1.18
���
(2.197 × 10 –3 /2800) 0.45
140 × 10 –6 × 0.02504 × 2800
���
2.197 × 10 –3
d�V0� �
�
2.83 × 10
Vt = 0.2 × 7.903
–5
��
0.001789
V t = 0.02504 m/s
d p 1.18
� (�/�) 0.45
Richards (1908) demonstrated that Stokes’s equation is inaccurate for particles with a
diameter larger than 0.2 mm (0.00787 in, mesh 70) and conducted extensive tests for

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
quartz particles (with a specific gravity of 2.65) in laminar, transitional, and turbulent
regimes. He derived the following equation for terminal velocity in mm/s:
8.925
Vt = ����
(3-12)
3 1/2 dg{[1 + 95(�S/�L – 1)d g] – 1}
Where dg, the diameter of the sphere, is expressed in mm. This equation covers the range
of particles between 0.15–1.5 mm (0.0059–0.059 in) at particle Reynolds numbers between
10 and 1000.
3-1-3-4 Large Spheres
For particles with a diameter in excess of 1.5 mm, Herbrich (1991) expressed the terminal
velocity by the following equation:
Vt = Kt�[d� ��� g( ��� S/ L�–� 1�)] � (3-13)
where Kt = an experimental constant = 5.45 for Rep > 800, according to Govier and Aziz
(1972).
Equation 3-13 is often called Newton’s law. In the regime of Newton’s law, the drag
coefficient of a sphere is approximately 0.44, as shown in Figure 3-2. Newton’s law applies
to turbulent flow regimes.
Other equations for terminal velocity of particles have been developed by various authors.
Four different equations are presented in Table 3-2.
Example 3-3
Using the Budyruck equation from Table 3-3, determine the terminal velocity of spheres
from 0.1 to1 mm.
A simple computer program is written in quickbasic as follows:
LPRINT
LPRINT “BUDRYCK AND RITTINGER EQUATION FOR TERMINAL
VELOCITY OF SPHERES IN WATER”
LPRINT
LPRINT DP0 = .1
FOR I=1 to 11
DP = I*DP0
VS= (8.925/DP)*(SQR(1+157*DP^30-1)
LPRINT USING “PARTICLE DIAMETER = ##.### mm TERMINAL
VELOCITY Vs = ##.### mm/s”;DP,VS
NEXT I
FOR J=12 TO 20
DP = J*DP0
TABLE 3-2 Equations for Terminal Speed of Large Spheres
Name Equation* Application
Budryck 3 1/2 Vt = 8.925[(1 + 157d g) – 1]/dg For dg < 1.1 mm
Rittinger Vt = 87(1.65dg) 1/2 For 1.2 < dg < 2 mm
*Where V t is expressed in mm/s and d g in mm.
3.7

3.8 CHAPTER THREE
TABLE 3-3 Calculation of Terminal Velocity of Spheres in Accordance with
Budryck’s Equation
Particle diameter Terminal velocity Particle diameter Terminal velocity
dp in mm Vs in mm/s dp in mm Vs in mm/s
0.1 6.75 0.7 81.63
0.2 22.4 0.8 89.49
0.3 38.34 0.9 96.64
0.4 51.85 1.0 103.26
0.5 63.21 1.1 109.45
0.6 73.02
VS= 87*SQR(1.65*DP)
LPRINT USING “PARTICLE DIAMETER = ##.### mm TERMINAL
VELOCITY Vs = ##.### mm/s”;DP,VS
NEXT J
END
The results are shown in Tables 3-3, 3-4, and Figure 3-3
Herbich (1968) measured drag coefficients for ocean nodules to be as high as 0.6 at
particle Reynolds numbers of 200. This high value is reached with spheres at a particle
Reynolds number of 1000.
3-1-4 Effects of Cylindrical Walls on Terminal Velocity
The previous paragraphs focused on the settling velocity of a single particle or widely
separated particles. The presence of a vessel or cylindrical walls tends to multiply the interaction
between particles and cause some collisions. Extensive tests have been conducted
on flows in vertical tubes. Brown and associates (1950) recommended multiplying the
terminal speed of a single particle by a wall correction factor Fw. For laminar flows they
proposed to use the Francis equation:
Fw = 1 – (d�/Di) 9/4 (3-14a)
They proposed to use the Munroe equation for a turbulent flow regime:
Fw = 1 – (d�/Di) 1.5 (3-14b)
where Di = the inner diameter of the tube
TABLE 3-4 Calculation of Terminal Velocity of Spheres in Accordance with
Rittinger’s Equation
Particle diameter Terminal velocity Particle diameter Terminal velocity
d p in mm V t in mm/s d p in mm V t in mm/s
1.1 117.21 1.6 141.36
1.2 122.42 1.7 145.71
1.3 127.42 1.8 149.93
1.4 132.23 1.9 154.04
1.5 136.87 2.0 158.04

in mm/s
Terminal velocity V
t
160
140
120
100
80
60
40
in mm/s
Terminal velocity V
t
MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
120
100
80
60
40
20
0
0 0.01 0.02 0.03 0.04 0.05
Sphere diameter d p in inches
0 0.2 0.4 0.6 0.8 1.0 1.2
Sphere diameter dp
in mm
0.04 0.05 0.06 0.07 0.08
1.0 1.2 1.4 1.6 1.8 2.0
Sphere diameter dp
in mm
(a)
Sphere diameter d p in inches
(b)
5
4
3
2
1
0
Terminal velocity V
t
in inch/sec
6
5
4
3
2
1
3.9
inch /sec
Terminal velocity V
t
in
FIGURE 3-3 Terminal velocity of spheres (a) in accordance with Budryck’s equation, (b) in
accordance with Rittinger’s equation.

3.10 CHAPTER THREE
Example 3-4
The flow described in Example 3-2 occurs in a 63 mm ID pipe. Determine the corrected
terminal velocity due to the wall effects.
Solution
The terminal velocity was determined to be 0.02504 m/s. The flow is in transition. Equation
3-14a for laminar flow is
Fw = 1 – (d�/DI) 9/4
F w = 1 – (0.140/63) 9/4
Fw = 0.999
Equation 3-14b for turbulent flow is
Fw = 1 – (0.14/63) 1.5 = 0.999.
More recently, Prokunin (1998) extended the analysis of the interaction of the wall
with the motion of a single particle by considering the angle of inclination and any rotation
that the particle may incur. His investigation included immersion in non-Newtonian
flows by testing with glycerin and silicone. He noticed from his tests that when the particle
approaches the wall, it develops a lift force. The lift force seems to increase with a reduction
of the gap that separates the particle from the wall. However, Prokunin could not
explain this lift force and recommended further research.
3-1-5 Effects of the Volumetric Concentration on the
Terminal Velocity
As the volumetric concentration of particles increases, it causes interactions and collisions,
and transfers momentum between the different (finer and coarser) units. The distance
between particles decreases. For spheres at 1% concentration by volume, the interparticle
distance is only 4 diameters. It shrinks to 2.5 diameters at 5% and to 2 diameters
at 10% concentration by volume. In an ideal laminar flow, the interaction is much simpler
than in a turbulent flow.
Worster and Denny (1955) published data on the terminal velocity of coal and gravel
particles, as shown in Table 3-5. The effect of the concentration is clearly marked by a
difference in terminal velocity between a single particle and a volumetric concentration of
30%.
Kearsey and Gill (1963) applied the Carman–Kozeney equation of flow through a
porous medium to determine the terminal velocity as
TABLE 3-5 Terminal Velocity for Coal and Gravel after Worster and Denny (1955)
Coal with a specific gravity of 1.5
________________________________
Gravel with a specific gravity of 2.67
________________________________
Particle size
____________
Single particle 30% Concentration
______________ ________________
Single particle
______________
30% Concentration
________________
mm Inches (cm/s) (ft/s) (cm/s) (ft/s) (cm/s) (ft/s) (cm/s) (ft/s)
1.59 1/16 4.6 0.15 3.0 0.10 9.1 0.30 3.0 0.10
6.4
12.7
1 –4
1 –2
15.2
30.5
1.50
1.00
10.7
21.3
0.35
0.70
30.5
61.0
1.00
2.00
10.7
21.3
0.35
0.70
25.4 1 51.8 1.70 36.6 1.20 106.7 3.50 36.6 1.20

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
(1 – Cv) 1 �P
� � 2 �s p L
where
sp = the specific surface expressed for as sphere as the surface area to volume ratio:
3
�2 KzC v
V c = � �� �� � (3-15)
�d g 2
3.11
sp = = 6/dg Kz = the Kozney constant, which is a function of particle shape, porosity, particle orientation,
and size distribution. The magnitude of Kz is between 3 and 6, but is
commonly assumed to be 5
�P/Li = the pressure gradient in the pipe due to the flow of the mixture
In the process of sedimentation, the pressure gradient is essentially due to the volumetric
concentration of the particles and is expressed as
= Cv(�s – �L)g (3-16)
In addition, the settling velocity due to a volumetric concentration is expressed as
Vc = � �� � (3-17)
For spheres with sp = 6/dg, the equation reduces to
Vc = � �� � (3-18)
As the volumetric concentration increases from 3% to 30%, the velocity drops drastically.
Assuming Kz to be equal to 5.0, the settling velocity for spheres reduces to a simple
equation:
(1 – Cv) = (3-19)
where V0 = the terminal velocity at very low volumetric concentration
Equation 3-19 does not apply to volumetric concentrations smaller than 8%. Equation
3-18 would apply to smaller concentrations.
3
(1 – Cv) (�s – �L) �
�
Vc � �
V0 10Cv
3 (1 – Cv) (�s – �L) �2 �s p
2 gd g
��
36KzCv 3 � 3 (�d g/6) �P
�
Li
g
��
KzCv Example 3-5
Assuming that the terminal velocity at a volumetric concentration of 8% is 100 mm/s,
apply Equation 3-18 from a volumetric concentration of 8–30%. Plot the results in Figure
3-4.
Thomas (1963) proposed the following empirical equation in the range of Vc/V0 of
0.08–1.0:
2.303 log10(Vc/V0) = –5.9CV (3-20)
Example 3-6
The free settling speed of solid particles is 22 mm/s at a volumetric concentration of 1%.
Using the Thomas equation 3-20, determine the settling speed at 25% volumetric concentration.

3.12 CHAPTER THREE
V c/
Vo
Solution
2.303 log 10(V c/V 0) = -5.9 × 0.25
V c/V 0 = 10 –0.64
V c/V 0 = 0.2288
V c = 0.2288 × 22 mm/s = 5.03 mm/s
1.0
0.8
0.6
0.4
0.2
0
0 0.1
0.2
Volumetric concentration
FIGURE 3-4 Effect of the volumetric concentration on the terminal velocity of spheres in
accordance with Equation 3-18.
The Kozney-based approach is limited to concentrations where the particles come into
contact with each other in a vertical flow. Beyond this point, the pressure gradient is
smaller than expressed by Equation 3-16. In the case of hard spheres, the settling process
completes when the particles come into contact with each other. In the case of flocculated
particles or clusters of flocculated fluid, stress may cause deformation and further settling
may occur by compaction.
Irregularly shaped particles and flocculates cause the development of a structure with
its own yield stress level. As the particles move closer, the yield stress increases until
equilibrium is reached. The weight of the overburden is then supported by the saturated
fluid and the compacted sediment.
3-2 GENERALIZED DRAG COEFFICIENT—
THE CONCEPT OF SHAPE FACTOR
Every day the slurry engineer has to deal with particles of all shapes and sizes. Although
the sphere represents a shape for reference, it is in the minority in the world of crushed or
naturally worn rocks.
Albertson (1953) conducted an extensive study on the effect of the shape of gravel
particles on the fall velocity in a vertical flow (Figure 3-5). He proposed a definition for a
shape factor:
where
a = the longest of three mutually perpendicular axes
b = the third axis
c = the shortest of three mutually perpendicular axes
0.3
c
�A = � (3-21)
�(a�b�)�

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
b
a
direction of fall
FIGURE 3-5 The axes of an irregularly shaped particle, according to Albertson.
3.13
Particles in a free fall tend to align themselves to expose the largest surface to the
flow. In other words, they act as free-falling leaves from a tree on an autumn day, where c
is taken as the dimension opposite to the direction of the fall. The projected area of the
particle is a function of the dimensions “a” and “b” but is often not equaled to such a
product as (ab) because particles are usually not rectangular in shape (see Table 3-6).
In a different approach, Clift et al. (1978) decided to compare the projected area of a
free-falling, irregularly shaped particle, with a sphere of equal projected area in order to
define a diameter:
da = �(4�S� ���)� f /
(3-22)
where
Sf = the projected area of the free-falling particle
However, Albertson (1953) preferred to define a different diameter base, dp, on the
fact that the actual volume of the free-falling particle could be equated to a sphere of the
TABLE 3-6 Clift Shape Factor of Various Particles
Isometric Typical mineral particles
____________________________________ _______________________________________
Particle � c Particle � c
Sphere 0.524 Sand 0.26
Cube 0.694 Sillimanite 0.23
Tetrahedron 0.328 Bituminous Coal 0.23
Irregular Rounded 0.54 Blast Furnace Slag 0.19
Cubic angular 0.47 Limestone 0.16
Tetrahedral 0.38 Talc 0.16
Plumbago 0.16
Gypsum 0.13
Flake Graphite 0.023
Mica 0.003
From Wilson et al. (1992).
c

3.14 CHAPTER THREE
same volume but with a diameter of dn. Albertson (1953) therefore proposed a Reynolds
number based on dn: dn�Vt Ren = � (3-23)
�
There may be a marked difference between naturally worn gravel and crushed gravel.
This is a fact that a slurry engineer should bear in mind when extrapolating data from lab
results.
Because Clift chose an equivalent diameter d a based on the projected area, he proposed
a different shape factor:
� c = particle volume/d a 3 (3-24)
Typical values are shown in Table 3-6. The Albertson and Clift shape factors are about
40 years apart in definition but can be related by a factor E:
� c = E� A
(3-25)
The logarithmic curves as shown in Figure 3-6 are sometimes difficult to read. Table
3-7 presents values of drag coefficient versus Reynolds number rounded off to the first
decimal point.
The work of Albertson was developed further by the Inter-Agency Committee on Water
Resources (1958), who developed the following two non-dimensional coefficients
(Figure 3-7):
and
Drag coefficient C D
10.0
1.0
0.1
C N = (� s/� L – 1)g�/V t 3 (3-26a)
C N = 0.75C D/Re n
(3-26b)
C S = �(� s/� L – 1)gd p 3 /(6� 2 ) (3-27a)
C S = 0.125�C DRe n 2 (3-27b)
ALBERTSON SHAPE FACTOR = a/ cb
0.3
0.5
0.7
0 10 100 10 3
10 4
10 5
10 6
0 10 100 103 104 105 106 Particle Reynolds number Re p
FIGURE 3-6 The drag coefficient versus Reynolds number and shape factor. (After Albertson,
1953.)
1.0

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
TABLE 3-7 Drag Coefficient versus Reynolds Number for Different Albertson Shape
Factors
Drag coefficient
3.15
Reynolds number Shape factor = 0.3 Shape factor = 0.5 Shape factor = 0.7 Shape factor = 1.0
7 7.0 6.0 4.7 4.0
8 6.5 5.5 4.3 3.7
9 6.1 5.1 4.0 3.4
10 5.8 4.74 3.75 3.15
15 4.64 3.7 3.0 2.4
20 3.95 3.2 2.55 2.0
32 3.0 2.6 2.1 1.55
40 2.7 2.28 1.84 1.3
50 2.5 2.08 1.67 1.12
60 2.3 1.94 1.56 1.0
70 2.25 1.74 1.4 0.94
80 2.2 1.67 1.35 0.844
100 2.08 1.62 1.3 0.8
150 1.87 1.44 1.16 0.68
200 1.75 1.36 1.11 0.6
300 1.74 1.33 1.08 0.5
400 1.8 1.34 1.09 0.44
500 1.9 1.38 1.1 0.4
600 1.94 1.42 1.12 0.38
700 1.988 1.47 1.14 0.36
800 2.0 1.51 1.15 0.34
900 2.07 1.54 1.16 0.334
1000 2.1 1.58 1.17 0.33
2000 2.3 1.72 1.22 0.3
3000 2.28 1.73 1.19 0.29
4000 2.48 1.69 1.16 0.294
5000 2.21 1.66 1.14 0.3
6000 2.2 1.62 1.13 0.31
7000 2.19 1.58 1.13 0.31
8000 2.183 1.55 1.14 0.32
9000 2.18 1.53 1.14 0.32
The drag coefficient C D is then plotted against the equivalent Reynolds number Re n to
determine the terminal velocity. On a logarithmic scale, C N and C S are superposed as
straight lines for reference (Figure 3-7).
In order to measure the Albertson shape factor, Wasp et al. (1977) developed a correlation
between the sieve diameter and the fall diameter d n (Figure 3-8).
The approach proposed by Albertson and Clift is limited to free fall of particles in a
fluid. However, turbulence can develop new forces. Whenever an engineering contract requires
the drag of particles to be measured, the engineer is well advised to conduct tests in
a fluid of similar dynamic viscosity as the one that will be used in the project. In addition
to the shape factor and drag coefficient, the slurry engineer must also determine the fluid
density, dynamic viscosity at the temperature of pumping, particle density (or specific
gravity of solids), nominal (or statistical average) diameter, and fall velocity.

Sieve diameter (mm)
3.16 CHAPTER THREE
FIGURE 3-7 C D and C W versus particle Reynolds number for different shape factors. Adapted
from the Inter-Agency Committee on Water Resources (1958).
1.0
0.8
0.6
0.4
0.2
spheres
S.F = 0.3
S.F = 0.5
S.F = 0.7
S.F= 0.9
0 0.2 0.4 0.6 0.8 1.0
Sieve diameter (mm)
7
6
5
4
3
2
1
S.F = 0.3
S.F =0.7
S.F =0.5
spheres
0 1 2 3 4
Fall diameter (mm) Fall diameter (mm)
FIGURE 3-8 Relationship between sieve and fall diameter after Wasp et al. (1977).
S.F=0.9
5

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
Example 3-7
A naturally worn particle has an Albertson shape factor of 0.7. It has a nominal diameter
of 250 �m. Its density is 3000 kg/m3 . It is allowed to free-fall in water at a temperature of
25° C.
Calculate the fall velocity for the single particle and the fall velocity if the volumetric
concentration of particles is increased to 20%.
Solution
Referring to Table 2-7 (or Table 2-8 for USCS units), the kinematic viscosity of water is
0.89 × 10 –6 m2 2 /s. We need to determine the coefficient CS = 0.125�CD/Ren. The curves
published by Inter-Agency Committee on Water Resources indicate that CS =
2 3 2 0.125�CD/Ren = 0.167�(�s/�L – 1)gd p/� = 203.
From Figure 3-6, at a shape factor of 0.7 and CS of 203, the Reynolds number would
be 7.2Vt = Re/(�dp) = 7.2/(890,000 × 0.00025) = 0.0324 m/s for a single particle.
Applying Equation 3-18 for a concentration of 20%, the velocity would be 0.256 ×
0.0324 = 0.0083 m/s.
3-3 NON-NEWTONIAN SLURRIES
3.17
Various models have been developed over the years to classify complex two- and threephase
mixtures (Table 3-8). In the case of mining, the following mixtures are often encountered:
� A fine dispersion containing small particles of a solid, which are uniformly distributed
in a continuous fluid and are found in copper concentrate pipelines and in slurry from
grinding after classification, etc.
TABLE 3-8 Regimes of Flows for Newtonian and Non-Newtonian Mixtures after
Govier and Aziz (1972)
Single-phase flows
___________________________
Multiphase flows (gas–liquid, liquid–liquid,
gas–solid, liquid–liquid)
___________________________________________________
Single-phase behavior
_____________________________________________________
Multiphase behavior
___________________________
Pseudohomogeneous
_______________________________
Heterogeneous
__________________
True homogeneous Laminar, transition, and
turbulent flow regime
Turbulent flow regime only
Purely viscous Newtonian flows
Purely viscous, non-Newtonian, Bingham plastic
and time-independent Dilatant
Pseudoplastic
Yield pseudoplastic
Purely viscous, non-Newtonian Thixotropic
and time-dependent Rheopectic
Viscoelastic Many forms

3.18 CHAPTER THREE
� A coarse dispersion containing large particles distributed in a continuous fluid and encountered
in SAG mills, cyclone underflows, and in certain tailings lines, etc.
� A macro-mixed flow pattern containing either a frothy or highly turbulent mixture of
gas and liquid, or two immiscible liquids under conditions in which neither is continuous.
Such patterns are found in flotation circuits in which froth is used to separate concentrate
from gangue.
� A stratified flow pattern containing a gas, liquid, two slurries of different particle sizes,
or two immiscible liquids under conditions in which both phases are continuous.
Designing a pipeline to operate in a non-Newtonian flow regime must be based on reliable
test data about the rheology and particle sizing (see Table 3-9). The engineer must
be cautious before venturing into generalizations about rheological properties.
In Figure 1-4 of Chapter 1, the relationship between dynamic viscosity and volumetric
concentration was presented. In fact, the industry has accepted the criterion that friction
losses are highly dependent on slurry viscosity in cases where the average particle diameter
is finer than 40–60 microns, and (depending on the specific gravity) at volumetric concentrations
in excess of 30%.
Fibrous slurries such as fermentation broths, fruit pulps, crushed meal animal feed,
tomato puree, sewage sludge, and paper pulp may not contain a high percentage of solids,
but may flow as non-Newtonian regimes. With these materials, the long fibers are flexible
and intertwine into a close-packed configuration and entrap the suspending medium. The
fibers may be flocculated or may form flocs with an open structure. Based on the volume
content of the flocs, the mixture may develop high dynamic viscosity. However, because
the flocs are compressible, they may deform with the flow.
Flocculated slurries are encountered in flotation cells circuits, thickeners, and various
processes in mineral extraction plants. With the formation of flocs, the slurry may develop
an internal structure. This structure may develop properties leading to a non-Newtonian
flow, shear thinning behavior (pseudoplastic), and sometimes thixotropic time-dependent
behavior. When shear stresses are applied to the slurry, the floc sizes may shrink and become
less capable of entrapping the carrier slurry. At higher shear stresses, the flocs may
shrink to the size of particles, and the flow may lose its non-Newtonian behavior.
3-4 TIME-INDEPENDENT NON-NEWTONIAN
MIXTURES
Certain slurries require a minimum level of stress before they can flow. An example is
fresh concrete that does not flow unless the angle of the chute exceeds a certain minimum.
Such a mixture is said to posses a yield stress magnitude that must be exceeded before
that flow can commence. A number of flows such as Bingham plastics, pseudoplastics,
yield pseudoplastics, and dilatant are classified as time-independent non-Newtonian fluids.
The relationship of wall shear stress versus shear rate is of the type shown in Figure
3-9 (a), and the relationship between the apparent viscosity and the shear rate is shown in
Figure 3-9 (b). The apparent viscosity is defined as
�a = Cw/(d�/dt) (3.28)
3-4-1 Bingham Plastics
For a Bingham plastics it is essential to overcome a yield stress � 0 before the fluid is set in
motion. The shear stress versus shear rate is then expressed as

TABLE 3-9 Examples of Bingham Slurries
MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
Coefficient
Yield of rigidity,
Particle size, Density, Stress, � mPa · s
Slurry d 50 kg/m 3 Pa (cP) Reference
3.19
54.3% Aqueous suspension 92% under 74 �m 1520 3.8 6.86 Hedstrom (1952)
of cement, rock
Flocculated aqueous China 80% under 1 �m 1280 59 13.1 Valentik &
clay suspension No. 1 Whitmore (1965)
Flocculated aqueous China 80% under 1 �m 1207 25 6.7 Valentik &
clay suspension No. 4 Whitmore (1965)
Flocculated aqueous China 80% under 1 �m 1149 7.8 4.0 Valentik &
clay suspension No. 6 Whitmore (1965)
Aqueous clay suspension I 1520 34.5 44.7 Caldwell &
Babitt (1941)
Aqueous clay suspension III 1440 20 32.8 Caldwell &
Babitt (1941)
Aqueous clay suspension V 1360 6.65 19.4 Caldwell &
Babitt (1941)
Fine coal @ 49% C W 50% under 40 �m 1 5 Wells (1991)
Fine coal @ 68% C W 50% under 40 �m 8.3 40 Wells (1991)
Coal tails @ 31% C W 50% under 70 �m 2 60 Wells (1991)
Copper concentrate @ 50% under 35 �m 19 18 Wells (1991)
48% C W
21.4% Bauxite < 200�m 1163 8.5 4.1 Boger & Nguyen
(1987)
Gold tails @ 31% C W 50% under 50 �m 5 87 Wells (1991)
18% Iron oxide < 50 �m 1170 0.78 4.5 Cheng &
Whittaker (1972)
7.5 % Kaolin clay Colloidal 1103 7.5 5 Thomas (1981)
Kaolin @ 32% C W 50% under 0.8 �m 20 30 Wells (1991)
Kaolin @ 53% CW with 50% under 0.8 �m 6 15 Wells (1991)
sodium silicate
Kimbelite tails @ 37% C W 50% under 15 �m 11.6 6 Wells (1991)
58% Limestone < 160 �m 1530 2.5 15 Cheng &
Whittaker (1972)
52.4% Fine liminite < 50 �m 2435 30 16 Mun (1988)
Mineral sands tails @ 50% under 160 �m 30 250 Wells (1991)
58% C w
13.9% Milicz clay < 70 �m 2.3 8.7 Parzonka (1964)
16.8% Milicz clay < 70 �m 5.3 13.6 Parzonka (1964)
19.6% Milicz clay < 70 �m 13 25 Parzonka (1964)
Phosphate tails @ 37% C W 85% under 10 �m 28.5 14 Wells (1991)
14% Sewage sludge 1060 3.1 24.5 Caldwell &
Babitt (1941)
Red mud @ 39% C W 5% under 150 �m 23 30 Wells (1991)
Zinc concentrate @ 75% C W 50% under 20 �m 12 31 Wells (1991)
Uranium tails @ 58% C W 50% under 38 �m 4 15 Wells (1991)

3.20 CHAPTER THREE
Shear Stress �
Apparent viscosity � a
Bingham Plastic
Dilatant
Newtonian
Rate of shear (� = du/dy)
Bingham Plastic
Yield Pseudoplastic
Pseudoplastic
Dilatant
Newtonian
Pseudoplastic
Rate of shear (� = du/dy)
(b)
FIGURE 3-9 (a) Shear stress versus shear rate; (b) viscosity versus shear rate of time-independent
non-Newtonian fluids.

�w – �0 = �d�/dt (3-29)
where
�w = shear stress at the wall
�0 = yield stress
� = the coefficient of rigidity or non-Newtonian viscosity
It is also related to a Bingham plastic limiting viscosity at infinite shear rate by the following
equation:
�0 � = � + �� (3-30)
(d�/dt)
The magnitude of the yield stress �0 may be as low as 0.01 Pascal for sewage sludge
(Dick and Ewing, 1967) or as high as 1000 MPa for asphalts and Bitumen (Pilpel, 1965).
The coefficient of rigidity may be as low as the viscosity of water or as high as 1000 poise
(100 Pa · s) for some paints and much higher for asphalts and bitumen. In the case of tarbased
emulsions or certain tar sands, it is customary to add certain chemicals to reduce the
dynamic viscosity of the emulsion or the coefficient of rigidity of the slurry. Tables 3-9
presents examples of Bingham slurries, magnitudes of yield stress, and coefficients of
rigidity � values.
Example 3-8
Samples of a mineral slurry with C w = 45% are examined in a lab. From the measurements
of the rate of shear (�) and shear stress (�), determine the yield stress and viscosity.
Rate of Shear � [s –1 ] 100 150 200 300 400 500 600 700 800
Shear Stress � (Pa) 10.93 12.27 13.49 15.68 17.66 19.49 21.2 22.84 24.43
� – � 0 (Pa) 4.11 5.45 6.67 8.87 10.85 12.67 14.39 16.03 17.61
The data is plotted in Figure 3-10. At a low shear rate < 100s – 1, the slope is
At high shear rate
� = 4.426/100 = 0.0443 Pa · s
4.426
�� = � = 0.0164 Pa · s
270
� =
Take a point at high shear rate (700 s –1 ):
Check at du/dy = 600
at du/dy = 800
MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
� =
� w – � 0
� du/dy
16.03
� 700
� = 0.0229 Pa · s
14.394
� = � = 0.02399
600
3.21

Shear stress (Pa)
3.22 CHAPTER THREE
17.61
� = � 0.022
800
An average � = 0.023 Pa · s is taken.
Alternative � = �0/(du/dy) + �a � = 6.82/700 + 0.0164 = 0.026 Pa · s
This example shows that at zero rate of shear the shear stress is 6.82 Pa. The yield
stress is therefore 6.82 Pa.
The yield stress increases as the concentration of solids augments. Thomas (1961) proposed
the following relationships between yield stress �0, coefficient of rigidity �, concentration
by volume Cv, and viscosity of the suspending medium �:
� 0 = K 1C v 3 (3-31)
�/� = exp(K2Cv) (3-32)
where K1 and K2 = constants and are characteristics of the particle size, shape, and concentration
of the electrolyte concentration.
These equations were derived from the work of Thomas (1961) on suspensions of titanium
dioxide, graphite, kaolin, and thorium oxide in a range of particle sizes from
0.35–13 micrometers and in volume concentration of 2–23%.
Thomas (1961) defined a shape factor �T1 for nonspherical particles as
�T1 = exp[0.7(sp/s0 – 1)] (3-33)
where
sp = the surface area per unit volume of the actual particles
s0 = the surface area per unit volume of a sphere of equivalent dimensions or 6/dg He indicated that the coefficient K 1 might then be expressed as
30
28
24
20
16
12
8
4
0
0
0 100 200 300
400 500 600 700
FIGURE 3-10 Plot of data for Example 3-8.
800 900
Rate of shear (sec
-1
)

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
u�T1 �2 d p
3.23
K 1 = (3-34)
Where K 1 is expressed in Pa (or lb f/ft 2 with u = 210 in), and the particle diameter d p is expressed
in microns.
Thomas defined a second shape factor � T2 = (s p/s 0) 1/2 to derive the equation:
K 2 = 2.5 + 14� T 2/�d� p� when 0.4 < d p < 20 microns (3-35)
Thomas (1963) extended his work to flocculated mixtures with dispersed fine and ultrafine
particles with overall dimensions up to 115 microns. He derived the following equations:
�/� = exp[(2.5 + �)Cv] (3-36)
where
� = �[( �d� f /d� ap� p) � –� 1�]� (3-37)
where
� = the ratio of immobilized dispersing fluid to the volume of solids related approximately
to the particle and floc apparent diameter
df = the apparent floc diameter
dapp = the apparent particle diameter
This particle diameter is shown by the following:
dapp = dp(s0/sp) exp(– 1 –
2 ln2 �) (3-38)
where
� = the logarithmic standard deviation
In general, and at a constant temperature, the following equations are applied to Bingham
plastic slurries:
�/� = A exp(BCv) (3-39)
�0 = E exp(FCv) (3-40)
The constants A, B, E, and F are derived from tests measuring particle size, shape, and the
nature of their surface.
Gay et al. (1969) proposed the following correlation for high concentrations of solids:
�/� = exp{[2.5 + [Cv/(Cv� – Cv)] 0.48 ](Cv/Cv�)} (3-41)
where
Cv� = the maximum packing concentration of solids
For a change in temperature in the order of 27°C (50°F). Parzonka (1964) developed
the following power law equation:
–n � = K3T a (3-42a)
where
n = an exponent
K3 = an exponent
Ta = absolute temperature
Govier and Aziz (1972) proposed an equation based on an exponential drop of Bingham
plastic viscosity with temperature:

3.24 CHAPTER THREE
� = A exp(B/T) (3-42b)
To obtain the viscosity, plot the curve of the shear stress (� – �0) in Pascals against the
shear rate � (s –1 ).
3-4-2 Pseudoplastic Slurries
Pseudoplastic fluids are time-independent non-Newtonian fluids that are characterized by
the following:
� An infinitesimal shear stress, which is sufficient to initiate motion
� The rate of increase of shear stress with respect to the velocity gradient decreases as
the velocity gradient increases
This type of flow is encountered when fine particles form loosely bound aggregates
that are aligned, stable, and reproducible at a given magnitude of shear rate.
The behavior of pseudoplastic fluids is difficult to define accurately. Various empirical
equations have been developed over the years and involve at least two empirical factors,
one of which is an exponent. For these reasons, pseudoplastic slurries are often
called power-law slurries. The shear stress is defined in terms of the shear rate by the following
equation:
�w = K[(d�/dt) n ] (3-43)
where
K = the power law consistency factor, expressed in Pa · sn n = the power law behavior index, and is smaller than unity
Examples of pseudoplastic slurries are shown in Table 3-10.
The apparent viscosity of a pseudoplastic is defined in terms of the ratio of the shear
stress to the shear rate:
� a = [� w/(d�/dt)] (3-44)
3-4-2-1 Homogeneous Pseudoplastics
Pseudoplastic slurries are another category of non-Newtonian slurries. Pseudoplastics are
divided into homogeneous and pseudohomogeneous mixtures. Whereas in the case of a
Bingham slurry, it was pointed out that the coefficient of rigidity was a linear function of
the shear rate, in the case of a pseudoplastic, the coefficient of rigidity is expressed by the
following power law:
� = K(d�/dt) n–1 (3-45)
The shear stress is plotted against the shear rate on a logarithmic scale at various volume
fractions. From the slope of such a plot, “K,” the power law consistency factor, and
“n,” the power law behavior index (smaller than unity) are derived as plotted in Figure 3-
11.
As indicated in Figure 3-12 the magnitude “K,” the power law consistency factor, and
the power law factor index n are dependent on the volumetric concentration of solids.
Example 3-9
A phosphate slurry mixture is tested using a rheogram. The following data describe the
relationship between the wall shear stress and the shear rate:

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
d�/dt 0 50 100 150 200 300 400 500 600 700 800
�w(Pa) 25 32 43 51 53 56 58 60 62 63.2 64.3
The mixture is non-Newtonian. If it is considered a power law slurry, derive the power
law exponent “n” and the power law coefficient K.
Solution
The first step is to plot the data on a logarithmic scale. In the equation for a pseudoplastic,
the coefficient of rigidity is expressed by equations (3.43) and (3.45), the values of “K”
and “n.” By using the logarithmic scale:
log �w = log K + n log (d�/dt)
log(d�/dt) 1.699 2 2.176 2.301 2.477 2.602 2.669 2.778 2.845 2.903
log(�w) 1.505 1.633 1.707 1.724 1.748 1.763 1.778 1.792 1.8 1.808
n — 0.425 0.592 0.136 0.136 0.12 0.154 0.112 0.13 0.14
log(d�/dt) 2 – log(d�/dt) 1
n = ���
(log�w) 2 – (log�w) 1
n � 0.132
1.8 = log K = 0.132 × 2.843
log K = 1.424
K = 26.5
TABLE 3-10 Examples of Power Law Pseudoplastics
Range of Range of Angle of
Particle weight consistency flow
size, concentration, coefficient K, behavior
Slurry d 50 % Ns n /m 2 index, n Reference
3.25
Cellulose acetate 1.5–7.4 1.4–34.0 0.38–0.43 Heywood (1996)
Drilling mud—barite 14.7 �m 1.0–40.0 0.8–1.3 0.43–0.62 Heywood (1996)
Sand in drilling mud 180 �m 1.0–15% 0.72–1.21 0.48–0.57 Heywood (1996)
sand using
drilling mud
with 18%
barite
Graphite 16.1 �m 0.5–5.0 Unknown Probably 1 Heywood (1996)
Graphite and 5 �m 32.2 total
magnesium (4.1 graphite 5.22 0.16 Heywood (1996)
hydroxide and 28.1
magnesium
hydroxide)
Flocculated kaolin 0.75 �m 8.9–36.3 0.3–39 0.117–0.285 Heywood (1996)
Deflocculated kaolin 0.75 �m 31.3–63.7 0.011–0.6 0.82–1.56 Heywood (1996)
Magnesium hydroxide 5 �m 8.4–45.3 0.5–68 0.12–0.16 Heywood (1996)
Pulverized fuel ash 38 �m 63–71.8 3.3–9.3 0.44–0.46 Heywood (1996)
(PFA-P)
Pulverized fuel ash 20 �m 70–74.4 2.12–9.02 0.48–0.57 Heywood (1996)
(PFA-P)

3.26 CHAPTER THREE
Shear stress
(in units of pressure)
1
0.1
0.01
0.001
0.0001
slope = y/x
Consider d�/dt = 700. Check �w = K(d�/dt) n .
62.9 = 26.5 × 7000.132 This is close to the measured stress of 63.2 Pa. Therefore, the equation of this phosphate
slurry is:
�w = 26.5(d�/dt) 0.132
The coefficient of rigidity is obtained as:
Power Law Consistency Factor K
Pa.s
n
/cm
2
10
8
6
4
2
0
0
x
0 1 10 100 1000 10,000
Shear rate (1/sec)
20
clays
magnetite
40
Volume Fraction of
solids, C V
y
0
n = y/x
FIGURE 3-11 Plotting the rheology on a logarithmic scale to obtain the consistency factor
“K” and the flow behavior index “n” of Pseudoplastics.
Flow Behavior Index "n"
1.0
0.8
0.6
0.4
0.2
0
0
magnetite
clays
20 40
K
Volume Fraction of
solids, CV
FIGURE 3-12 Effect of volumetric concentration on the consistency factor “K” and the flow
behavior index “n” of Pseudoplastics (after Aziz and Govier, 1972).
n

at d�/dt = 700
at d�/dt = 600.
MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
� = K(d�/dt) n–1
� = 26.5(d�/dt) –0.878
� = 26.5 × (700) –0.878
� = 0.084 Pa · s
� = 26.5 × 600 = 0.096 Pa · s
3.27
3-4-2-2 Pseudohomogeneous Pseudoplastics
Pseudohomogeneous pseudoplastics behave similarly to their homogeneous counterparts.
Clay suspensions and magnetite-based slurries demonstrate an exponential relationship
between n and C v as shown in Figure 3-12. The power law factor K has a more complex
relationship with C v, as shown in Figure 3-12.
Various equations have been derived to solve the power law factor of pseudoplastics.
These equations are presented to help the reader appreciate the rheological constants that
must be determined by testing, as will be described in Section 3-6.
The Prandtl–Eyring equation is based on Dahlgreen’s (1958) discussion of the study
conducted by Eyring and Prandtl on the kinetic theory of liquids:
� = A sinh –1 [(d�/dt)/B] (3-46)
where
A and B = the rheological constants
sinh = the hyperbolic function
From Equation 3-44, the apparent viscosity is derived as
�a = {A/(d�/dt)}{sinh –1 [(d�/dt)/B]} (3-47)
The Ellis equation is more flexible but is an empirical equation and uses three rheological
constants. Skelland (1967) demonstrates how the equation is based on the work of Ellis
and Round and is explicit with respect to the velocity gradient rather than the shear rate:
(d�/dt) = (A0 + A1� (�–1) )�w (3-48)
where A0, A1, and � are the rheological coefficients of the slurry material.
The apparent viscosity is expressed as
(�–1) �a = 1/(A0 + A1�w ) (3-49)
When A1 = 0, the equation takes on a Newtonian form where A0 = 1/�.
The equation reduces to the conventional power law equation with � = 1/n and A1 =
(1/k) 1/n . When � > 1, the equation approaches a Newtonian flow at low shear stresses, and
when � < 1, it tends to approach a Newtonian flow at high shear stress.
The Cross equation (Cross, 1965) is a versatile equation that is based on measurements
of viscosity, �0 at zero shear rate and �� at infinite shear rates.
�� – �0 �a = �0 + ��
(3-50)
2/3
1 + �(d�/dt)
where � is a coefficient used to express to the shear stability of the mixture.
This equation has been tested and has successfully predicted the behavior of a wide

3.28 CHAPTER THREE
variety of pseudoplastic mixtures, such as suspensions of limestone, non-aqueous polymer
solutions, and nonaqueous pigment paste.
3-4-3 Dilatant Slurries
Dilatant fluids are time-independent non-Newtonian fluids and are characterized by the
following:
� An infinitesimal shear stress is sufficient to initiate motion.
� The rate of increase of shear stress with respect to the velocity gradient increases as the
velocity gradient increases.
Dilatant fluids, therefore, use similar equations as pseudoplastic fluids. They are much
less common than pseudoplastics. Dilatancy is observed under specific conditions such as
certain concentrations of solids, shear rates, and the shape of particles. Dilatancy is due to
the shift, under shear action, of a close packing of particles to a more open distribution in
the liquid.
Govier et al. (1957) observed the phenomena of dilatancy in suspensions of magnetite,
galena, and ferrosilicon in a range of particle sizes from 5 microns to 70 microns.
It is observed that the slope of the shear stress versus the shear rate increases, particularly
in the range of shear rates from 80 to 120 sec –1 . Metzener and Whitlock (1958) explained
the phenomenon of dilatancy as follows.
Two mechanisms account for the inflection and subsequent increase in the slope of
the curve. Initially, the shear stress approaches a magnitude at which the size of flowing
particles and aggregates is at a minimum and a Newtonian behavior develops (at
the inflection of the curve). As the level of stress rises, the mixture expands volumetrically,
and entire layers of particles start to slide or glide over each other. In the interim,
the slurry acts as a pseudoplastic until the shear stress is high enough to cause dilatancy.
The phenomenon of dilatancy is not easy to model. According to Metzener and Whitlock
(1958), it is observed at volumetric concentration in excess of 27–30% and at shear
rates in excess of 100 s –1 .
3-4-4 Yield Pseudoplastic Slurries
Yield pseudoplastic fluids are time-independent non-Newtonian fluids and are characterized
by the following:
� An infinitesimal shear stress is sufficient to initiate motion.
� The rate of increase of shear stress, with respect to the velocity gradient, decreases as
the velocity gradient increases.
� A yield stress must be overcome at zero shear rate for motion to occur.
Examples of yield pseudoplastics are shown in Table 3-11.
Equation 3-44 is then modified to account for the yield stress as follows:
�w – �0 = K[(d�/dt) n ] (3-51)
Equation 3-51 is known as the Herschel–Buckley equation of yield pseudoplastics and
is accepted by most slurry experts to describe the rheology of yield pseudoplastics with

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
TABLE 3-11 Examples of Yield Pseudoplastics
Range of Angle of
consistency flow
Density, Yield stress coefficient K, behavior
Slurry kg/m 3 � 0, Pa Ns n /m 2 index, n Reference
3.29
Sewage sludge 1024 1.268 0.214 0.613 Chilton and Stainsby (1998)
Sewage sludge 1011 0.727 0.069 0.664 Chilton and Stainsby (1998)
Sewage sludge 1013 2.827 0.047 0.806 Chilton and Stainsby (1998)
Sewage sludge 1016 1.273 0.189 0.594 Chilton and Stainsby (1998)
Kaolin slurry 1071 1.880 0.010 0.843 Chilton and Stainsby (1998)
Kaolin slurry 1061 1.040 0.014 0.803 Chilton and Stainsby (1998)
Kaolin slurry 1105 4.180 0.035 0.719 Chilton and Stainsby (1998)
low to moderate concentration of solids. At high shear rates, certain complex phenomena
such as dilatancy may develop. Certain bentonite clays develop a yield pseudoplastic rheology
at 20% concentration by volume.
Krusteva (1998) investigated the rheology of a number of inorganic waste slurries
such as drilling fluids in petroleum output, residue mineral materials in tailing ponds, filling
of abandoned mine galleries, etc. In the case of clay containing industrial wastes, he
indicated that colloidal forces of attraction or repulsion are ever present with Brownian
forces and may cause thermodynamic instability. Waste materials such as blast furnace
slag, fly ash, and material from mine filling exhibited various forms of a yield pseudoplastic
rheology.
The behavior of yield pseudoplastics can be expressed by the Carson model as described
by Lapasin et al. (1998):
�n = � n n
0 +�� (d�/dt) (3-52)
By binary system, Lapasin meant a mixture of two sizes of particles above the colloidal
range and by ternary, three sizes. Alumina powders with average d50 diameters of 0.9
�m, 1.4 �m, and 3.9 �m, and different specific surface areas (8.23 m2 /cm3 , 5.74
m2 /cm3 , and 2.65 m2 /cm3 ) were investigated. A dispersing agent was used. Appreciable
time-dependent effects were only noticed at a concentration of the dispersing agent below
a critical value. Multicomponent suspensions were found to have a viscosity that
was dependent on the total volume concentration of solids Cv and on the composition of
the dispersed phase expressed as a volume fraction. It was also dependent on the shear
rate of the mixture.
Vlasak et al. (1998) investigated the addition of peptizing agents to kaolin–water mixtures.
These mixtures were described as yield pseudoplastics that follow the
Bulkley–Herschel rheological model (these will be discussed in Chapter 5). The addition
of peptizing agents initially achieved a rapid drop of viscosity down to 8–10% of the original
value up to an optimum concentration. As the concentration of the peptizing agent is
increased beyond an optimum value, its effects are neutralized and the viscosity of the
slurry increases again. Soda Water-GlassTM as a peptizing agent seemed to achieve the
best reduction in viscosity when added at a concentration of 0.4%. The effect was a drastic
drop of viscosity by 92% of its original value (without the peptizing agent). The optimum
concentration of sodium carbonate, another peptizing agent, was 0.1%. The viscosity
was reduced by 90%. These narrow bands of concentration of peptizing agents can
effectively reduce the cost of hydro-transporting kaolin–water mixtures by reducing viscosity
and therefore the coefficient of friction.

3.30 CHAPTER THREE
3-5 TIME-DEPENDENT NON-NEWTONIAN
MIXTURES
Because crude oils and slurries of tar sands from certain Canadian mining projects develop
a time-dependent non-Newtonian behavior in cold temperatures, a section of this chapter
will pay attention to these complex thixotropic properties.
In time-dependent non-Newtonian flows, the structure of the mixture and the orientation
of particles are sensitive to the shear rates. Due to structural changes and reorientation
of particles at a given shear rate, the shear stress becomes time-dependent as the particles
realign themselves to the flow. In other words, the shear stress takes time to readjust
to the prevailing shear rate. Some of these changes may be reversible when the rate of reformation
is the same as the rate of decay. However, in the case of flows in which the deformation
is extremely slow, the structural changes or particle reorientation may be irreversible
(see Figure 3-13).
3-5-1 Thixotropic Mixtures
When the shear stress of a fluid decreases with the duration of shear strain, the fluid is
called thixotropic. The change is then classified as reversible and structural decay is observed
with time under constant shear rate. Certain thixotropic mixtures exhibit aspects of
permanent deformation and are called false thixotropic.
When the rate of structural reformation exceeds the rate of decay under a constant sustained
shear rate, the behavior is classified as rheopexy (or negative thixotropy).
One typical example of a thixotropic mixture is a water suspension of bentonitic
clays. These difficult slurries are produced by mud drilling associated with the use of
positive displacement diaphragm or hose pumps. The reader may find throughout literature
considerable discussion about “hysterisis.” This function is used to measure the
behavior of the mixture by gradually increasing the shear rate and then by decreasing it
back in steps. These curves are interesting but are of limited help to the designer of a
pumping system.
Shear Stress ( )
Thixotropic
Rheopectic
Rate of shear ( = du/dy)
FIGURE 3-13 Rheology of time-dependent fluids.

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
3.31
Moore (1959) proposed expressing the complex behavior of a thixotropic fluid that
does not possess a yield stress value in terms of six parameters:
� = (�0 + c�)(d�/dt)
d�/d� = a – �(a + bd�/dt)
where
� = duration of the shear for a time-dependent fluid
a, b, c, and �0 = materials constants
� = a structural parameter that has two values (0 and 1) at the limits where
the material is fully broken down or fully developed
Fredrickson (1970) discussed the modeling of thixotropic mixtures of suspensions of
solids in viscous liquids and proposed that rheological tests be conducted to measure four
constants to understand the qualitative nature of the mixture.
Ritter and Govier (1970) proposed representing the behavior of thixotropic fluids as
follows:
� The formation of structures, networks, or agglomerates is similar to a second-order
chemical reaction.
� The breakdown of the structure is similar to a series of consecutive first-order chemical
reactions where formation is meant by behavior that is time-dependent, whereas the
breakdown occurs when the viscosity of the fluid acts as a Newtonian mixture that is
independent of both the shear rate and the duration of shear (Figure 3-14).
Shear stress, +0.01, lb /ft
-1 10
8
6
4
10
4
2
2
-2
10
Duration of
shear, min
0
1
10
100
100 1000
-1
Rate of Shear, d /dt + 10 in sec
FIGURE 3-14 Rheology of Pembina crude oil at 44.5°F at constant duration of shear. (After
Govier and Aziz, 1972.)

3.32 CHAPTER THREE
Ritter and Govier (1970) therefore proposed to express the shear stress of the fluid in
terms of structural stress � s and � �, a component of shearing stress due to the Newtonian
component of the fluid:
� = � s + � �
(3-53)
log� � = –KD� �log � – log K �s – �s� �s0 + �s� ��
DR (3-54)
where
�s0, �s� = structural stresses at a given shear rate after zero and infinite duration of shear
�s0 = �0 – �(d�/dt)
�s� = �� – �(d�/dt)
KD = a constant that is independent of shear rate but is related to the first-order structural
decay process and is expressed in the minutes –1 .
KDR = a dimensionless measure of the interaction between the network or structure decay
and the reestablishment processes
The coefficient KDR is evaluated as
�
KDR = (3-55)
where �s1 is measured after a lapse of 1 minute. In Equations 3-54 and 3-55, KDR, KD, �s0, �s1, and �s� are determined from rheology tests.
Kherfellah and Bekkour (1998) examined the thixotropy of suspensions of montmorillonite
and bentonite clays. Montmorillonite clays are used as thickening agents for
drilling fluids, paints, pesticides, cosmetics, pharmaceuticals, etc. Commercial bentonite
suspensions exhibited thixotropic properties for concentrations higher than 6% by weight.
Rheopectic or negative thixotropic mixtures are not common in mining and will not be
examined in this chapter.
2 s0 – �s1�s� ��
�s1�s� – � 2 � 2 (� s0/�s�) – �s �s0 – �s� s�
3-6 DRAG COEFFICIENT OF SOLIDS
SUSPENDED IN NON-NEWTONIAN FLOWS
Some solids may be transported by highly viscous fluids in a non-Newtonian flow
regime. One such example includes solids transported in the process of drilling a tunnel in
a sandy soil rich with clay or bentonite. Other examples of solids suspended in non-Newtonian
flows are energy slurries, which are mixtures of fine coal and crude oils. In such
circumstances, the drag coefficient of the coarse components is of interest.
Brown (1991) reviewed the literature for settling of solids in non-Newtonian flows,
but cautioned that the studies have been limited to single particles. Considerably more research
is needed in this field.
3-7 MEASUREMENT OF RHEOLOGY
In the proceeding sections of this chapter, the concepts of Newtonian and non-Newtonian
fluids were explored. Measuring the viscosity of a slurry mixture is recommended for ho-

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
mogeneous flows, mixtures with a high concentration of particles, and for fibrous and
flocculated slurries.
Subsieve particles are defined as particles with an average diameter smaller than
35–70 �m (depending on whose reference book you consult). Slurry flows with subsieve
particles at a relatively high concentration by volume (C v � 30%) are strongly rheologydependent.
Heterogeneous flows, flows without subsieve particles, or flows with subsieve
particles at a very low concentrations, are not governed by the rheology of the slurry.
Flocculation or the addition of flocculates in the process of mixing slurries tends to result
in non-Newtonian rheology.
Rheology in simple layman’s terms is the relationship between the shear stress and the
shear rate of the slurry under laminar flow conditions. Although this relationship extends
to transitional and turbulent flows, most tests are conducted in a laminar regime, often in
tubes or between parallel plates.
3-7-1 The Capillary-Tube Viscometer
The purpose of the capillary tube viscometer is to measure the rheology of a laminar flow
under controlled velocity conditions. Tubes are used in a range of diameters from 0.8–12
mm (1/32–1/2 in). The length of the tube is accurately cut to account for entrance effects
and end effects. Typically, the length may be as much as 1000 times the inner diameter.
The capillary tube viscometer is used to plot the average rate versus the shear stress at
the wall of the tube. This is called the pseudoshear diagram, as defined by the
Mooney–Rabinovitch equation:
(du/dr) w = �0.75 + 0.2 8
d[ln(8V/Di)] � ���
(3-56)
Di
d[ln(�P/4Li)]
where
(du/dr) w = rate of shear at the wall
�P = pressure drop due to friction over a length Li of pipe of inner diameter Di V = average velocity of the flow
d = derivative
The data is then plotted on a logarithmic scale as per Figure 3-15.
The use of capillary-like viscometers is complicated by the “effective slip” of non-
Newtonian fluid-suspended material, which tends to move away from the wall, leaving an
attached layer of liquid. The result is a reduction in the measurements of effective viscosity.
Therefore, it is often recommended to conduct such tests in a number of tubes of different
diameters.
Measuring the pressure loss between two points well away from the entrance and end
effects gives the shear stress at the wall as:
�w = Ri�P/(2Li) (3-57)
By considering that the velocity profile at a height y above the wall is a function of the
shear stress we obtain
–(du/dy) w = f (�)
It may be possible to establish a relationship between the flow rate Q and the shear stress
� as
Q
� �R 3
1
= � �w �3 � w 0
3.33
� 2 f (�)d� (3-58)

3.34 CHAPTER THREE
8V
D
shear rate
For a Newtonian flow:
or � = � w/(8V/D i).
For a Bingham flow:
for � > � 0, where � 0 is the yield stress.
The velocity profile is expressed as
2V �w = � = � (3-59)
Di 4�
� = �(du/dr) w + � 0
= = � �2 2V � �w (� – �0)d� (3-60)
By integration of this equation and by multiplying by 4, the shear rate is derived as
8V
� DI
100
10
1.0
Q
� �R 3
0
0
� Di
� w
� �
water
Q
� �R 3
increasing tube diameter
Shear Stress
� � w 3
4
�
3
� 0
� �w
= � 1 – � � + � �� (3-61)
Equation 3-61 is called the Buckingham equation. This equation cannot be solved
without long iterations. Many engineers prefer to simplify the Buckingham equation by
ignoring the term (� 0/� w) 4 , as this term is of negligible magnitude compared with the other
terms:
� �0
1.0 10
1
�
3
D
2
D 1
D 3
4 � 0
�4 � w
D 4
D P
4 L
FIGURE 3-15 Pseudoshear diagram of a non-Newtonian mixture tested in a capillary tube
rheometer.

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
�w � 8V�/Di + 4/3�0 The modified equation is plotted in Figure 3-16.
(3-62)
For a pseudoplastic slurry or power law fluid, the shear stress is expressed by Equation
3-43. By analogy with the method developed for a Bingham flow in a tube, the following
equation is expressed:
= = � � 2 (�/K) 1/n Q 2V
�3 �R
1
�3 �w
�
d� (3-63)
� Di
or
= � �w (3-64)
0
which once integrated is expressed as
= � � (3-65)
The effective viscosity is expressed as
�e = �w/(8V/Di) = K(8V/Di) (n–1) [4n/(3n + 1)] n �
1/n
2V n � w
� � �1/n Di 3n + 1 K
(3-66)
Unfortunately, Equation 3-66 is of no value when n < 1.0, which is the case for many
power law slurries. It would mean that as the shear rate increases, the effective viscosity
decreases to zero. This is contradictory to nature. For power law exponents smaller than
1.0, alternative equipment should be used to measure rheology.
It is tricky to avoid errors with the use of capillary effect viscometers. A particular
source of errors is the end effect. At the entrance exit of the tube, contraction and expansion
of the flow cause additional pressure losses.
(3+1/n)
Q
� ��
3
1/n
�R (3 + 1/n)K
w
Shear Stress
Velocity profile
w
2 r 0
� �0
shear rate
FIGURE 3-16 Pseudoshear diagram for a Bingham plastic.
dV
�
dy
dU
dy
3.35

3.36 CHAPTER THREE
3-7-2 The Coaxial Cylinder Rotary Viscometer
A more practical instrument to use when measuring rheology is the coaxial cylinder viscometer.
In basic terms, it is a device used to measure the resistance or torque when rotating
a cylinder in a viscous fluid (Figure 13-17). The moment of inertia in the cylinder is
established by the manufacturer.
The torque is due to the force the fluid exerts tangentially to the outside surface of the
cylinder:
T = 2�R0h�wR0 (3.67)
where
T = (surface area) (shear stress) (radius)
R0 = outside radius of the rotating cylinder
h = height of the cylinder
�w = shear stress at the wall
The shear stress at any radius r in the fluid can be expressed as
T du
�w = � = ��
2 2�r h dy
If the liquid is rotating at an angular velocity �, then
(du/dy) w = –rd�/dr
r
R c
scale to measure torque
rotation of bob at
speed
R
FIGURE 3-17 The rotating concentric viscometer.
0
slurry
(3.68)

and
MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
� = –�rd�/dr
d�
� dr
=
� �
d� = � 0
Rc R0 –T
� 2�h�r
–T
� dr 3 2�h�r
3.37
or
� = � – � (3.69)
where Rc is the radius of the outside cylinder.
2 This is known as the Margulus equation. It is obvious that R 0 can be related to the moment
of inertia Ik of the rotating bob cup.
Since for a Bingham slurry, the rate of shear is expressed as du/dr = (� – �0); the Margulus
equation can be demonstrated as
� = � – � – ln� � (3.70)
This equation is known as the Reiner–Rivlin equation.
For a Pseudoplastic:
2 1/n � = n[T/(2�R 0hK)] [1 – (R0/Rc) 2/n T 1 1
� � � 2 2
4�h� R 0 R c
T 1 1 �0 Rc � � � � � 2 2
4�h� R 0 R c � R0
] (3.71)
At the wall:
2 �w = T/(2�R b h) (3.72)
A plot of log �w versus log � can be constructed. The slope gives the flow index n and, by
substituting Equation 3-45, the value of K can be calculated.
Heywood (1991) discussed errors with the use of rotating viscometers. Particular
sources of errors are the end effects from both cylinders and the possible deformation of
the laminar layer under the effect of high rotational speed. Heywood recommended the
use of cylinders with a long length to diameter ratio. Wall slip effects can be detected by
using cylinders of different radius but same length. The vendors of rheometers publish
equations to correct for wall slip and end effects.
One important problem about the use of rheometers is that they do not distinguish between
Bingham and Carson slurries. This can lead to grave mistakes in the design of a
pipeline. Certain slurries have a course of fractions that could also precipitate during a
rheometer test. Unfortunately, this would give false readings. When there is doubt, the
safest approach is to conduct a proper pump test in a loop.
Whorlow (1992) published a book on rheological techniques that includes dynamic
tests and wave propagation tests. In the appendix, he listed a number of rheological investigation
equipment manufacturers. Some of the techniques apply more to polymers and
are not relevant to our discussion. Dynamic vibration tests have been extended to fresh
concrete (Teixera et al., 1998). Concord and Tassin (1998) described a method to use
rheo-optics for the study of thixotropy in synthetic clay suspensions. A rheometer optical
analyzer was used on laponite, a synthetic hectorite clay. Laponite was mixed with water
and tests were conducted at various intervals for up to 100 days. Rheo-optics seems to be

3.38 CHAPTER THREE
FIGURE 3-18 Stresstech rheometer, courtesy of ATS Rehosystems. The rheometer was developed
for the pharmaceutical and cosmetics industries, where materials consistency may
vary from fluid to solid.
a new technique based on the ability of solids to reorient themselves by applying to them
a negative electrical charge.
3-8 CONCLUSION
In this chapter, it was demonstrated that mixtures of solids and liquids are complex systems.
The size of the particles, the diameter of the pipe, the interaction with other particles,
the viscosity of the carrier, and the temperature of the flow all interact to yield Newtonian
or non-Newtonian flows.
In the next three chapters, the principles discussed in the present chapter will be applied
to calculate the velocity of deposition, the critical velocity, the stratification ratio,
and the friction loss in closed and open conduits for heterogeneous and homogeneous
mixtures.
3-9 NOMENCLATURE
a The longest axis of a particle in Albertson’s model
A Parameter used to express viscosity of non-Newtonian flows

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
3.39
A0 Coefficient
A1 Coefficient
b Axis of a particle in Albertson’s model
B Parameter used to express viscosity of non-Newtonian flows
c The shortest axis of a particle in Albertson’s model
C Parameter used to express viscosity of non-Newtonian flows
CD Drag coefficient of an object moving in a fluid
CDo Profile drag coefficient of an object moving in a fluid
CL Lift coefficient of an object moving in a fluid
CN CS Cv Cv� Cw da Coefficient based on equivalent Ren Coefficient based on equivalent Ren Concentration by volume of the solid particles in percent
Maximum packing concentration of solids
Concentration by weight of the solid particles in percent
Diameter of a sphere with a surface area equal to the surface area of the irregularly
shaped particle
dapp Apparent particle diameter
df Apparent flocculant diameter
dg Sphere diameter
dn Diameter of a sphere with a volume equal to the volume of the irregularly
shaped particle in Albertson’s model
d� Particle diameter
D Drag force
Di Tube or pipe inner diameter
E Factor between Albertson and Clift shape factors
f( ) Function of
FBF Buoyancy force
Fw Wall effect correction factor for free-fall speed of a particle
g Acceleration due to gravity (9.78–9.81 m/s2 )
gc Conversion factor, 32.2ft/s2 if U.S. units between lbms and slugs
h Height of the cylinder
Ik Moment of inertia
K Consistency index or power law coefficient for a pseudoplastic
KD A constant that is independent of shear rate but is related to the first-order
structural decay process and is express in minutes –1
KDR A dimensionless measure of the interaction between the network or structure
decay and the reestablishment processes
Kt Coefficient for terminal velocity
Kz Kozney constant
K1, K2, K3 Coefficients
ln natural logarithm
L Lift force
Lc Characteristic length
LI Length of pipe or tube
n Flow behavior index, or exponent for a pseudoplastic (

3.40 CHAPTER THREE
Ren Reynolds Number of a particle based on dn Rep Reynolds Number of a sphere particle based on its diameter
Ri Inner radius of a pipe or tube
R0 Radius of the bob in the coaxial cylinder rotary viscometer
sp The surface area per unit volume of a sphere of equivalent dimensions or
6/dg, also called specific surface of a particle
Sf Front area of a particle orthogonal to the direction of flow
Sw Surface area of a wing along the direction of flight
T Applied torque for the cylinder rotary viscometer
Ta Absolute temperature
V Average velocity of the flow
V0 Terminal velocity at very low volume concentration of solids
Vc Terminal velocity at given volume concentration of solids
Vt The terminal (or free settling) speed
W Weight
� the ratio of immobilized dispersing fluid to the volume of solids related approximately
to the particle and floc apparent diameter
� A coefficient used to express to the shear stability of a pseudoplastic mixture
� Concentration by volume in decimal points
� Shear strain
d�/dt Wall shear rate or rate of shear strain with respect to time
� Coefficient of rigidity of a non-Newtonian fluid, also called Bingham viscosity
� A structural parameter for thixotropic fluids, which do not possess a yield
stress value
� Carrier liquid absolute viscosity
�a Apparent viscosity of a pseudoplastic fluid
�e Effective viscosity
�0 Apparent viscosity of a pseudoplastic fluid at zero shear rate
�� Bingham plastic limiting viscosity, or apparent viscosity of a pseudoplastic
fluid at very high shear rate
� Pythagoras number (ratio of circumference of a circle to its diameter)
� Duration of the shear for a time-dependent fluid
� Density
� Shear stress at a height y or at a radius r
�0 Yield stress for a Bingham plastic or yield pseudoplastic
�s Structural stress of a thixotropic fluid
�w Wall shear stress
� Kinematic viscosity
� Angular velocity of particle
� Angular velocity of complete system
� The logarithmic standard deviation
�A Albertson shape factor
�c Clift shape factor
Thomas shape factor
� T
Subscripts
g Equivalent sphere
L Liquid
m Mixture
p Particle
s Solids

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MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
3.41
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mixtures. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia:
University of Ljubljana.
Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid-Liquid Flow—Slurry Pipeline Transportation.
Trans-Tech Publications.
Wells, P. J. 1991. Pumping non-Newtonian slurries. Technical Bulletin 14. Sydney, Australia: Warman
International.
Whorlow, R. W. 1992. Rheological Techniques, 2d. ed. New York: Ellis Horwood.
Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New
York: Elsevier Applied Sciences.
Worster, R. C., and D. E. Denny. 1955. Hydraulic transport of solid materials in pipes. Proceedings
of the Institute of Mechanical Engineers (UK), 38, 230–234.
Further Reading:
Caldwell, D. H., and H. E. Babitt. 1942. Pipeline flow of solids in suspension. Symp. Am. Soc. of Civ.
Eng. Proc., 68, 3 (March), 480–482.
Goodrich, J. E., and R. S. Porter. 1967. Rheological interpretation of torque—Rheometer data. Polymer
Eng & Science, 7 (January), 45–51.

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS
3.43
Lazerus, J. H., and P. T. Slatter. 1988. A method for the rheological characterization of tube viscometer
data. Journal of Pipelines, 7, 165–176.
Thomas, D. G. 1960. Heat and momentum transport characteristics of non-Newtonian aqueous thorium
oxide. AIChE Journal, 7, 431.
Wilson, K. C. 1991. Pipeline design for settling slurries. In Slurry Handling. Edited by N. P. Brown,
and N. I. Heywood. New York: Elsevier Applied Sciences.
Wilson, K. C. 1991. Slurry transport in flumes. In Slurry Handling. Edited by N. P. Brown, and N. I.
Heywood. New York: Elsevier Applied Sciences.

APPENDIX A
SPECIFIC GRAVITY AND
HARDNESS OF MINERALS
The following abbreviations are used in the table below
A Amphibole group M Mica group
B Bauxite component O Orthoclase
C Clay or clay-like Ov Olivine group
D Diopside series P Pyroxene
E Enstatite group R Rare earth group
F Feldspar group S Spinel group
Fp Feldspathoid group Sc Scapolite series
G Garnet group W Wolframite
H Hornblende Z Zeolite group
J Jamesonite
Name Description or composition Specific gravity Mohr hardness
Acanthite Ag2S 7.2–7.3 2–2.5
Achroite Colorless tourmaline
Acmite NaFe(SiO3) 2 3.4–3.6 6–6.5
Actinolite Ca2(MgFe) 5(Si8O22)(OH) 2 3.0–3.2 5–6
Adularia Clear orthoclase
Aergite Acmite, aegrinine
Agate Banded chalcedony
Alabandite MnS 4.03.4–4
Alabaster Fine-grained gypsum
Albite Na(AlSi3O8) Alexandrite Chrysoberyl, gemstone
Allanite (Ce,Ca,Y)(Al,Fe) 3(SiO4) 3(OH) 3.5–4.2 5.5–6
Allemontite AsSb 5.8–6.2 3–4
Allophane Al2O3·SiO2·nH2O 1.8–1.9 3
Almandite Fe3Al2(SiO4) 3, red 4.25 7
Altaite PbTe 8.16 3
Alunite KAl3(SO4) 2(OH) 6 2.6–2.8 4
Amazonstone Green microcline
Amblygonite (Li,Na)AlPO4(F,OH) 3.0–3.1 6
Amethyst Purple quartz
A.1

A.2 APPENDIX A
Name Description or composition Specific gravity Mohr hardness
Amphibole A group of minerals
Analcime Na(AlSi2O6)·H2O 2.27 5–5.5
Anastase TiO2 3.9 5.5–6
Anauxite Silicon-rich kaolinite
Andalusite Al2SiO5 3.1–3.2 7.5
Andesine Ab70An30—Ab50An50 2.69 6
Andradite Ca3Fe2(SiO4) 3 3.75 7
Anglesite PbSO4 6.2–6.4 3
Anhydrite CaSO4 2.8–3.0 3–3.5
Ankerite Ca(Fe,Mg,Mn)(CO3) 2 2.9–3.0 3.5
Annabergite (Ni,Co) 3(AsO4) 2·8H2O 3.0 3–3.5
Anorthite CaAl2Si2O8 2.76 6
Anorthoclase (Na,K)AlSi3O8 2.58 6
Antigorite Serpentine
Antimony Sb 6.7 3
Anterite Cu3SO4(OH) 4 3.9 3.5–4
Apatite Ca5(PO4,CO3) 3(F,OH,Cl) 3.1–3.2 5
Apophylite KCa4Si8O20(F,OH)·8H2O 2.3–2.4 4.5–5
Aquamarine Green-blue beryl, gemstone
Aragonite CaCO3 2.95 3.5–4
Arfvedsonite Na2-3(Fe,Mg,Al) 5Si8O22(OH) 2 3.45 6
Argentite Ag2S 7.3 2–2.5
Arsenic As 5.7 3.5
Arsenopyrite FeAsS 5.9–6.2 5.5–6
Asbestos A group of minerals
Altacamite Cu2Cl(OH) 3 3.7–3.8 3–3.5
Augite (Ca,Na)(Mg,Fe,Al)(Si,Al) 2(O) 6 3.2–3.4 5–6
Aurichalcite (Zn,Cu) 5(CO3) 2(OH) 6 3.2–3.7 2
Autunite Ca(UO2) 2(PO4) 2·10H2O 3.1–3.2 2–2.5
Awaruite FeNi2 7.7–8.1 5
Axinite (Ca,Mn,Fe) 3Al2BSi4O15(OH) 3.2–3.4 6.5–7
Azurite Cu3(CO3) 2(OH) 2 3.77 3.5–4
Balas ruby Red Spinel, gemstone
Barite BaSO4 4.5 3–3.5
Barytes Barite
Bastnaesite (Ce,La)(CO3)(F,OH) 4.9–5.2 4–4.5
Bauxite Aluminum hydroxide mixture
Beidellite Al8(Si4O10) 3(OH) 12·12H2O 2.6 1.5
Bentonite Montmorillonite clay
Beryl Be3Al2(Si6O18) 2.7–2.8 7.5–8
Biotite K(Mg,Fe2+ )3(Al,Fe3+ )Si3O10(OH) 12 2.8–3.2 2.5–3
Bismite Bi2O3 8 4.5
Bismuth Bi 9.8 2–2.5
Black Jack Sphalerite
Blende Sphalerite
Bloodstone Heliotrope
Blue vitriol Chalcanthite
Boehmite AlO(OH) 3.0–3.1
Boracite Mg3B7O13Cl 2.9–3.0 7

SPECIFIC GRAVITY AND HARDNESS OF MINERALS
Name Description or composition Specific gravity Mohr hardness
Borax Na2B4O7·10H2O 1.7 2–2.5
Bornite Cu5FeS4 5.0–5.1 3
Boulangerite Pb5Sb4S11 6–6.3 2.5–3
Boumonite PbCuSbS3 5.8–5.9 2.5–3
Brannerite (U,Ca,Ce)(Ti,Fe) 2O6 4.5–5.4 4.5
Braunite 3Mn2O3·MnSiO3 4.8 6–6.5
Bravoite (Ni,Fe)S2 4.66 5.5–6
Brochantite Cu4(OH) 6SO4 3.9 3.5–4
Bromyrite Ag(Br,Cl)—Br,Cl 6–6.5 2.5
Bronzite (Mg,Fe)SiO3 3.1–3.3 5.5
Brookite TiO2 3.9–4.1 5.5–6
Brucite Mg(OH) 2 2.39 2.5
Bytownite Ab30An70-Ab10An90 2.74 6
Caimgom Quartz, black/smoky
Calamine Hemimorphite
Calaverite AuTe2 9.35 2.5
Calcite CaCO3 2.73 3
Calomel Hg2Cl2 7.2 1.5
Cancrinite (Fp) (Na2,Ca) 4(AlSiO4) 6CO3nH2O 2.45 5–6
Carnalite KMgCl3·6H2O 1.6 1.0
Carnolite K(UO2) 2(VO4) 2·3H2O 4.1 soft
Cassiterite SnO2 6.8–7.1 6–7
Cat’s eye Chrysoberyl or quartz, gemstone
Celestite SrSO4 3.9–4.0 3–3.5
Celsian (F) BaAl2Si2O8 3.37 6
Cerargyrite Ag(Cl,Br)—Cl, Br 5.5–6 2.5
Cerussite PbCO3 6.55 3–3.5
Cervanite Sb2O4 4.0–5.0 4–5
Chabazite (Z) Ca(Al2Si4O12)·6H2O 2.0–2.2 4–5
Chalcanthite CuSO4·5H2O 2.1–2.3 2.5
Chalcedony Cryptocrystaline quartz 2.6–2.7
Chalcocite Cu2S 5.5–5.8 2.5–3
Chalcopyrite Cu2FeS2 4.1–4.3 3.5–4
Chalcotrichite Cuprite, fibrous
Chalk Calcite, fine-grained
Chalybite Siderite
Chert SiO2, cryptocrystalline quartz 2.65
Chessylite Azurite
Chiastolite Andalusite
Chloanthite Skutterudite, nickel variety
Chlorite (MgFe2+ ,Fe3+ ) 6AlSi3O10(OH) 8 2.6–2.9 2–2.5
Chloritoid (M) Fe2Al4Si2O10(OH) 4 3.5 6–7
Chondrodite (Mg,Fe)3SiO4(OH,F) 2 3.1–3.2 6–6.5
Chromite (Fe,Mg)O.(Fe,Al,Cr) 2O3 4.3–4.6 5.5
Chrysoberyl BeAl2O4 3.6–3.8 8.5
Chrysocolla Cu2H2(Si2O5)(OH) 4 2.0–2.4 2 -4
Chrysolite Olivine
Chrysoprase Chalcedony, green
Chrysolite Serpentine asbestos
A.3

A.4 APPENDIX A
Name Description or composition Specific gravity Mohr hardness
Cinnabar HgS 8.10 2.5
Cinnamon stone Grossularite garnet
Citrine Quartz, pale yellow
Clay A family of minerals
Cleavelandite Albite, white
Cliachite Aluminum hydroxide in bauxite
Clinochlore Chlorite
Clinoclase Cu3(AsO4)(OH) 3 4.38 2.5–3
Clinoenstatite (E) (Mg,Fe)SiO3 3.19 6
Clinoferrosilite (P) (Fe,Mg) SiO3 3.6 6
Clinohumite Mg9Si4O16(F,OH) 2 3.1–3.2 6
Clinozoisite Ca2Al3Si3O12(OH) 3.2–3.4 6–6.5
Cobalite CoAsS 6.33 5.5
Colemanite Ca2B6O11·5H2O 2.42 4–4.5
Collophane Apatite
Columbite (Fe,Mn)(Nb,Ta) 2O6-NbTa 5.2–6.7 6
Copper Cu 8.9 2.5–3
Copper glance Chalcocite
Copper pyrites Chalcopyrite
Cordierite (Mg,Fe) 2Al4Si5O18 2.6–2.7 7–7.5
Corundum Al2O3 4.0 9
Covellite CuS 4.6–4.7 1.5–2
Cristobalite SiO2, high-temperature quartz 2.3 7
Crocidolite Na3Fe2+ 3Fe3+ 2(SiO23)(OH) 3.2–3.3 4
Crocoite PbCrO4 5.9–6.1 2.5–3
Cryolite Na3AlF6 2.9–3 2.5
Cubanite CuFe2S3 4.0–4.2 3.5
Cummingtonite (Fe,Mg) 7(Si8O22)(OH) 2 3.1–3.6 6
Cuprite Cu2O Cyanite Kyanite
Cymophane Chrysoberyl
Danaite (Fe,Co)As 5.9–6.2 5.5–6
Danburite CaB2(SiO4) 2 2.9–3.0 7.0
Datolite CaB(SiO4)(OH) 2.8–3.0 5.0–5.5
Davidite Brannerite, Th variety
Demantoid (G) Andradite garnet, green gemstone
Diallage Diopside
Diamond C 3.5 10
Diaspore AlO(OH) 3.3–3.4 6.5–7.0
Diatomite Diatoms, siliceous 0.4–0.6 2
Dichroite Cordierite
Dickite (C) Al2Si2O5(OH) 4, kaolin 2.6 2.0–2.5
Digenite Cu9S5 5.6 2.5–3.0
Diopside (P) CaMg(SiO3) 2 3.2–3.3 5.0–6.0
Dioptase CuSiO2(OH) 2 3.3 5.0
Disthene Kyanite
Dolomite CaMg(CO3) 2 2.85 3.5–4
Dry bone ore Smithsonite
Dumortierite (Al,Fe) 7O3(BO3)(SiO4) 3 3.2–3.4 7.0

SPECIFIC GRAVITY AND HARDNESS OF MINERALS
Name Description or composition Specific gravity Mohr hardness
Endenite (H) Ca2NaMg5(AlSi7O22) 3.0 6.0
Electrum Au, Ag, natural alloy 13.5–17 3.0
Eleolite Nepheline
Embolite Ag(Cl,Br)—Cl=Br 5.6 1.0–1.5
Emerald Beryl, green gemstone
Emery Corundum with magnetite
Enargite Cu3AsS4 4.4–4.5 3.0
Endichite Vanadite, arsenic variety
Enstatite (P) MgSiO3 3.2–3.5 5.5
Epidote Ca2(Al,Fe) 3Si3O12(OH) 3.3–3.5 6–7
Epsomite MgSO4·7H2O, Epsom salt 1.75 2.0–2.5
Erythrite Co3(AsO4)·8H2O 2.95 1.5–2.5
Essonite (G) Grossularite
Euclase BeAlSiO4(OH) 3.1 7.5
Eucryptite (Y,Ce,Ca,U,Th) 2(Ti,Nb,Ta,Fe) 2O6 5.0–5.9 5.5–5.6
Fahlore Tetrahedrite
Fayalite (OV) Fe2SiO4 4.14 6.5
Feather ore Jamesonite
Feldspar (F) A group of minerals
Feldspathoid A group of minerals
Ferberite (W) FeWO4 7.5 5
Fergusonite (R) (RE,Fe)(Nb,Ta,Ti) O4 4.2–5.8 5.5–6.5
Ferrimolybdite Fe2(MoO4) 3·8H2O 3 1.5
Ferrosilite (P) FeSiO3 3.6 6
Fibrolite Sillimanite
Flint SiO2, cryptocrystalline quartz 2.65 7
Flos ferri Aragonite, arborescent
Fluorite CaF2 Fool’s gold Pyrite
Formanite (R) Fergusonite with TaNb
Forsterite (Ov) Mg2SiO4 3.2 6.5
Fowlerite Rhodonite, zinc bearing
Franklinite (Fe2+ ,Zn,Mn2+ ) (Fe3+ , Mn3+) 2O4 5.15 6
Freibergite Tetrahedrite, silver bearing
Gadolinite (R) Be2FeY2Si2O10 4.0–4.5 6.5–7.0
Gahnite (S) ZnAl2O4 4.55 7.5–8.0
Galaxite (S) MnAl2O4 4.03 7.5–8.0
Galena PbS 7.4–7.6 2.5
Garnet (G) A group of minerals 3.5–4.3 6.5–7.5
Gamierite (Ni,Mg) 3Si2O5(OH) 4 2.2–2.8 2.0–3.0
Gaylussite Na2Ca(CO3) 2·5H2O 1.99 2.0–3.0
Gedrite (A) Anthophylite, Al variety
Geocronite Pb5(Sb,As) 2S8 6.6–6.5 2.5
Gersdorffite NiAsS 5.9 5.5
Geyserite Opal
Gibbsite Al(OH) 3 2.3–2.4 2.5–3.5
Glauberite Na2Ca(SO4) 2 2.7–2.8 2.5–3.0
Glaucodot Danaite
Glauconite (M) (K,Na)(Al,Fe3+ ,Mg) 2(Al,Si) 4O10(OH) 2 2.3 2
A.5

A.6 APPENDIX A
Name Description or composition Specific gravity Mohr hardness
Glaucophane (A) Na2(Mg,Fe2+ ) 3Al2Si8O22(OH) 2 3.0–3.2 6.0–6.5
Gmelinite (Z) (Na2·Ca)Al2Si4O12·6H2O 2.0–2.2 4.5
Goethite FeO(OH) 4.37 5.0–5.5
Gold Au 15–19.3 2.5–3.0
Goslarite ZnSO4·7H2O 1.98 2.0–2.5
Graphite C 2.3 1.0–2.0
Greenockite CdS 4.9 3.0–3.5
Grossularite (G) Ca3Al2(SiO4) 3 3.53 6.5
Gummite UO3·nH2O 3.9–6.4 2.5–5
Gypsum CaSO4·2H2O 2.32 2.0
Halite NaCl, common salt 2.16 2.5
Hallosite (C) Al2Si2O5(OH)·nH2O 2.0–2.2 1.0–2.0
Harmotome (Z) (Ba,K)(Al,Si) 2Si6O16·6 H2O 2.45 4.5
Hastingsite (H) NaCa2(Fe,Mg) 5Al2Si6O22(OH) 2 3.2 6.0
Hausmannite Mn3O4 4.8 5.5
Hauynite (Fp) (Na,Ca) 4·8Al6Si6O24·(SO4·S) 1-2 2.4–2.5 5.5–6
Hectorite (C) (Mg,Li) 6Si8O20(OH) 4 2.5 1.0–1.5
Hedenbergite (P) CaFe(Si2O6) 3.6 5.0–6.0
Heliotrope Chalcedony, green and red
Helvite (Mn,Fe,Zn) 4Be3(SiO4) 3S 3.2–3.4 6.0–6.5
Hematite Fe2O3 5.3 5.5–6.5
Hemimorphite Zn4(Si2O7)(OH) 2·H2O 3.4–3.5 4.5–5.0
Hercynite (S) FeAl2O4 4.4 7.5–8.0
Hessite Ag2Te 8.4 2.5–3.0
Heulandite (Na,Ca) 4—6Al6(Al,Si) 4Si26O72·24H2O 2.2 3.5–4.0
Hiddenite Spodumene, green
Holmquisite (A) Glaucophane, Li variety
Homblende Ca2Na(MgFe2+ ) 4(AlFe3+ ,Ti)Al Si8O22 (O,OH) 2
3.2 5.6
Hom silver Cerargyrite
Huebnerite (W) MnWO4 7.0 5.0
Humite Mg7(SiO4) 3(F,OH) 2 3.1–3.2 6.0
Hyacinth Zircon
Hyalite Opal
Hyalophane (O) (K,Ba)Al(Al,Si) 3O8 2.8 6.0
Hydromica (M) Illite
Hydrozincite Zn5(CO3) 2(OH) 6 3.6–3.8 2.0–2.5
Hypersthene (P) (MgFe)SiO3 3.4–3.5 5.0–6.0
Ice H2O 0.95 1.5
Iddingsite H8Mg9Fe2Si3O14 3.5–3.8 3.0
Idocrase Ca10(Mg,Fe) 2Al4(SiO4) 5(Si2O7) 2(OH) 4 3.3–3.4 6.5
Illite (C) Al,K,Ca,Mg
Ilmenite FeTiO3 4.7 5.5–6.0
Ilvaite CaFe2+ 2Fe3+ (SiO4) 2(OH) 4.0 5.5–6.0
Indicolite Tourmaline
Iodobromite Ag(Cl,Br,I) 5.7 1.0–1.5
Iodyrite AgI 5.7 1.0–1.5
Iolite Cordierite, gemstone
Iridium Ir, platinoid 22.7 6.0–7.0

SPECIFIC GRAVITY AND HARDNESS OF MINERALS
Name Description or composition Specific gravity Mohr hardness
Iridosmine Ir,Os, platinoid 19.3–21.0 6.0–7.0
Iron pyrite Pyrite
Jacinth Hyacinth, zircon
Jacobsite (S) (Mn2+ ,Fe2+ ,Mg)(Fe3+ ,Mn3+ ) 2O4 5.1 5.5–6.5
Jade Jadeite or nephrite
Jadeite (P) Na(Al,Fe)Si2O6 3.3–3.5 6.5–7.0
Jamesonite Pb4FeSb6S14 5.5–6.0 2.0–3.0
Jarcon Zircon
Jarosite KFe3(SO4) 2(OH) 6 2.9–3.3 3.0
Jasper Quartz
Kainite MgSO4·KCl.3H2O 2.1 3.0
Kalinite Alum, potash
Kallophilite K(AlSiO4) 2.61 6.0
Kalsilite Nephelines
Kaolin Clay mineral
Kaolinite Al2(Si2O5)(OH) 4 2.6–2.7 2.0–2.5
Kemite Na2B4O7·4H2O 1.95 3.0
Krennerite AuTe2 8.62 2.0–3.0
Kunzite Spudomene, pink
Kyanite Al2SiO5 3.6–3.7 5.0–7.0
Labradorite (P) Ab50An50·Ab30An70 2.71 6.0
Langbeinite K2Mg2(SO4) 3 2.83 2.5–3.5
Lapis lazuli Impure lazurite
Larsenite (Ov) PbZnSiO4 5.9 3.0
Laumontite (Z) (Ca,Na)Al2Si4O12·4H2O 2.3 4.0
Lawsonite CaAl2(Si2O7)(OH) 2·H2O 3.1 8.0
Lazulite (Mg,Fe3+ )Al2PO4) 2(OH) 2 3.0–3.1 5.0–5.5
Lazurite (Na,Ca) 4(AlSiO4) 3(SO4,S,Cl) 2.4–2.5 5.0–5.5
Lechatelierite SiO2, fused silica 2.2 6.0–7.0
Lepidocrocite FeO(OH) 4.1 5.0
Lepidolite (M)
Leucite
K(LiAl) 3(Si,Al) 4O10(F,OH) 2 2.8–3.0 2.5–4.0
Libethenite Cu2(PO4)(OH) 4.0 4.0
Limonite FeO(OH)·nH2O 3.6–4.0 5.0–5.5
Linarite PbCu(SO4)(OH) 2 5.3 2.5
Linnaeite Co3S4 4.8 4.5–5.5
Lithium mica Lepidolite
Lithiophilite Li(Mn2+ ,Fe2+ )PO4 3.5 5.0
Loellingite FeAs2 7.4–7.5 5.0–5.5
Magnesite MgCO3 3.0–3.2 3.5–5.0
Magnetite (S) (Fe,Mg)Fe2O4 5.2 6.0
Malachite Cu2CO3(OH) 2 3.9–4.0 3.5–4.0
Manganite MnO(OH) 4.3 4.0
Manganosite MnO 5.0–5.4 5.5
Marcasite FeS2, white iron pyrite 4.9 6.0–6.5
Margarite (M) CaAl2(Al2Si2O10)(OH) 12 3.0–3.1 3.5–5.0
Marialite (Sc) 3NaAlSi3O8·NaCl 2.7 5.5–6.0
Marmatite Sphalerite, iron bearing 3.9–4.0
Martite Hematite after magnetite
A.7

A.8 APPENDIX A
Name Description or composition Specific gravity Mohr hardness
Meerschaum Sepiolite
Meionite (Sc) 3CaAl2Si2O8·CaCO3 2.7 5.0–5.6
Melaconite Tenorite
Melanite (G) Andradite garnet (black)
Melanterite FeSO4·7H2O 1.9 2.0
Melilite (Na,Ca) 2(Mg,Al)(Si,Al) 2O7 2.9–3.1 5.0
Menaccanite Limenite
Menaghinite (J) CuPb13Sb7S24 6.36 2.5
Mercury Hg 13.6
Miargyrite AgSbS2 5.2–5.3 2.5
Mica (M) A group of minerals
Microcline (F) K(AlSi3O8), K feldspar 2.5–2.6 6.0
Microlite (Na,Ca) 2(Ta,Nb) 2O6 6.33 5.5
Microperthite (F) Microcline and albite
Milerite NiS 5.3 -5.7 3.0–3.5
Mimetite Pb5Cl(AsO4) 3 7.0–7.2 3.5
Minium Pb3O4 8.9–9.2 2.5
Mispickel Arsenopyrite
Molybdenite MoS2 4.6–4.7 1.0–1.5
Monazite (Ce,La,Y,Th)(PO4,SiO4) 5.0–5.3 5.0–5.5
Monticellite CaMgSiO4, rare olivine 3.2 5.0
Montmorillonite (C) (Al,Mg) 8(Si4O10) 3(OH) 10·10H2O 2.5 1.0–1.5
Moonstone (O) Opalescent albite or orthoclase
Morganite Beryl, rose color
Mullite Al6Si2O13 3.2 6.0–7.0
Muscovite (M) KAl2(AlSi3O10)(OH) 2 2.7–3.1 2.0–2.5
Nacrite (C) Al2(Si2O5)(OH) 2, kaolin group 2.6 2.0–2.5
Nagyagite Pb5Au(Te,Sb) 4S5-8 7.4 1.0–1.5
Natroalunite Alunite with NaK
Natrolite Na2(Al2Si3O10)·2H2O 2.3 5.0–5.5
Nepheline (Fp) (Na,K)AlSiO4 2.5–2.7 5.5–6.0
Nephrite Tremolite, similar to jade
Niccolite NiAs 7.8 5.0–5.5
Nickel bloom Annabergite
Nickel iron Ni,Fe, meteorite alloy 7.8–8.2 5.0
Ni skutterudite (Ni,Co,Fe)As3 6.1–6.9 5.5–6.0
Nitre KNO3, saltpeter 2.0–2.1 2.0
Nontronite (C) Fe(AlSi) 8O20(OH) 4 2.5 1.0–1.8
Norbergite Mg3(SiO4)(F,OH) 2 3.1–3.2 6.0
Noselite (Fp) Na8Al6Si6O24(SO4) 2.2–2.4 6.0
Octahedrite Anatase
Oligoclase (P) Ab90An10·Ab70An30 2.5 6.0
Olivine (Ov) (Mg,Fe) 2SiO4 3.3–4.4 6.5–7.0
Onyx Chalcedony, layered structure
Opal SiO2·nH2O 1.9–2.2 5.0–6.0
Orpiment As2S3 3.5 1.5–2.0
Orthite Allanite
Orthoclase (F) K(AlSi3O8), K feldspar 2.6 6.0
Osmiridium Iridosmine

SPECIFIC GRAVITY AND HARDNESS OF MINERALS
Name Description or composition Specific gravity Mohr hardness
Ottrelite (M) (Fe2+ ,Mn)(Al,Fe3+ )Si3O10·H2O 3.5 6.0–7.0
Palladium Pd 11.9 4.5–5.0
Paragonite (M) NaAl2(AlSi3O10)(OH) 12 2.85 2.0
Pargasite (H) NaCa2Mg4Al3Si6O22(OH) 2 3.0–3.5 5.5
Peacock ore Bomite
Pearceite (Ag,Cu) 16As2S11 6.15 3.0
Pectolite NaCa2Si3O8(OH) 2.7–2.8 5.0
Penninite Chlorite
Pentlandite (Fe,Ni) 9S8 4.6–5.0 3.5–4.0
Peridot (Ov) Olivine, gemstone
Perovskite CaTiO3 4.03 5.5
Perthite (F) Mixture of microcline and albite
Petalite (Fp) Li(AlSi4O10) 2.4 6.0–6.5
Petzite Ag3AuTe2 8.7–9.0 2.5–3.0
Phenacite Be2SiO4 2.9–3.0 7.5–8.0
Phillipsite (Z) (K2,Na2,Ca)Al2Si4O12·H2O 2.2 4.5–5.0
Phlogopite (M) K(Mg,Fe) 3AlSi3O10(OH,F) 2 2.86 2.5–3.0
Phosgenite Pb2Cl2CO3 6.0–6.3 3.0
Phosphuranylite Ca(UO2) 4(PO4) 2(OH) 4·7H2O Picotite (S) Spinel, chromium
Piedmontite Epidote, Mn2+ 3.4 6.5
Pigeonite (P) (Ca,Mg,Fe)SiO3 3.2–3.4 5.0–6.0
Pinite (M) Muscovite mica
Pitchblende Uraninite
Plagioclase (P) A group of aluminum silicates
Plagionite (J) Pb5Sb8S17 5.6 2.5
Platinum Platinum metal alloy 14–19 4.0–4.5
Pleonaste (S) Spinel, iron
Plumbago Graphite
Polianite MnO2, pyrolusite 5.0 6.0–6.5
Pollucite (Cs,Na) 2Al2Si4O12·H2O 2.9 6.5
Polybasite (Ag,Cu) 16Sb2S11 6.0–6.2 2.0–3.0
Polycrase (R) Y,Ce,Ca,U,Th,Ti,Nb,Ta,Fe oxide 4.7–5.9 5.5–6.5
Polyhalite K2Ca2Mg(SO4) 4·2H2O 2.78 2.5–3.0
Potash aluminum KAl(SiO4) 2·11H2O 1.75 2.0–2.5
Potassium feld KalSi3O8, see orthoclase
Potash mica (M) Muscovite
Powellite CaMoO4 4.2 3.5–4.0
Prase Jasper, green
Prehnite Ca2Al2(Si3O10)(OH) 12 2.8–2.9 6.0–6.5
Prochlorite Chlorite group
Proustite Ag3AsS3 5.6 2.0–2.5
Psilomelane Manganese mineral group
Pyrargyrite Ag3SbS3 5.9 2.5
Pyrite FeS2 5.02 6.0–6.5
Pyrochlore (Na,Ca) 2(Nb,Ta) 2O6(OH,F) 4.2–4.5 5.0
Pyrolusite MnO2 4.8 1–2
Pyromorphite Pb5(PO4) 3Cl 6.5–7.1 3.5–4.0
Pyrope (G) (Mg,Fe) 3Al2(SiO4) 3 3.5 7.0
A.9

A.10 APPENDIX A
Name Description or composition Specific gravity Mohr hardness
Pyrophylite Al2Si4O10(OH) 2 2.8–2.9 1.0–2.0
Pyroxene (P) A group of minerals
Pyrrhotite Fe1-x S where x = 0.0 to 0.2 4.6 4.0
Quartz SiO2 2.7 7
Rammelsbergite NiAs2 7.1 5.5–6.0
Rasorite Kemite
Realgar AsS 3.5 1.5–2.0
Red ochre Hematite
Rhodochrosite MnCO3 3.5–3.6 3.5–4.5
Rhodolite (G) 3(Mg,Fe)O,Al2O3·3SiO2 3.8 7.0
Rhodonite MnSiO3 3.6–3.7 5.5–6.0
Riebeckite (A) Na2(Fe,Mg) 5Si8O22(OH) 2 3.4 4.0
Rock salt Halite
Roscoelite (M) K(U,Al,Mg) 3Si3O10(OH) 2 3.0 2.5
Rubellite Tourmaline, red or pink
Ruby Corundum, red, gemstone
Ruby copper Cuprite
Ruby silver Pyrargyrite or proustite
Rutile TiO2 4.2–4.3 6.0–6.5
Samarskite (Y,Ce,U,Ca,Fe,Pb,Th)(Nb,Ta,Ti,Sn) 2O6 4.1–6.2 5.0–6.0
Sanadine (O) Orthoclase, high temperature
Saponite (C) (Mg,Al) 6(Si,Al) 8O20(OH) 4 2.5 1.0–1.5
Sapphire Corundum, blue, gemstone
Satin spar Gypsum, fibrous
Scapolite (Na or Ca) 4Al3(Al,Si) 3Si6O24(Cl,CO3,SO4) 2.6–2.7 5.0–6.0
Scheelite CaWO4 5.9–6.1 4.5–5.0
Schorlite Tourmaline, black
Scolecite (Z) Ca(Al2Si3O10)·3H2O 2.2–2.4 5.0–5.5
Scorodite FeAsO4·2H2O 3.1–3.3 3.5–4.0
Scorzalite (Fe,Mg)Al2(PO4) 2(OH) 2 3.4 5.5–6.0
Selenite Clear and crystalline gypsum
Semseyite Pb9Sb8S21 5.8 2.5
Sepiolite Mg4(Si2O5) 3(OH) 2·6H2O, Meerschaum 2.0 2.0–2.5
Sericite (M) Muscovite mica, fine grained
Serpentine (Mg,Fe) 3SiO5(OH) 4 2.2 2.0–5.0
Siderite FeCO3 3.8–3.9 3.5–4.0
Siegenite (Co,Ni) 3S4 4.8 4.5–5.5
Sillimanite Al2SiO5 3.2 6.0–7.0
Silver Ag 10.5 2.5–3.0
Silver glance Argentite
Sklodowskite Mg(UO2) 2Si2O7·6H2O 3.5
Skutterudite (Co,Ni,Fe)As3 6.1–6.9 5.0
Smaltite Skutterudite variety
Smithsonite ZnCO3 4.3–4.4 5.0
Soapstone Talc
Sodalite (Fp) Na4Al3Si3O12Cl 2.2–2.3 5.5–6.0
Soda nitre NaNO3 2.3 1.0–2.0
Specular iron Hematite, foliated
Sperrylite PtAs2 10.5 6.0–7.0

SPECIFIC GRAVITY AND HARDNESS OF MINERALS
A.11
Name Description or composition Specific gravity Mohr hardness
Spessartite (G) Mn3Al2(SiO4) 3, red, brown 4.2 7.0
Sphalerite (Zn,Fe)S 3.9–4.1 3.5–4.0
Sphene CaTiO(SiO4) 3.4–3.5 5.0–5.5
Spinel group (Mg,Fe,Zn,Mn)Al2O4 3.6–4.0 8.0
Spodumene (P) LiAl(Si2O6) 3.1–3.2 6.5–7.0
Stannite Cu2FeSnS4 4.4 4.0
Staurolite (Fe,Mg) 2Al2Si4O23(OH) 3.6–3.8 7.0–7.5
Steatite Talc
Stephanite Ag5SbS4 6.2–6.3 2.0–2.5
Stembergite AgFe2S3 4.1–4.2 1.0–1.5
Stibnite Sb2S3 4.5–4.6 2.0
Stilbite (Z) NaCa2Al5Si3O36·14H2O 2.1–2.2 3.5–4.0
Stillwellite (Ce,La,Ba)BSiO5 4.6
Stolzite PbWO4 8.3–8.4 2.5–3.0
Stromeyerite (Cu,Ag)S 6.2–6.3 2.5–3.0
Strontianite SrCO3 3.7 3.5–4.0
Sulphur S 2.0–2.1 1.5–2.5
Sunstone (F) Oliglase, translucent
Sylvanite (Au,Ag)Te2 8.0–8.2 1.5–2.0
Sylvite KCl 2.0 2.0
Talc Mg3(Si4O10)(OH) 2 2.7–2.8 1.0
Tantalite (Fe,Mn)(Ta,Nb) 2O6,TaNb 6.2–8.0 6.0–6.5
Tennantite (Cu,Fe,Zn,Ag) 12As4S13 4.6–5.1 3.0–4.5
Tenorite CuO 5.8–6.4 3.0–4.0
Tephroite (Ov) Mn2(SiO4) 4.1 6.0
Tetrahedrite (Cu,Fe,Zn,Ag) 12Sb4S13 4.6–5.1 3.0–4.5
Thenardite Na2SO4 2.68 2.5
Thomsonite (Z) NaCa2Al5Si5O20·6H2O 2.3 5.0
Thorianite ThO2 9.7 6.5
Thorite Th(SiO4) 5.3 5.0
Thulite Zoisite, pink to red
Tiger’s eye Form of quartz
Tin Sn 7.3 2.0
Tinstone Cassiterite
Titanite Sphene
Topaz Al2(SiO4)(F,OH) 2 3.4–3.6 8.0
Torbernite Cu(UO2) 2(PO4) 2·8H2O 3.2 7.0–7.5
Tourmaline (Na,Ca)(Al,Fe,Li,Mg) 3Al6(BO3) 3(Si6O22) (OH) 4
3.0–3.2 7.0–7.5
Tremolite (A) Ca2Mg5(Si8O22)(OH) 2 3.0–3.3 5.0–6.0
Tridymite SiO2 2.3 7.0
Triphylite Li(Fe,Mn)PO4 3.4–3.6 4.0–5.5
Troilite Pyrrhotite
Trona Na2CO3·NaHCO3·2H2O 2.1 3.0
Troostite Manganiferous willemite
Tungstite WO3·nH2O 2.5
Turgite 2Fe2O3·nH2O 4.2–4.6 6.5
Turquoise CuAl6(PO4) 4(OH) 8·5H2O 2.6–2.8 6.0
Tyuyamunite Ca(UO2) 2(VO4) 2·5H2O 3.7–4.3 2.0

A.12 APPENDIX A
Name Description or composition Specific gravity Mohr hardness
Ulexite NaCaB5O9·8H2O 1.96 1.0
Uralite (H) Homblende after pyroxene
Uraninite UO2 to UO3 9.0–9.7 5.5
Uranophane Ca(UO2) 2Si2O7·6H2O 3.8–3.9 2.0–3.0
Uvarovite (G) Ca3Cr2(SiO4) 2, green 3.5 7.5
Vanadinite Pb5(VO4) 3Cl 6.7–7.1 3.0
Variscite Al(PO4)·2H2O 2.4–2.6 3.5–4.5
Vermiculite (M) Biotite, altered 2.4 1.5
Vesuvianite Idocrase
Violarite Ni2FeS4 4.8 4.5–5.5
Vivianite Fe3(PO4) 2·8H2O 2.6–2.7 1.5–2.0
Wad Manganese oxides
Wavelite Al3(OH) 3(PO4) 2·5H2O 2.3 3.5–4.5
Wemerite (Sc) Scapolite
White pyrite Marcasite
White mica (M) Muscovite
Wilemite Zn2SiO4 3.9–4.2 5.5
Witherite BaCO3 4.3 3.5
Wolframite (Fe,Mn)WO4 7.0–7.5 5.0–5.5
Wollastonite Ca(SiO3) 2.8–2.9 5.0–5.5
Wood tin Cassiterite
Wulfenite PbMoO4 6.5–7.5 3.0
Wurtzite (Zn,Fe)S 4.0 4.0
Xenotime YPO4 4.4–5.1 4.0–5.0
Zeolite A group of minerals
Zincite ZnO 5.7 4.0–4.5
Zinc spinel Gahnite
Zinkenite (J) Pb6Sb14S27 5.3 3.0–3.5
Zinnwaldite Fe,Li, mica 3.0 2.5–3.0
Zircon ZrSiO4 4.7 7.5
Zoisite Ca2Al3Si3O12(OH) 3.3 6.0
Adapted from T. J. Glover, Pocket Ref, Second Edition. Littleton, CO: Sequoia.

CHAPTER 10
MATERIALS SCIENCE FOR
SLURRY SYSTEMS
10-0 INTRODUCTION
Slurry is essentially a mixture of solids and liquids. Abrasion, erosion, and corrosion are
so associated with pumping slurry that manufacturers of slurry pumps sometimes spend
more money dealing with wear issues than with developing new hydraulics.
Wear is very complex and too often oversimplified. It depends on many factors such as
the microstructure of the surface of the pump part, the hardness and shape of the crushed
or milled minerals, the speed of flow, the scaling of pipes, etc. Dredge and slurry pumps,
ball mill liners and shells, magnetic separators, and agitators are made from metals that
combine the ability to resist stress and impact loads, erosion, and corrosion. Excellent
books on materials science are available to the reader but, unfortunately, they too often
dedicate just few lines or one or two paragraphs to the white irons or polymers used by
the designers of slurry systems. These materials are too often classified as materials for
special applications. This chapter will therefore make an effort to expand on this topic, as
a good understanding of it can save on maintenance costs to the operator and is necessary
for the successful design of a slurry system.
10-1 THE STRESS–STRAIN RELATIONSHIP
OF METALS
The stress–strain relationship of metals under tension (Figure 10-1) is often represented in
the form of a graph of stress versus strain. Stress is essentially the load force per unit area.
It is called direct stress � when the load is normal to the force, and shear stress � when the
load is parallel to the surface:
� = FN/A (10-1)
where
� = direct stress
� = tangential force
A = area
� = F N/A (10-2)
10.1

10.2 CHAPTER TEN
Stress
u
Yield y
Elastic
Limit
e
e
E
Utimate Tensile Strength
2% offset
strain
FIGURE 10-1 Stress–strain relationship of metals.
fracture
FN = normal force
FT = tangential force
When a specimen bar of steel is subject to a tension load at its ends (Figure 10-2), it
stretches. Within a certain range, the elongation is elastic, and this means that if the load
is removed, the specimen will return to its original length. The maximum stress in this
elastic range is called the elastic limit and is shown in Fig 10-1 as �e. The elongation �L
is divided by the original length L0 of the specimen to define the strain �:
� = �L/L0 (10-3)
It is a nondimensional measure, often expressed in percentage of original length, and mistakenly
called “elongation” instead of direct strain. The direct strain under tension is correlated
to the direct stress in the elastic range up to �e by the Young modulus E:
� = �/E (10-4)
Steels, but not all metals, exhibit a further nonlinear elongation up to a value called the
yield stress. The elongation becomes permanent as the materials yield. Beyond the yield
point, the elongation continues to grow, until a value called the ultimate tensile strength
�u is reached. This is the point at which the specimen can withstand the greatest load and
beyond which fracture is likely to occur rather quickly.
Direct stress by itself induces a degree of secondary shear stresses. For each material
there is a Poisson ratio �. For steel it is 0.30 and for gray cast iron it is 0.26. Shear strain
correlates with shear strain by the modulus of rigidity G (also called shear modulus). It is
related to the Young modulus by the Poisson ratio:
E
G = � (10-5)
2(1 + �)
Due to fatigue considerations, steel shafts of rotating equipment are designed for maximum
stresses of less than 18% to the ultimate strength or 30% of yield strength. This is
due to the combination of direct stress, torsion, and bending moments. Components of a
slurry mill or pump must be able to absorb the energy of impact without fracture. The energy
due to impact involves both loads and deflections. The capacity to absorb such ener-
f

gy is called the resilience. Its measure is the modulus of resilience. For steels, it is the area
under the stress-to-strain triangle in the elastic range.
10-2 IRON AND ITS ALLOYS FOR THE
SLURRY INDUSTRY
Cast iron is an alloy of iron, carbon, silicon, and manganese. Carbon is in the range of 2 to
4%. The cooling rate after casting of cast iron and subsequent heat treatment determine its
mechanical properties. Carbon is very important to the properties of cast iron.
10-2-1 Grey Iron
MATERIALS SCIENCE FOR SLURRY SYSTEMS
10.3
FIGURE 10-2 Hardness of minerals. (From Wilson, 1985. Reproduced by permission from
McGraw-Hill.)
Grey iron is cast iron with carbon precipitated in the form of graphite flakes. Graphite
flakes weaken the casting in tension, and grey iron is considered to have a compressive

10.4 CHAPTER TEN
strength three to five times as much as in tension. Grey iron has good damping properties
and is used in the base of machinery, bearing assembly of slurry pumps, parts essentially
under compression such as certain low-pressure pump casings, engine blocks, gears, flywheels,
and brake disks.
10-2-2 Ductile Iron
The addition of magnesium as an alloying element precipitates excess carbon in the form
of small nodules. These nodules do not disturb the structure of cast iron, as is the case
with the graphite flakes. Ductile iron, also called nodular iron, has better properties in tension,
and has better ductility, impact resistance, and stiffness than grey iron. Ductile iron
is used for the casting of high-pressure pump casings or metal- and rubber-lined pumps,
as well as unlined casings for mild slurries. Ductile iron is available in different grades
such as 60-40-18 or 60,000 psi (413 MPa) ultimate tensile strength, 40,000 psi (276 MPa)
yield, and 18% elongation. Such an elongation is not required for slurry pumps, and following
heat treatment and alloying, other grades are used. For pumps, ductile iron that
meets the standard ASTM A536-84 or SAE-J434C is available with a tensile strength of
552 MPa (80,000 psi), a yield strength of 414 MPa (60,000 psi), elongation of 3%, and
with a minimum hardness of 187 BHN.
10-3 WHITE IRON
Wear-resistant alloy irons are essentially “white irons.” They have found widespread application
in the mining industry for the manufacturing of crushers, mill liners, and slurry
pumps as well as for shot blasting. White irons used to be considered a very useless byproduct
of the melting of irons, with all the carbon precipitating in the form of carbides in
the pearlitic matrix. Up to the beginning of the 20th century, they were discarded because
they are extremely brittle and impossible to machine.
10-3-1 Malleable Iron
Malleable iron is made from white iron by a two-stage heat treatment process. The resultant
structure contains excess graphite in the form of tempered nodules. Because white
iron is used, castings can be thinner than 76 mm (3�). Malleable iron has found applications
for the bearing surfaces of heavy parts of farm equipment, trucks, railroad equipment,
and to a certain extent in some slurry applications.
Malleable cast iron has a structure that consists of ferrite, pearlite, and graphite. Its ultimate
tensile strength is in the range of 400 to 500 MPa (58,000–72,500 psi). Its ductility
and toughness decrease as the quantity of pearlite increases. Zakharov (1962) described
the conversion of white iron to malleable cast iron by a two-step heat treatment process.
To be properly heat-treated, the carbon content must be low and must not exceed 2.5
to 2.8%. The lower the carbon content, the less graphite forms. Silicon must not exceed
1% and manganese must not exceed 0.5%. If the silicon content exceeds 1%, it prevents
the transformation of graphite flakes into nodules. Although the presence of manganese
facilitates the casting of white iron, excessive amounts tend to stabilize the carbides during
heat treatment. Stabilized carbides increase the resistance to wear but they also make
it very difficult to machine the cast component.

In the first step of heat treatment, the white iron is heated to 900°C–950°C
(1650–1742°F) and then allowed to cool. During the second step, it is annealed at 720°C
to 760°C (1330°F–1400°F). Before annealing, white iron consists essentially of two phases:
austenite and cementite (a constituent of lebedurite eutectic). During annealing,
austenite is not affected, but the cementite decomposes to form iron and graphite:
Fe3C = 3Fe + C (graphite)
After this first step of graphitization, the malleable cast iron consists of austenite and
graphite. The carbon content of austenite is about 1% at 900°C (1650°F). If the white iron
is allowed to cool after this first step of heat treatment, secondary cementite or ferrite
forms, depending on the applied cooling rate. If cementite and ferrite are considered to be
undesirable from a point of view of machining, a second step of heat treatment is applied
to decompose the secondary and pearlite cementite and to form nodular carbon.
Conventionally annealing white iron into malleable form used to be a very lengthy
process that could take three days or more in a heat treatment furnace. Various methods
have been developed over the years to accelerate the process, such as first-step heat treatment
at 1000°C (1832°F), hardening before annealing, etc.
10-3-2 Low-Alloy White Irons
The British Standard BS 4844:1986 defines three grades of low-alloy white irons shown
in Tables 10-1 and 10-2. These alloys have been superseded by alloyed irons.
10-3-3 Ni-Hard
MATERIALS SCIENCE FOR SLURRY SYSTEMS
10.5
The International Nickel Company developed special alloys of white iron with nickel.
These are called Ni-hard and a number of alloys such as Ni-hard 1 to Ni-hard 4 are now
produced (Tables 10-3 to 10-6). The presence of nickel increases the hardness but it also
ensures the transformation of the austenite to martensite after proper heat treatment. The
selection of alloying elements is based on the intended use and on the thickness of the cast
part. The maximum carbon content is 3.2–3.6%, but when impact resistance is important,
the carbon should be trimmed to 2.7–3.2%.
The composition of Ni-hard 1 to 4 is not exactly the same from one country to another,
as shown in table 10-3 to 10-6.
Ni-hard 1, or ASTM A532 Class 1, Type A, is a martensitic white iron. It is used in
relatively mild erosive applications were impact forces are low. It is heat treated for stress
relief. Due to the limitations on thickness to 200 mm (8�), as indicated in Table 10-3, Nihard
1 has found limited applications in wastewater plants and mild slurries.
Ni-hard 4 (ASTM A532 Class I, Type D) has a tensile strength in the range of
420–700 MPa (60,000 to 100,000 psi). To increase its hardness, some manufacturers of
slurry pumps conduct cryogenic or heat treatment. Its excellent fluidity makes it a suitable
TABLE 10-1 Composition of Low-Alloy White Irons
BS 4844:1986 C Si Mn Cr
Grade 1A, 1B, 1C 2.4–3.4 0.5–1.5 0.2–0.8 2.0 max

10.6 CHAPTER TEN
TABLE 10-2 Hardness of Low-Alloy White Irons
alloy for casting fairly complex shapes of wear-resistant liners for mills, grinding balls,
and pump parts.
10-3-4 High-Chrome–Molybdenum Alloys
By adding chrome in the range of 12 to 28%, together with nickel and molybdenum, allows
the casting of abrasion-resistant alloys that are tough and can be cast in large sizes to
match the needs of the mining industry. Eutectic carbides in the form of M 7C 3 in combination
with an austenitic, martensitic, or pearlitic matrix gives a full range of alloys. Some
of the components are cast pearlitic to allow machining, then heat treated to obtain an
abrasion-resistant martensitic structure. Tables 10-7 and 10-8 compare two alloys in this
family as cast in France, Germany, the United Kingdom, and the United States. They are
often called by the foundries 16% and 27% chrome.
High-chrome white irons cannot be welded. They are often very difficult to machine
TABLE 10-3 Composition of Ni-Hard 1 White Iron
France Germany United Kingdom United States
Standard NF 32-401 (1980) DIN 1695 (1981) BS 4844 Pt2:1972 ASTM A532
(1982)
Grade FB Ni4 Cr2 HC G-X330 NiCr4 2 2B Class 1 Type A
Ni-Cr LC
C (%) 3.2–3.6 3.0–3.6 3.2–3.6 3.0–3.6
Si (%) 0.2– 0.8 0.2–0.8 0.3–0.8 0.8 max
Mn (%) 0.3–0.7 0.3–0.7 0.2–0.8 1.3 max
Ni (%) 3.0–5.5 (may be
replaced by Cu)
3.3–5.0 3.0–5.5 3.0–5.0
Cr (%) 1.5–2.5 1.4–2.4 1.5–2.5 1.4–4.0
Cu (%) — — — —
Mo (%) 0.0–1.0 0.5 0.5 max —
P max (%) — 0.3 —
S max (%) — 0.15 0.15
Minimum hardness
as cast
— 450 (430) HB 550 (500) HB 550 HB
Minimum hardness
as cast
500–700 HB 550 (500) 550
Annealed —
Maximum section 200 mm (8�)
Data from Brown (1994).
Hardness (minimum) HB
Grade Thickness � 50 mm Thickness > 50 mm
1A 400 350
1B 400 350
1C 250 200

MATERIALS SCIENCE FOR SLURRY SYSTEMS
TABLE 10-4 Composition of Ni-Hard 2 White Iron
10.7
France Germany United Kingdom United States
Standard NF 32-401 (1980) DIN 1695 (1981) BS 4844 Pt2:1972 ASTM A532
(1982)
Grade FB Ni4 Cr2 BC G-X260 NiCr4 2 2A Class 1 Type B
Ni-Cr LC
C (%) 2.7–3.2 2.6–2.9 2.7–3.2 2.5–3.0
Si (%) 0.2– 0.8 0.2–0.8 0.3–0.8 1.3 max
Mn (%) 0.3–0.7 0.3–0.7 0.2–0.8 0.8 max
Ni (%) 3.0–5.5 (may be
replaced by Cu)
3.3–5.0 3.0–5.5 3.0–5.0
Cr (%) 1.5–2.5 1.4–2.4 1.5–2.5 1.4–4.0
Cu (%) — — — —
Mo (%) 0.0–1.0 0.5 max 1.0 max
P max (%) — 0.3 0.3
S max (%) — 0.15 0.15
Minimum hardness
as cast
— 450 (430) HB 550 HB
Minimum hardness
as cast
450–650 HB 520 (480) 500
Annealed —
Maximum section 200 mm (8�)
Data from Brown (1994).
except by grinding. An electric induction furnace is used to achieve the high temperature
needed for melting. The Cr–Fe oxides that form during the melting process attack the silica
lining of furnaces. Alumina linings should be used for these furnaces.
Scrap steel, foundry return, spent impeller, and SAG mill liners are melted with addition
of high- and low-carbon ferrochromium to obtain the iron at the required carbon con-
TABLE 10-5 Composition of Ni-Hard 3 White Iron
France United States
Standard NF 32-401 (1980) ASTM A532 (1982)
Grade FB A Class 1 Type C Ni-Cr-GB
C (%) 2.7–3.9 2.9–3.7
Si (%) 0.4–1.5 1.3 max
Mn (%) 0.2–0.8 0.8 max
Ni (%) 0.3–3.0 2.7–4.0
Cr (%) 0.2–2.0 1.1–1.5
Cu (%) — —
Mo (%) 0.0–1.0 1.0
P max (%) — 0.3
S max (%) — 0.15
Minimum hardness as cast — 550 HB
Minimum hardness as cast
Annealed
500–700 HB
Maximum section 75 mm (3�) diameter ball
Data from Brown (1994).

10.8 CHAPTER TEN
TABLE 10-6 Composition of Ni-Hard 4 White Iron
France Germany United Kingdom United States
Standard NF 32-401
(1980)
DIN 1695
(1981)
BS 4844 Pt2:1972
ASTM A532
(1982)
Grade FB Cr9 Ni5 GX 300 CrNSi 2C 2D 2E Class 1 Type D
9 5 2 Ni-Hi-Cr
C (%) 2.5–3.6 2.5–3.5 2.4–2.8 2.8–3.2 3.2–3.6 2.5–3.6
Si (%) 1.5–2.2 1.5–2.2 1.5–2.2 1.5–2.2 1.5–2.2 1.3 max
Mn (%) 0.3– 0.7 0.3– 0.7 0.2– 0.8 0.2– 0.8 0.2– 0.8 1.0–2.2
Ni (%) 4.0–6.0 8.0–10.0 4.0–6.0 4.0–6.0 4.0–6.0 4.5–7.0
Cr (%) 5.0–11.0 4.5–6.5 8.0–10.0 8.0–10.0 8.0–10.0 7.0–10.0
Cu (%) — —
Mo (%) 0.5 max 0.5 0.5 max 0.5 max 0.5 max 1.0 max
P max (%) — 0.3 0.3 0.3 0.3 0.1
S max (%) — 0.15 0.15 0.15 0.15 0.15
Minimum
hardness
as cast
450 (430) HB 550 HB
Minimum
hardness
as cast
550–750 HB 600 (535) HB 500 550 600 600
Annealed 400
Maximum 300 mm (12�)
section diameter ball
Data from Brown (1994).
tent. Ferromolybdenum is added. About 5% of the chrome is lost during melting. The
melting temperature for the low-carbon, high-chrome iron is in the range of 1600 to
1650°C (or 2912 to 3002°F), but for the higher carbons it is in the range of 1550 to
1600°C (or 2822 to 2912°F).
During the process of casting, a viscous oxide film forms on the surface of the molten
TABLE 10-7 Composition of 16% High-Chrome Abrasion-Resistant White Iron
France Germany United Kingdom United States
Standard NF 32-401
(1980)
DIN 1695
(1981)
BS 4844 Pt2:1972 ASTM A532 1982
Grade FB Cr9 Ni5 GX 300 CrMo 3A 3B Class II Class II
15 3 Type B Type C
C (%) 2.0–3.6 2.3–3.6 2.4–3.0 3.0–3.6 2.4–2.8 2.8–3.6
Si (%) 0.2–0.8 0.2–0.8 1.0 max 1.0 max 1.0 max 1.0 max
Mn (%) 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0
Ni (%) 0.0–2.5 0.7 0.0–1.0 0.0–1.0 0.5 max 0.5 max
Cr (%) 14.0–17.0 14.0–17.0 14.0–17.0 14.0–17.0 14.0–18.0 14.0–18.0
Cu (%) — 0–1.2 0–1.2 1.2 max 1.2 max
Mo (%) 0.5–3.0 1.0–3.0 0.0–2.5 1.0–3.0 1.0–3.0 2.3–3.5
P max (%) — 0.3 0.3 0.1 0.1
S max (%) — — 0.1 0.06 0.06
Data from Brown (1994).

MATERIALS SCIENCE FOR SLURRY SYSTEMS
TABLE 10-8 Composition of 27% High-Chrome Abrasion-Resistant White Iron
10.9
France Germany United Kingdom United States
Standard NF 32-401
(1980)
DIN 1695 (1981) BS 4844 Pt2:1972
ASTM A532
(1982)
Grade FB Cr26MoNI GX 260 Cr27 GX300CrMo 3D 3E Class III
27 1 Type A
C (%) 1.5–3.5 2.3–2.9 3.0–3.5 2.4–2.8 2.8–3.2 2.3–3.0
Si (%) 0.2–1.2 0.5–1.5 0.2–1.0 1.0 max 1.0 max 0.5–1.5
Mn (%) 0.5–1.5 0.5–1.5 0.5–1.0 0.5–1.5 0.5–1.5 1.0 max
Ni (%) 0.0–2.5 1.2 1.2 0.0–1.0 0.0–1.0 1.5 max
Cr (%) 22.0–28.0 24.0–28.0 23.0–28.0 22.0–28.0 22.0–28.0 23.0–28.0
Cu (%) 0.0–1.5 2.0 max 2.0ax 1.2 max
Mo (%) 0.5–3.0 1.0 1.0–2.0 0–1.5 0–1.5 1.5 max
P (%) — 0.1 0.1 0.1
S (%) — — 0.1 0.06 0.06
Data from Brown (1994).
metal in the furnace or ladle. It must be skimmed before pouring and the running system
must be designed to trap oxide dross. Metal filters are used wherever possible. Gating
should be designed to provide rapid filling of the mold with minimum turbulence.
The alloy shown in Table 10-7 is sometimes called 15/3 Chrome/Moly Iron, or 16%
chrome. It is a martensitic white iron of moderate erosion resistance. It is used for the
casting of pumps for a number of applications such as carbon in pulp circuits, coal transfer
pumps, sewage treatment, and some newspaper recycling pumps.
The alloy known in the industry as 20% chrome, also known as ASTM A532-87 Class
II Type D, is available with a tensile strength of 595 to 875 MPa and with suitable multistep
heat treatment can have a minimum hardness of 650 HBN or 59 HRC. It is used for
the manufacture of mill liners, slurry pump parts, pulverizer parts, and other wear-resistant
castings. Its composition includes carbon in the range of 2 to 3.3%, chromium in the
range of 18 to 23%, and molybdenum at a maximum of 3%. The structure consists of eutectic
and secondary carbides in a martensitic matrix.
The alloy known in the industry as 27% chrome, whose composition is shown in
Table 10-8, is very popular with manufacturers of slurry pumps, as it offers abrasion resistance,
erosion resistance, mild corrosion resistance, and mild heat resistance. It consists
of a microstructure of chromium carbides in a martensitic–austenitic matrix. Its tensile
strength is in the range of 525 to 700 MPa (75,000–100,000 psi) and can be heat treated to
a minimum hardness of 650 HBN/58 HRC. Table 10-9 summarizes the classes of white
iron.
Corrosion can have very detrimental effects on the wear resistance of certain white
irons. If the matrix wears out and exposes the carbides to a corrosive environment, they
may quickly deteriorate, causing a further loss of wear resistance. Manufacturers of slurry
pumps have gone beyond the specifications of ASTM A532-87 to increase the chromium
contents to 30% with 50% chromium carbides in a martensitic matrix. These materials
have found applications in phosphate matrix and phosphoric acid pumping. These new alloys
are intermediary between white irons and superalloys. Walker (1990) reported that
the use of these alloys leads to a substantial increase of the wear life over conventional
white iron castings (Ni-hard, 27% high chrome).
The corrosion properties of stainless steel and duplex stainless are enhanced by adding
high-chromium carbides in the matrix to produce a range of materials suitable for flue gas
desulfurization. The matrix of the alloy is a balance between ferritic stainless steel rich in

10.10 CHAPTER TEN
TABLE 10-9 Comparison of White Iron Classes
Hypoeutectic Eutectic Hypereutectic
Example Ni-hard 27% chrome Very high chrome (>30%)
Matrix and carbides Soft primary ferrous White iron matrix with White iron austenitic/
dendrites with eutectic very fine dispersion of martensite matrix with
carbides eutectic carbides coarse, discrete primary
carbides
Note on casting Refer to ASTM 532 Solidification during
for maximum sizes of casting must be controlled
castings to avoid cracking
soluble chromium, nickel, and other alloys to resist corrosion and a substantial volume of
chromium carbides to resist erosion wear.
When designing components of pumps subject to torsion and bending loads, such as
shaft sleeves and pump casings, it is important to appreciate that the harder the material,
the narrower the limit on strain or elongation. A range of elongation of 0.5% to 1.5% is a
minimum for bowls, mantle liners, pump-wetted parts, ball mill grates, and components
subject to impact loads.
The author was once asked to express an opinion on the failure of a large number of
stainless steel shaft sleeves at start-up. The slurry pumps used very hard shaft sleeves.
Upon investigation it was noticed that the crack was longitudinal. A modeling of the shaft
sleeve by finite element analysis revealed that the strain due to torsion exceeded the allowed
maximum value. These large pumps were using motors with 250% starting torque.
That initial torque screwed the pump impeller tightly against the shaft sleeve and applied
FIGURE 10-3 Mictograph showing grain structure of Moballoy, a proprietary high-chrome,
abrasion-resisting alloy. (Courtesy of Mobile Pulley and Machine Works, Inc.)

a strong torsion moment. The shaft sleeve, due to its excessive hardness, simply cracked.
It had to be replaced by shaft sleeves with a lower hardness. The manufacturer eventually
developed a special shaft sleeve with a hard ceramic face, but with a more ductile base to
take the torsion loads.
The resistance of the matrix and carbides to wear is a complex phenomenon due to
hardness of the individual components. Ferrite matrix has a mere hardness of 150–250
HV. Austenite, as found in many stainless steels, has a hardness of 300–500 HV. Martensite,
which is sometimes obtained by very fast solidification or chilling of the casting, has
a hardness of 500–1000 HV. The iron carbides Fe 3C have a hardness in the range of
850–1000 HV, Chromium carbide (FeCr) 7C 3 has a hardness of 1400–1600 HVas primary
carbides and a hardness of 1200–1400 as eutectics (Huggett and Walker, 1992).
Dredge pumps and ball mill liners must be able to absorb the energy of impact without
fracture. The energy due to impact involves both loads and deflections. The capacity to absorb
such energy is called the resilience. Its measure is the modulus of resilience. For steels,
it is the area under the stress-to-strain triangle in the elastic range shown in Figure 10-1.
The fracture toughness is a relative measure of the ability to absorb shock loads. This
is particularly important with dredge pumps. Certain 16% high-chrome alloys (ASTM
A532 Class II Type B) have a fracture toughness in the range of 14–28 KSI-in. 20% highchrome
alloys (ASTM A532 Class II Type E) have a fracture toughness in the range of
21–28 KSI-in. 27% high chrome alloys (ASTM A532 Class III Type A) have a fracture
toughness in the range of 22–27 KSI-in. Because different manufacturers and foundries
apply different levels of heat treatments, these values should be taken as indicative. The
manufacturer should always be consulted.
10-4 NATURAL RUBBERS
MATERIALS SCIENCE FOR SLURRY SYSTEMS
10.11
Elastomers, particularly different grades of rubber, are widely used in the lining of slurry
handling equipment as well as piping. Natural rubber is preferred for solids with a diameter
smaller than 6 mm or 1 – 4 in for pump impellers and smaller than 15 mm or 5 – 8 in for pump
liners. Natural rubber resists well a number of acids with a pH less than 6.0, but is not
suitable for pumping oils or solvents. Liquid temperature must be lower than 65°C, or
150°F.
The abrasion resistance of natural rubber exceeds the resistance of all other elastomers.
A number of forms of natural rubber are available such as:
TABLE 10-10 Properties of Natural Aashto
Durometer (Shore A) hardness 50 (±5) to 70 (±5)
Tensile strength (MPa) 22
Tensile strength (psi) 3190
Elongation (%) 450
Specific gravity 1.07
Tear resistance Excellent
Abrasion resistance Good
Impact resistance Excellent
Flame resistance Poor
Heat aging at 100°C (212°F) Average
Weather resistance Good

10.12 CHAPTER TEN
� Pure tan gum natural rubber
� Natural aashto
� Carbon-filled natural rubber
� White food-grade natural rubber
� Carbon-filled and silica-filled natural rubber
� Hard natural rubber/butadiene styrene compound filled with graphite
10-4-1 Natural Aashto
Natural aashto is a natural rubber of excellent resistance to impact and to tear, with good
resistance to abrasion. It is weather-resistant and is widely used as a certified bridge construction
material. It is available in 50, 60, and 70 durometer hardness. Its temperature
range for use is from –50°C to 100°C (–60°F to 212°F). The mechanical properties of natural
aashto are listed in Table 10-10.
10-4-2 Pure Tan Gum
Pure tan gum is a natural rubber of excellent resistance to abrasion and high tensile
strength. It is excellent as protection against impinging abrasion in chutes, troughs, and
hoppers. It is available in 35 to 45 durometer shore A hardness. Its temperature range for
use is from –50°C to 65°C (–60°F to 150°F). It has poor resistance to ozone. The mechanical
properties of pure tan gum are listed in Table 10-11.
10-4-3 White Food-Grade Natural Rubber
A special high-quality odorless natural rubber is available with a Shore A hardness of 40.
It is limited to a temperature of 65°C (150°F).
TABLE 10-11 Properties of Pure Tan Gum
Durometer (Shore A) hardness 40 (±5)
Tensile strength (MPa) 17
Tensile strength (psi) 2500
Elongation (%) 600
Tear resistance Excellent
Abrasion resistance Excellent
Impact resistance Excellent
Flame resistance Poor
Ozone resistance Poor
Heat aging at 100°C (212°F) Good
Resistance to oil and petroleum products Poor, oil and petroleum products cause swelling
and degradation
Typical applications Sand and gravel
Tip speed limit for pump impellers 25 m/s (4920 ft/min)

10-4-4 Carbon-Black-Filled Natural Rubber
This is an industrial grade of natural rubber with better tear resistance but slightly lower
abrasion resistance than natural rubber. It typically has durometer hardness 45-Shore
hardness A. Its temperature limit is 82°C (180°F). This form of filled natural rubber is
widely used in the industrial, mining, and utilities industries.
10-4-5 Carbon-Black- and Silicon-Filled Natural Rubber
These fillers are added to natural rubber to slightly improve the resistance to traces of oil
and to stabilize the matrix against thermal degradation by friction. It typically has durometer
hardness 55 Shore hardness A. Its temperature limit is 82°C (180°F). The tip speed of
the impeller can be a maximum of 28 m/s (5500 ft/sec).
10-4-6 Hard Natural Rubber/ Butadiene Styrene Compound Filled
with Graphite
This compound is resistant to chlorine and a wide range of acids. It is particularly hard after
curing, which allows machining. Its temperature limit is 93°C (200°F). Nominal
durometer hardness is 60 Shore D.
Rubber-lined products that are subject to long storage must be stabilized against storage-based
degradation by suitable antioxidants and antidegradants.
10-5 SYNTHETIC RUBBERS
In addition to natural rubber, a number of synthetic elastomers are available for handling
slurries.
10-5-1 Polychlorene (Neoprene)
MATERIALS SCIENCE FOR SLURRY SYSTEMS
10.13
Neoprene, a dupont trademark for polychlorene, is superior to natural rubber in its resistance
to oils and chemicals, and for higher-temperature applications. Although Neoprene
TABLE 10-12 Properties of Polychlorene (Neoprene)
Durometer (Shore A) hardness 40 ± 5 50 ± 5 60 ± 5 70 ± 5 80 ± 5
Tensile strength (MPa) 8 8 8 8 8
Tensile strength (psi) 1160 1160 1160 1160 1160
Elongation (%) 560 500 350 250 200
Tear resistance Good
Abrasion resistance Good
Impact resistance Good
Flame resistance Good
Ozone resistance Good
Typical temperature range –34°C to 100°C
–30°F to 212°F

10.14 CHAPTER TEN
TABLE 10-13 Properties of EPDM
Durometer (Shore A) Hardness 60 ± 5
Tensile strength (MPa) 7
Tensile strength (psi) 1000
Elongation (%) 400
Tear resistance Average
Abrasion resistance Good
Impact resistance Good
Flame resistance Poor
Ozone resistance Excellent
Weather resistance Excellent
Typical temperature range –50°C to 177°C
–60°F to 350°F
has a lower resistance to abrasion than narural rubber, it is the next most common material
for pump parts. The tip speed for impellers is 33 m/s (or 6496 ft/sec). The mechanical
properties of Neoprene in different hardness grades are presented in Table 10-12.
A form of hard Neoprene is available for upper and lower screens of vertical sump
pumps and for cap nuts and flingers that protect the shaft seals. Its nominal hardness is 80
Shore A, but it can be postcured to a hardness of 85 Shore A.
Because of its superior resistance to oils and hydrocarbons, Neoprene is preferred to
natural rubber for pumping oil (tar) sand slurries, oil shales, pulp and paper, and phosphate
and in moly circuits when oils are used as flotation reagents. A special curing
process is applied when molding equipment parts out of Neoprene to give it excellent resistance
to water absorption and swelling, and suitable thermal resistance for use up to a
temperature of 120°C (248°F) with a nominal hardness of 60 Shore A.
Neoprene is affected or attacked by strong oxidizing acids, acetic acid, ketones, esters,
and chlorinated and nitro hydrocarbons.
TABLE 10-14 Properties of Jade Green Aramound
Durometer (Shore A) Hardness 60 ± 5
Tensile strength (MPa) 22
Tensile strength (psi) 3190
Elongation (%) 700
Tear resistance Excellent
Abrasion resistance Excellent
Impact resistance Excellent
Flame resistance Poor
Ozone resistance Poor
Weather resistance Average
Typical temperature range –50°C to 66°C
–60°F to 150°F

10-5-2 Ethylene Propylene Terpolymer (EPDM)
EPDM has a generally acceptable resistance to most moderate chemicals, alcohol, ozone,
and organic acids. It is, however, attacked by strong acids, solvents, most hydrocarbons,
chloroform, and aromatic solvents. The minimum temperature for use is –51°C (–60°F)
and its maximum temperature is 177°C or 350°F. EPDM offers better abrasion resistance
than butyl rubber. It is also used as a high-temperature gasket. Mechanical properties are
presented in Table 10-13.
10-5-3 Jade Green Armabond
This product is highly resilient and abrasion resistant. It is a used as a soft facing and protective
lining for material handling equipment such as chutes, launders, ball mills, and cement
mixers. Goodyear manufactures this material. Its mechanical properties are presented
in Table 10-14.
10-5-4 Armadillo
Armadillo is a synthetic rubber produced by Goodyear. It is highly resistant to abrasion. It
is used to line chutes, launders, and troughs and provides excellent noise abatument. The
material may be supplied with a heavy nylon backing to be self-supporting and bridge
gaps in chutes and launders. It has a nominal hardness 60 shore A.
10-5-5 Nitrile
MATERIALS SCIENCE FOR SLURRY SYSTEMS
10.15
Nitrile offers moderate resistance to abrasion with excellent ability to handle oils and hydrocarbons.
A special food grade is available for the food and beverage industry. The
properties of nitrile are presented in Table 10-15. Nitrile may be reinforced with heavy
nylon fabric for high-pressure hoses. Nitrile is used for seals, flingers, and screens. It is
attacked by ozone, ketones, esters, aldehydes, and chlorinated and nitro-hydrocarbons.
TABLE 10-15 Properties of Nitrile—Food Grade
Durometer (Shore A) Hardness 60 ± 5
Tensile strength (MPa) 5
Tensile strength (psi) 700
Elongation (%) 450
Tear resistance Good
Abrasion resistance Good
Impact resistance Fair
Flame resistance Poor
Gas impermeability Good
Typical temperature range –35°C to 121°C
–30°F to 250°F

10.16 CHAPTER TEN
TABLE 10-16 Properties of Hypalon—Food Grade
Durometer (Shore A) Hardness 60 ± 5
Tensile strength (MPa) 10
Tensile strength (psi) 1400
Elongation (%) 500
Tear resistance Average
Abrasion resistance Good
Impact resistance Good
Flame resistance Good
Gas impermeability Good
Heat aging Good
Weather resistance Excellent
Ozone resistance Excellent
TABLE 10-17 Properties of Thermoset Polyurethane
Casting resins
50–60% Mineral filled
potting and casting
Properties Liquid Unsaturated compounds
Processing temperature 85–121°C 121°C–250°F
185–250°F (casting)
Molding pressure 0.7–35 MPa
100–5000 psi
Mold (Linear) shrinkage (in/in) 0.020 0.001–0.002
Tensile strength at break 1.2–69 MPa 69–76 MPa 6.9–48.3 MPa
0.175–10 ksi 10–11 ksi 1–7 ksi
Elongation at break (%)
Compressive strength
100–1000 3–6 5–55
(yield/rupture) 138 MPa
20,000 psi
Flexural strength 4.8–31 MPa 131 MPa
(yield/rupture) 700–4,500 psi 19,000 psi
Tensile modulus 69–689 kPa
10–100 ksi
Compressive modulus 69–689 kPa
10–100 ksi
Flexural modulus at 23°C, 73°F 69–689 kPa 4200 kPa
10–100 ksi 610
Hardness Shore A10 Barcol 30-35 Shore A90
D 90 D 52-85
Specific gravity 1.03–1.5 1.05 1.37–2.1
Water absorption over 24 hr in
a 3.2 mm (1/8 in) specimen
0.2–1.5% 0.1–0.2% 1.37–2.1%
Data from Harper, 1992.

10-5-6 Carboxylic Nitrile
Carboxylic nitrile combines excellent oil resistance with abrasion resistance. It is available
in nominal hardness of 80 Shore A. It exhibits a high tensile modulus, low elongation,
improved hot tear and tensile resistance, and better resistance to hot oil and air aging
than most nitriles.
10-5-7 Hypalon
MATERIALS SCIENCE FOR SLURRY SYSTEMS
10.17
Hypalon is a tradename for CSM chlorosulfonated polyethylene. It offers good resistance
to moderate chemicals, ozone, alkaline solutions, hydrogen, Freon, alcohols, aliphatic hydrocarbons,
as well as ultraviolet degradation from sunrays. Strong oxidizing acids, ketones,
esters, acetic acid, and chlorinated and nitro-hydrocarbons attack Hypalon. Its temperature
range for applications is from –40°C to 150°C (–40°F to 300°F). Its physical
properties are presented in Table 10-16.
TABLE 10-18 Properties of Thermoplastic Polyurethane
Unreinforced
10–20%
glass-fiber-reinforced Long glass-reinforced
Properties molding compound molding compounds molding compound
Melting temperature 24–58°C
75–137°F
75°F
Processing temperature 221–266°C 182–210°C 182–232°F
430–510°F 360–410°F 360–450°F
Molding pressure 55–76 MPa
8–11 ksi
Tensile strength at break 50–62 MPa 69–76 MPa 186–227 MPa
7.2–9 ksi 4.8– 7.5 ksi 27–33 ksi
Elongation at break (%) 60–180 3–70 2
Tensile yield strength, psi 53.7–76 MPa 186–227 MPa
7.8–11.0 ksi 27–33 ksi
Compressive strength 35 MPa
(yield/rupture) 5000 psi
Flexural strength 4.8–31 MPa 11.7–42 MPa 310–393 MPa
(yield/rupture) 10.2–15 ksi 1.7–6.2 ksi 45–57 ksi
Tensile modulus 1.31–2.07 GPa 4.1–9.6 MPa 11.7–17.2 GPa
190–300 ksi 0.6–1.40 ksi 1700–2500 ksi
Flexural modulus at 1.62–2.14 GPa 0.28–0.62 GPa 10.3–15.16 GPa
23°C, 73°F 235–310 ksi 40–90 ksi 1500–2200 ksi
Hardness R > 100, M48 R 45-55
Specific gravity 1.2 1.22–1.38
Water absorption over 24 0.17–0.19% 0.4–0.55%
hr in a 3.2 mm (1/8 in)
specimen
Data from Harper, 1992.

10.18 CHAPTER TEN
10-5-8 Fluoro-elastomer (Viton)
This elastomer offers exceptional resistance to oils and many chemicals at elevated temperature,
but limited resistance to erosion wear.
10-5-9 Polyurethane
Polyurethane is a castable synthetic rubber. It is elected for applications where tramp is a
problem. It offers excellent resistance to high tear and high tensile strength. Its resistance
to erosion is, however, lower than natural rubber.
Polyurethane pump liners and pump parts are selected for fine slurries with an average
particle diameter of (minus 230 (m, minus 65 mesh). Impellers may operate up to a tip
speed of 31 m/s (6100 ft/sec). Polyurethane lined pumps are used for flue gas desulfurization
applications as well as for pumping fairly fine tailings from copper and gold plants
and for moly and flotation circuits where oils are used as reagents. The upper limit for use
of polyurethane is 82°C. Nominal hardness is 80 Shore A.
Polyurethane is subject to hydrolic degradation at its higher temperature limits.
Polyurethane is machinable. It does not seal as well as rubber, and additional O-rings and
gaskets are needed when switching from rubber to polyurethane lined parts. Polyurethane
is available in thermoset and thermoplastic forms. A new generation of glass-reinforced
polyurethane thermoplastic materials are available from different manufacturers. The
properties of polyurethane are presented in Tables 10-17 and 10-18.
10-6 WEAR DUE TO SLURRIES
A predominant factor in wear life and the selection of the proper material for the manufacture
of slurry handling equipment and piping is the abrasiveness of slurries. This is often
related to the hardness of the particles, their degree of roundness, their concentration,
the angle of impingement, and corrosion. Wilson (1985) plotted the hardness of different
materials on a scale (Figure 10-2) and developed a chart for the selection of materials for
the construction of centrifugal pumps (Table 10-19). Metals with high elastic limits are
used to absorb abrasion associated with impact. Hard metals are particularly brittle and
are selected where there is low impact but relatively hard particles flowing in parallel to
the surface of the equipment.
The rate of wear or mass loss is a function of the speed of the flow raised to the power
of 2.5 to 4.0 (Wilson, 1985). Wear is also accentuated when the solids are trapped between
a rotating and a fixed surface. This occurs, for example, between the back wear
plate and the impeller (Figure 10-4).
The American Society for Testing of Materials (Miller, 1987) has adopted the concept
of the Miller number as a measure of the abrasivity of slurries. Standard G-75 describes
the relative abrasivity based on the mass loss of a standard 27% chrome white
iron block when rubbed in slurry for a particular period of time. To conduct the test and
to measure the slurry abrasion response (SAR) number, also called Miller number, a
special machine (Figure 10-5) is used. It consists of a wear block driven at 48 strokes
per minute with a stroke length of 200 mm (8 in). A dead weight of 2.27 kg [5 lb] is
applied to the wear block. At the bottom of the tray, a piece of Neoprene is installed to
act as a lap. The tray takes the form of a flat bottom between V-shaped walls that confine
the slurry around the block. The tray is filled with sand. The mounted blocks are
lowered in the trays. Four, 4-hour tests are conduced and the weight of the block is

MATERIALS SCIENCE FOR SLURRY SYSTEMS
TABLE 10-19 Classification of Pumps According to Solid Particle Size
a Theoretical value. From Pump Handbook, reprinted by permission of McGraw-Hill.
10.19

10.20 CHAPTER TEN
FIGURE 10-4 Wear for the wear plate of a concentrate charge pump.
measured after each run to obtain the mass loss as a form of wear. The data is plotted
to obtain the following correlation between mass loss and time:
mass loss = A·tB where
t = time
A = coefficient
B = power number
The Miller is determined at the end of the first 2 hours of the test as
Miller number = 18.18 [A·B· 2B–1 ] (10-6)
The Miller number is directly related to the abrasivity of the slurry. In other words, slurry
FIGURE 10-5 Machine for measuring the Miller number. (From Miller, 1987. Reproduced
by permission of ASTM.)

with a Miller number of 150 will be twice as destructive as slurry with a Miller number of
75. Examples of Miller numbers are presented in Table 10-20. Typically, slurries with a
Miller number smaller than 50 are considered to cause mild wear. The Miller number is
also a function of the weight concentration (Figure 10-6).
The ASTM has developed a new standard (G75-01) to determine the slurry abrasivity
(Miller) number and the slurry response of materials (SAR number).
10-7 CONCLUSION
MATERIALS SCIENCE FOR SLURRY SYSTEMS
TABLE 10-20 Miller Number of Certain Slurries
Material Miller number
Alundum, 400 mesh 241
Alundum, 200 mesh 1058
Ash 127
Fly ash 83, 14
Bauxite 9, 33, 50, 134
Calcium carbonate 14
Carbon 16
Carborundum, 220 mesh 1284
Clay 36
Coal 6, 10, 21, 28, 57
Copper concentrate 19, 37, 68, 128
Dust, blast furnace 57
Gypsum 41
Iron ore (or concentrate) 28, 64, 122, 234
Kaolin 7, 30
Limestone 22, 39, 46
Limonite 113
Magnetite 64, 71, 134
Mud, drilling 10
Phosphate 68, 74, 84, 134
Pyrite 194
Sand, silica 51, 68, 116, 246
Sewage (digested) 15
Sewage (raw) 25
Sulfur 1
Tailings (all types) 24, 91, 217, 644
Data from Miller, 1987.
10.21
A variety of materials are available for the lining and manufacture of slurry handling
equipment and piping. Various selections are available based on the degree of abrasiveness,
corrosion, temperature, and presence of oils or solvents. In terms of measurement of
wear, the American Society for Testing of Materials has developed a slurry abrasion response
(SAR) number, also called Miller number. It is extremely useful for determining
the need to line pipelines and pumps.

10.22 CHAPTER TEN
FIGURE 10-6 The Miller number as a function of the weight concentration of solids. This
example uses 70 mesh sand. (From Miller, 1987. Reproducted by permission of ASTM.)
REFERENCES
ASTM. 1999. ASTM A532/A532M. Standard Specification for Abrasion-Resistant Cast Irons.
ASTM. 2001. Test Method G75-01. Standard Test Method for Determination of Slurry Abrasivity
(Miller Number) and Slurry Response of Materials (SAR Number).
BSI. 1986—withdrawn. Specification for Abrasion-Resisting White Cast iron.
Brown, J. R. 1994. FOSECO Foundryman’s Handbook. London: Butterworth Heinemann.
Farag, M. M. 1997. Minerals Selection for Engineering Design. Upper Saddle River, NJ: Prentice
Hall.
Harper, A. C. 1992. Handbook of Plastics, Elastomers, and Composites. New York: McGraw-Hill.
Juvinall, R. C. 1983. Fundamentals of Machine Component Design. New York: Wiley.
Miller, J. E. 1987. Slurry Abrasivity Determination. The American Society for Testing of Materials
Standardization News, July.
Wilson, G. 1985. Construction of Solids-handling Centrifugal Pumps. Chapter 9-17-2 in The Pump
Handbook, I. J. Karassik, W. C. Krutzch, W. H. Fraser, and J. P. Messina Eds. New York:
McGraw-Hill.
Zakharov, B. 1962. Heat Treatment of Metals. Moscow: Peace Publishers.
Further readings. The following technical bulletins may be obtained directly from Weir Slurry
Group for further readings on white chrome irons:
Dolman, K. F. 1992. Ultrachrome—Corrosion Resistant Alloys. Warman Technical Bulletin No. 19,
Sydney, Australia.
Huggett, P. G. and Walker, C. I. 1992. White Iron Microstructure and Wear. Warman Technical Bulletin
No 18, Sydney, Australia.
Walker, C. I. 1990. Hyperchrome White Irons. Warman Technical Bulletin No 4, Sydney, Australia.

CHAPTER 2
FUNDAMENTALS OF
WATER FLOWS IN PIPES
2-0 INTRODUCTION
The mechanics of pipe flow is a topic well dealt with in the scientific literature. In this
chapter, some of the concepts are reviewed in a simplified manner as they relate to slurry
flows.
The friction factor for single phase flows is presented in terms of boundary layer theory.
Losses in pipes are summarized in terms of fittings and conduits commonly used on
large engineering projects. Numerous books have been written for single-phase flows.
This chapter limits itself to a brief introduction.
The equations in this chapter are based on SI units for consistency. They can be readily
used in USCS (United States Customary System) units if the reader assumes the use of
slugs and not pounds for mass units. There is considerable confusion about the use of
slugs, which are units of mass, and pounds, which are units of force. The conversion factor
between these is often called g c and is equal to 32.2 ft/sec. Many other references input
g c in their equations to achieve such a conversion, but it was not deemed necessary in
this book. The worked examples show how slugs should be used as a unit of mass.
Certain models of Newtonian slurry flows are attempts to apply correction factors to
the friction losses of the carrier liquid on the basis of the volumetric concentration of
solids. The reader is encouraged to examine this chapter before proceeding with more
complex themes.
2-1 SHEAR STRESS OF LIQUID FLOWS
Modern fluid mechanics is based on the concept of a controlled volume. Mass momentum
and energy must be conserved when a particle enters and leaves the volume.
Considering flow through a section of pipe of a constant diameter between two locations
1 and 2 as in Figure 2-1, the hydraulics force associated with the drop of pressure
is
�
F12 = � 2 D i (P1 – P2) (2-1)
4
2.1

2.2 CHAPTER TWO
This force is balanced by the friction force F r
F r = � w�D iL (2-2)
where L is the distance between points 1 and 2 and � w is the wall shear stress
or
P
1
�
� 4
D i 2 (P1 – P 2) – � w�DL = 0
Di(P1 – P2) Ri(P1 – P2) �w = �� = ��
(2-3)
4L 2L
The shear stress at any radius and from the center of the pipe is
Ri�P r
� = � = �w �
2L R
where
L = length
Ri = the pipe inner radius (at the inside wall of the pipe)
r = local radius
The shear stress � is calculated. At the center of the pipe there is no shear stress.
L
Example 2-1
Homogeneous slurry is tested in a pipe with an inner diameter of 53 mm (2.086 in). The
pressure drop due to friction is measured between two points (A and B), which are separated
by a distance of 1.8 m (5.9 ft). The pressure drop is recorded as 3000 Pa (0.435 psi).
To appreciate the shear stress distribution from the wall to the center of the pipe, determine
the shear stress distribution at the wall and at three points: at a radius of 20 mm
(0.787�), at a radius of 12 mm (0.472�), and at the center of the pipe.
Solution
This problem will be solved in SI units [Système International (metric)] and in USCS
units.
w
P
2
FIGURE 2-1 Shear stress and pressure for flow in a pipe.

Solution in SI Units
Using Equation 2-3, the shear stress at the wall is
0.053(3000)
�w = �� = 22 Pa
4 × 1.8
since the inner radius of the pipe is 26.5 mm.
At a radius of 20 mm the shear stress is
r 20
� = �w� � = 22� �
At a radius of 12 mm the shear stress is
r 12
� = �w� � = 22� �
At the center of the pipe the shear stress is
r 0
� = �w� � = 22� �
= 16.6 Pa
= 10 Pa
� = 0 Pa
26.5
Solution in USCS Units
Using Equation 2-3, the shear stress at the wall is
2.086 in (0.435 psi)
�w = ��� = 0.0032 psi
4 × 5.9 × 12
since the inner radius of the pipe is 1.043 in.
At a radius of 20 mm (0.787 in) the shear stress is
r
0.787
� = �w� � = 0.0032� �
� = 0.0024 psi
1.043
At a radius of 12 mm (0.472 in) the shear stress is
0.472
� = 0.0032��� = 0.00145 psi
1.043
At the center of the pipe � = 0
FUNDAMENTALS OF WATER FLOWS IN PIPES
2-2 REYNOLDS NUMBER AND
FLOW REGIMES
� Ri
� Ri
� Ri
� Ri
� 26.5
� 26.5
Determining the magnitude of friction was historically a controversial topic until the end
of the 19th century. The great disagreement was between the practical engineers and the
theory of hydrodynamics.
Hager (in 1839) and Poiseville (in 1840) demonstrated that under certain conditions
friction was a linear function of the speed of flow. In 1858, Darcy demonstrated that under
other conditions friction was in fact proportional to the square of the mean speed of the
flow. By 1883, Reynolds had demonstrated that both Poiseville and Darcy were correct, as
the mechanics of flows were fundamentally different at very low speeds and at high speeds.
2.3

2.4 CHAPTER TWO
Through nondimensional analysis, Reynolds demonstrated that under certain fixed
conditions, the transition from a laminar Poiseville flow to a turbulent Darcy flow was
based on the ratio of the inertia forces to the viscous forces. In his honor, such a ratio is
now called the Reynolds number:
Re = = = (2-4)
where
� = density of the fluid
� = absolute or “dynamic” viscosity
� = kinematic viscosity (defined as the absolute viscosity divided by the density of the
liquid)
U = average velocity of the flow
The kinematic viscosity is the absolute (or dynamic viscosity) divided by the density.
Its unit of measurement are, strictly speaking, m2 /s or ft2 Inertia forces �UDi UDi �� � �
Viscous forces � �
/sec. Another unit used is the
centistokes, which is obtained by dividing the absolute viscosity in centipoises by the specific
gravity of the fluid. A centipoise is equivalent to one milli-Pascal-second.
One unit for kinematic viscosity used in the oil industry (but of limit use in the mining
industry) is the seconds Saybolt universal (or SSU). For values of kinematic viscosity
larger than 70 centistokes (cst), the following formula is recommended by the Hydraulic
Institute (1990):
SSU = centistokes × 4.635
In simplified terms, it may be said that two geometrically similar bodies immersed in a
fluid will develop inertia and viscous forces in a constant ratio when body forces are negligible.
Since Reynolds developed his theory, his approach has been has been extended to other
fluids. Modern aerodynamics uses the chord of the wing aerofoil instead of the pipe diameter
as the distance parameter for Equation 2-4. In Chapter 3, the concept of the particle
Reynolds number based on a characteristic particle diameter shall be introduced.
Figure 2.2 presents the equations of the Reynolds number for different shapes and flows.
For pipe flows, the critical Reynolds number is considered to be between 2300 and
2800. In many pipes, flow becomes unstable above a Reynolds number of 2300 and slides
into a transition regime before converting into turbulent motion.
2-3 FRICTION FACTORS
The Fanning friction factor is a nondimensional number defined as the ratio of the wall
shear stress to the dynamic pressure of the flow:
fN = � (2-5)
2 �U /2
Users of USCS units should use slugs/ft3 for density and not the more commonly used
units of pounds per cubic feet. The conversion between these two is gc or 32.2 ft/sec2 .
Substituting Equation 2-3 into Equation 2-5
�U 2
�PDI � 4L
� w
� 2
f N = � � /� � (2-6)

c
Airfoil chord "c"
Re = U c /
Annular flow
Re= ( U/ )*2 r + r - r
2 2 2
0 i m
2 2
0 i 10 0 I
where r = (r - r )/2.3 log (r /r )
m
FUNDAMENTALS OF WATER FLOWS IN PIPES
Close parallel plates
Equation 2-6 clearly indicates that the friction factor is dependent on the flow. It will
be demonstrated that there is a relationship between the friction factor and the Reynolds
number of the flow. In USCS units, density is expressed in slugs/ft 3 .
Example 2-2
The slurry in Example 2-1 has a specific gravity of 1.2 , or a density of 1200 kg/m3 . If the
speed of the flow is 2 m/s (6.56 ft/s), determine the Fanning friction factor.
Solution in SI Units
Using the shear stress calculated in Example 2-1 as 22 Pa and inserting it into Equation 2-
5, the Fanning friction factor can be calculated as
22
fN = �� = 0.0092
2 1200 × 2 × 0.5
Solutions in USCS Units
Assume the density of water to be 62.3 lbm/ft3 . Since specific gravity equals 1.2, the density
of the slurry is 1.2 × 62.3 = 74.76 lbs/ft3 . To convert lbs/ft3 , into slugs/ft3 complete
the following equation:
= 2.32 slugs/ft3 74.76
�
32.2
In Example 2-1, the wall shear stress was determined to be 0.0032 psi. Using equation
2.5
0.0032 × 144
fN = �� = 0.0092
2 2.32 × 6.56 × 0.5
h
2.5
Use inner diameter for sphere
for pipe flow use diameter
Re= U D /
i
Re = [U h/2 ] 32/3
FIGURE 2-2 Definition of Reynolds number for various shapes.
D I
Re= U d /
p
d p

2.6 CHAPTER TWO
2-3-1 Laminar Friction Factors
The friction or resistance of a body to motion is confined to a viscous layer at the wall or
surface of the body. This layer is called the boundary layer. Understanding this phenomenon
remained illusive until the end of the 19th century. At the turn of the 20th century,
Prandtl developed the principles of boundary layer theory.
To understand the basic principles of the boundary layer, let us start by examining the
flow between closely related plates at low speed. With one plate stationary and the other
moving at a speed U, a linear velocity profile is established.
The wall shear stress �w is established as
�U
�w = �
h
by Newton’s law, where h is the spacing between plates and
U
u = y� �
� (2-7)
h
With y as the vertical ordinate from the stationary plate, the velocity gradient is defined as
du U
� = � = rate of shearing strain or shear rate (2-8)
d� H
Thus, the dynamic viscosity is
� Shear Stress
� = � = ���
(2-9)
du/d� Rate of Shear Strain
Equation 2-8 is the basis for boundary layer theory. Instead of a moving plate at a velocity
U, the velocity of the flow outside the boundary layer is studied (see Figure 2-3).
The shear rate is not necessarily linear. For example flow around an aircraft airfoil can
be attached, stagnant at a point, or can even reverse flow after separation.
Laminar flow in a pipe is described by the Hagen–Poiseville equation:
�P 32�U
� = � (2-10)
2 L D i
Upper plate moves at speed U
h u=U
y
y
h w
FIGURE 2-3 Linear velocity distribution due to a plate moving at a speed U above a stationary
plate.

which can be rearranged as
FUNDAMENTALS OF WATER FLOWS IN PIPES
� = � �� � = 2 �P D i Di�P Di � � ������ (2-11)
L 32U 4L 8U
� = � (2-12)
8U/Di
or the ratio of the wall shear stress and the mean velocity gradient.
The term (8U/Di) is often called the pipe (or pipeline) shear rate in laminar flow. From
Equation 2-12
�8U
�w = (2-13a)
� Di
Substituting Equation 2-12 into Equation 2-5, the Fanning friction factor for a laminar
flow can be expressed as
�U
fN = � �/� � (2-14)
2
�8U
� �
Di 2
16� 16
fN = � = � (2-15)
�VDi Re
The Fanning friction factor is more commonly found in reference publications on
chemical engineering. Another friction factor used by mechanical engineers is the Darcy
friction factor:
f Darcy = 4 × f Fanning
To avoid confusion, in the book the symbols f D will be used for Darcy friction factor
and f N will be used for Fanning friction factor.
Flow in a laminar regime is considered to be independent of pipe roughness.
Example 2-3
A viscous fluid is flowing in a laminar flow at a speed of 1m/s (or 3.28 ft/s) in a pipe
with an inner diameter of 336.6 mm (13.25 in). The measured pressure drop over a distance
of 200 m (656 ft) is 8400 Pa (1.22 psi). The density of the fluid is 855 kg/m 3 (SG
= 0.855).
Determine an equivalent viscosity for the pipeline fluid, the Reynolds number, and the
friction factor.
Solution in SI Units
From Equation 2-3, the shear stress at the wall is
0.3366 × 8400
�w = �� = 3.53 Pa
4 × 200
The Fanning Friction Factor from Equation 2.5 is
2�w 2 × 3.53
fN = � = � = 0.0083
2
2
�V 855 × 1
� w
2.7

2.8 CHAPTER TWO
The equivalent pipeline viscosity from Equation 2-12 is
�w 3.53 × 0.3366
� = � = �� = 0.148 Pa-s
8U/D 8 × 1
The Reynolds Number from Equation 2-4
�UD 855 × 1 × 0.3366
Re = � = �� = 1938
� 0.148
Check on friction factor:
16
fN = � = 0.00823
1938
Solution in USCS Units
density = = 1.654 slugs/ft3 Since it was stated that the pressure drop = 1.215 psi, over a distance of 656.2 ft the
shear stress is:
�w = = 0.0005 psi
The Fanning Friction Factor from Equation 2.5 is
fN = = = 0.0081
The equivalent pipeline viscosity converting the shear stress from psi to lbf/ft2 , 0.0005
× 144 = 0.072 lbf/ft2 , is
� = = = 0.003 lbf-sec/ft2 0.8555 × 62.3
��
32.2
13.25 × 1.2
��
4 × 656.2 × 12
2�w 2 × 0.0005 × 144
� ��
2
2
�V 1.654 × 3.28
Di�w (13.25/12) × 0.072
� ��
8V 8 × 3.28
The Reynolds number is
1.654 × 3.28 × 13.25/12
Re = ��� = 1997
0.003
Check on friction factor:
16
fN = � = 0.008
1997
2-3-2 Transition Flow Friction Factor
The transition from a laminar to a turbulent flow is difficult to describe.
For Reynolds numbers up to 3 × 106 , Wasp et al. (1977) recommended the use of Nikuradse
equation for the Fanning coefficient:
0.0553
fN = 0.0008 + � (2-16)
0.237 Re

FUNDAMENTALS OF WATER FLOWS IN PIPES
2-3-3 Friction Factor in Turbulent Flow
In turbulent flow, the roughness of the pipe becomes an important factor in determining
the friction factor. The roughness � is measured in units of length. Up to a certain limiting
Reynolds number Re, the friction factor can be expressed by the following Colebrook
equation:
= 4 loge� � + 3.48 – 4 loge�1 + 9.35 � (2-17)
A more simplified equation for the Darcy factor is called the Prandtl–Colebrook equation:
= –2 log10� + � (2-18)
A popular equation used by mechanical engineers (Lindeburg, 1997) as it is explicit
and does not require tedious iterations is the Swamee–Jain equation. It is suitable for the
range of Reynolds numbers between 5000 and 100,000,000:
0.25
fD = ���
(2-19)
��� log10��� / D
� + (5.74/Re
3.
7
0.9 )�� 2
1
Di Di � � ��
�fN� 2�
2� Re�fN�
1
2.51 �
� � �
�fD�
Re�fN� 3.7 Di
Because the slurry flows occur at Reynolds Numbers smaller than 100,000,000, Equation
2.19 is satisfactory in the context of this handbook.
Example 2-4
Using the Swamee–Jain equation, determine the friction factor for a flow of 3500 US gpm
in an 18� OD pipe with a wall thickness of 0.375 in. The fluid has a specific gravity of
1.02 and a dynamic viscosity of 2.7 × 10 –5 lbf-sec/ft 2 .
Solutions in SI Units (For conversion factors refer to the Appendix at the end of this
book.)
Q = = 0.221 m3 3500 × 3.785
�� /s
60000
ID of pipe = (18 – 2 × 0.375) = 17.25 in (0.438 m)
Area of flow = � × 0.25 × 0.438 2 = 0.1506 m 2
0.221
Average velocity of flow = � = 1.467m/s
0.1506
Density = 1.02 × 1000 = 1020 kg/m 3
� = 2.7 × 10 –5 × 47.88 = 0.00129 Pa.s
1020 × 1.467 × 0.438
Re = ��� = 506975
0.00129
2.9

2.10 CHAPTER TWO
The absolute roughness � of steel pipes is 6 × 10 –5 m. Relative roughness is
6 × 10
�/Di = = 0.000137
–5
�
0.438
fD = = 0.01486
Solutions in USCS Units
Q = 3500 × 0.1337 = 467.95 ft3 0.25
�����
[log10(0.000137/3.7 + 5.74/506975
/min
0.9 )] 2
ID of pipe = (18 – 2 × 0.375) = 17.25 in or 1.4375 ft
Area of flow = � × 0.25 × 1.4375 2 = 1.623 ft 2
467.95
Velocity of flow = � = 255.3 ft/min or 4.80 ft/s
1.623
Density = = 1.973 slugs/ft3 1.02 × 62.3
��
32.2
1.973 × 4.80 × 1.4375
Re = ��� = 504,779
2.7 × 10
The absolute roughness of steel is 0.0002 ft. Relative roughness is
–5
0.0002
� 1.4375
= 0.000139
fD = 0.25/[log10(0.000139/3.7 + 5.74/5047790.9 )] 2 = 0.0149
The Colebrook and Prandtl–Colebrook formulas are limited to a certain range of
Reynolds number magnitude. At high Reynolds numbers, the friction factor becomes independent
of the Reynolds number. The value of the Reynolds number beyond which the
friction factor is independent is calculated using the following equation:
Re = 70 �� = 70 Di 2 Di 8
� � � �
� fN � ��fD
The region for which equation 2-19 applies is shown on the Moody diagram to be to
the right of the the dashed curve (Figure 2-4).
The equations established so far have been developed for clear water and do not apply
for plastic fluids or liquids carrying coarse particles. They can apply for any other singlephase
Newtonian liquids. (These terms will be explained in Chapter 3).
Most fluid dynamics books publish data for linear roughness based on Moody’s work.
Such values are applicable to water. However, tests conduced on slurry pipelines can
yield different values, due to the erosion of pipe, wear and tear of rubber linings, etc.
The Moody diagram is a general graph for the Darcy factor versus the Reynolds number.
It is applicable to a very large number of different pipe materials. There are four principal
pipe materials associated with slurry flows: plastic pipes [high-density polyethylene
(HDPE)], plain steel pipes, rubber-lined steel pipes, and concrete pipes. The absolute

2.11
FIGURE 2-4 Moody diagram for the friction factor versus the Reynolds number for pipe flow (Reproduced from V. L. Streeter, Fluid
Mechanics, McGraw-Hill, 1971. Reproduced by permission of McGraw-Hill, Inc.)

2.12 CHAPTER TWO
roughness of these materials is presented in Table 2-1. Plain steel pipes and rubber-lined
steel pipes are the most common, but HDPE and HDPE-lined steel pipes have gained in
importance in the last quarter of the 20th century. One of the concentrate pipelines used in
Escondida, Chile featured a long section of gravity flow in HDPE pipe.
Certain reference books show a roughness of steel of 0.045–0.05 mm. This is difficult
to maintain in steel pipes carrying slurries as they are often subject to erosion and corrosion.
For this reason, the author recommends the use of a slightly higher roughness of the
order of 0.06 mm in friction calculations.
The dimensions of plain steel pipes, their pressure ratings, and relative roughness are
presented in Table 2-2. It is obvious that steel pipes are limited in pressure rating to
3000 psi. This criterion is essential when considering location of booster pump stations
or chokes. In the case of slurry pipelines, the thickness of the pipes is selected on the
basis of
� Pressure
� Corrosion allowance
� Wear allowance
Because of the wear allowance, erosion, and corrosion, the rating of slurry pipes is
lower than presented in Table 2-2. Steel pipes may be hardened for carrying coarse slurry
particles (larger than 6.5 mm or 1 – 4�), or sacrificial thickness is used to resist abrasion.
The pressure rating as presented in Table 2-2 should be used as a starting point for the
design calculations, and the appropriate allowance should be made for wear. Reclaimed
water pipelines use the pressure ratings as in Table 2-2. The roughness and inner diameter
of reclaimed water pipes may change due to scaling and deposition of lime.
Steel pipes are rubber lined to a typical thickness of 6 mm (or 0.25�) for small sizes of
pipes [< 150 mm (6�), 9.5 mm ( 3 – 8�)], and 13 mm ( 1 – 2�) for pipe sizes up to 24�. Larger pipes
may be custom lined. Lining is done in an autoclave and the rubber is cured under steam.
Rubber lining is limited to pumping coarse material up to a size of 6 mm (� 1 – 4�). Rubber
does not contribute to the pressure rating of steel pipes. Table 2-3 presents the dimensions
and relative roughness of rubber-lined steel pipes.
The dimensions of plain HDPE pipes (not HDPE-lined steel) for pressures up to 110
psi (760 kPa) are listed in Table 2-4. The dimensions of HDPE pipes for pressures in the
range of 125 to 300 psi (863–2070 kPa) are presented in Table 2-5. These dimensions are
slightly different than metric pipes. HDPE is not a magic material but can withstand the
abrasion of taconite and some coarse laterites. As with rubber, there must be a cut-off size
of particle size beyond which the use of HDPE is not acceptable. Very little has been published
on this subject.
The use of concrete pipes is often associated with gravity flows.
TABLE 2-1 Absolute Roughness of New Materials Used in Slurry Pipes
Description Roughness (m) Roughness (ft)
Plastic pipes, PVC, ABS, HDPE 1.5 × 10 –6 0.000004921
Steel pipes 6.0 × 10 –5 0.000197
Rubber-lined pipes 0.00015 0.000492
Concrete pipes 0.0012 0.00394

FUNDAMENTALS OF WATER FLOWS IN PIPES
TABLE 2-2 Size, Rating, and Relative Roughness of Plain Steel Pipes to
U.S. Dimensions*
2.13
Pipe Allowed Relative
Outside Wall internal working roughness
Denomination diameter thickness diameter pressure (psi) for new
(inch) (inch) (inch) (inch) to 650 °F pipe
2� Sch 40 2.375 0.154 2.067 1159 0.001143
2� Sch 80 0.218 1.939 2038 0.001212
2� Sch 160 0.344 1.687 3890 0.001400
XX 0.436 1.503 5356 0.001572
3� Sch 40 3.500 0.216 3.068 1341 0.000770
3� Sch 80 0.300 2.900 2129 0.000815
3� Sch 160 0.438 2.624 3495 0.000900
XX 0.600 2.300 5252 0.001027
4� Sch 40 4.500 0.237 4.026 1191 0.000587
4� Sch 80 0.337 3.826 1905 0.000617
4� Sch 120 0.438 3.624 2663 0.000652
4� Sch 160 0.531 3.438 3387 0.000687
XX 0.674 3.152 4553 0.000749
5� Sch 40 5.633 0.257 5.117 1071 0.000461
5� Sch 80 0.375 4.883 1950 0.000484
5� Sch 120 0.500 4.633 2502 0.000510
5� Sch 160 0.625 4.383 3284 0.000539
XX 0.750 4.133 4098 0.000572
6� Sch 40 6.625 0.280 6.065 1000 0.000389
6� Sch 80 0.432 5.761 1739 0.000410
6� Sch 120 0.562 5.501 2394 0.000429
6� Sch 160 0.719 5.187 3215 0.000455
XX 0.864 4.897 4004 0.000482
8� Sch 20 8.625 0.250 8.125 655 0.000291
8� Sch 30 0.277 8.071 752 0.000293
8� Sch 40 0.322 7.981 916 0.000296
8� Sch 60 0.406 7.813 1225 0.000302
8� Sch 80 0.500 7.625 1577 0.000310
8� Sch 100 0.594 7.437 1935 0.000318
8� Sch 120 0.719 7.187 2422 0.000329
8� Sch 140 0.812 7.001 2792 0.000337
XX 0.875 6.875 3046 0.000344
8� Sch 160 0.906 6.813 3173 0.000347
10� Sch 20 10.75 0.250 10.250 523 0.000230
10� Sch 30 0.307 10.136 688 0.000233
10� Sch 40 S 0.365 10.020 856 0.000236
10� Sch 60 X 0.500 9.750 1255 0.000243
10� Sch 80 0.594 9.562 1537 0.000247
10� Sch 100 0.719 9.312 1918 0.000254
10� Sch 120 0.844 9.062 2308 0.000261
10� Sch 140 XX 1.000 8.750 2804 0.000270
10� Sch 160 1.125 8.500 3211 0.000278
(continued)

2.14 CHAPTER TWO
TABLE 2-2 Continued
Pipe Allowed Relative
Outside Wall internal working roughness
Denomination diameter thickness diameter pressure (psi) for new
(inch) (inch) (inch) (inch) to 650 °F pipe
12� S 12.750 0.250 12.250 440 0.000193
12� Sch 40 0.330 12.090 634 0.000195
12� X 0.375 12.000 744 0.000197
12� Sch 60 X 0.406 11.938 820 0.000198
12� Sch 80 0.500 11.750 1052 0.000201
12� Sch 100 0.562 11.626 1207 0.000203
12� Sch 120 XX 0.688 11.374 1526 0.000208
12� Sch 140 0.844 11.062 1927 0.000214
12� Sch 160 1.00 10.750 2337 0.000220
1.125 10.500 2672 0.000225
1.312 10.126 3183 0.000233
14� Sch 10 14.000 0.250 13.500 401 0.000175
14� Sch 20 0.330 13.376 537 0.000177
14� Sch 30 S 0.375 13.250 676 0.000178
14� Sch 40 0.400 13.124 817 0.00180
14� X 0.500 13.000 956 0.000182
14� Sch 60 0.594 12.812 1169 0.000184
14� Sch 80 0.750 12.500 1528 0.000189
14� Sch 100 0.938 12.124 1969 0.000195
14� Sch 120 1.062 11.876 2265 0.000199
14� Sch 140 1.250 11.500 2724 0.000205
14� Sch 160 1.406 11.188 3112 0.000211
16� Sch 10 16.000 0.250 15.500 350 0.000152
16� Sch 20 0.312 15.376 469 0.000154
16� Sch 30 S 0.375 15.250 590 0.000155
16� Sch 40 X 0.500 15.000 834 0.000157
16� Sch 60 0.656 14.688 1142 0.000161
16� Sch 80 0.844 14.312 1520 0.000165
16� Sch 100 1.031 13.938 1903 0.000169
16� Sch 120XX 1.219 13.562 2296 0.000176
16� Sch 140 1.438 13.124 2764 0.000180
16� Sch 160 1.594 12.812 3104 0.000184
18� Sch 10 18.000 0.250 17.500 312 0.000135
18� Sch 20 0.312 17.376 416 0.000136
18� S 0.375 17.250 524 0.000137
18� Sch 30 0.438 17.124 632 0.000138
18� X 0.500 17.000 739 0.000139
18� Sch 40 0.562 16.876 847 0.000140
18� Sch 60 0.750 16.500 1178 0.000143
18� Sch 80 0.938 16.124 1514 0.000147
18� Sch 100 1.156 15.688 1911 0.000151
18� Sch 120 1.375 15.225 2318 0.000155
18� Sch 140 1.562 14.876 2673 0.000159
18� Sch 160 1.781 14.438 3096 0.000164

TABLE 2-2 Continued
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.15
Pipe Allowed Relative
Outside Wall internal working roughness
Denomination diameter thickness diameter pressure (psi) for new
(inch) (inch) (inch) (inch) to 650 °F pipe
20� Sch 10 20.000 0.250 19.500 280 0.000121
20� Sch 20 S 0.375 19.250 471 0.000123
20� Sch 30 X 0.500 19.000 664 0.000124
20� Sch 40 0.594 18.812 811 0.000126
20� Sch 60 0.812 18.376 1155 0.000129
20� Sch 80 1.031 17.938 1507 0.000132
20� Sch 100 1.281 17.438 1917 0.000135
20� Sch 120 1.500 17.000 2284 0.000139
20� Sch 140 1.750 16.500 2710 0.000143
20� Sch 160 1.969 16.082 3091 0.000147
24� Sch 10 24.000 0.250 23.500 233 0.0001005
24� Sch 20 S 0.375 23.250 392 0.0001016
24� X 0.500 23.000 552 0.0001027
24� Sch 30 0.562 22.876 635 0.000103
24� Sch 40 0.688 22.624 795 0.000104
24� Sch 60 0.969 22.062 1165 0.000107
24� Sch 80 1.219 21.562 1500 0.000109
24� Sch 100 1.513 20.938 1927 0.000113
24� Sch 120 1.812 20.376 2319 0.000116
24� Sch 140 2.062 19.875 2674 0.000119
24� Sch 160 2.344 19.312 3083 0.000122
30� Sch 10 30.000 0.312 29.376 254 0.0000804
30� S 0.375 29.250 313 0.0000807
30� Sch 20 X 0.500 29.00 440 0.0000814
30� Sch 30 0.625 28.750 568 0.0000822
36� Sch 10 36.000 0.312 35.376 207 0.0000668
36� S 0.375 35.250 260 0.0000670
36� Sch 20 X 0.500 35.000 366 0.0000675
36� Sch 30 0.625 34.750 473 0.0000679
36� Sch 40 0.750 34.500 580 0.0000685
42 S 42.000 0.375 41.250 223 0.0000573
0.500 41.000 313 0.0000576
48 S 48.000 0.375 47.250 195 0.0000499
0.500 47.000 274 0.0000503
*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of steel is taken as 60 �m.

2.16 CHAPTER TWO
TABLE 2-3 Size and Relative Roughness of Rubber-Lined Steel Pipes to U.S.
Dimensions*
Relative
Outside Wall Rubber Pipe internal roughness for
Denomination diameter thickness thickness diameter rubber-lined
(inch) (inch) (inch) (inch) (inch) pipe
2� Sch 40
2� Sch 80
2� Sch 160
XX
3� Sch 40
3� Sch 80
3� Sch 160
XX
4� Sch 40
4� Sch 80
4� Sch 120
4� Sch 160
XX
5� Sch 40
5� Sch 80
5� Sch 120
5� Sch 160
XX
6� Sch 40
6� Sch 80
6� Sch 120
6� Sch 160
XX
8� Sch 20
8� Sch 30
8� Sch 40
8� Sch 60
8� Sch 80
8� Sch 100
8� Sch 120
8� Sch 140
XX
8� Sch 160
10� Sch 20
10� Sch 30
10� Sch 40 S
10� Sch 60 X
10� Sch 80
10� Sch 100
10� Sch 120
10� Sch 140 XX
10� Sch 16
2.375
3.500
4.500
5.633
6.625
8.625
10.75
0.154
0.218
0.344
0.436
0.216
0.300
0.438
0.600
0.237
0.337
0.438
0.531
0.674
0.257
0.375
0.500
0.625
0.750
0.280
0.432
0.562
0.719
0.864
0.250
0.277
0.322
0.406
0.500
0.594
0.719
0.812
0.875
0.906
0.250
0.307
0.365
0.500
0.594
0.719
0.844
1.000
1.125
0.25
0.25
0.25
0.25
0.25
0.375
0.375
1.559
1.431
1.179
0.995
2.560
2.392
2.116
1.792
3.518
3.138
3.116
2.930
2.664
4.609
4.375
4.125
3.875
3.625
5.557
5.253
4.993
4.679
4.389
7.375
7.732
7.231
7.063
6.875
6.687
6.437
6.251
6.125
6.063
9.500
9.386
9.27
9.00
8.812
8.562
8.312
8.000
7.750
0.003788
0.004127
0.005009
0.005935
0.002307
0.002469
0.002790
0.003295
0.001679
0.00178
0.001895
0.002015
0.002233
0.001281
0.00135
0.001431
0.001524
0.001629
0.001063
0.001124
0.001183
0.001262
0.001345
0.000801
0.000807
0.000817
0.000836
0.000859
0.000883
0.000917
0.000945
0.000964
0.000974
0.000621
0.000629
0.000637
0.000656
0.000670
0.000689
0.000710
0.000738
0.000762

TABLE 2-3 Continued
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.17
Relative
Outside Wall Rubber Pipe internal roughness for
Denomination diameter thickness thickness diameter rubber-lined
(inch) (inch) (inch) (inch) (inch) pipe
12� S
12� Sch 40
12� X
12� Sch 60 X
12� Sch 80
12� Sch 100
12� Sch 120 XX
12� Sch 140
12� Sch 160
14� Sch 10
14� Sch 20
14� Sch 30 S
14� Sch 40
14� X
14� Sch 60
14� Sch 80
14� Sch 100
14� Sch 120
14� Sch 140
14� Sch 160
16� Sch 10
16� Sch 20
16� Sch 30 S
16� Sch 40 X
16� Sch 60
16� Sch 80
16� Sch 100
16� Sch 120XX
16� Sch 140
16� Sch 160
18� Sch 10
18� Sch 20
18� S
18� Sch 30
18� X
18� Sch 40
18� Sch 60
18� Sch 80
18� Sch 100
18� Sch 120
18� Sch 140
18� Sch 160
12.750
14.000
16.000
18.000
0.250
0.330
0.375
0.406
0.500
0.562
0.688
0.844
1.00
1.125
1.312
0.250
0.330
0.375
0.400
0.500
0.594
0.750
0.938
1.062
1.250
1.406
0.250
0.312
0.375
0.500
0.656
0.844
1.031
1.219
1.438
1.594
0.250
0.312
0.375
0.438
0.500
0.562
0.750
0.938
1.156
1.375
1.562
1.781
0.375
0.375
0.375
0.375
11.500
11.340
11.250
11.188
11.000
10.870
10.624
10.312
10.000
9.750
9.376
12.750
12.626
12.500
12.374
12.250
12.062
11.750
11.374
11.126
10.750
10.438
14.750
14.626
14.500
14.250
13.938
13.562
13.188
12.668
12.374
12.062
16.750
16.626
16.500
16.374
16.250
16.126
15.750
15.374
14.938
14.500
14.126
13.688
0.000513
0.000521
0.000525
0.000528
0.000537
0.000543
0.000556
0.000572
0.000591
0.000606
0.000629
0.000463
0.000468
0.000472
0.000477
0.000482
0.000489
0.000503
0.000519
0.000531
0.000549
0.000566
0.000400
0.000404
0.000407
0.000414
0.000424
0.000435
0.000448
0.000466
0.000477
0.000489
0.000353
0.000355
0.000358
0.000361
0.000363
0.000366
0.000375
0.000384
0.000395
0.000407
0.000418
0.000431
(continued)

2.18 CHAPTER TWO
TABLE 2-3 Continued
Relative
Outside Wall Rubber Pipe internal roughness for
Denomination diameter thickness thickness diameter rubber-lined
(inch) (inch) (inch) (inch) (inch) pipe
20� Sch 10
20� Sch 20 S
20� Sch 30 X
20� Sch 40
20� Sch 60
20� Sch 80
20� Sch 100
20� Sch 120
20� Sch 140
20� Sch 160
24� Sch 10
24� Sch 20 S
24� X
24� Sch 30
24� Sch 40
24� Sch 60
24� Sch 80
24� Sch 100
24� Sch 120
24� Sch 140
24� Sch 160
*Dimensions for steel are based on ANSI B36.1 and B31.1. Absolute roughness of rubber is taken
as 150 �m.
2-3-4 Hazen–Williams Formula
In the United States, the Hazen–Williams formula is often used by civil engineers because
it is independent of the Reynolds number.
Speed is calculated as:
0.63 0.54 U = 1.319 CRH S in ft/s (2-20)
S = Hv/L = ���
(2-21)
1.67 × C 1.85 1.17 × RH where Q gpm is expressed in US gallons per minute, and
S = slope or head loss per unit length
U = average velocity of fluid in ft/sec
R H = hydraulic radius = area of pipe/perimeter of pipe
C = Surface roughness coefficient (refer to Table 2-6)
In SI units:
20.000
24.000
0.250
0.375
0.500
0.594
0.812
1.031
1.281
1.500
1.750
1.969
0.250
0.375
0.500
0.562
0.688
0.969
1.219
1.513
1.812
2.062
2.344
0.375
0.375
1.85 Qgpm 18.750
18.500
18.250
18.062
17.626
17.188
16.688
16.250
15.750
15.312
22.750
22.500
22.250
22.126
21.874
21.312
20.812
20.224
19.626
19.126
18.562
0.000315
0.000319
0.000324
0.000327
0.000335
0.000344
0.000354
0.000363
0.000375
0.000386
0.000259
0.000260
0.000263
0.000265
0.000267
0.000270
0.000277
0.000284
0.000292
0.000309
0.000318

FUNDAMENTALS OF WATER FLOWS IN PIPES
TABLE 2-4 Dimensions of North American HDPE Pipes for Pressure Ratings 50 psi
to 110 psi at a Temperature of 23°C (73.4 °F)
2.19
DR 32.5 DR 26 DR 21 DR 17 DR 15.5
Pressure rating 50 psi 64 psi 80 psi 100 psi 110 psi
Average Average Average Average Average Average
outside inside inside inside inside inside
Pipe size diameter diameter diameter diameter diameter diameter
(inch) (inch) (inch) (inch) (inch) (inch) (inch)
3 3.500 3.214 3.147 3.063 3.021
4 4.500 4.133 4.046 3.938 3.885
5 5.563 5.109 5.001 4.870 4.802
6 6.625 6.193 6.084 5.957 5.798 5.720
7 7.125 6.661 6.544 6.406 6.237 6.150
8 8.625 8.063 7.921 7.754 7.550 7.446
10 10.750 10.048 9.874 9.665 9.410 9.279
12 12.750 11.919 11.711 11.463 11.160 11.005
13 13.375 12.502 12.285 12.025 11.707 11.545
14 14.000 13.086 12.859 12.586 12.253 12.086
16 16.000 14.967 14.696 14.385 14.005 13.812
18 18.000 16.826 16.533 16.183 15.755 15.539
20 20.000 18.696 18.370 17.982 17.507 17.265
22 22.000 20.565 20.206 19.778 19.257 18.992
24 24.000 22.435 22.043 21.577 21.007 20.718
26 26.000 24.304 23.880 23.375 22.759 22.445
28 28.000 26.173 25.717 25.174 24.508 24.171
30 30.000 28.043 27.554 26.971 26.258 25.898
32M 31.594 29.541 29.054 28.414 27.663 27.288
36 36.000 33.651 33.054 32.366 31.510 31.075
40M 39.469 35.898 36.225 35.469 34.561
42 42.000 39.261 38.576 37.760 36.761
48M 47.382 44.302 43.526 42.616 41.489
U = 0.8492 CR H 0.63 S 0.54 in m/s (2-22)
All parameters in Equation 2-22 must be in SI units. There is no consistency in using the
Hazen–Williams formula from small to large pipes.
Despite the fact that commercial publications from pipe suppliers sometimes use
Hazen–Williams equations to determine pressure loss of slurries, this method is highly
discouraged for long pipelines.
2.4 THE HYDRAULIC FRICTION GRADIENT
OF WATER IN RUBBER-LINED STEEL PIPES
Equations 2-15 to 2-19 have established a relationship between the friction factor and the
Reynolds number. To compute the latter, the properties of water as a carrier fluid are presented
in Tables 2-7 and 2-8.

2.20 CHAPTER TWO
TABLE 2-5 Dimensions of North American HDPE Pipes for Pressure Ratings 128 psi
to 300 psi at a Temperature of 23°C (73.4 °F)
DR 13.5 DR11 DR 9 DR 7.3 DR6.3
Pressure rating 128 psi 160 psi 200 psi 254 psi 300 psi
Average Average Average Average Average Average
outside inside inside inside inside inside
Pipe size diameter diameter diameter diameter diameter diameter
(inch) (inch) (inch) (inch) (inch) (inch) (inch)
3 3.500 2.951 2.826 2.675 2.485 2.321
4 4.500 3.795 3.633 3.440 3.194 2.986
5 5.563 4.690 4.490 4.253 3.948 3.690
6 6.625 5.584 5.349 5.065 4.700 4.395
7 7.125 6.006 5.751 5.446 5.056 4.727
8 8.625 7.270 6.693 6.594 6.119 5.723
10 10.750 9.062 8.679 8.219 7.627 7.133
12 12.750 10.749 10.293 9.745 9.046 8.459
13 13.375 11.274 10.797 10.225 9.491 8.674
14 14.000 11.802 11.301 10.701 9.934 9.289
16 16.000 13.488 12.915 12.231 11.853 10.615
18 18.000 15.174 14.532 13.760 12.772 —
20 20.000 16.880 16.145 15.289 14.191 —
22 22.000 18.544 17.780 16.819 — —
24 24.000 20.231 19.374 18.346
26 26.000 21.917 20.988 19.875
28 28.000 23.603 22.606 21.405
30 30.000 25.289 24.219 22.934
32M 31.594 26.645 25.527
TABLE 2-6 Hazen–Williams Roughness Coefficients
Description Roughness coefficient
Extremely smooth pipe 140
Very smooth pipe 130
Concrete pipe 120
Riveted new pipes and tiled channels 110
Normal cast pipes, 10 year old steel pipes, masonry channels 100
Very rough pipes 60

FUNDAMENTALS OF WATER FLOWS IN PIPES
TABLE 2-7 Physical Properties of Water in SI Units
2.21
Dynamic Kinematic Surface Vapor
viscosity, viscosity, tension pressure
kinematic dynamic in contact at atmospheric
Temperature Density �L viscosity � viscosity � with air � pressure
T (°C) (kg/m3 ) (mPa·s) (km2 /s) (N/m) (kN/m2 )
0 999.8 1.781 1.785 0.0756 0.61
5 1000 1.518 1.519 0.0749 0.87
10 999.7 1.307 1.306 0.0742 1.23
15 999.1 1.139 1.139 0.0735 1.70
20 998.2 1.002 1.003 0.0728 2.34
25 997.0 0.890 0.893 0.0720 3.17
30 995.7 0.798 0.800 0.0712 4.24
40 992.2 0.653 0.658 0.0696 7.38
50 988.0 0.547 0.553 0.0679 12.33
60 983.2 0.466 0.474 0.0662 19.92
70 977.8 0.404 0.413 0.0644 31.16
80 971.8 0.354 0.364 0.0626 47.34
90 965.3 0.315 0.326 0.0608 70.10
100 958.4 0.282 0.294 0.0589 101.33
TABLE 2-8 Physical Properties of Water in USCS Units
Vapor pressure
Temperature Density Kinematic Dynamic Surface tension in at atmospheric
T � L viscosity � viscosity � contact with air � pressure
(°F) (slug/ft 3 ) (lbf-sec/ft 2 ) (ft 2 /sec) (lbf/ft) (psia)
32 1.940 0.00003746 0.00001931 0.00518 0.09
40 1.940 0.00003229 0.00001664 0.00614 0.12
50 1.940 0.00002735 0.00001410 0.00509 0.18
60 1.938 0.00002359 0.00001217 0.00504 0.26
70 1.936 0.0000205 0.00001059 0.00498 0.36
80 1.934 0.00001799 0.00009300 0.00492 0.51
90 1.931 0.00001595 0.00008260 0.00486 0.70
100 1.927 0.00001424 0.00007390 0.00480 0.95
110 1.923 0.00001284 0.00006670 0.00473 1.27
120 1.918 0.00001168 0.00006090 0.00467 1.69
130 1.913 0.00001069 0.00005580 0.00460 2.22
140 1.908 0.00000981 0.00005140 0.00454 2.89
150 1.902 0.00000905 0.00004760 0.00447 3.72
160 1.896 0.00000838 0.00004420 0.00441 4.74
170 1.890 0.0000078 0.00004130 0.00434 5.99
180 1.883 0.00000726 0.00003850 0.00427 7.51
190 1.876 0.00000678 0.00003620 0.00420 9.34
200 1.868 0.00000637 0.00003410 0.00413 11.52
212 1.860 0.00000593 0.00003190 0.00404 14.70

2.22 CHAPTER TWO
The friction losses are expressed by the following equation:
H = (2-23)
where
H = head due to losses (in meters for SI units, in ft for USCS units)
L = length of the pipe (in meters for SI units, in ft for USCS units)
fD = Darcy friction factor
DI = pipe inner diameter
V = speed of flow in the pipe (m/s or ft/s)
g = acceleration due to gravity (9.81 m/s or 32.2 ft/s)
The hydraulic friction gradient is defined as the head loss per unit length. It is defined as
fDV iw = = (2-24)
This is a very important parameter that will be used in Chapter 4 to evaluate the friction
loss of Newtonian flows.
Calculations of the friction factor by Equation 2-18 require iterations. The Moody diagram
is a logarithmic scale that is rather difficult to use and prone to reading errors. With
modern computers, a simple program will give more accurate numbers for the Darcy friction
factor than from reading the Moody curve on a difficult logarithmic scale. The following
program was written for plain steel and rubber-lined steel pipes. It uses standard
US pipe sizes and applies the Swamee–Jain equation (2-19).
2
fDV H
� �
L 2gDI
2L �
2gDI
DIM PIP(300), t(300), a(300), q(300), ep(300)
pi = 4 * ATN(1)
DEF fnlog10 (X) = LOG(X) * .4342944
‘ p refers to nominal size
‘ outside diameter is stated
p2 = 2.375
p3 = 3.5
p4 = 4.5
p5 = 5.633
p6 = 6.625
p8 = 8.625
p10 = 10.75
p12 = 12.75
p14 = 14
p16 = 16
p18 = 18
p20 = 20
p24 = 24
p30 = 30
p36 = 36
p42 = 42
p48 = 48
INPUT “choose between steel (1) and rubber (2)”, ch
IF ch = 1 THEN tr1 = 0
IF ch = 1 THEN tr2 = 0

IF ch = 2 THEN tr1 = .254
IF ch = 2 THEN tr2 = .375
IF ch = 1 THEN e = .00006
IF ch = 2 THEN e = .00015
ef = e / .0254
IF ch = 1 THEN LPRINT “steel pipe”
IF ch = 2 THEN LPRINT “rubber lined pipe”
‘2
FOR I = 1 TO 4
t(1) = .154
t(2) = .218
t(3) = .344
t(4) = .436
PIP(I) = p2 - 2 * t(I) - 2 * tr1
ep(I) = ef / PIP(I)
LPRINT USING “ppe od = ##.### in pip id = ##.#### in thick =
#.### in k/d =
#.########”; p2; PIP(I); t(I); ep(I)
GOSUB swamee
NEXT I
LPRINT
LPRINT
The program is repeated for all US sizes of pipes (not shown here)
swamee:
d1 = PIP(I) * .0254
a = .25 * pi * d1 ^ 2
FUNDAMENTALS OF WATER FLOWS IN PIPES
FOR K = 1 TO 5
q = a * K * 1000
qus = (q * 60 / 3.7854)
fd = .4
em = ep(I)
Re = 1000 * K * d1 / .001
110
z = (em / 3.7) + (5.74 / Re ^ .9)
y = fnlog10(z)
fd = .25 / y ^ 2
1111 hl = fd * K ^ 2 / (2 * 9.81 * d1)
PRINT “revised swamee factor “; fd
LPRINT USING “veloc = ##.### m/s; flow q = ####.#### L/s;
flow = ######.## gpm “; K;
q; qus
LPRINT USING “RE = #########; fd = #.#####; hL =
#####.####”; Re; fd; hl
LPRINT
PRINT “iteration error in swamee friction factor “; dg
2.23

2.24 CHAPTER TWO
TABLE 2-9 Flow and Hydraulic Friction Gradient for Steel Pipes at a Speed of 1 m/s to
5m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m 3 (62.3 lbs/ft 3 ) and a
Kinematic Viscosity of 1 cP (2.09 × 10 –5 lbf-sec/ft 2 ) Using the Swamee–Jain Equation*
Relative Friction
Wall roughness Speed Speed Flow Darcy gradient
Denomination thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
2� Sch 40
2� Sch 80
2� Sch 160
3� Sch 40
3� Sch 80
3� Sch 160
4� Sch 40
4� Sch 80
4� Sch 160
0.154
0.218
0.344
0.216
0.300
0.438
0.237
0.337
0.531
0.001143
0.00121
0.001400
0.000770
0.000815
0.000900
0.000587
0.000617
0.000687
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
2.2
4.3
6.5
8.6
10.8
1.9
3.8
5.7
7.6
9.5
1.4
2.9
4.3
5.8
7.2
4.77
9.5
14.3
19.08
23.8
4.3
8.5
12.8
17.1
21.3
3.5
7
10.5
13.9
17.5
8.21
16.4
24.6
32.8
41.1
7.4
14.8
22.3
29.7
37.1
6
12
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
34.3
69
103
137
172
30.2
60.4
90.6
121
151
22.9
45.7
68.6
91.4
114
75.6
151
227
303
378
67.5
135
203
270
337
55.3
111
166
221
277
130.2
260.4
391
521
651
117
253
353
470
588
95
190
52,502
105,004
157505
210007
262509
49,251
98,501
147,752
197002
246253
42,850
85,700
128,549
171,399
214249
77,927
155,854
233,782
311709
389636
73,660
147,320
202,980
294,640
368,300
66,650
133,299
199,949
266,598
332,248
102,260
204,521
306,781
409,042
511,302
97,180
194,361
291,541
388,722
485,902
87,325
174,650
0.0244
0.0227
0.0221
0.0217
0.0214
0.0248
0.0231
0.0224
0.0220
0.0218
0.0258
0.0239
0.0232
0.0228
0.023
0.02213
0.0206
0.0200
0.0197
0.0195
0.0224
0.0209
0.0203
0.0199
0.0197
0.0230
0.0214
0.0208
0.0204
0.0202
0.0207
0.0193
0.0188
0.0185
0.0183
0.021
0.01957
0.019
0.0187
0.0185
0.0215
0.0201
0.0237
0.0883
0.1927
0.3367
0.5204
0.0257
0.0956
0.2087
0.3648
0.5637
0.0306
0.114
0.249
0.434
0.671
0.0145
0.054
0.118
0.206
0.319
0.0155
0.0579
0.1264
0.2209
0.3415
0.0176
0.0655
0.1431
0.2501
0.3866
0.0103
0.0386
0.0843
0.1473
0.2278
0.011
0.041
0.09
0.157
0.243
0.0126
0.047

TABLE 2-9 Continued
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.25
Relative Friction
Wall roughness Speed Speed Flow Darcy gradient
Denomination thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
4� Sch 160
5� Sch 40
5� Sch 80
5� Sch 160
6� Sch 40
6� Sch 80
6� Sch 160
8� Sch 40
8� Sch 80
0.257
0.375
0.625
0.280
0.432
0.719
0.322
0.500
0.000461
0.000484
0.000539
0.000389
0.000410
0.000455
0.000296
0.000310
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
18
24
30
13.3
26.5
39.8
53
66.3
12.1
24.1
36.2
48.4
60.4
9.7
19.5
29.2
38.9
48.7
18.7
37.3
55.9
74.6
93.2
16.8
33.6
50.5
67.3
84
13.6
27.3
40.9
54.5
68.2
32.3
65.6
96.8
129.1
161.4
29.4
58.9
88.4
117.8
147.3
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
285
380
475
210
421
631
841
1052
192
383
575
766
958
154.3
308.6
463
617
772
295.4
591
886
1182
1477
267
533
800
1066
1333
216
432
648
864
1,080
512
1,023
1,535
2,046
2,558
467
934
1,401
1,868
2335
261,976
349,301
436,626
129,972
259,944
389,915
519,887
649,859
124,028
248,056
372,085
496,113
620,141
111,328
222,656
333,985
445,313
556,641
154,051
308,102
462,153
616,204
770,255
146,329
292,659
438,988
585,318
731,647
131,750
263,500
395,249
526,999
658,749
202,717
405,435
608,152
810,870
1,013,587
193,675
387,350
581,025
774,700
968,375
0.0195
0.0192
0.019
0.0196
0.0183
0.0178
0.0175
0.0173
0.1981
0.0185
0.018
0.0177
0.0175
0.0203
0.0189
0.0184
0.0181
0.0179
0.0189
0.0176
0.0171
0.0169
0.0167
0.0191
0.0178
0.0173
0.0170
0.0169
0.0195
0.0183
0.0177
0.0174
0.0173
0.0177
0.0166
0.0161
0.0159
0.0157
0.0179
0.0168
0.0163
0.0160
0.0158
0.102
0.179
0.277
0.0077
0.0287
0.0628
0.1098
0.1697
0.0081
0.0304
0.0665
0.1163
0.1797
0.0093
0.0347
0.0759
0.1327
0.2052
0.0062
0.233
0.051
0.0892
0.138
0.0066
0.0248
0.0543
0.095
0.147
0.0076
0.0282
0.0617
0.108
0.167
0.0045
0.0167
0.0365
0.0639
0.0988
0.0047
0.0176
0.0386
0.0675
0.1044
(continued)

2.26 CHAPTER TWO
TABLE 2-9 Continued
Relative Friction
Wall roughness Speed Speed Flow Darcy gradient
Denomination thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
8� Sch 160
10� Sch 40 S
10� Sch 60 X
10� Sch 120 XX
12� S
12� X
12� Sch 120 XX
14� S
14� X
0.906
0.365
0.500
1.000
0.375
0.500
1.000
0.375
0.500
0.00347
0.000236
0.000243
0.000269
.000197
0.000201
0.00022
0.000178
0.000182
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
23.5
47
70.6
94.1
117.6
51
102
153
204
254
48
96
145
193
241
39
78
116
155
194
73
146
219
292
365
59
117
177
235
293
59
117
177
235
293
89
178
267
356
445
86
171
257
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
373
746
1,118
1,491
1,864
806
1,613
2,419
3,226
4,032
764
1,527
2,291
3,054
3,818
615
1,230
1,845
2,460
3,075
1,157
2,313
3,470
4,626
5,783
928
1,856
2,784
3,713
4,641
928
1856
2784
3713
4641
1,410
2,820
4,231
5,640
7,050
1,357
2,175
4,072
173,050
346,100
519,151
692,201
865,251
254,508
509,106
763,524
1,018,032
1,272,540
247,650
495,300
742,950
990,600
1,238,250
222,250
444,500
666,750
889,000
1,111,250
304,800
609,600
914,400
1,219,200
1,524,000
273,050
546,100
819,150
1,099,000
1,365,250
273,050
546,100
819,150
1,099,000
1,365,250
336,550
673,100
1,009,650
1,346,200
1,682,750
330,200
660,400
990,600
0.0184
0.0172
0.0167
0.0164
0.0163
0.0169
0.0158
0.0154
0.0151
0.0150
0.017
0.0159
0.0155
0.0152
0.0151
0.0174
0.0163
0.0158
0.0156
0.0154
0.0163
0.0152
0.0148
0.0146
0.0144
0.0166
0.0156
0.0152
0.0149
0.0148
0.0166
0.0156
0.0152
0.0149
0.0148
0.0159
0.0149
0.0145
0.0143
0.0141
0.016
0.015
0.0146
0.0054
0.0202
0.0443
0.0774
0.1197
0.0034
0.0127
0.0277
0.0485
0.0750
0.0035
0.0131
0.0286
0.0501
0.0775
0.004
0.0149
0.0327
0.0571
0.0883
0.0027
0.0102
0.0223
0.0390
0.0603
0.0031
0.0116
0.0255
0.0445
0.0689
0.0031
0.0116
0.0255
0.0445
0.0689
0.0024
0.0090
0.0198
0.0346
0.0536
0.0025
0.0093
0.0202

TABLE 2-9 Continued
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.27
Relative Friction
Wall roughness Speed Speed Flow Darcy gradient
Denomination thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
14� X
14� Sch 120
16� Sch 30 S
16� X
16� Sch 120
18� S
18� X
18� Sch 120
20� Sch 20S
1.062
0.375
0.500
1.291
0.375
0.500
1.375
0.375
0.000531
0.000155
0.0001575
0.000176
0.000137
0.000139
0.000155
0.000123
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
343
428
72
143
214
286
357
118
236
354
471
589
114
228
342
456
570
91
182
274
365
456
151
302
452
603
754
146
293
439
586
732
118
236
354
471
589
188
375
563
751
939
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
5,429
6,787
1,133
2,266
3398
4531
5664
1,868
3,736
5,604
7,471
9,339
1,807
3,614
5,421
7,228
9,035
1,446
2,892
4,338
5784
7230
2,390
4,780
7,170
9,560
11,949
2,321
4,642
6,963
9,284
11,605
1,868
3,734
5,604
7,471
9340
2,976
5,952
8,929
11,905
14,881
1,320,800
1,651,000
301,650
603,301
904,951
1,206,602
1,508,251
387,350
774,700
1,162,050
1,549,400
1,936,750
381,000
762,000
1,143,000
1,524,000
1,905,000
340,817
681,634
1,022,452
1,363,269
1,704,086
438,150
876,300
1,314,450
1,752,600
2,190,750
431,800
863,600
1,295,400
1,727,200
2,159,000
387,350
774,700
1,162,050
1,549,000
1,966,750
488,950
977,900
1,466,850
1,955,800
2,444,750
0.0144
0.0142
0.0163
0.0153
0.01485
0.0146
0.0145
0.0155
0.0145
0.0141
0.0139
0.0138
0.0155
0.0146
0.0142
0.0139
0.0138
0.0159
0.0149
0.0145
0.0143
0.0141
0.0151
0.0142
0.0138
0.0136
0.0134
0.0151
0.0142
0.0138
0.0136
0.0135
0.0155
0.0145
0.0141
0.0139
0.0138
0.0148
0.0139
0.0135
0.0133
0.0131
0.0354
0.0548
0.0028
0.0103
0.0226
0.0395
0.0611
0.002
0.0076
0.0167
0.0292
0.0452
0.0021
0.0078
0.0170
0.0298
0.0461
0.0024
0.0089
0.0195
0.0341
0.0527
0.0018
0.0066
0.0144
0.0252
0.0390
0.0018
0.0067
0.0147
0.0257
0.0397
0.0020
0.0076
0.0167
0.0292
0.0452
0.0015
0.0058
0.0126
0.0221
0.0342
(continued)

2.28 CHAPTER TWO
TABLE 2-9 Continued
Relative Friction
Wall roughness Speed Speed Flow Darcy gradient
Denomination thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
20� Sch 30 X
24� Sch 20 S
24� Sch S
0.500
0.375
0.500
0.000124
0.000102
0.000102
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
183
366
549
732
915
274
548
822
1096
1370
268
536
804
1072
1340
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of steel is taken as 60 �m.
‘INPUT “hit any key to continue”; m$
CLS
NEXT K
RETURN
2,899
5,799
8,698
11,598
14,497
4,341
8,683
13,025
17,366
21,707
4,249
8,497
12,746
16,995
21,243
482,600
965,200
1,447,800
1,930,400
2,413,000
590,550
1,181,100
1,771,650
2,362,200
2,952,750
584,200
1,1684,000
1,752,600
2,336,800
2,921,000
0.0148
0.0139
0.0135
0.0133
0.0132
0.0142
0.0134
0.013
0.0128
0.01267
0.01425
0.01338
0.01302
0.01282
0.01269
0.0016
0.0059
0.0128
0.0225
0.0348
0.0012
0.0046
0.0101
0.0177
0.0273
0.0012
0.0047
0.0102
0.0179
0.0277
The range of speeds used to carry solids in Newtonian flows is typically between 1.5
m/s and 5m/s.
Tables 2-9 and 2-10 present friction factor and head losses for water as a carrier fluid
for plain and rubber-lined steel pipes. There is no point in tabulating other fluids here as
they are rarely used for slurry mixtures.
The hydraulic friction gradient of water in rubber-lined pipes in the range of 2� to 18�
is presented in Figures 2-5 to 2-13. Rubber thickness of 6.4 mm (0.25�) was assumed for
2�, 3�, 4�, and 6� (up to 150 mm) pipes. Rubber thickness of 9.5 mm (0.375�) was assumed
for 8� to 24� (200 to 610 mm NB) pipes. HDPE friction head was plotted for similar sizes
at SDR11 (suitable for 100 psi pressure), to mark the advantages of reduced friction at
these sizes using HDPE instead of rubber-lined pipes, wherever it may be appropriate.
The design engineer must take in account the pressure limitations of HDPE pipes versus
rubber-lined steel pipes.
The hydraulic friction gradient for HDPE pipes up to a size of 20� (508 mm), and for
speeds in the range of 1 to 5 m/s (3.3 to 16.5 ft/sec) is presented in Table 2-11
These curves and tables allow an easier and accurate determination of the hydraulic
friction gradient of water than the Moody diagram.

FUNDAMENTALS OF WATER FLOWS IN PIPES
2.29
TABLE 2-10 Flow and Hydraulic Friction Gradients for Rubber-Lined Steel Pipes at a
Speed of 1 m/s to 5 m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m 3 (62.3
lbs/ft 3 ) and a Kinematic Viscosity of 1 cP (2.09 × 10 –5 lbf-sec/ft 2 ) Using the Swamee–Jain
Equation*
Relative Energy
Wall roughness Speed Speed Flow Darcy gradient
Pipe size thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
2� Sch 40
2� Sch 80
2� Sch
160
3� Sch 40
3� Sch 80
3� Sch 160
4� Sch 40
4� Sch 80
Steel 0.154
Rubber 0.250
Steel 0.218
Rubber 0.250
Steel 0.344
Rubber 0.250
Steel 0.216
Rubber 0.250
Steel 0.300
Rubber 0.250
Steel 0.438
Rubber 0.250
Steel 0.237
Rubber 0.250
Steel 0.337
Rubber 0.250
0.003788
0.004123
0.005008
0.002487
0.002791
0.001679
0.00178
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1.2
2.5
3.7
4.9
6.2
1
2.1
3.1
4.2
5.2
0.7
1.4
2.1
2.8
3.5
3.3
6.6
10
13.3
16.6
2.9
5.8
8.7
11.6
14.5
2.3
4.5
6.8
9.1
11.3
6.3
12.5
18.8
25.1
31.4
5.6
11.2
16.7
23.3
27.8
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
19.5
39
58.6
78
97.6
16.5
33
49.4
65.8
82.2
11.2
22.3
33.5
44.7
55.8
53
105
158
211
263
46
92
138
184
230
36
72
108
144
180
99
199
298
398
497
88.4
177
265
354
442
39,599
79,197
118,796
158,394
197,993
36,347
72,695
109,042
145,390
181,737
29,947
59,893
89,840
119,786
149,733
65,024
130,048
195,072
260,096
325,120
60,757
121,514
182,270
243,027
303,784
53,746
107,493
161,239
214,986
268,732
89,357
178,714
268,072
357,429
446,786
84,277
168,554
252,832
337,109
421,386
0.0309
0.0297
0.289
0.0287
0.0287
0.0317
0.0299
0.0296
0.0295
0.0337
0.0337
0.0322
0.0317
0.0314
0.0312
0.0269
0.0257
0.0253
0.0251
0.0250
0.0274
0.0262
0.0258
0.0256
0.0255
0.0283
0.0272
0.0267
0.0265
0.0263
0.0247
0.0237
0.0233
0.0231
0.0229
0.0251
0.0240
0.0236
0.0234
0.0233
0.0399
0.153
0.338
0.595
0.925
0.045
0.171
0.377
0.665
1.033
0.0574
0.2195
0.4854
0.8551
1.3284
0.0211
0.0808
0.1788
0.3150
0.4895
0.0230
0.088
0.1949
0.3435
0.5337
0.0269
0.103
0.228
0.402
0.624
0.0141
0.054
0.1195
0.2106
0.3273
0.0151
0.0581
0.1287
0.2268
0.352
(continued)

2.30 CHAPTER TWO
TABLE 2-10 Continued
Relative Energy
Wall roughness Speed Speed Flow Darcy gradient
Pipe size thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
4� Sch
160
5� Sch 40
5� Sch 80
5� Sch
160
6� Sch 40
6� Sch 80
6� Sch
160
8� Sch 40
8� Sch 80
Steel 0.531
Rubber 0.250
Steel 0.257
Rubber 0.250
Steel 0.375
Rubber 0.250
Steel 0.625
Rubber 0.250
Steel 0.280
Rubber 0.250
Steel 0.432
Rubber 0.250
Steel 0.719
Rubber 0.250
Steel 0.322
Rubber 0.375
Steel 0.500
Rubber 0.375
0.002015
0.001281
0.001349
0.00152
0.001063
0.001124
0.001262
0.000817
0.000859
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
04.4
8.7
13.1
17.4
21.8
10.8
21.5
32.3
43.1
53.8
9.7
19.4
29.1
38.8
48.5
7.61
15.2
22.8
30.4
38.0
15.6
31.3
47
62.5
78.2
14
28
42
56
70
11.1
22.2
33.3
44.4
55.5
26.5
53
79.5
106
132
24
48
72
03.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
069
138
207
276
345
171
341
512
683
853
154
308
461
615
769
121
241
362
482
603
248
496
744
992
1240
222
443
665
886
1108
176
352
528
703
879
420
840
1,260
1,680
2,100
380
760
1,139
074,422
148,844
223,266
297,688
372,110
117,069
234,137
351,206
468,274
585,343
111,125
222,250
333,375
444,500
555,625
98,425
196,850
295,275
393,700
492,125
141,148
282,296
423,443
564,591
705,739
133,426
266,852
400,279
533,705
667,131
118.847
237,693
356,540
475,386
594,233
183,667
367,335
551,002
734,670
918,337
174,625
349,250
523,875
0.0259
0.0248
0.0244
0.0242
0.0241
0.023
0.0221
0.0217
0.0215
0.0214
0.0233
0.0224
0.022
0.0218
0.0217
0.024
0.0231
0.0227
0.0225
0.0224
0.0219
0.0211
0.0207
0.0206
0.0204
0.0222
0.0214
0.0210
0.0208
0.0207
0.0229
0.0219
0.0215
0.0215
0.0213
0.0205
0.0198
0.0195
0.0193
0.0192
0.0208
0.0199
0.0197
0.0178
0.068
0.151
0.265
0.412
0.01
0.038
0.085
0.150
0.233
0.0107
0.0410
0.0901
0.160
0.249
0.0124
0.0478
0.1058
0.1865
0.2898
248
496
744
992
1240
0.0085
0.0326
0.0723
0.1274
0.198
0.0098
0.0377
0.0835
0.1472
0.2288
0.0057
0.0219
0.0486
0.0856
0.1331
0.006
0.0233
0.0517

TABLE 2-10 Continued
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.31
Relative Energy
Wall roughness Speed Speed Flow Darcy gradient
Pipe size thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
8� Sch 80
8� Sch
160
10� Sch
40 S
10� Sch
60 X
10� Sch
120 XX
12� S
12� X
12� Sch
120 XX
14� S
Steel 0.906
Rubber 0.375
Steel 0.365
Rubber 0.375
Steel 0.500
Rubber 0.375
Steel 1.000
Rubber 0.375
Steel 0.375
Rubber 0.375
Steel 0.500
Rubber 0.375
Steel 1.000
Rubber 0.375
Steel 0.375
Rubber 0.375
0.000974
0.000637
0.000656
0.000738
0.000525
0.000537
0.000591
0.000472
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
96
120
19
37
6
75
93
43.5
87
131
174
218
41
82
123
164
205
32
65
97
130
162
64.1
128.3
192.4
256.5
320.7
61
123
184
245
307
51
101
152
203
253
79
158
238
317
396
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
1,519
1898
295
590
886
1,181
1,476
690
1,380
2,071
2,761
3,451
651
1,301
1,952
2,602
3,253
514
1,028
1,542
2,056
2,570
1,017
2,033
3,049
4,066
5,082
972
1,943
2,915
3,887
4,859
803
1606
2409
3212
4015
1,255
2,510
3,765
5,020
6,275
698,500
873,125
154,000
308,000
462,001
616,001
770,001
235,458
470,916
706,374
941,832
1,177,290
228,600
457,200
658,800
914,400
1,143,000
203,200
406,400
609,600
812,800
1,016,000
285,750
571,500
857,250
1,143,000
1,428,750
279,400
558,800
838,200
1,117,600
1,397,000
254,000
508,000
762,000
1,016,000
1,270,000
317,500
635,000
952,500
1,270,000
1,587,500
0.0195
0.0194
0.0215
0.0206
0.0203
0.0201
0.020
0.0194
0.0186
0.0183
0.0182
0.0181
0.0195
0.0188
0.0185
0.0183
0.0182
0.0201
0.0198
0.0189
0.0188
0.0187
0.01853
0.0178
0.01755
0.0174
0.0173
0.0186
0.0179
0.0176
0.0175
0.0174
0.0190
0.0183
0.0180
0.0179
0.0179
0.0181
0.0174
0.0171
0.0170
0.0169
0.0912
0.142
0.0071
0.0273
0.0604
0.1065
0.1656
0.0042
0.0161
0.0357
0.063
0.098
0.0044
0.0167
0.0371
0.0653
0.1016
0.0050
0.0193
0.0429
0.0756
0.1174
0.0033
0.0127
0.0282
0.0497
0.0722
0.0034
0.0131
0.029
0.051
0.079
0.038
0.0147
0.0326
0.0574
0.0892
0.0029
0.0112
0.0248
0.0437
0.0679
(continued)

2.32 CHAPTER TWO
TABLE 2-10 Continued
Relative Energy
Wall roughness Speed Speed Flow Darcy gradient
Pipe size thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
14� X
14� Sch
120
16� Sch
30 S
16� X
16� Sch
120
18� S
18� X
18� Sch
120
20� Sch
20S
Steel 0.500
Rubber 0.375
Steel 1.062
Rubber 0.375
Steel 0.375
Rubber 0.375
Steel 0.500
Rubber 0.375
Steel 1.291
Rubber 0.375
Steel 0.375
Rubber 0.375
Steel 0.500
Rubber 0.375
Steel 1.375
Rubber 0.375
Steel 0.375
Rubber 0.375
0.000482
0.000531
0.000407
0.000414
0.000466
0.000358
0.000363
0.000407
0.000319
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
76
152
228
304
380
63
125
188
251
314
107
213
319
426
532
103
206
309
412
515
81
162
243
324
405
138
276
414
552
690
134
268
401
535
669
107
213
320
426
533
173
346
520
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
1,205
2,410
3,616
4,820
6,026
997
1988
2983
3977
4971
1,689
3,377
5,066
6,755
8,443
1,631
3,262
4,893
6,524
8,155
1,289
2,578
3867
5155
6444
2,187
4,373
6,560
8,746
10,933
2,121
4,241
6,362
8,483
10,604
1,689
3,377
5,066
6,755
8,443
2,7495
497
8,246
311,150
622,300
933,450
1,244,600
1,555,750
282,600
565,201
847,801
1,130,402
1,413,002
368,300
736,600
1,104,900
1,473,200
1,841,500
361,950
713,900
1,085,850
1,447,800
1,809,750
321,767
643,534
965,302
1,287,069
1,608,836
419,100
838,200
1,257,300
1,676,400
2,095,500
412,750
825,500
1,238,250
1,651,000
2,063,750
368,300
736,600
1,104,900
1,473,200
1,841,500
469,900
939,800
1,409,700
0.0182
0.0175
0.0172
0.0171
0.0170
0.0186
0.0179
0.0176
0.0175
0.0174
0.0175
0.0168
0.0166
0.0165
0.0164
0.0176
0.0169
0.0167
0.0165
0.0164
0.0180
0.0174
0.0171
0.0169
0.0169
0.0170
0.0164
0.0161
0.0160
0.0159
0.0171
0.0164
0.0162
0.0160
0.0159
0.0175
0.0168
0.0166
0.0165
0.0164
0.0166
0.0159
0.0157
0.003
0.0115
0.0254
0.0447
0.0695
0.0034
0.0129
0.0286
0.0503
0.0783
0.0024
0.0093
0.0207
0.0364
0.0566
0.0025
0.0095
0.0211
0.0372
0.0578
0.0029
0.011
0.0244
0.0429
0.0668
0.002
0.008
0.0176
0.0311
0.0484
0.0021
0.0081
0.0180
0.0317
0.0493
0.0024
0.0093
0.0207
0.0364
0.0566
0.0018
0.0069
0.0154

TABLE 2-10 Continued
2-5 DYNAMICS OF THE BOUNDARY LAYER
Boundary layer theory has been extensively covered by a number of authors. A book by
Schlichting (1968) is considered one of the classical references on this subject. When a
uniform flow approaches a plate, the particles at the wall of the plate are slowed down by
the dynamic viscosity of the fluid. A layer called the boundary layer develops. When the
flow enters a pipe, effects develop at the entrance until the flow is uniform.
2-5-1 Entrance Length
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.33
Relative Energy
Wall roughness Speed Speed Flow Darcy gradient
Pipe size thickness for new of flow Flow of flow US Reynolds friction (m/m)
(inch) (inch) pipe (m/s) L/s (ft/s) gpm number factor or (ft/ft)
20� Sch
20S
20� Sch
30 X
24� Sch
20 S
24� Sch S
Steel 0.500
Rubber 0.375
Steel 0.375
Rubber 0.375
Steel 0.500
Rubber 0.375
0.000324
0.000262
0.000265
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
694
867
169
338
506
675
844
257
513
780
1,026
1,283
251
502
753
1,003
1,254
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
10,995
13,744
2,675
5,350
8,025
10,670
13,375
4,066
8,132
12,198
16,284
20,330
3,976
7,952
11,930
15,904
19,880
1,879,600
2,349,500
463,550
927,100
1,390,650
1,854,200
2,317,750
571,500
1,143,000
1,714,500
2,286,000
2,857,500
565,150
1,130,300
1,695,450
2,260,600
2,825,700
0.0156
0.0155
0.0166
0.0160
0.0158
0.0156
0.0155
0.0159
0.0153
0.0151
0.0150
0.0149
0.0159
0.0154
0.0151
0.0150
0.0149
0.0271
0.0421
0.0018
0.0070
0.0156
0.0275
0.0428
0.0014
0.0055
0.0121
0.0214
0.0332
0.0014
0.0055
0.0123
0.0216
0.0336
*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of rubber is input as 150 �m
(0.000492 ft).
Flow in a pipe at relatively low speed when the Reynolds number is smaller than 2500 is
characterized by a certain distance called “entrance length,” over which the velocity profile
takes the final parabolic shape shown in Figure 2-14. The length Le is expressed as
Le = 0.028 DiRe For turbulent flows, the entrance length is equivalent to 50 times the inner diameter.

2.34 CHAPTER TWO
rs (0.000492 ft)
Hydraulic friction gradient (m/m)
Hydraulic friction gradient (ft/ft)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
sch 160
1 m/s
2 m/s
sch 80
sch 40
3 m/s
4 m/s
0.0 2.0 4.0 6.0
Flow Rate (L/s)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
sch 160
6.6 ft/sec
3.3 ft/sec
5 m/s
8.0
rometers (0.000492 ft)
sch 80
sch 40
9.9ft/sec
13.2 ft/sec
16.5 ft/sec
0.0 20 40 60 80 100 120
Flow Rate (US gallons/min)
FIGURE 2-5 Hydraulic friction gradient for water in a rubber-lined 2� pipe, Sch 40, Sch 80
and Sch 160. Caculations for 2� steel pipe. Rubber thickness = 0.250� (6.4 mm). Rubber roughness
= 150 �m (0.000492 ft).

Hydraulic friction (ft/ft) gradient (ft/ft) Hydraulic friction (m/m) gradient (m/m)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.7
0.6
0.5
0.4
0.3
0.2
1 m/s
2 m/s
FUNDAMENTALS OF WATER FLOWS IN PIPES
3 m/s
0.0 2.0 4.0 6.0
0.1
3.3 ft/sec
0.0
0
6.6 ft/sec
9.9ft/sec
4 m/s
8.0
13.2 ft/sec
2 m/s
sch 160
5 m/s
2.35
2-5-2 Friction Velocity
Prandtl proposed a concept of friction velocity:
Uf = = U (2-25)
�� ��
Blasius conducted tests on turbulent flows in pipes and developed an equation for the
shear wall stress in terms of the maximum velocity outside the boundary layer.
�w = 0.0225 �(Umax) 7/4� � 1/4
(2-26)
The local magnitude of the velocity in a boundary layer at a height y above the wall is
= 8.73(y + ) n (2-27)
where n = 1/7 to 1/9 and where the nondimensional height parameter y + �w fN � �
� 2
r
�
R
u
�
Uf
is defined as the
relative distance from the wall:
sch 80
sch 40
10.0 12.0 14.0 16.0 18.0 20.0
Flow Rate (L/s)
16.5 ft/sec
sch 160
6.6 ft/sec
3 m/s
sch 80
9.9 ft/sec
sch 40
4 m/s
Rubber Lined
Steel Pipes
20 40 60 80 100 120 140 160 180 200 220 240 260 220
Flow Rate (US gallons/min)
HDPE SDR 11
5 m/s
Rubber Lined
Steel Pipes
HDPE SDR 11
16.5 ft/sec
13.2 ft/sec
FIGURE 2-6 Hydraulic friction gradient for water in a rubber-lined 3� pipe, Sch 40, Sch 80,
and Sch 160. Rubber thickness = 0.25� (6.4 mm). Rubber roughness = 150 �m (0.000492 ft).
2-HDPE line 3� SDR11 roughness 1.5 �m (0.00000492 ft).
240
260

2.36
Energy gradient (m/m)
0.4
0.3
0.2
0.1
2 m/s
1 m/s
0.0
0.0
4 m/s
sch 160
sch 80
sch 40
5 m/s
3 m/s 5 m/s
2 m/s
Rubber Lined
Steel Pipes
3 m/s
4 m/s
10 20 30 40
Flow Rate (L/s)
HDPE
SDR
11
Energy gradient (ft/ft)
0.4
0.3
0.2
0.1
0.0
0.0
9.9 ft/sec
sch 160
13.2 ft/sec
Rubber Lined
Steel Pipes
sch 80
sch 40
9.9 ft/sec
6.6 ft/sec
3.3 ft/sec
100 200 300 400 500
Flow Rate (US gallons/min)
16.5 ft/sec
6.6 ft/sec 13.2 ft/sec
16.5 ft/sec
HDPE
SDR
11
FIGURE 2-7 Hydraulic friction gradient for water in a rubber-lined 4� pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.25� (6.4
mm). Rubber roughness = 150 �m (0.000492 ft). 2-HDPE line 3� SDR11 roughness 1.5 �m (0.00000492 ft).

Energy gradient (ft/ft)
Energy gradient (m/m)
0.30
0.25
0.20
0.15
0.10
0.05
0.0
0.30
0.25
0.20
0.15
0.10
0.05
0.0
0.0
0.0
1 m/s
FUNDAMENTALS OF WATER FLOWS IN PIPES
2 m/s
3 m/s
4 m/s
Rubber Lined Steel
5 m/s
sch 160
2.37
y + = (2-28)
and where u = velocity of the flow at distance y.
The boundary layer can be divided into a number of sections. At small values of y + up
to 5, the velocity profile is linear in a sublayer (see Figure 2-15). The flow is considered
to be laminar in the sublayer. Above y + = 5, a buffer zone develops up to y + Uf� �
�
= 50 and turbulence
develops. The thickness of the boundary viscous sublayer � is usually expressed
as
11.6 � 11.6 � fD � = � = � � (2-29)
�(���w�)� �U �� 8
In the turbulent region, the velocity profile is established as
u
Ufy � = 5.75 log10��� + 5.5 (2-30)
Uf
�
sch 80
Flow Rate (L/s)
sch 40
10 20 30 40 50 60 70 80
9.9 ft/sec
6.6 ft/sec
3.3 ft/sec
13.2 ft/sec
sch 160
200 400 600 800 1000 1200
Flow Rate ( US gallons/sec)
16.5 ft/sec
HDPE SDR11
FIGURE 2-8 Hydraulic friction gradient for water in a rubber-lined 6� pipe, Sch 40, Sch 80,
and Sch 160. Rubber thickness = 0.25� (6.4 mm). Rubber roughness = 150 �m (0.000492 ft).
2-HDPE line 3� SDR11 roughness 1.5 �m (0.00000492 ft).
sch 80
sch 40
HDPE SDR11

2.38
Hydraulic friction gradient (m/m)
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.0
2 m/s
1 m/s
3 m/s
0.0 20 40 60
4 m/s
sch 160
80
5 m/s
sch 80
sch 40
Rubber Lined Steel
0.18
0.16
0.14
Hydraulic friction gradient (ft/ft)
0.12
0.10
0.08
13.2 ft/sec
0.06
9.9 ft/sec
HDPE SDR11 0.04
HDPE SDR11
0.02
6.6 ft/sec
0.0
3.3 ft/sec
100 120 140
0 400 800 1200 1600 2000
Flow Rate (L/s)
Flow Rate (US gallons/min)
sch 160 16.5 ft/sec
sch 80
sch 40
Rubber Lined Steel
FIGURE 2-9 Hydraulic friction gradient for water in a rubber-lined 8� pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.375� (9.5 mm). Rubber
roughness = 150 �m (0.000492 ft). 2-HDPE roughness 1.5 �m (0.00000492 ft).

3.3 ft/sec
Hydraulic friction gradient (m/m)
Hydraulic friction gradient (ft/ft)
0.12
0.10
0.08
0.06
0.04
0.02
0.0
0.12
0.10
0.08
0.06
0.04
0.02
0.0
0.0
FUNDAMENTALS OF WATER FLOWS IN PIPES
2 m/s
1 m/s
40
80
3 m/s
9.9 ft/sec
6.6 ft/sec
10" sch 40
1 m/s
120
10" sch 40
2 m/s
160
5 m/s
16.5 ft/sec
4 m/s
4 m/s
3 m/s
12" sch 40
200 240 280
Flow Rate (L/s)
9.9 ft/sec
320
0.0 800 1600 2400 3200 4000 4800
Flow Rate ( US gallons/sec)
6.6 ft/sec
13.2 ft/sec
12" sch 40
13.2 ft/sec
5 m/s
16.5 ft/sec
2.39
FIGURE 2-10 Hydraulic friction gradient for water in a rubber-lined 10� pipe, Sch 40 and
12�, Sch 40 pipes. Rubber thickness = 0.375� (9.5 mm). Rubber roughness = 150 �m
(0.000492 ft). 2-HDPE roughness 1.5 �m (0.00000492 ft).

2.40 CHAPTER TWO
Hydraulic friction Hydraulic friction
gradient (ft/ft)
gradient (m/m)
0.08
0.06
0.04
0.02
0.0
0.08
0.06
0.04
0.02
0.0
0.0
0.0
The reader is encouraged to review the work of Schlichling (1968) for details of
boundary layer theory.
Example 2-5
Determine the boundary viscous sublayer thickness and friction velocity of Example 2-4.
Solution in SI Units
In Example 2-4, the Darcy friction factor was determined to be 0.0178. In Section 2.31 it
was stated that fD = 4fN, therefore
0.0178
fN = � = 0.00445
4
Since the velocity of the flow is 1.467 m/s,
ff Uf = U�� �
2
10" SDR 11
5 m/s
12" SDR 11
5 m/s
40 80 120 160 200 240 280 320
Flow Rate (L/s)
10" SDR 11
16.5 ft/sec
16.5 ft/sec
12" SDR 11
800 1600 2400 3200 4000 4800
Flow Rate ( US gallons/sec)
FIGURE 2-11 Hydraulic friction gradient of water in 10� and 12� SDR11 HDPE pipes.
Roughness 1.5 �m (0.00000492 ft).
0.00445
= 1.467��
� = 0.0692 m/s
2
From Equation 2.28 the thickness of the viscous sublayer is calculated as
� = �� f 11.6 � D
� �8
�U

Hydraulic friction gradient (m/m)
Hydraulic friction gradient (ft/ft)
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.0
FUNDAMENTALS OF WATER FLOWS IN PIPES
2 m/s
1 m/s
3 m/s
4 m/s
14" S (Sch 30)
� = �� = (4.72) 10–7 11.6 (0.00129) 0.0178
�� � m
1020 (1.467) 8
Solution in USCS Units
0.0178
fN = � = 0.00445
4
Since the average velocity flow in 4.8 ft/s,
0.00445
Uf = 4.8 � 0.2264 ft/s
��2 2.41
FIGURE 2-12 Hydraulic friction gradient for water in rubber-lined 14� pipe, Sch 40 and 16�
S and 18� S pipes. Wall thickness = 0.375� (9.5 mm). Rubber thickness = 0.375� (9.5 mm).
Rubber roughness = 150 �m (0.000492 ft).
16" S
18" S
5 m/s
0.0 100 200 300 400 500 600 700
009
Flow Rate (L/s)
0.09
0.08
0.07
0.06
Flow Rate (L/s)
0.05
0.04
13.2 ft/sec
16.5 ft/sec
0.03
0.02
0.01
0.0
9.9 ft/sec
6.6 ft/sec
3.3 ft/sec
0.0 2000 4000 6000 8000 10000
Flow Rate (US gallons/min)
14" S (Sch 30)
16" S
18" S

2.42 CHAPTER TWO
Hydraulic friction gradient (m/m)
Hydraulic friction gradient (ft/ft)
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.0
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.0
0.0 100 200 300
0.0
14" SDR 11
14" SDR 11
400
5 m/s
16" SDR 11
5 m/s
500 600 700
Flow Rate (L/s)
16" SDR 11
2000 4000 6000 8000 10000
Flow Rate (US gallons/min)
g 2-
18" SDR 11
5 m/s
18" SDR 11
16.5 ft/sec
FIGURE 2-13 Hydraulic friction gradient of water in 14�, 16� and 18� SDR 11 HDPE pipes.
Absolute roughness 1.5 �m (0.00000492 ft).

FUNDAMENTALS OF WATER FLOWS IN PIPES
2.43
TABLE 2-11 Flow and Hydraulic Friction Gradient for HDPE Pipes SDR11 at a Speed
of 1 m/s to 5 m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m 3 (62.3 lbs/ft 3 ) and
a Kinematic Viscosity of 1 cP (2.09 × 10 –5 lbf-sec/ft 2 ) Using the Swamee–Jain Equation
Pipe Friction
inner Speed Speed Flow Darcy gradient
Pipe size diameter Relative of flow Flow of flow US Reynolds friction (m/m)
(in) (in) roughness (m/s) L/s (ft/s) gpm number factor or (ft/ft)
3
4
5
6
7�
8�
10�
12�
2.826
3.633
4.49
5.349
5.7510
7.270�
8.675�
10.293�
0.0000053
00000041
0.0000033
0.0000028
0.0000026
0.0000022
0.0000017
0.0000015
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
4.0
8.0
12.1
16.2
20.2
6.7
13.4
20.1
26.8
33.4
10.2
204
306
408
510
15
29
44
58
73
17
34
51
67
84
25
49
74
98
123
38
76
114
153
191
54
107
161
215
268
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
64
128
192
257
321
106
212
318
424
530
162
324
486
648
810
230
460
689
919
1,150
266
531
797
1,063
1,328
389
779
1,168
1,558
1,947
605
1,210
1,815
2,420
3,025
851
1,702
2,552
3,404
4,254
71,780
143,561
215,341
287,122
358,902
92,278
184,556
276,835
369,113
461,391
114,046
228,092
342,138
456,184
570,230
135,865
271,729
407,594
543,458
679,323
146,075
292,151
438,226
584,302
730,377
176,860
353,720
530,581
707,441
884,301
220,447
440,893
661,340
881,786
1,102,233
261,442
522,884
784,327
1,045,769
1,307,211
0.01917
0.01659
0.01532
0.0145
0.01391
0.0182
0.0158
0.0146
0.0138
0.01329
0.01738
0.01514
0.01403
0.01331
0.01279
0.01677
0.01465
0.01359
0.01290
0.01241
0.01653
0.01445
0.01341
0.01274
0.01225
0.0159
0.0139
0.01296
0.01232
0.01186
0.0152
0.0134
0.0125
0.0119
0.0114
0.01475
0.0130
0.0121
0.0115
0.0111
0.0136
0.0471
0.0979
0.167
0.247
0.01
0.035
0.0726
0.1223
0.1835
0.0078
0.0271
0.0564
0.0592
0.1429
0.0063
0.0220
0.0459
0.0774
0.1164
0.0058
0.0202
0.0421
0.0711
0.1069
0.0046
0.0161
0.0336
0.0568
0.0854
0.0035
0.0124
0.0259
0.0439
0.0660
0.0029
0.0101
0.0212
0.0359
0.0541
(continued)

2.44 CHAPTER TWO
TABLE 2-11 Continued
Pipe Friction
inner Speed Speed Flow Darcy gradient
Pipe size diameter Relative of flow Flow of flow US Reynolds friction (m/m)
(in) (in) roughness (m/s) L/s (ft/s) gpm number factor or (ft/ft)
13�
14�
16�
18�
20�
10.797�
11.301�
12.915�
14.532�
16.146�
0.0000014
0.0000013
0.0000012
0.000001
0.0000009
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
59
118
177
236
295
65
129
194
259
324
85
169
253
338
423
107
214
321
428
535
132
264
396
528
660
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
3.3
6.6
9.9
13.2
16.5
936
1,872
2,808
3,745
4,681
1,026
2,051
3,077
4,103
5,129
1,340
2,679
4,020
5,359
6,698
1,696
3,392
5,088
6,784
8,480
2,094
4,187
6,281
8,375
10,469
From the Equation 2.28, the thickness of the viscous sub-layer is
� = �� = 1.56 × 10–6 11.6 × 2.7 × 10 0.0178
� ft
8
–5
��
1.973 × 4.8
2-6 PRESSURE LOSSES DUE TO CONDUITS
AND FITTINGS
274,244
548,488
822,731
1,096,975
1,371,219
287,045
574,091
861,136
1,148,182
1,435,227
328,041
656,082
984,123
1,312,164
1,640,205
369,113
738,226
1,107,338
1,476,451
1,845,564
410,108
820,217
1,230,325
1,640,434
2,050,542
0.0146
0.0129
0.0120
0.0114
0.0110
0.0145
0.0128
0.0119
0.0113
0.0109
0.0141
0.0125
0.0116
0.0110
0.0107
0.0138
0.0122
0.0114
0.0109
0.0105
0.0136
0.0120
0.0112
0.0107
0.0103
0.0027
0.0096
0.0201
0.0340
0.0512
0.0026
0.0091
0.0190
0.0322
0.0485
0.0022
0.0078
0.0163
0.0276
0.0416
0.0019
0.0068
0.0142
0.0240
0.0362
0.0017
0.0060
0.0125
0.0213
0.0321
The sizing of pumps is based on determining pressure losses between the starting point
(A) and the final delivery point (B) (Figure 2-16). It is important to know that the static
pressure at (A) includes atmospheric pressure or the pressure of any pressurizing gas, and

D
i
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.45
the pressure due to the height of liquid above the centerline of the first impeller or impeller
at suction. It is also important to know that static pressure at (B) includes any pressurizing
gas at (B) and the height of liquid at (B) above the centerline of the pump’s impeller.
The additional pressure losses between (A) and (B) include the friction losses and
pressure losses in all the pipe fittings such as valves, elbows, expansions, contraction
branches, and bypasses. Pressure is also lost at entry and exit as well. Such pressure losses
are expressed in terms of the Darcy–Weisbach equation and in terms of pressure loss
factors for each fitting.
Total pressure loss due to friction:
�U
Hf = fD + �Kf (2-31)
2
�U Lj � �
Dij 2
2
�
2
where
L j = length of the conduit j
K f = pressure loss of the fitting f
+
+
L e
FIGURE 2-14 Entrance length for flows in pipes.
V
V
Turbulent layer
Buffer layer
Viscous sublayer
FIGURE 2-15 Boundary layer of flow over a plate.

g 6
2.46 CHAPTER TWO
H
1
5 Pipe Diameters
FIGURE 2-16 Simplified pumping system between two tanks.
TABLE 2-12 Equivalent Length of Valves for Friction Loss of Calculations for
Single-Phase Turbulent Flow
Equivalent
length/diameter
Minimum recommended
speed for full disc lift
Fitting ratio m/s Ft/s
Gate valves 8
Globe valves 340
Angle valves 55
Ball valves 3
Butterfly valves 16
Plug valves—straightway 18
Plug valves—3 way through flow 30
Plug valve—branch flow 90
Stop check valve—straight through 400 2.12 6.96
Stop check valve—angular 90 deg 200 2.89 9.49
Swing check valve 300 2.32 7.59
Lift check valve 55 5.4 17.7
Tilting disc check valve 5–15 1.16–3.08 3.80 to 10.13
Foot valve with strainer—poppet disc 420 0.58 1.90
Foot valve with strainer—hinged disc 75 1.35 4.43
H
2

FUNDAMENTALS OF WATER FLOWS IN PIPES
Loss Factor K
0.5
0.4
0.3
0.2
0.1
0.0
0 2 4
6 8 10
Ratio R/D
(bend radius/pipe diam)
The differential head that a pump must deliver to pump a liquid between (A) and (B) is
therefore
TDH = (P B – P A)/�g + (Z B – Z A) + H fOB + H fOA
where
H fOA = the pressure losses due to conduits and fittings between the tank (A) and the
pump, including entry loss
H fOB = the losses due to conduits and fittings between the pump and tank (B), including
exit losses
Table 2-12 presents examples of loss coefficients for fittings. Some practical considerations
limit the use of fittings in slurry circuits. For example, elbows should have a minimum
radius of three pipe diameters to avoid short turns (see Figure 2-17). Such an approach
minimizes wear.
Example 2-6
The fluid of Example 2-4 is pumped at a flow rate of 3500 gpm through a 16 × 14 pump.
The steel fittings include 14� × 18� reducer, an 18� knife gate valve, three long radius 90°
elbows with a diameter to radius ratio of 3. The length of the pipe is 355 ft. Determine the
total dynamic head if the liquid level in the suction tank is 10 ft and the level in the discharge
tank is 50 ft above the centerline of the impeller. Ignore the length of the suction
pipe as negligible.
Solution in SI Units
Net static head:
(50 ft – 10 ft) 0.3048 = 12.2 m
Dynamic head at the entry of the pump:
Di = 15.25 in or 0.387 m
Suction Area = 0.1178 m 2
2.47
FIGURE 2-17 Loss Factors for rough wall bends for pipes 1–7� (after Crane Technical Bulletin
No 410).

2.48 CHAPTER TWO
Speed = = 1.875 m/s
Dynamic head at suction:
1.875
= = 0.18 m
For a sharp entrance pipe, the recommended K factor is 0.5. Friction losses at the entry
to the pump are calculated as follows:
0.5 × 0.18 = 0.09 m
On the discharge of the pump for a 14 × 18 reducer, the loss factor is calculated from
the area ratio as
2
U
�
2 × 9.81
2
0.221
�
0.1178
�
2g
K = �1 – � 2
= �1 – � 2
� � = 0.1681
2
2
d 2 17.25
� For an18 ft full-bore gate valve, the loss factor K = 0.10
� For the elbow r/D = 3, the loss factor K = 0.14
� For the reentry pipe L/D = 65
Total friction losses:
355 × 0.3048 + 65 × 0.438
0.09 m + � ��� × 0.0178 + (0.1681 + 0.10 + 3 × 0.14)�
0.438
1.467 2
× � = 0.774 m
2 × 9.81
13.25 2
TDH = 0.774 m + 12.2 m = 12.97 m
Solution in USCS Units
Net static head = 50 ft – 10 ft = 40 ft
Speed in suction pipe is calculated as follows:
ID = 15.25� = 1.27 ft
d 1 2
Flow Rate = 7.79 ft 3 /s
Area = 1.267 ft 2
Velocity = = 6.15 ft/s
Dynamic head at suction:
6.15
= 0.587 ft
The K factor for sharp entrance in 0.5. Loss during suction is 0.5 × 0.587 = 0.293 ft.
On the discharge of the pump the K factor is determined as in the SI unit solution. Total
friction losses are:
2
7.79
�
1.267
�
2 × 32.2

FUNDAMENTALS OF WATER FLOWS IN PIPES
0.293 ft + �0.0178� � + 0.1681 + 0.10 + 0.42�
355 + 65 × 1.27
�� � = 2.44 ft
1.27
2 × 32.2
TDH = 40 + 2.44 = 42.44 ft
2-7 ORIFICE PLATES, NOZZLES, AND
VALVE HEAD LOSSES
The flow through an orifice plate, a nozzle, or a valve is reduced from the ideal theoretical
value by a discharge coefficient:
Q = CdQideal (2-32)
The ideal theoretical flow is considered the product of speed and area at the opening of
the orifice, valve, or nozzle.
The theoretical velocity through the orifice is calculated as
2�P
Vth = �2�g�h� = � (2-33)
�� R
However, the velocity is typically smaller than the theoretical velocity:
V o = C veV th
where C ve = velocity coefficient.
The flow through an opening contracts from the full area. This is known as the vena
contracta effect.
Reentrant Sharp
tube Edged
V V
Square
Edged
Length = 1/2 to
Stream clears
1 diameter sides
4.8 2
Reentrant
tube
C =0.52
C =0.61 C =0.61 C =0.73
d d d
d
V
2.49
Length = 2-1/2
diameters
FIGURE 2-18 Discharge coefficients of orifice plates and nozzles.

2.50 CHAPTER TWO
For a thin plate or sharp-edged orifice, the vena contracta is assumed to be one half of
an orifice diameter d1 downstream from the orifice, but in reality the distance may be from
30% to 80% of d1. For flow of water at a high Reynolds number through a small orifice diameter,
Lindeburg (1998) reported that the contracted area is approximately 61% to 63%.
The coefficient of contraction is defined as
area of vena contracta
Cc = ���
(2-34).
orifice area
The total discharge is the product of the reduced velocity and the contracted area:
Q = C veV th(C c A 1)
The product of the velocity coefficient by the area contraction coefficient is called the discharge
coefficient:
C d = C veC c
(2-35)
Q = C d A 1�(2�g�h�)� (2-36a)
Q = Cd A1�(2���P�/��)� (2-36b)
actual discharge
Cd = ���
theoretical discharge
Typical values for discharge coefficients from nozzles and orifices are shown in Figure
2-18. The Cameron Hydraulic Handbook (1977) recommends a further correction for
large openings when d2/d1 > 0.30:
2gh
Q = Cd A �� �� 1 – (d1/d2) 4
(2-37)
This equation works for liquids with a dynamic viscosity similar to the viscosity of water.
The discharge vena contracta and velocity coefficient presented in Figure 2-18 are
based on controlled flow conditions upstream. Flow disturbances can affect the magnitude
of these coefficients.
Manufacturers of valves in North America have developed a valve coefficient to relate
flow rate to pressure drop as Cv, which is defined as:
�Ppsi Qgpm = Cv�� � (2-38)
S.G.
This coefficient is not dimensionally homogeneous and is not equal to the discharge
coefficient from orifices and nozzles. Although the flow coefficient Cv was developed for
control valves, a relationship is often established for other fittings in terms of the K factor:
(29.9)(d in) 2
Cv = ��
(2-39)
�K�
The reader should be very careful not to confuse C v (the flow coefficient commonly
used in North America) with the discharge coefficient C d more commonly used in the rest
of the world. Such a mix-up can lead to serious errors. C v is not used outside North America
and has no relationship to the terms defined in Equations 2-34 to 2-37. The reader
should avoid the common confusion that it sometimes creates.

FUNDAMENTALS OF WATER FLOWS IN PIPES
2.51
FIGURE 2-19 Cross-section of a Series 39 slurry check valve. (Courtesy of Red Valve
Company, Carnegie, PA, U.S.A.)
FIGURE 2-20 Front view of a Series 39 slurry check valve. (Courtesy of Red Valve Company,
Carnegie, PA, U.S.A.)

2.52 CHAPTER TWO
FIGURE 2-21 Slurry knife-gate valve cross-sectional drawing. (Courtesy of Red Valve
Company, Carnegie, PA, U.S.A.)
FIGURE 2-22 Slurry knife-gate valve. (Courtesy of Red Valve Company, Carnegie, PA,
U.S.A.)

Manufacturers of slurry valves have developed very specific designs to meet the requirements
of wear and operation without plugging. These include:
� Rubber-lined check valves
� Rubber-lined knife-gate valves
� Rubber-lined pinch valves
� Ceramic ball valves
� Plug valves
FUNDAMENTALS OF WATER FLOWS IN PIPES
2.53
FIGURE 2-23 Slurry pinch valve, showing cut through the rubber sleeve. (Courtesy of Red
Valve Company, Carnegie, PA, U.S.A.)
Special check valves are available for sewage and slurry flows. The Red Valve Company
Series 39 valves (Figures 2-19 and 2-20) feature a special reinforced elastomer check
sleeve. The valve check sleeve seals under reverse flow or back-pressure and opens under
pressure from the pump. It does not incorporate any discs that may wear on contact with
slurry. This type of valve is therefore different in design than the type shown in books on
water flows. The consultant engineer should therefore request from the manufacturer of
the slurry check valves the estimated K factor for pressure losses. The Red Valve Company
Series 39 slurry check valves are available in sizes up to 48� (1220 mm), with a choice
of elastomers such as pure gum rubber, neoprene, Hypalon, chlorobutyl, Buna-N, EPDM,
and Viton.
Knife-gate valves for slurry flows (Figures 2-21 and 2-22) feature a metal gate sand-

2.54 CHAPTER TWO
FIGURE 2-24 Principles of operation of a pinch valve, pinched by a roller. (Courtesy of Red
Valve Company, Carnegie, PA, U.S.A.)
wiched between two rubber linings (or cartridges). They are often installed on the suction
side of slurry pumps to provide a method of isolating them during repairs and maintenance.
Most knife-gate valves are rated to a maximum of 1 MPa (150 psi), but some manufacturers
offer valves rated at 2 MPa (300 psi).
Globe valves are not suitable for slurry applications because they wear rather rapidly.
To control slurry flows, a rubber pinch valve is recommended (Figure 2-23). The valve
features a special reinforced sleeve. The sleeve is closed by pinching using a special roller
(mechanical pressure) (Figure 2-24) or by the use of air pressure (Figure 2-25).
Ceramic ball valves are used as shut-off valves for pipelines, particularly to close under
high pressure.
2-8 PRESSURE LOSSES THROUGH
FITTINGS AT LOW REYNOLDS NUMBERS
Certain slurry flows, particularly those of a non-Newtonian regime, do occur at relatively
moderate Reynolds numbers and in laminar conditions (Tables 2-13 to 2-14). For many
years, the method using the K factor and the equivalent length has been the most widely

FUNDAMENTALS OF WATER FLOWS IN PIPES
2.55
FIGURE 2-25 Principles of operation of a pneumatically actuated pinch valve. (Courtesy of
Red Valve Company, Carnegie, PA, U.S.A.)
accepted method. It is based on experimental data obtained usually in steel pipes at very
high Reynolds numbers. As the Reynolds number is reduced closer to laminar flow, the K
factor becomes inversely proportional to it. Since certain homogeneous slurries are sometimes
pumped at relatively low Reynolds numbers, even quite close to the critical value, it
is important to emphasize an alternative approach. Hooper (1992) emphasized the limitations
of this method and proposed a two-K method:
K1 K� K = � + ��
(2-40)
Re 1 + 1/D1-in

2.56 CHAPTER TWO
TABLE 2-13 Equivalent Length of Fittings for Friction Loss of Calculations for
Single-Phase Turbulent Flow*
Fitting Type Length/Diameter Ratio
Standard threaded elbow 90 degree 30
Standard threaded elbow 45 degree 16
Standard threaded elbow Long radius 90 degree—5 diameter
bend as used in slurry plants
16
Mitre bend 15 degree bend 4
30 degree bend 8
45 degree bend 15
60 degree bend 25
75 degree bend 40
90 degree bend 60
Standard tee Through flow 20
Through branch 60
*Data from Ingersoll Rand (1977).
where
K1 = value of K at a Reynolds number of 1
K� = value of K at high Reynolds numbers
DI-in = internal pipe diameter in inches.
Values of these two constants are presented in Table 2-15.
Regarding the equivalent length method, Hooper (1992) wrote:
TABLE 2-14 Dynamic Loss Factor K for Expansions and Contractions, where Loss =
KV 2 /2g*
Fitting Description Loss factor K
Pipe exit Projecting sharp edged, rounded 1.0
Pipe entrance Inward projecting 0.78
Pipe entrance (flush) Sharp edged 0.5
Bellmouth fillet/diameter = 0.02 0.28
Bellmouth fillet/diameter = 0.04 0.24
Bellmouth fillet/diameter = 0.06 0.15
Bellmouth fillet/diameter = 0.10 0.09
Bellmouth fillet/diameter = 0.15
and up
0.04
Reentry pipe L/D = 65
Sudden enlargements in pipes 2 2 K = (1 – d 1/d 2) Sudden contractions in pipes 2 2 K = 0.5(1 – d 1/d 2) Gradual enlargements in pipes � Less than 45 degrees 2 2 2
K = 2.6 sin (�/2)(1 – d 1/d 2) � Larger than 45 degrees 2 2 2
K = (1 – d 1/d 2) Gradual contractions in pipes � Less than 45 degrees 2 2 K = 0.8 sin �(1 – d 1/d 2) � Larger than 45 degrees 2 2 K = 0.5(1 – d 1/d 2)�(s�in� ��/2�)�
*Data from Ingersoll Rand (1977).

FUNDAMENTALS OF WATER FLOWS IN PIPES
TABLE 2-15 Constants for the Two-K Method* (after Hooper 1992)
The equivalent-length method concept contains a booby trap for the unwary. Every
equivalent length method has a specific friction factor ( f ) associated with it, because
the equivalent lengths were originally developed from the K factor in the formula
Le = KD/f. This is why the latest version of the equivalent length method (the
1976 edition of the Crane Technical Paper 410 ...properly requires the use of two
friction factors. The first is the actual friction factor for the pipe ( f ), and the second
is a “standard” friction factor for the particular fitting ( f T). Thus the two-K method
is as easy to use and as accurate as the updated equivalent-length method.
The two-K method will be explored further in Chapter 5.
2.57
K1 at K� at very
Fitting Description Type Re = 1 high Re
Elbows 90° Standard R/D = 1, screwed 800 0.40
Standard R/D = 1, flanged/welded 800 0.25
Long radius (R/D = 1.5), all types 800 0.20
Mitered Elbow R/D = 1.5 1 weld 90° 1000 1.15
2 welds 45° 800 0.35
3 welds 30° 800 0.30
4 welds 22.5° 800 0.27
5 welds 18° 800 0.25
45° Standard (R/D = 1.0), all types 500 0.20
Long radius (R/D = 1.5), all types 500 0.15
Mitered, 1 weld, 45° angle 500 0.25
Mitered, 2 welds, 22.5° angle 500 0.15
180° Standard R/D = 1, screwed 1000 0.60
Standard R/D = 1, flanged/welded 1000 0.35
Long radius (R/D = 1.5), all types 1000 0.30
Tees Used as elbows Standard, screwed 500 0.70
Long radius, screwed 800 0.40
standard, flanged/welded 800 0.80
Stub-in-type branch 1000 1.00
Run-through tee Screwed 200 0.10
Flanged or welded 150 0.05
Stub-in-type branch 100 0.00
Valves Gate, ball, plug Full line size, � = 1 300 0.10
Reduced trim, � = 0.9 500 0.15
Reduced trim, � = 0.85 1000 0.25
Globe Standard 1500 4.00
Globe Angle or Y-type 1000 2.00
Diaphragm Dam type 1000 2.00
Butterfly 800 0.25
Check Lift 2000 10.0
Swing 1500 1.50
Tilting check 1000 0.50
*Use R/D = 1.5 values for R/D = 5 pipe bends, 45° to 180°. Use appropriate tee values for flowthrough
crosses.

2.58 CHAPTER TWO
2-9 THE BERNOULLI EQUATION
The last few sections of this chapter examined the concept of friction and pressure losses.
The presence of friction forces, changes in elevation between one point and another along
the piping, the presence of a pump to add energy to the fluid, or a turbine to extract energy
can all be expressed in terms of the extended Bernoulli’s equation:
(E p + E v + E z) 1 + E A = (E p + E v + E z) 2 + E E + E f + E m
2 U 1 P2 + � + Z2g + EA = � + � + Z2g + EE + Ef + Em 2
� 2
where subscripts 1 and 2 refer to points 1 and 2.
Ep = P1/� = energy due to static pressure per unit mass
2 U 1/2 = energy due to dynamic pressure per unit mass
Z = location of point above a reference datum
EA = energy added (e.g., by a pump) per unit mass
EE = energy extracted (e.g., by a turbine) per unit mass
Ef = Energy per unit mass due to friction losses
Em = Energy lost due to fittings, per unit mass
In USCS units.
2-10 ENERGY AND HYDRAULIC GRADE
LINES WITH FRICTION
(2-41)
When the total energy for flow in a pipeline is plotted against distance, a profile called the
energy gradient line is obtained. The energy drops with friction or extraction through a
turbine, and increases by absorption from a pump.
The hydraulic gradient is the sum of the pressure and the potential energies. The hydraulic
gradient is therefore smaller than the energy gradient by the dynamic head (Figure
2-26).
2-11 FUNDAMENTAL HEAT TRANSFER
IN PIPES
In many areas of the world, mining is done in cold climates (Figure 2-27). Long tailing
pipelines are exposed to wind, snow, and freezing conditions. In some oil–sand processes,
temperature is used to facilitate the pumping or separation of tar from sand. In other
processes, hot slurries are fed to autoclave furnaces. The field of heat transfer is immense,
but in the following paragraphs, some fundamentals will be reviewed. There are three
main phenomena of heat transfer:
1. Conduction
2. Convection
3. Radiation
P 1
� �
U 2 2

EGL
HGL
Energy and Hydraulic Gradients
For a pump
FUNDAMENTALS OF WATER FLOWS IN PIPES
EA 2
E v=
V /2
EGL
2
V 1 /2
HGL
Energy and Hydraulic Gradients
For an expansion
FIGURE 2-26 Energy and hydraulic gradients.
2
V 2 /2
2.59
FIGURE 2-27 The construction of mines may require pipelines that operate in extremely
cold environments. This water pipeline was insulated and heat-traced for an Arctic environment.

2.60 CHAPTER TWO
TABLE 2-16 Examples of Conductivity Range
Range of conductivity K, Range of conductivity K,
Material W/m °K Btu-ft/hr-ft 2 °F
Insulators 0.03–0.21 0.02–0.12
Nonmetallic Liquids 0.09–0.70 0.05–0.40
Nonmetallic Solids 0.03–2.6 0.02–1.5
Liquid metals 8.7–78 5.0–45
Metallic alloys 14–120 8.0–70
Pure metals 52–420 30–240
2-11-1 Conduction
Heat transfer by conduction occurs essentially by molecular vibration and movement of
free electrons. As metals have more free electrons than nonmetals, they are better conductors
of heat. Thermal conductivity, also known as thermal conductance, is a measure of
the rate of heat transfer per unit thickness. Examples of conductivity range are presented
in Table 2-16
Thermal conductivity is a function of temperature. For metals it decreases with temperature,
whereas for insulators it increases with temperature. To simplify matters,
it is common to assume the thermal conductivity at the average temperature of 1 – 2(T 1 +
T 2).
2-11-2 Thermal Resistance
Defining heat transfer power as Q t, thermal resistance is defined as
R th = (2-42)
where Qt is expressed in watts or Btu/hr.
For a flat plate with a thickness path length L and an area A, and if heat transfer occurs
by conduction and kth is the thermal conductivity of the material, the resistance factor Rth is:
L
Rth = (2-43)
For a layer of insulation around a pipe, this equation is expressed in terms of the inner
and outer radius of the insulation layer:
ln(RO/RI) Rth = � (2-44)
2�kthL
2-11-3 The R Value
T1 – T2 �
Qt
� kthA
One term commonly used by the industry is the thermal resistance per unit area or R value.

R Value = � RthA (2-45)
Qt/A
2-11-4 The Specific Heat or Heat Capacity C ��
The specific heat capacity is defined as the energy required to increase the temperature of
a unit mass by a unit degree and is calculated as
Qt = C�m�T (2-46).
2-11-5 Characteristic Length
Characteristic length is defined as the ratio of the volume to its surface area and is calculated
as
V
Lc = (2-47)
2-11-6 Thermal Diffusivity
Thermal diffusivity is a measure of the speed of propagation of a specific temperature
into a solid. The higher the diffusivity, the faster the material will reach a certain temperature.
Thermal diffusivity is calculated as
where
� e = thermal resistivity (�-cm or �-in)
� = diffusivity (m 2 /s or ft 2 /hr)
K th = conductivity (W/m-°K or Btu-ft/hr-ft 2 -°F)
C � = specific heat capacity (J/kg°K – Btu/lbm-°F)
2-11-7 Heat Transfer
FUNDAMENTALS OF WATER FLOWS IN PIPES
T 1 – T 2
� As
k th
2.61
� = � (2-48)
�eC� Heat transfer is essentially a transmission of energy from one body to another in a period
of time. For this reason, it has the same unit as power in SI units, i.e., the watt. In USCS
units Btu/hr is used. However, many equations ignore the time factor. Heat transfer per
unit area qta is often used so that the total heat transfer Qt over an area A is calculated as
Qt = qtaA Qt = mC��T (2-49)
where
m = the mass of the body
�T = the temperature change or power or rate of heat transfer
The rate of heat transfer or power associated with the flow is expressed as
Pwt = �QC��T

2.62 CHAPTER TWO
Heat transfer can take different forms when slurry is stored in tanks, varies in thickness,
or flows in pipes. In the northern climates, loss of heat can lead to frozen pipelines.
In the hot climates, the heat absorbed from the sun leads to expansion of plastic lines and
significant pipe stresses.
2-12 CONCLUSION
In this chapter, some very important principles regarding water flows were introduced.
Since water is the principal carrier of slurry mixtures, the tools developed in this chapter
such as hydraulic friction gradients and methods to correlate the friction velocity with the
friction factor will be extensively used for pipe flow and open channel flow of heterogeneous
mixtures (Chapters 4 and 6).
This chapter discussed some specific valve types such pinch, rubber sleeve, and check
valves. These valves have their own experimental loss coefficients, which need to be obtained
from manufacturers. This chapter presented the conventional K and the new two-K
loss factors. The two-K factor as developed by Hooper is of particular importance for
slurry flows at low Reynolds numbers. The engineer should therefore avoid the common
pitfall of using published data on turbulent water flows for conventional waterworks
valves when estimating the losses in a slurry system.
2-13 NOMENCLATURE
A Cross-sectional area of the flow
As Surface area
C Hazen–Williams roughness factor
Cc Coefficient of contraction
Cd Discharge coefficient
C� Specific heat or heat capacity
Cv Valve coefficient
Cve Velocity coefficient
din Pipe diameter expressed in inches
DH Hydraulic diameter = 4A/P
Di Conduit inner diameter (m)
Dij Inner diameter of the pipe j
E Energy per unit mass
EA Energy added per unit mass
EE Energy extracted per unit mass
Ef Energy due to friction loss per unit mass
Em Energy lost due to fittings per unit mass
Ep Energy due to static pressure per unit mass
Ev Energy due to dynamic pressure per unit mass
Ez Potential energy per unit mass due to elevation above a reference point
fD Darcy friction factor
fN Fanning friction factor
Fr Friction force
F12 Force between points 1 and 2
g Acceleration due to gravity (9.8 m/s2 )

FUNDAMENTALS OF WATER FLOWS IN PIPES
2.63
gc Conversion factor between slugs and lbm or 32.2 ft/sec2 h Spacing between plates
Hf Head loss due to friction
Hv Head loss in the Hazen–Williams formula
kth Conductivity
Kf Pressure loss of the fitting f
L Length of conduit or pipe
Lc Characteristic length
Le Entrance length
Lj Length of the conduit j
m The mass of the body
P Pressure
Ppsi Pressure in psi
Pwt Rate of heat power transfer
Q Flow rate (m3 /s)
Qgpm Flow rate expressed in US gallons per minute
Qideal Ideal flow rate through an orifice as product of area and velocity
qth Heat transfer per unit area
r local radius
RI Radius at the inner wall of the pipe, or inner radius in an annular flow
RH Hydraulic radius = area/perimeter
R Resistance factor for thermal insulation
Ri is the pipe inner radius (at the inside wall of the pipe)
RO Outer radius in an anuular flow
Rth thermal factor
Re Reynolds number
S Slope or head per unit length
S.G. Specific gravity
T Average temperature
TDH Total dynamic head that a pump is required to develop
u Velocity of the flow at distance y
U Average speed of a flow outside the boundary layer
Uf Friction velocity
Umax Maximum speed in the boundary layer
VO Practical velocity across an orifice due to vena contracta
Vth Theoretical velocity across an orifice
W weight
y + The relative distance from the wall in the boundary layer
ZA Elevation of a point above a reference grade
� Shear strain
d�/dt Wall shear rate or rate of shear strain with respect to time
� Diffusivity
� The thickness of the boundary layer �
� Linear roughness (m)
� Carrier liquid absolute or dynamic viscosity (usually expressed in Pascal-seconds
or poise)
� Pythagoras number (ratio of circumference of a circle to its diameter)
� Duration of the shear for a time-dependent fluid
� Density in kg/m3 or slug/ft3 �e Thermal resistivity
� Shear stress at a height y or at a radius r

2.64 CHAPTER TWO
�w Wall shear stress
� kinematic viscosity (defined as absolute viscosity divided by density)
2-14 REFERENCES
Hooper W. B. 1992. Fittings, Number and Types. pp. 391–397 of The Piping Design Handbook,
Edited by J. J. McKetta. New York: Marcel Dekker.
Ingersoll Rand. 1977. The Cameron Hydraulic Handbook. Ner Jersey: The Ingersoll Rand Company.
Johnson, M. 1982. Non-Newtonian Fluid System Design. Some Problems and Their Solutions. Paper
read at the 8th International Conference on the Hydraulic Transport of Solids in Pipe, Johannesburg,
South Africa.
Lindeburg, M. R. 1997. Mechanical Engineering Reference Manual. Belmont, CA: Professional
Publications Inc.
Schlichting, H. 1968. Boundary Layer Theory, 6th ed. New York: McGraw-Hill.
The Hydraulic Institute.1990. Engineering Data Book. Cleveland, OH: The Hydraulic Institute.
Wasp E., J. Penny, and R. Handy. 1977. Solid–Liquid Flow Slurry Pipeline Transportation. Aedermannsdorf,
Switzerland: Trans Tech Publications.

Index terms Links
A
ABB wrap-around motors 1.25
Abrasion, (see also white cast iron) 8.3
ACME thread 8.49 8.50
Activators 7.39
Adsorption 7.38
Affinity laws 8.60
Agitators (see also mixers)
anchor 7.43 7.44
biological treatment 7.44
critical speed 7.45
forces 7.45 7.47 7.54 7.55 7.56 7.57
7.59
helicoidal 7.44
pump 8.56
propeller-type 7.43 7.47
Air release valve 11.22 11.23
Albertson shape factor 3.12 3.13 3.14 3.15 3.16 3.17
Alkination 11.10
Allen’s equation 3.6
Alundum 10.21
Anderson 8.6
Andesite 7.22
Angularity test 1.11
Antemina 11.31
Anti-dunes 6.30 6.32 6.33
Apatite 7.39 11.17
API (American Petroleum Institute)
Grade 5LX65 11.29
Grade 5LX70 11.29
RP1102 11.28
RP1109 11.28
RP1111 11.28
Std 1104 11.29
Std 1130 11.29
Archimedean number 4.11 4.13 4.51
Area ratio 8.7
Argentina 1.3 1.34 11.22
ASARCO 1.25
Ash 10.21
ASTM Standards
A532 10.6 10.10
B3 1.11 11.28

Index terms Links
D422 12.21
D698 12.22
D854 12.21
D1556 12.21
D2488 1.13
D3017 12.21
D3155 1.8
D4767 12.22
Aswan High Dam 1.35
Athabasca region 11.23
Austenitic 10.9
Australia 1.34 1.35 11.10 11.11
Australian standard AS2576 11.19
B
Backfill 11.24 11.25 11.26
Bajo Alumbrera 1.3 11.22
Ball mill (see mill)
Barite 7.17 7.22
Basalt 7.22
Batu Hijau 1.34
Bauxite 3.19 5.3 5.4 5.7 5.35 5.36
5.38 7.17 7.22 10.21 11.1
Bearings, pump 8.2 8.3
Bed layer 6.18
Bedforms (see dunes)
Belonovo.Novosibirsk (Pipeline) 1.34 11.10
Bentonite 11.22
Bernoulli’s equation 2.58
BHRA or BHR Group 4.2
Bingham plastics 3.17 3.18 3.19 3.20 3.21 3.22
5.2
5.11 6.4
Bitumen (see also oil sands, tar) 11.24
Black Mesa Pipeline 1.34 11.2 11.7
Blasius’s equation 2.35
Blockage
Bolt adjustement for pump bearing
4.43 4.45 4.46
cartridge 8.3
Bond equation 7.1
Bougainville Pipeline 1.34
Boundary Layer 2.45
Brazil 1.34
Brazilian split test
British Standard
1.11
BS 812 1.11

Index terms Links
BS 5930 1.8
Buckingham equation 3.34 5.5 5.6
Budryck’s equation 3.7 3.8 3.9
Buffer layer 2.45
Buoyancy 3.3
C
Calavaras Pipeline 1.34
Calcium carbonate 14
California 1.3
Cameron 2.50
Canada 11.23 11.24
Canal permissible velocity 6.25
Carbides 10.10
Carbon 10.21
Carbonate 1.8
Carborundum 10.21
Carman, Kozeney equation 3.10
Carson slurries 3.29 3.37
Cascades 6.54 6.55 6.56
Cast iron 10.3 10.4 11.10
Cavitation
Celik and Rodi, method for dilute
8.18 8.20 8.22
8.24 8.25
flows in launders 6.18 6.21
Cellulose acetate 3.21
Cement, kiln feed slurry 1.16 3.19 5.3 5.4 5.7
Cement clinker 7.22
Centrifuges 7.12 7.62 7.63 7.64
Chalcopyrite 7.38
Chemineer scale, for agitation 7.49 7.50 7.51
Chemisorption 7.39
Chevron Phosphate Pipeline 1.34 11.19 11.20
Chezy number 6.4 6.9 6.32
Chile
Chilton and Stainsby method for
1.3 11.22
non-Newtonian flows 5.19 5.20 5.21 5.22
China 1.33 1.34
Chokes 2.12 11.20 12.22
Chongin Pipeline 1.34
Churchill’s equation 4.52 5.30
Clarification 1.5 1.31
Classification
Classifier
1.25 1.26
hydraulic
hydrocyclone (see
hydrocyclone)
7.32

Index terms Links
mechanical 1.26 7.33
Clay 1.12 1.15 1.16 1.18 1.34 3.19
3.32 5.4
5.7 5.31 5.38 5.39
Clift shape factor 3.13 3.14 3.15
Closure and reclamation plan 12.23 12.24
Coal 1.17 1.32 1.34 3.10 3.19 3.32
4.26 7.17 10.21 11.1
degradation 11.3 11.4
dewatering 11.6 11.7
ship loading 11.8 11.9
size of particles for pumping 11.2 11.3
tailings 3.19
terminal velocity 3.10
Coal-magnetite mixture 11.4 11.5
Coal-oil mixture
11.5 11.6
Coal-water mixture, combustion 11.8 11.9 11.10
Coalescence 1.19
Cobbles 6.25
Codelco 11.22
Coefficient of curvature 1.13
Coefficient of uniformity 1.13
Coke 7.22
Colebrook’s equation 2.9 2.10
Collahausi Pipeline 1.34
Colorado School of Mines 1.32
Compound mixtures 4.33
Concentrate
Concentration
1.27 1.30
volume 1.19 1.20 1.21 1.22
weight 1.19 1.21
Conduction 2.60
Conductivity 2.60
Conduits, pressure losses 2.44 2.46 2.47
Consolidated Coal Company 1.32 4.30
Consolidation Pipeline 1.34
Contact load
Copper
4.51
concentrate slurry 1.16 1.32 1.34 3.19 10.21 11.21
ore 1.34 7.22
tailings 1.34
Cost estimates 12.24 12.27
Coulombic friction 4.51
Critical depth, in open channels 6.59
Critical slope 4.59
Critical speed 11.14

Index terms Links
Critical velocity, in open channels 6.23
Crushing 1.3 1.24 1.25 7.3 7.11
Crushers
cone
gyratory 7.7 7.8
impact 7.8 7.10
jaw 7.5 7.6 7.7
primary 1.25 7.4
roll 7.11
secondary 1.25 7.9
tertiary 1.25
Cuajone 1.30
Cunningham Creek Dam 1.35
Cutwater, pump 8.29
Cyclone (see also hydrocyclones) 1.25 1.28 1.29
D
Dallas White Rock Pipeline 1.34
Darcy, Weisbach equation 2.45
Darby
method for Bingham plastics 5.9 5.10
method for yield-pseudoplastics 5.24 5.25 5.26 5.27 5.28
Density 1.8 1.19 1.20
Deposited bed 4.56
Dewatering 1.30 11.6 11.7 11.10
Diffuser 8.13
Diffusivity, thermal 2.48
Dilatancy 1.12
Dilatant slurries 3.17 3.28
Diorite 7.22
Discharge, coefficient of 2.49 2.50
Dithiosphophate 7.39
Dodge and Metzner model 5.22
Dolomite 7.17
Dominguez’s equation 6.26 6.27
Drag
coefficient 3.2 3.3 3.4 3.5 5.39 5.40
force 3.1 3.2 7.45
reduction in non-Newtonian
flows 5.39
Dredge, arm 1.7
Dredgeability 1.15
Dredging 1.5 1.7 1.34 1.36 8.6 12.6
sand cutter 1.5 1.9
Drillability, test 1.11
Drive end, pump 8.2 8.42 8.52
Drop boxes 6.55 6.56 6.71

Index terms Links
Duboy’s equation 6.11 6.36
Ductile iron 10.4
Dunes 6.30 6.32 6.33
Durand and Condolios’s equations 1.32 4.1 4.20
velocity factor 4.8
Dust, blast furnace 10.21
Dynamic seal (see expeller)
E
Efficiency, hydraulic 8.5 8.18 8.22
Efficiency ratio, pump 8.64 8.72
Egypt 1.3 1.6 1.32 1.34
Einstein’s equation 1.21 6.18 6.36
Einstein, Brown’s equation 6.36
Elasticity, static modulus 1.11
Electrometallurgy 1.24
Emergency pond 12.9
Emulsions, 1.16 1.17 1.19
Entrance length 2.45
Equivalent length of fittings 2.45 2.56
Erosion, corrosion 12.19
Escondida 1.3 1.34 11.22
Eskay Creek 11.2
Ethiopia 1.34
Ethyl xanthate 7.38
ETSI Pipeline 1.34 11.2
Eutectic 10.10
Expeller 8.2 8.5 8.34
F
Fanning friction (see friction factor)
Feldspar 7.17 7.22
Ferrite 10.10
Ferrochrome 7.22 10.7
Ferro-manganese 7.22
Ferro-molybdenum 10.6
Ferro-silicon 7.22
Filtering 1.27 1.30
Fittings 2.54 2.55 2.56 2.57
5.3 5.4 5.75
Flange loads 8.52
Flint 7.17
Flocculation launders 6.44
Flocculants 3.24 7.38 12.5
Flotation 1.3 1.25 1.26 1.27 1.28
1.29 7.12
froth 1.28
7.38

Index terms Links
Flows
heterogeneous 1.16
homogeneous 1.16
laminar 1.19
transition 2.3
turbulent 2.3
Flue gas desulphurisation (FGD) 10.9
Flyash 3.21 10.21
Fluorspar 7.17 7.22
Fracture toughness 10.10
Francis’s equation 3.8
Freeport 1.34
Friction
factor for single-phase fluids 1.19 2.4 2.12
factor for heterogeneous flows 4.15
force 2.2
gradient 2.24 2.44
Of heterogeneous flows 4.19 4.41 6.29 6.39
velocity 2.35 4.30 6.30 6.34
Frost and Nielsen’s equation 8.22
Froth (see also flotation) 3.18 7.39 8.2 8.56 8.57 8.68
8.70
8.72
Frothers 7.39
Froude number 4.11 6.45 6.67
G
G-gradient 6.44
Gabbro 7.17 7.22
Gangue 1.24 1.26
Garbett and Yiu 11.10
Geho pumps 9.8 9.13 11.22
Geller and Gray 4.25
Gilles equation 4.11 4.13 4.15
Gioasfertil Phosphate Pipeline 11.20
Gladstone Pipeline 11.10 11.13
Gland, pump 8.2
Glass 7.17 7.22
Gneiss 7.22
Gold
ore 7.22
tailings 3.19 11.2
Gradient
energy (see friction gradient) 2.59
hydraulic 2.59
Graf and Acaroglu’s equation 4.45 4.46 6.33 6.34
Granite 7.17 7.22

Index terms Links
Graphite 1.28 3.21 7.22
Gravel 1.15 3.10 6.25 7.22
Grease cup, pump 8.2
Green 6.45
Grey iron 10.3
Grinability work index 7.14 7.17
Grinding 1.24 1.25 1.26 1.30 7.12 7.31
Gypsum rock 7.17 7.22 10.21
H
Hagen, Poiseville equation 2.6
Hanks and Dadia equation 5.9
Hanks and Pratt equation 5.8
Hanks and Ricks, method for yield
pseudoplastics 5.17 5.18
Hardness of particles 1.3 1.10 10.3 11.17 A.1
Hatch and Associates 1.30
Hayden and Stelson equation 4.29
Hazen, William’s method 2.18 2.19 2.20
Head coefficient 8.14
Head loss, in open channels 6.3
Head, pump 8.5 8.8 8.10 8.25 8.26
Head ratio, pump 8.64 8.72
Heat
capacity 2.61
transfer 2.61
Hedstrom number 5.1 5.2 5.6
Hematite (see also iron ore) 7.22 7.39
Herschel, Bulkley model 5.19 5.21
viscosity 5.19
Heterogeneous flows 1.16
Heywood’s equation 5.14 5.18
High chrome alloy (see also 8.4 10.5 10.10
white cast iron)
High-density polyethylene (HDPE) 1.31 2.19 2.20 11.8 11.22
Hindustan Zinc Phosphate Pipeline 11.31
Homogeneous flows 1.16
Hoses 10.15
Hunt’s equation 4.31
Hydraulic diameter 6.3
Hydraulic friction gradient of
heterogeneous flow 4.9 4.19 4.25
Hydraulic friction gradient of water 2.19 2.20 2.22 2.44 4.9
Hydraulic grade line 2.58 2.59
Hydraulic jump 6.60 6.63
Hydraulic radius 6.2 6.3

Index terms Links
Hydrocyclone 1.25 1.28 1.29 7.12 7.23 7.33
7.38 11.7 11.25
Hydrolic degradation 10.17
Hydrometallurgy 1.22
Hyperchrome 11.19
Hypereutectic
I
10.10
Impactors 11.22
Impeller 8.2 8.7 8.18 8.31 8.34 8.42
Inclined flows 4.58 4.61
India
Inter-Agency Committee on Water
11.21
Resources 3.14
International Society for Rock
Mechanics
1.11
Inverted U-flowmeter
Iron (see also cast iron)
4.57
concentrate 1.28 1.32 1.34 10.21
ore 7.22 10.21 11.12
ore, hematite 7.22
ore, oolitic 7.22
ore, specular 7.22
ore, taconite 7.22
oxide 5.3 5.4 5.7
sand 11.1
Irvine, method for pseudoplastics 5.16
Ismail equation for sediment
distribution
J
4 30
Jian Shan Phosphate Pipeline
K
K-factor
1.34
for conduits in turbulent flows 2.45 2.46 2.47 2.56
at low Reynolds Numbers 2.54
5.74
2.55 2.56 2.57 5.3 5.4
Kaolin (see also clay) 3.19 3.21 3.22 3.23 5.38 5.39
9.7 10.21
Karasev function 6.41
Kenya 1.4
Kimbelite tails 3.19
Khan and Richardson model for
two
layers 4.53
Knelson, concentrators 7.63
Koorawatha dam 1.35

Index terms Links
Korrumbyn Creek Dam 1.35
Kozney’s equation 3.11 3.12
KSB-GIW, slurry research lab 1.19
L
Laminar flows, single phase 2.6 2.8
La Parla, Hercules Pipeline 1.34 11.15
Larox valves 11.22
Las Truchas 1.34
Launders (see also open 10.14 10.15
channel flows)
Leaching
acid 11.27
alkaline 11.27
Lead, ore 7.22
Lead, zinc, ore 7.22
Length, equivalent 2.56 2.57
Lift
force 3.1 3.2 3.10 4.25 7.45
coefficient 3.2
Limestone slurry 1.16 1.34 3.19 5.3 5.7 7.17
7.22 10.21 11.1 11.1 1.13
Liminite 3.19 5.3 5.4 5.7
Limonite 10.21
Liners
clay 11.27
geological 11.27
mantle 10.10
pump (see also rubber, white 8.2 8.27 10.10 10.17
cast iron)
synthetic 11.27
Los Bronces 1.34
M
Magnesite 7.22
Magnesium hydroxide 3.21
Magnetite 7.17 10.21
Malleable iron 10.4
Manganese ore 7.22
Manning number 6.4 6.5 6.6 6.7
Martensitic 10.9
Materials selection 10.1 10.21 10.19
Mazdak pump 8.3 8.30 8.54 8.56 8.58 8.59
Mazdak, Slurry Research Lab 1.19
Melbourne University, Australia 1.19
Metallurgy, extractive 1.22
Metals, stress, strain 10.1 10.3

Index terms Links
Metso Minerals (formerly the 7.4 7.13
companies of Nordberg and
Svedala)
Metzner 1.22 1.24
Metzner, Reed, method for
pseudoplastics 5.11 5.13
Meyer Peter curve 6.36
Meyer Peter Muler curve 6.36
Michigan Limestone Tailings
pipeline 1.34
Mill
autogeneous 1.25 1.30
ball 1.25 1.30 7.12 7.21 7.23 7.26
critical speed 7.25
grate 7.24
hammer 7.24 7.28 7.31
liners 7.21 7.23 7.25
overflow 7.24
pebble 1.25 7.23 7.24
rod 1.25 7.19 7.23 7.26
roller 7.28
semi. Autogenous (SAG) 1.25 1.26 1.29 7.23 7.27
spindle 7.24 7.28
tower 7.24 7.28
trommels 7.25
trunnions 7.23
tumbling 7.23
vertical 1.30 7.24 7.28
vibrating 7.24 7.28
Miller number 9.1 10.20 10.22 11.18
Milling (see also mills) 1.3 1.24 1.24
Mixers 7.4 7.59
correction factor 7.49 7.50
diameter sizing 7.51
equivalent volume 7.48
flow coefficient 7.48 7.49
power coefficient 7.48 7.49
Reynolds number 7.48
Mixing length 6.9
Moballoy, white iron 10.10
Mobile Pulley and Machine Works 1.7 1.8 1.9 10.10
Mogas valves 11.22
Molerus diagram 5.26
Molybdenum 1.28 7.22
Montuori number 6.55 6.56 6.57

Index terms Links
Moody diagram 2.10 2.11 5.29
Motor, wrap.around 1.25
Mud (see also red mud)
drilling 1.16 3.21 10.21 11.22
Munroe’s equation 3.8
N
Natural slurry flows 1.4 1.5
Newtonian flow 3.17
New Zealand Sands 1.34
Newitt 4.1 4.2 4.28 4.29 4.44
Nickel 4.28 7.22
Ni-hard alloys (see also white 8.4 10.5 10.8 11.10
cast iron)
Nikrudase roughness 5.3 5.4 5.71 6.34
Nile River 1.3 1.5
Non-Newtonian flows 1.16 2.49 3.17 3.38 5.1 5.42
7.54 11.12
Nordberg (see Metso Minerals)
Nordstrom valves 11.22
Nozzle 2.49
NPSH 8.18 8.20 8.22 8.23 8.24 8.25
O
Ohio Pipeline 11.7
Oil sands, (see also tar sands) 11.4 11.23 11.24
Oil shale 7.22
OK Tedi 1.34 11.21
Open channel flows 1.29
Ore, classification 1.24 1.26
Orifice plate 2.49 2.50
Orimulsion 1.17 5.29
P
Particle sizes conversion 1.14
Peat 1.15
Pechuka tanks 7.42
PDVSA, Bitor 1.17
Pena pipeline 1.34
Permanent International
Association 1.5 1.10 1.11 1.12
of Navigation Congresses
Pipe
concrete 2.10 2.12
high-density polyethylene 2.10 2.19 2.20
rubber-lined 2.10 2.12 2.16 2.17 2.18
steel 2.10 2.13
2.15
Pipelines
Pinto Valley
1.27
1.34
1.29 1.31 1.34 11.1 11.31

Index terms Links
Phosphate
maton 11.18
matrix 10.9 10.21 11.15
ore concentrate 1.34 11.1 11.15
rock 7.22 12.6
tailings 3.19
Phosphoric acid 10.9 11.15 11.19
Placer mining 1.15
Plasticity index 1.15
Point load test 1.11
Poiseville flow 2.3 2.6
Porosity 1.8
Potash ore 7.22
Prandtl, Colebrook’s equation 2.10
Pressure
dynamic 2.4
gradient for heterogeneous
flows 4.19 4.41
loss due to friction 2.45
loss in fittings at low Reynolds
numbers 5.3 5.4 5.74 5.75
pump 8.28 8.29
start-up 5.2 5.5
Pseudohomogeneous flows 4.18 4.19 4.47 4.48
Pseudoplastics 3.17 5.11 5.17
yield 3.17 5.17 5.22 5.24 5.28
Pulley, radial force 8.51 8.52
Pulp and paper, friction calculations 5.40 5.41
Pumps 1.28 1.36 8.1 8.72 9.1 9.16
chopper 8.59
diaphragm 9.8 9.9 9.10 9.11 9.12 9.13
dredge 8.2 8.59 8.60 10.10
froth-handling 8.2
lockhopper 9.15 9.16
low-head 8.2
mill discharge 8.2
performance corrections 8.61 8.62 8.72
peristaltic 9.13
piston 9.1 9.2 9.3 9.4 9.5
plunger 9.6 9.7 9.8 11.15 11.20 11.21
positive displacement 1.31 9.1 9.16 11.24
rotary lobe 9.14 9.15
submersible 8.2 12.9 12.10
tailings 8.2
tank 8.57

Index terms Links
unlined 8.3
vertical cantilever 8.2 8.53 8.59
Pyrhotite, ore 7.22
Pyrite 7.22 10.21
Pyrometallurgy 1.24
Q
Quartz 7.17 7.22
Quartzite 7.17 7.22
Quipolly Reservoir 1.35
R
R-value 2.60 2.61
Rabinowitsch, Mooney, 5.11
method for pseudoplastics
Radial, force 7.47
Radium 11.27
Reagents 1.28
Recirculation, in impellers 8.12
Recirculation load 7.21
Reclaim water 12.6
Red mud 3.19 5.35 5.37
Repeller (see expeller)
Resistance, thermal 2.60
Reynolds Number
single phase fluids 2.3 2.5
critical 2.4 5.8 5.14 5.15 5.18
modified 5.11 5.14 5.20 5.25
particle
Rheology 3.32 3.38 12.5 12.6
Rheopectic slurries 3.17 3.30
Richards’s equation 3.6
Rittinger’s equation 3.7 3.9
Rod mills 7.19
Roughness
absolute 2.10 2.11 2.12 6.6 6.7 6.31
effects on non-Newtonian
flows 5.29 5.3 5.4 5.73
equivalent 6.25
piping material 2.10 2.11 2.12 6.31
relative 2.13 2.18
sand 4.52
Rubber
armadillo 10.14
Buna-N 2.53
carbon-black-filled 10.13
carbon-black and silicon-filled 10.13
carboxylic nitrile 10.17

Index terms Links
chlorobutyl 2.53
EPDM 2.53 10.15
food grade 10.12 10.15
fluoro-elastomer 10.15
Hypalon 2.53 10.15 10.17
jade green armabond 10.14
lining 7.19 7.20 7.35 8.4
natural 10.11 10.13
natural aashto 10.11
Neoprene 2.53 10.14
nitrile 10.15
polychlorene 10.13
polyurethane 10.16 10.18 11.26
pure tan gum 2.53 10.11
synthetic 10.13 10.17
viton 2.53 10.18
white food grade 10.12
Rubber/butadiene styrene, 10.12
graphite-filled
Rugby Pipeline 1.34
Russia 1.34 11.10
Rutile, ore
Ryan and Johnson, method for
7.22
pseudoplastics
S
SAG (see semiautogeneous mill)
5.14 5.15
Saltation 4.3 4.43 4.47
Samarco 1.34 11.12
Sand 1.15 1.18 6.25 10.21
mineral 1.28
Sandvik 7.7 7.8
Sandstone 7.17
Saskatchewan Science Research
Center
1.19 4.53 11.24
Savage River 1.34
SCADA 11.29 12.1 12.21
Schiller’s equation
Screens
4.9 4.11 4.46
banana 7.32
shaking 7.32
trommel 7.32
vibrating 7.32
Screening devices 7.31
Sedimentation 1.5 1.34 7.59
gravity 7.60 7.62

Index terms Links
Seismic velocity test 1.11
Semiautogeneous mill 7.12 7.18
Separation
electrostatic 1.2 1.26 1.27 1.28 1.29
gravity 1.25 1.26 1.27
magnetic 1.25 1.26 1.27 1.28 7.12 7.38
Settling design speed for mixers 7.49
Sewage 3.11 5.3 5.4 5.7 7.53 10.21
Shale 6.25 7.17 7.23
Shear rate 1.15 2.7 3.18 3.38 5.2 5.9
5.14 5.19 5.3 5.4 5.75 7.53
Shear stress of flows 2.2 2.8
Shook’s bimodal model 4.50
Shut-down of pipeline 4.45
Siemens wrap-around motors 1.25
Sierra Grande 1.34
Sieve diameter 3.16
Silica 7.22 7.39
sand 7.23 10.21
Silicon carbide 7.23
Silt 1.4 1.5 1.12 1.13 1.15 1.18
1.36 6.25
Siltation 1.3 1.34 1.35 1.36 1.37
Simplot 1.34
Slack flow (see open channel flows)
Slag 7.22
Sleeve, shaft 8.2
Slip of coarse materials 6.35
Slippage, wall 5.3 5.4 5.73
Slip factor, in pumps 8.11 8.13
Slip of coarse materials 6.35 8.61 8.64
Slug flow 6.55 6.56
Sodium silicate 7.23
SOGREAH 1.32 4.2
Soils
classification 1.5 1.6 1.11
coefficient of curvature 1.13
coefficient of uniformity 1. 13
cohesive 1.5 1.12
composition tests 1.8
liquid limit 1.13
non.cohesive 1.5
organic 1.12
particle sizes 1.13
plasticity 1.13 1.15

Index terms Links
stratification 1.10
strength 1.10 1.12
testing 1.8 1.12 1.15
textures 1.13
Southern Peru Copper 1.34
Specific heat 1.22
Specific speed, pump 8.14 8.18
Spodumene, ore
Stainless steel 10.9
Stepanoff 8.6
Steward 11.24 11.25
Stockpile 1.24 7.11
Stoke’s equation 3.5 3.6
Stratification velocity 4.49
Stratified flows 4.48 4.50
Stuffing box, pump 8.2
Suction mouthpiece, for dredging
boats 1.8
Suction plate, pump 8.2
Sudan 1.4
Suez Canal 1.32
Sulfur 10.21
Sulzer 1.18
Svedala (see Metso Minerals)
Swamee, Jain friction factor 2.9
Swirl number 7.33
Syenite 7.23
T
Taconite (see also iron ore) 1.16 1.24 4.14
7.17 7.22 11.15
Tailings 1.3 1.10 1.25 1.27 1.28 1.30
8.7 10.21 12.1 12.3 12.28
dam 12.11 12.18 12.21
submerged disposal 12.15 12.17
Tar (see also bitumen) 5.29
Texas A&M University 1.19
Thermal conductivity 1.22
Thickeners 7.60 7.61 7.62 12.2 12.5 12.6
Thickening 1.26 1.27 1.30
Thixotropic slurries 3.17 3.30 3.32 5.28 5.29
Thomas’s equation 1.21 1.22
Thorium oxide 3.23
Thread pull force 8.48 8.51
Throatbush, pump
Thrust
8.29 8.30
axial 8.42 8.48

Index terms Links
radial 8.42 8.48
Tin, ore 7.23
Titanium
ore 7.23
dioxide 3.23
Tomb chart, pumps 8.6
Tomita, method for pseudoplastics 5.13 5.16
Torrance, method for yield 5.18 5.19 5.29
pseudoplastics
Trap rock 7.23
Transition flows (single phase 2.8 2.9
liquids)
Turbine, slurry 1.31
Turbulent layer 2.45
Turton and Levenspiel equation 3.4
Two-layer model 4.50 4.56
U
Uganda 1.4
Ultrasonic velocity test 1.11
Unconfined compressive strength 1.5 1.10
Uranium tailings 3.19 11.27
V
Vallentine blockage factor 4.45 4.46
Valves
ball 2.46 2.53
check 2.46 2.51 2.53
coefficient 2.46 2.50
equivalent length 2.46 2.47
foot valve 2.46
knife gate 2.46 2.52 2.53
lift check valve 2.46
pinch 2.53 2.54 2.55
plug 2.53
sw D422 12.21
D698 12.22
D854 12.21
ing check valve 2.46
tilting check valve 2.46
Vedernikov number 6.55 6.56 6.57
Velocity
critical 1.17 6.23 6.26
deposition in open channels 6.28 6.29
sinking (see also terminal
below) 1.17 1.18
terminal 3.2 3.3 3.17 4.28

Index terms Links
transition 5.9
Viscous transition 1.19
Vena contracta 2.50
Vertimill 1.30
Viscometer 3.33 3.38
Viscosity
correction for volumetric 1.23
concentration
dynamic 1.15 1.19 1.21 1.22
eddy 6.9
effect on pumps 8.61 8.64
effective in launders 6.31
kinematic 2.4
Viscous sublayer 2.45
Volcanic ash 6.25
Volute 8.2 8.6 8.30
Von Karman constant 4.30 6.9
Vortex flow 8.7 8.8
W
Wall effect on terminal velocity 3.8 3.10
Water physical properties 2.21
Waterfalls 6.58 6.71
Wear 8.4 10.16
Wear plate, pumps 8.2
Weirs, for plunge pools 6.66 6.71
Wenglu Pipeline 1.34
West Irian 1.34
Wet end pump 8.2 8.53
White cast iron 8.3 8.6 10.4 10.11
Wilson and Judge correlation 4.11
Wilson, Snyder pumps 9.7 11.12 11.20
Wilson- Thomas method for
non-Newtonian flows 5.22 5.25 5.30 5.32
Wirth pumps 9.2 9.13
X
Xanthates 7.39
Z
Zandi and Govatos’s equation 4.16 4.21 4.22
Zinc
concentrate 3.19
ore (see also lead, zinc) 7.23

CONTENTS
Preface xvii
PART ONE HYDRAULICS OF SLURRY FLOWS
1 General Concepts of Slurry Flows 1.3
1-0 Introduction 1.4
1-1 Properties of Soils for Slurry Mixtures 1.5
1-1-1 Classifications of Soils for Slurry Mixtures 1.5
1-1-2 Testing of Soils 1.8
1-1-3 Textures of Soils 1.13
1-1-4 Plasticity of Soils 1.13
1-2 Slurry Flows 1.15
1-2-1 Homogeneous Flows 1.16
1-2-2 Heterogeneous Flows 1.16
1-2-3 Intermediate Flow Regimes 1.16
1-2-4 Flows of Emulsions 1.16
1-2-5 Flows of Emulsions - Slurry Mixtures 1.17
1-3 Sinking Velocity of Particles, and Critical Velocity of Flow 1.17
1-3-1 Sinking or Terminal Velocity of Particles 1.17
1-3-2 Critical Velocity of Flows 1.17
1-4 Density of a Slurry Mixture 1.19
1-5 Dynamic Viscosity of a Newtonian Slurry Mixture
1-5-1 Absolute (or Dynamic) Viscosity of Mixtures with Volume
1.21
Concentration Smaller Than 1%
1-5-2 Absolute (or Dynamic) Viscosity of Mixtures with Solids
1.21
with Volume Concentration Smaller Than 20%
1-5-3 Absolute (or Dynamic) Viscosity of Mixtures with High
1.21
Volume Concentration of Solids 1.22
1-6 Specific Heat 1.22
1-7 Thermal Conductivity and Heat Transfer 1.22
1-8 Slurry Circuits in Extractive Metallurgy 1.24
1-8-1 Crushing 1.24
1-8-2 Milling and Primary Grinding 1.25
1-8-3 Classification 1.26
1-8-4 Concentration and Separation Circuits 1.26
1-8-5 Piping the Concentrate 1.30
1-8-6 Disposal of the Tailings 1.30
1-9 Closed and Open Channel Flows, Pipelines Versus Launders 1.31
1-10 Historical Development of Slurry Pipelines 1.32
vii

viii CONTENTS
1-11 Sedimentation of Dams—A role for the Slurry Engineer 1.33
1-12 Conclusion 1.37
1-13 Nomenclature 1.37
1-14 References 1.38
2 Fundamentals of Water Flows in Pipes 2.1
2-0 Introduction 2.1
2-1 Shear Stress of Liquid Flows 2.1
2-2 Reynolds Number and Flow Regimes 2.3
2-3 Friction Factors 2.4
2-3-1 Laminar Friction Factors 2.6
2-3-2 Transition Flow Friction Factor 2.8
2-3-3 Friction Factor in Turbulent Flow 2.9
2-3-4 Hazen–Williams Formula 2.18
2.4 The Hydraulic Friction Gradient of Water in Rubber-Lined
Steel Pipes 2.19
2-5 Dynamics of the Boundary Layer 2.33
2-5-1 Entrance Length 2.33
2-5-2 Friction Velocity 2.35
2-6 Pressure Losses Due to Conduits and Fittings 2.44
2-7 Orifice Plates, Nozzles and Valves Head Losses 2.49
2-8 Pressure Losses Through Fittings at Low Reynolds Number 2.54
2-9 The Bernoulli Equation 2.58
2-10 Energy and Hydraulic Grade Lines with Friction 2.58
2-11 Fundamental Heat Transfer in Pipes 2.58
2-11-1 Conduction 2.60
2-11-2 Thermal Resistance 2.60
2-11-3 The R Value 2.60
2-11-4 The Specific Heat or Heat Capacity Cp 2.61
2-11-5 Characteristic Length 2.61
2-11-6 Thermal Diffusivity 2.61
2-11-7 Heat Transfer 2.61
2-12 Conclusion 2.62
2-13 Nomenclature 2.62
2-14 References 2.64
3 Mechanics of Suspension of Solids in Liquids 3.1
3-0 Introduction 3.1
3-1 Drag Coefficient and Terminal Velocity of Suspended Spheres
in a Fluid 3.1
3-1-1 The Airplane Analogy 3.1
3-1-2 Buoyancy of Floating Objects 3.3
3-1-3 Terminal Velocity of Spherical Particles
3-1-3-1 Terminal Velocity of a Sphere Falling in a
3.3
Vertical Tube 3.3
3-1-3-2 Very Fine Spheres 3.5
3-1-3-3 Intermediate Spheres 3.6
3-1-3-4 Large spheres 3.7
3-1-4 Effects of Cylindrical Walls on Terminal Velocity 3.8

CONTENTS
3-1-5 Effects of the Volumetric Concentration on the
Terminal Velocity 3.10
3-2 Generalized Drag Coefficient—The Concept of Shape Factor 3.12
3-3 Non-Newtonian Slurries 3.17
3-4 Time-Independent Non-Newtonian Mixtures 3.18
3-4-1 Bingham Plastics 3.18
3-4-2 Pseudoplastic Slurries 3.25
3-4-2-1 Homogeneous Pseudoplastics 3.25
3-4-2-2 Pseudohomogeneous Pseudoplastics 3.27
3-4-3 Dilatant Slurries 3.28
3-4-4 Yield Pseudoplastic Slurries 3.28
3-5 Time-Dependent Non-Newtonian Mixtures 3.30
3-5-1 Thixotropic Mixtures 3.30
3-6 Drag Coefficient of Solids Suspended in Non-Newtonian Flows 3.32
3-7 Measurement of Rheology 3.32
3-7-1 The Capillary-Tube Viscometer 3.33
3-7-2 The Coaxial Cylinder Rotary Viscometer 3.36
3-8 Conclusion 3.38
3-9 Nomenclature 3.38
3-10 References 3.41
4 Heterogeneous Flows of Settling Slurries 4.1
4-0 Introduction 4.1
4-1 Regimes of Flow of a Heterogeneous Mixture in Horizontal Pipe 4.2
4-1-1 Flow with a Stationary Bed 4.3
4-1-2 Flow with a Moving Bed 4.3
4-1-3 Suspension Maintained by Turbulence 4.4
4-1-4 Symmetric Flow at High Speed 4.4
4-2 Hold Up 4.5
4-3 Transitional Velocities 4.5
4-3-1 Transitional Velocities V1 and V2 4.7
4-3-2 The Transitional Velocity V3 or Speed for Minimum
Pressure Gradient 4.8
4-3-3 V4: Transition Speed between Heterogeneous and
Pseudohomogeneous Flow 4.18
4-4 Hydraulic Friction Gradient of Horizontal Heterogeneous Flows 4.19
4-4-1 Methods Based on the Drag Coefficient of Particles 4.21
4-4-2 Effect of Lift Forces 4.25
4-4-3 Russian Work on Coarse Coal 4.26
4-4-4 Equations for Nickel–Water Suspensions 4.28
4-4-5 Models Based on Terminal Velocity 4.28
4-5 Distribution of Particle Concentration in Compound Systems 4.30
4-6 Friction Losses for Compound Mixtures in Horizontal
Heterogeneous Flows 4.33
4-7 Saltation and Blockage 4.43
4-7-1 Pressure Drop Due to Saltation Flows 4.43
4-7-2 Restarting Pipelines after Shut-Down or Blockage 4.45
4-8 Pseudohomogeneous or Symmetric Flows 4.47
4-9 Stratified Flows 4.48
4-10 Two-Layer Models 4.50
ix

x CONTENTS
4-11 Vertical Flow of Coarse Particles 4.57
4-12 Inclined Heterogeneous Flows 4.58
4-12-1 Critical Slope of Inclined Pipes 4.59
4-12-2 Two-Layer Model for Inclined Flows 4.61
4-13 Conclusion 4.62
4-14 Nomenclature 4.63
4-15 References 4.66
5 Homogeneous Flows of Nonsettling Slurries 5.1
5-0 Introduction 5.1
5-1 Friction Losses for Bingham Plastics 5.2
5-1-1 Start-up Pressure 5.2
5-1-2 Friction Factor in Laminar Regime 5.5
5-1-3 Transition to Turbulent Flow Regime 5.8
5-1-4 Friction Factor in the Turbulent Flow Regime 5.9
5-2 Friction Losses for Pseudoplastics 5.11
5-2-1 Laminar Flow 5.11
5-2-1-1 The Rabinowitsch–Mooney Relations 5.11
5-2-1-2 The Metzner and Reed Approach 5.11
5-2-1-3 The Tomita Method 5.13
5-2-1-3 Heywood Method 5.14
5-2-2 Transition Flow Regime 5.14
5-2-3 Turbulent Flow 5.14
5.3 Friction Losses for Yield Pseudoplastics 5.17
5-3-1 The Hanks and Ricks Method 5.17
5-3-2 The Heywood Method 5.18
5-3-3 The Torrance Method 5.18
5-4 Generalized Methods 5.19
5-4-1 The Hershel–Bulkley Model 5.19
5-4-2 The Chilton and Stainsby Method 5.19
5-4-3 The Wilson–Thomas Method 5.22
5-4-4 The Darby Method: Taking into Account Particle Distribution 5.24
5-5 Time-Dependent Non-Newtonian Slurries 5.28
5-6 Emulsions 5.29
5-7 Roughness Effects on Friction Coefficients 5.29
5-8 Wall Slippage 5.33
5-9 Pressure Loss through Pipe Fittings 5.34
5-10 Scaling up From Small to Large Pipes 5.35
5-11 Practical Cases of Non-Newtonian Slurries 5.35
5-11-1 Bauxite Residue 5.35
5-11-2 Kaolin Slurries 5.38
5-12 Drag Reduction 5.39
5-13 Pulp and Paper 5.40
5-14 Conclusion 5.41
5-15 Nomenclature 5.42
5-16 References 5.44
6 Slurry Flow In Open Channels and Drop Boxes 6.1
6-0 Introduction 6.1
6-1 Friction for Single-Phase Flows in Open Channels 6.2

CONTENTS
6-2 Transportation of Sediments in an Open Channel 6.9
6-2-1 Measurements of the Concentration of Sediments 6.12
6-2-2 Mean Concentrations for Dilute Mixtures (C v < 0.1) 6.18
6-2-3 Magnitude of � 6.22
6-3 Critical Velocity and Critical Shear Stress 6.23
6-4 Deposition Velocity 6.27
6-5 Flow Resistance and Friction Factor for Heterogeneous Slurry Flows 6.29
6-5-1 Flow Resistances in Terms of Friction Velocity 6.30
6-5-2 Friction Factors 6.31
6-5-2-1 Effect of Roughness 6.31
6-5-2-2 Effect of Particle Concentration on Slurry Viscosity 6.31
6-5-2-3 Effects of Particle Sizes on the Chezy Coefficient 6.32
6-5-2-4 Effect of Bed Form on the Friction 6.33
6-5-3 The Graf–Acaroglu Relation 6.33
6-5-4 Slip of Coarse Materials 6.35
6-5-5 Comparison between Different Models 6.36
6-6 Friction Losses and Slope for Homogeneous Slurry Flows 6.39
6-6-1 Bingham Plastics 6.40
6-7 Flocculation Launders 6.44
6-8 Froude Number and Stability of Slurry Flows 6.45
6-9 Methodology of Design 6.45
6-10 Slurry Flow in Cascades 6.54
6-11 Hydraulics of the Drop Box and the Plunge Pool 6.56
6-12 Plunge Pools and Drops Followed by Weirs 6.67
6-13 Conclusion 6.71
6-14 Nomenclature 6.71
6-15 References 6.74
PART TWO EQUIPMENT AND PIPELINES
7 Components of Slurry Plants 7.3
7-0 Introduction 7.3
7-1 Rock Crushing 7.3
7-1-1 Primary Crushers 7.4
7-1-1-1 Jaw Crushers 7.5
7-1-1-2 Gyratory Crushers 7.7
7-1-1-3 Impact Crushers 7.8
7-2 Secondary and Tertiary Crushers 7.9
7-2-1 Cone Crushers 7.9
7-2-2 Roll Crushers 7.11
7-3 Grinding Circuits 7.11
7-3-1 Single-Stage Circuits 7.21
7-3-2 Double-Stage Circuits 7.23
7-4 Horizontal Tumbling Mills 7.23
7-4-1 Rod Mills 7.26
7-4-2 Ball Mills 7.26
7-4-3 Autogeneous and Semiautogeneous Mills 7.26
7-5 Agitated Grinding 7.27
7-5-1 Vertical Tower Mills 7.28
xi

xii CONTENTS
7-5-2 Vertical Spindle Mills 7.28
7-5-3 Roller Mills 7.28
7.5.4 Vibrating Ball Mills 7.28
7.5.5 Hammer Mills 7.31
7-6 Screening Devices 7.31
7-6-1 Trommel Screens 7.32
7-6-2 Shaking Screens 7.32
7-6-3 Vibrating Screens 7.32
7-6-4 Banana Screens 7.32
7-7 Slurry Classifiers 7.32
7-7-1 Hydraulic Classifiers 7.32
7-7-2 Mechanical Classifiers 7.33
7-7-3 Hydrocyclones 7.33
7-7-4 Magnetic Separators 7.38
7-8 Flotation Circuits 7.38
7-9 Mixers and Agitators 7.40
7-10 Sedimentation 7.59
7-10-1 Gravity Sedimentation 7.60
7-10-2 Centrifuges 7.62
7-11 Conclusion 7.64
7-12 Nomenclature 7.64
7-13 References 7.66
8 The Design of Centrifugal Slurry Pumps 8.1
8.0 Introduction 8.1
8.1 The Centrifugal Slurry Pump 8.2
8.2 Elementary Hydraulics of the Slurry Pump 8.6
8.2.1 Vortex Flow 8.7
8-2-2 The Ideal Euler Head 8.8
8-2-3 Slip of Flow Through Impeller Channels 8.11
8-2-4 The Specific Speed 8.14
8-2-5 Net Positive Suction Head and Cavitation 8.18
8-3 The Pump Casing 8.25
8-4 The Impeller, the Expeller and the Dynamic Seal 8.34
8-5 Design of the Drive End 8.42
8-5-1 The Radial Thrust Due To Total Dynamic Head 8.43
8-5-2 The Axial Thrust Due to Pressure 8.43
8-5-3 Thread Pull Force 8.48
8-5-4 Radial Force on the Drive End 8.51
8-5-5 Total Forces from the Wet End 8.51
8-5-6 Flange Loads 8.52
8-6 Adjustment of the Wet End 8.53
8-7 Vertical Slurry Pumps 8.53
8-8 Gravel and Dredge Pumps 8.59
8-9 Affinity Laws 8.60
8-10 Performance Corrections for Slurry Pumps 8.61
8-10-1 Corrections for Viscosity and Slip
8-10-2 Concepts of Head Ratio and Efficiency Ratio Due to
8.61
Pumping Solids 8.64

CONTENTS
8-10-3 Concepts of Head Ratio and Efficiency Ratio Due to
Pumping Froth 8.68
8-11 Conclusion 8.72
8-12 Nomenclature 8.72
8-13 References 8.75
9 Positive Displacement Pumps 9.1
9-0 Introduction 9.1
9-1 Solid Piston Pumps 9.1
9-2 Plunger Pumps 9.6
9-3 Diaphragm Piston Pumps 9.8
9-4 Accessories for Piston and Plunger Pumps 9.13
9-5 Peristaltic Pumps 9.13
9-6 Rotary Lobe Slurry Pumps 9.14
9-7 The Lockhopper Pump 9.15
9-8 Conclusion 9.16
9-9 References 9.17
10 Materials Science for Slurry Systems 10.1
10.0 Introduction 10.1
10-1 The Stress- Strain Relationship of Metals 10.1
10-2 Iron and Its Alloys for the Slurry Industry 10.3
10-2-1 Grey Iron 10.3
10-2-2 Ductile Iron 10.4
10.3 White Iron 10.4
10-3-1 Malleable Iron 10.4
10-3-2 Low-Alloy White Irons 10.5
10-3-3 Ni-Hard 10.5
10-3-4 High-Chrome–Molybdenum Alloys 10.6
10.4 Natural Rubbers 10.11
10-4-1 Natural Aashto 10.12
10-4-2 Pure Tan Gum 10.12
10-4-3 White Food-Grade Natural Rubber 10.12
10-4-4 Carbon-Black-Filled Natural Rubber 10.13
10-4-5 Carbon-Black- and Silicon-Filled Natural Rubber
10-4-6 Hard Natural Rubber/ Butadiene Styrene Compound
10.13
Filled with Graphite 10.13
10-5 Synthetic Rubbers 10.13
10-5-1 Polychlorene (Neoprene) 10.14
10-5-2 Ethylene Propylene Terpolymer (EPDM) 10.15
10-5-3 Jade Green Armabond 10.15
10-5-4 Armadillo 10.15
10-5-5 Nitrile 10.15
10-5-6 Carboxylic Nitrile 10.17
10-5-7 Hypalon 10.17
10-5-8 Fluoro-elastomer (Viton) 10.18
10-5-9 Polyurethane 10.18
10-6 Wear Due to Slurries 10.18
10-7 Conclusion 10.21
10-8 References 10.22
xiii

xiv CONTENTS
11 Slurry Pipelines 11.1
11.0 Introduction 11.1
11-1 Bauxite Pumping 11.1
11-2 Gold Tailings 11.2
11-3 Coal Slurries 11.2
11-3-1 Size of Coal Particles 11.2
11-3-2 Degradation of Coal During Hydraulic Transport 11.3
11-3-3 Coal–Magnetite Mixtures 11.4
11-3-4 Chemical Additions to Coal–Water Mixtures. 11.5
11-3-5 Coal–Oil Mixtures 11.5
11-3-6 Dewatering Coal Slurry 11.6
11-3-7 Ship Loading Coarse Coal 11.8
11-3-8 Combustion of Coal–Water Mixtures (CWM) 11.8
11-3-9 Pumping Coal Slurry Mixtures 11.10
11-4 Limestone Pipelines 11.10
11-5 Iron Ore Slurry Pipelines 11.12
11-6 Phosphate and Phosphoric Acid Slurries 11.16
11-6-1 Rheology 11.17
11-6-2 Materials Selection for Phosphate 11.18
11-6-3 The Chevron Pipeline 11.19
11-6-4 The Goiasfertil Phosphate Pipeline 11.20
11-6-5 The Hindustan Zinc Phosphate Pipeline 11.21
11-7 Copper Slurry and Concentrate Pipelines 11.21
11-8 Clay and Drilling Muds 11.22
11-9 Oil Sands 11.23
11-10 Backfill Pipelines 11.24
11-11 Uranium Tailings 11.27
11-12 Codes and Standards for Slurry Pipelines 11.27
11-13 Conclusion 11.30
11-14 References 11.31
12 Feasibility Study for A Slurry Pipeline
and Tailings Disposal System 12.1
12-0 Introduction 12.1
12-1 Project Definition 12.2
12-2 Rheology, Thickeners Performance, Pipeline Sizing 12.5
12-3 Reclaim Water Pipeline 12.8
12-4 Emergency Pond 12.9
12-5 Tailings Dams 12.11
12-5-1 Wall Building by Spigotting 12.11
12-5-2 Deposition by Cycloning 12.12
12-5-2-1 Mobile Cycloning by the Upstream Method 12.14
12-5-2-2 Mobile Cycloning by the Downstream Method 12.14
12-5-2-3 Deposition by Centerline 12.15
12-5-2-4 Multicellular Construction 12.15
12-6 Submerged Disposal 12.15
12-6-1 Subsea Deposition Techniques 12.17
12-7 Tailings Dam Design 12.17
12-8 Seepage Analysis of Tailings Dams 12.18

CONTENTS
12-9 Stability Analysis for Tailings Dams 12.18
12-10 Erosion and Corrosion 12.19
12-11 Hydraulics 12.19
12-12 Pump Station Design 12.19
12-13 Electric Power System 12.20
12-14 Telecommunications 12.21
12-15 Tailings Dam Monitoring 12.21
12-16 Choke Stations and Impactors 12.22
12-17 Establishing an Approach for Start-up and Shutdown 12.22
12-18 Closure and Reclamation Plan 12.23
12-19 Access and Service Roads 12.24
12-20 Cost Estimates 12.24
12-20-1 Capital Costs 12.24
12-20-2 Operation Cost Estimates 12.25
12-21 Project Implementation Plan 12.27
12-22 Conclusion 12.27
12-23 References 12.28
Appendix A Specific Gravity and Hardness of Minerals A.1
Appendix B Units of Measurement B.1
Index I.1
xv

PART ONE
HYDRAULICS OF
SLURRY FLOWS

CHAPTER 1
GENERAL CONCEPTS OF
SLURRY FLOWS
1-0 INTRODUCTION
Slurry is essentially a mixture of solids and liquids. Its physical characteristics are dependent
on many factors such as size and distribution of particles, concentration of
solids in the liquid phase, size of the conduit, level of turbulence, temperature, and
absolute (or dynamic) viscosity of the carrier. Nature offers examples of slurry flows
such as seasonal floods that carry silt and gravel. Every year during the flood season,
the Nile transports massive amounts of silt over thousands of miles to the Saharan
desert. To rephrase Herodotus, who once said “Egypt is the gift of the Nile,” one may
consider that one of the most ancient civilizations was dependent on natural slurry flows
for its survival.
Dredging is one of the most common and ancient processes involving slurry flows; the
dredged materials contain a wide range of particles, tree debris, rocks, etc. Mining has
employed the concept of slurry flows in pipelines since the mid-nineteenth century, when
the technique was used to reclaim gold from placers in California. Long-distance slurry
pipelines have evolved in all continents since the mid 1950s. Some slurry mixtures consist
of very fine solids at high concentration, such as those in the copper concentrate
pipelines of Escondida, Chile, and Bajo Alumbrera, Argentina. Other mixtures are based
on coarse particles up to a size of 150 mm (6�), such as those pumped from fields of phosphate
matrix.
This chapter introduces some of the basic principles of slurry mixtures and flows. The
slurry engineer has to appreciate the properties of the soil to be mined, dredged, or mixed
with water. Original rock sizes, hardness, and plasticity play a major role in the selection
of the equipment for crushing, milling, flotation, tailings disposal, or soil reclamation.
Understanding sinking and critical speeds are essential when sizing the pipeline. A brief
introduction to slurry flows in extractive metallurgy serves the purpose of focusing on the
essentials of the application of slurry flows to engineering.
Natural slurry flows, even in very dilute forms, can have negative effects on the environment
if not properly managed. Some of the great dams of the world built in the
twentieth century are starting to suffer from siltation. Behind such dams, large lakes are
often man-made. The river flow is brought to a sufficiently slow speed for the silt to deposit
at the bottom. Engineers in the twenty-first century will have to learn to manage
the siltation of large man-made lakes using the science of dredging and piping slurry
flows.
1.3

1.4 CHAPTER ONE
1-1 PROPERTIES OF SOILS FOR
SLURRY MIXTURES
Slurry flows occur in nature in different ways. They are often associated with the transportation
of silt from one region to another. Strong rains lead to soil erosion, mud slides,
and the eventual drainage of slurries toward rivers. These are dilute slurries, in the sense
that the soils mix naturally at a weight ratio of solids to liquids smaller than 15%.
One very interesting river is the Nile. It may be said that during two months of the
year it becomes a massive slurry flow. Torrential tropical rains over Lake Victoria in
Uganda and Kenya are the source of the White Nile. Torrential tropical rains over the
Ethiopian plateau are the source of the Blue Nile. On their way to the Sudan, both
branches of this longest river in the world transport silt and soils. The White Nile seems
to lose a lot of its water as it enters the swamps of the Bahr El Ghazal in Sudan. What
is left of the White Nile joins the Blue Nile near Khartoum in Sudan. The Nile pursues
its trip to the north and gradually enters the Saharan desert through Nubia and Egypt. As
the flood season terminates, the silt transported by the Nile sediments by gravity. The
silt has deposited for thousands of years, creating a narrow strip of rich farmland. Out
of this silt grew the towns and states in Nubia and Egypt. The Pharaohs built an advanced
civilization on the silt brought to them by the Nile’s natural slurry flows. The
“gift of the Nile” was silt that would not have been deposited without a form of natural
slurry flow.
A simplified flow sheet (Figure 1-1) of the Nile illustrates this natural slurry flow. The
steps in the process are:
� Water from the rains is the carrier liquid.
� The flow of water from the mountains of Uganda and Kenya moves fast enough during
the flood season to scour the ground of silt and transport it in the form of a dilute slurry.
(This is a step of slurry formation.)
Uganda/Kenya
Ethiopia
floods
floods
torential
rains
rains
silt transported
by the White Nile
silt transported
by the Blue Nile
Sedimentation
at Bahr El Ghazal
Sudan
Nubia
The Saharan Desert
Egypt
sedimentation by gravity
of the silt after the flood
(Egypt is the Gift of the Nile)
FIGURE 1-1 There is no better example of the importance of slurry to civilization than the
land of Egypt. For thousands of years, the Nile has transported massive quantities of silt over
thousands of kilometers to cover by its floods a narrow stretch of land. From these silt layers, a
civilization grew.

GENERAL CONCEPTS OF SLURRY FLOWS
� As the waters from the rains over the mountains of Uganda and Kenya join, they form
the White Nile. (This step is natural hydrotransport.)
� As the White Nile enters the Bahr El Ghazal in Sudan, it spreads and stagnates, forming
swamps. A nomadic life has long flourished around these swamps. (This step involves
partial sedimentation by stagnation in the swamps.)
� In another region (in Ethiopia), rains form the Blue Nile. The flow of water from the
mountains of Ethiopia move fast enough during the flood season to scour the ground of
silt and transport it in the form of a dilute slurry. (This is another step of slurry formation.)
� The Blue and White Nile merge near Khartoum, Sudan, and continue their flow to the
north.
� As the floods enter Nubia and Egypt, they overflow the banks of the Nile and transport
speed of the slurry mixture drops.
� Sedimentation of silt occurs, with Egypt acting as a massive clarifier for the waters of
the Nile, particularly at its delta with the Mediterranean Sea. (This step is natural gravity
sedimentation.)
For thousands of years the Pyramids and the Sphinx have stared at this immense natural
slurry clarifier that is the Valley of the Nile in the middle of the Saharan Desert (Figure 1-
2).
Dredging is an important engineering activity in which gravel is moved in the form of
slurry into a hopper on a specially constructed boat (Figure 1-4). A special pump is often
used in a drag arm (Figure 1-3), and a special suction mouthpiece (Figure 1-5) is used at
the tip of the drag arm.
To complete dredging and form the slurry, it is essential to cut through the sand layers,
rocks, and debris, using special cutters for sand (Figure 1-6a) and for rocks (Figure 1-6b)
with very hard, replaceable blades.
The composition of a slurry mixture depends on many factors such as particle size and
distribution. Particles may be found in nature as soils or may be created by the processes
of crushing, milling, and grinding. For applications such as dredging, natural soils are
pumped without any crushing or grinding. For mining processes, an understanding of the
physical properties of soils is essential for sizing equipment, crushing and milling, slurry
preparation, mixing, and pumping (see Figure 1-7).
1-1-1 Classifications of Soils for Slurry Mixtures
There are a variety of methods used to classify soils. Two main classes are:
1. Cohesive soils such as certain silts and clays with a median particle diameter smaller
than 0.0625 mm (less than 0.0025 in, or mesh 250)
2. Noncohesive soils such as certain silts and clays with a median particle diameter larger
than 0.0625 mm (larger than 0.0025 in, or mesh 250)
For underwater dredging, the rock’s strength is determined by its core, and this property
has a very important effect on the efficiency of dredging. Herbrich (1991) proposed a
classification of soils in terms of unconfined compressive strength (see Table 1-1).
The Permanent International Association of Navigation Congresses (1972) adopted a
system of classification of soils, reviewed by Sargent (1984) and summarized in Tables
1.5

1.6 CHAPTER ONE
FIGURE 1-2 For five thousand years, the Sphinx and the Pyramids have stared from the
Gizeh plateau in the desert at history and at the Nile, which transforms itself every summer into
a natural slurry transporter, bringing silt and life to the desert.
1-2, 1-3, and 1-4, that is recommended for use in dredging. In these tables, visual inspection
is mentioned as a quick way to determine the nature of soils. This method does not
relieve the engineer from the responsibility of conducting a proper size distribution test
and rheology test before any design.
The Standard D2488 of the American Society for Testing of Materials (ASTM)
(1993) also offers a classification of soils, with a range of particle sizes as presented in
Table 1-5. This standard is widely used in North America.

GENERAL CONCEPTS OF SLURRY FLOWS
discharge pipe
electric cable
pump
bottom of lake
drag arm column
FIGURE 1-3 Dredging boat and dredge arm.
FIGURE 1-4 Special dredger boat.
hopper for solids
1.7

1.8 CHAPTER ONE
FIGURE 1-5 Suction mouthpiece for boat dredger. Courtesy of Mobile Pulley and Machine
Works.
1-1-2 Testing of Soils
Various soil tests are recommended before mixing the soil with water in the early stages
of designing a dredging or slurry transportation system. Particle size distribution should
be established. Table 1-6 presents conversion factors between the three most common
scales for measuring particle size.
A number of tests are recommended to determine the dredgeability of soils and their
behavior in placer mining or slurry mixing (Table 1-7). In nature, silts may be found in
association with clays; thus, the parameters for both silts and clays should be assessed.
The following testing parameters are accepted by the industry.
Composition Tests
� Visual inspection: For the purpose of assessment of the rock mass. Such a test indicates
the in situ state of the rock mass. Tests may be conducted in situ or under lab
conditions in accordance with British Standard Institute Standard BS 5930 (1999).
� Section thickness test: A lab test conducted for the purpose of geotechnical identification
and as a tool to determine mineral composition of the rock mass.
� Bulk density: Wet and dry tests are conducted under laboratory conditions to assess the
weight and volume relationship. (International Journal of Rock Mechanics and Mineral
Sciences, 1979).
� Porosity: This is a calculation of voids as a percentage of total volume and is based on
lab tests on bulk density.
� Carbonate content: This lab test should be conducted in accordance with American Society
for Testing Materials (ASTM) Standard D3155 (1983) to measure lime content,
particularly in limestone and chalks.

(a)
(b)
FIGURE 1-6 (a) Special dredging sand cutter. The blades are replaceable. Courtesy of Mobile
Pulley and Machine Works. (b) Special dredging rock cutter. Courtesy of Mobile Pulley
and Machine Works.
1.9

1.10 CHAPTER ONE
FIGURE 1-7 Mineral process plants can reject fairly coarse material that is left after crushing
and milling mineral rocks. In this case, the coarse material is transported by piping in the
form of a tailings slurry and used to build a tailings dam.
Strength, Hardness, and Stratification Tests
� Surface hardness: This lab test should be conducted to determine hardness in terms of
the Mohr’s scale (from 0 for talc to 10 for diamonds). Appendix I presents a tabulation
of density and Mohr hardness of minerals. The hardness of minerals is critical to the
wear life of equipment associated with slurry flows.
� Uniaxial compression: This lab test measures ultimate strength under uniaxial stress.
These tests should be done on fully saturated samples. The dimensions of the test sample
and the directions of stratification influence stress direction. Cylinder samples
TABLE 1-1 Classification of Soils in Terms of Unconfined Compressive Strength.
(After Herbrich, 1991)
Unconfined compressive strength
Characteristic MPa 10 3 psi
Very weak < 1.25 < 0.145
Weak 1.25–5.0 0.15–0.73
Moderately weak 5.0–12.5 0.73–1.8
Moderately strong 12.5–50.0 1.8–7.3
Strong 50–100 7.3–14.6
Very strong 100–200 14.6–29.2
Extremely strong > 200 > 29.2

GENERAL CONCEPTS OF SLURRY FLOWS
TABLE 1-2 Classification of Noncohesive Dredged Soils after the Permanent
International Association of Navigation Congresses (1972, 1984)
should have a length-to-diameter ratio of 2:1, as per The International Society for Rock
Mechanics (1978).
� Brazilian split: This is a lab test to measure strength as derived from uniaxial testing.
This procedure is similar to the uniaxial compression test but with a different lengthto-diameter
ratio. For further details, consult The International Society for Rock Mechanics
(1977).
� Point load test: This is a quick lab test to measure strength. It should be conducted with
the uniaxial compression test as described by Broch and Franklin (1972).
� Seismic velocity test: This field in situ test is conducted to check on the stratigraphy
and fracturing of rock masses. It is useful for extrapolating field and lab measurements
to rock mass behavior.
� Ultrasonic velocity test: This lab test is conducted on cores in the longitudinal direction.
� Static modulus of elasticity: This lab test measures stress/strain rate and gives an indication
of the brittleness of rock.
� Drillability: This in situ test measures penetration rate, torque, feed force, fluid pressure,
depth of layers, etc., and is used to establish the drill techniques and specification
for placer mining or dredging.
� Angularity: This lab test is conducted to assess the shape of particles by visual inspection
in accordance with British Standard Institute BS 812 (1999).
The expertise of a geologist is essential for mining or dredging large areas.
1.11
Type
Identification of particle sizes
Strength and
of soils mm BS sieve units Identification structural properties
Boulders > 200 6 Visual
and 60–200 examination and
cobbles measurement
Gravel Fine 2–6 mm Fine No. 7— 1 Medium 6–20 mm
–
4 in
Medium
Visual May be found loose
1 –
4– 3 Coarse 20–80 mm
–
4 in
Coarse
examination in some fields, or in
3 –
4–3 in cemented beds, or
may appear as weak
conglomerate beds or
hard packed gravel
intermixed with sand
Sands Fine 0.06–0.2 mm Fine mesh 72–200 Visual Strength varies
Medium 0.2–0.6 mm Medium mesh 25–72 examination. No between compacted,
Coarse 0.6–2 mm Coarse mesh 7–25 cohesion when loose and cemented.
dry Homogeneous or
stratified structures.
Intermixture with silt
or clay may produce
hardpacked sands

TABLE 1-3 Classification of Cohesive Natural Soils after the Permanent International
Association of Navigation Congresses (1972, 1984)
Type
Identification of particle sizes
Strength and
of soils mm BS sieve Identification structural properties
Silts Fine 0.002–0.006 Passing No. Individual particles Coarse and sandy particles are
Medium 0.006–0.02 200 are invisible. Wet nonplastic but similar
Coarse 0.02–0.06 lumps or coarse are characteristics to sands. Fine
visible. Determination silts are plastic and similar to
by testing for clays. They are often found in
dilatancy*. Silt can nature intermixed with sand
be dusted off fingers and clay. They may be homoafter
drying and dry geneous or stratified and their
lumps are powdered consistency may vary from
by finger pressure fluid silt to stiff silt or siltstone
Clays Finer than 0.002 N/A Clays are very
Strength Shear strength
cohesive and are Very soft: may < 20kN/m2 plastic without be squeezed < 2.9 psi
dilatancy. Moist easily between
samples stick to fingers
fingers with smooth,
Soft: easily 20–40 kN/m2 greasy touch. Dry
lumps do not powder.
molded by
fingers
2.9–5.8 psi
Clays shrink and Firm: requires 40–75 kN/m2 crack by drying and strong pressure 5.8–10.9 psi
develop high strength to mold by
fingers
Structure of clays may Stiff: can not 75–150 kN/m2 be fissured, intact, be molded by 10.9–21.8 psi
homogeneous, fingers, dent
stratified, or by thumbnail
weathered.
Hard: tough, Above 150
intended with kN/m2 difficulty by
thumbnail
21.8 psi
*Dilatancy is a property exhibited by silt when shaken, and is due to high permeability of silt. When
a moistened sample is shaken in the open hand, water appears on the surface, giving it a glossy appearance.
TABLE 1-4 Classification of Organic Soils after the Permanent International
Association of Navigation Congresses (1972, 1984)
Type
Identification of particle sizes
Strength and
of soils mm BS sieve Identification structural properties
Peat and N/A N/A It is generally identified It may be firm or spongy in
organic as brown or black with a nature and its strength is
soils strong organic smell and different in horizontal and
contains wood and fibers. vertical directions.
1.12

1-1-3 Textures of Soils
Granular soils are found in nature as a mixture of particles of different sizes. Two coefficients
are used to express such texture:
1. The coefficient of curvature, C c (equation 1-1)
2. The coefficient of uniformity, C u (equation 1-2)
C c = (1-1)
Cu = � (1-2)
D10
Where D10, D30, and D60 are defined as the grain size at which 10%, 30%, and 60% of the
soil is finer. According to Herbrich (1991)
If 1 < C c < 3, the grain size distribution will be smooth
If C u > 4 for gravels then there is a wide range of sizes
If C u > 6 for sands then there is a wide range of sizes
Alternatively, the soil is said to contain very little fines and is well graded.
1-1-4 Plasticity of Soils
GENERAL CONCEPTS OF SLURRY FLOWS
D2 30
�
(D60D10) D 60
1.13
For clays and silts, an additional test for the liquid limit (L L) and the plastic limit (P L) are
recommended.
The liquid limit is defined as the moisture content in soil above which it starts to act as
a liquid and below which it acts as a plastic. To conduct a test, a sample of clay is thoroughly
mixed with water in a brass cup. The number of bumps required to close a groove
cut in the pot of clay in the cup is then measured. This test is called the Atterberg test.
The plastic limit is defined as the limit below which the clay will stop behaving as a
plastic and will start to crumble. To measure such a limit, a sample of the soil is formed
into a tubular shape with a diameter of 3.2 mm (0.125 in) and the water content is measured
when the cylinder ceases to roll and becomes friable.
TABLE 1-5 Range of Particle Sizes of Soils According to ASTM D2488 (1993)
Material Range of sizes in mm Range of sizes in inches
Boulders > 300 > 12
Cobbles 75–300 3–12
Coarse gravel 19–75 0.75–3
Fine gravel 4.75–19 0.019–0.75
Coarse sand 2.00–4.75 0.08–0.0188
Medium sand 0.43–2.00 0.017–0.08
Fine sand 0.08–0.43 0.003–0.017
Silts and clays < 0.075 < 0.003

TABLE 1-6 Conversion between Scales of Particle Size
Sieve opening Sieve opening
U.S. no. Tyler mesh (micrometers) (inches) Grade of soils
3
2
1.50
Screen shingle gravel
26670 1.050
22430 0.883
18850 0.742
15850 0.624
13330 0.525
11200 0.441
9423 0.371
2.5 2.5 7925 0.321
3 3 6680 0.263
3.5 3.5 5613 0.221
4 4 4699 0.185
5 5 3962 0.156
6 6 3327 0.131
7 7 2794 0.110
8 8 2362 0.093 Very coarse sand
9 9 1981 0.078
10 10 1651 0.065
12 12 1397 0.055
14 14 1168 0.046 Coarse sand
16 16 991 0.039
20 20 833 0.0328
24 24 701 0.0276
28 28 589 0.0232 Medium sand
32 32 495 0.0195
35 35 417 0.0164
42 42 351 0.0138
50 50 297 0.0117
60 60 250 0.01 Fine sand
70 70 210 0.0823
80 80 177 0.07
100 100 149 0.06
120 120 125 0.05
140 140 105 0.041
170 170 88 0.034 Silt
200 200 74 0.029
230 250 63 0.025
270 53 0.02
325 43 0.017 Pulverized silt
400 38 0.015
500 25 0.01
625 20 0.008
1250 10 0.004
2500 5 0.002
12500 1 0.0004

The difference between the liquid and plastic limits is defined as the plasticity index:
PI = LL – PL (1-3)
1-2 SLURRY FLOWS
GENERAL CONCEPTS OF SLURRY FLOWS
TABLE 1-7 Testing Parameters on Soils to Determine
Dredgeability, Suitability for Placer Mining, or Slurry Preparation
Type of soil Testing parameters
Sand Density
Water content
Specific gravity of grains
Grain size
Water permeability
Frictional properties
Lime content
Organic content
Silt Density
Water content
Water permeability
Shear strength or sliding resistance
Plasticity
Lime content
Organic content
Clay Density
Water content
Sliding resistance
Consistency ranges (plasticity)
Organic content
Peat Same parameters as clay
Gravel Same parameters as sand
1.15
A slurry mixture is essentially a mixture of a carrying fluid and solid particles held in suspension.
The most commonly used fluid is water, but over the years, attempts have been
made to use crude oils with milled coal, and even air in pneumatic conveying.
The flow of slurry in a pipeline is much different from the flow of a single-phase liquid.
Theoretically, a single-phase liquid of low absolute (or dynamic) viscosity can be allowed
to flow at slow speeds from a laminar flow to a turbulent flow. However, a twophase
mixture, such as slurry, must overcome a deposition critical velocity or a viscous
transition critical velocity. The analogy can be made here in terms of an airplane: if the
speed drops excessively, the airplane stalls and stops flying. If the slurry’s speed of flow
is not sufficiently high, the particles will not be maintained in suspension. On the other
hand, in the case of highly viscous mixtures, if the shear rate in the pipeline is excessively
low, the mixture will be too viscous and will resist flow.
Sections 1-2-1 and 1-2-2 define the two basic slurry flows.

1.16 CHAPTER ONE
1-2-1 Homogeneous Flows
Solids are uniformly distributed throughout the liquid carrier. An example of homogeneous
flows is copper concentrate slurry after undergoing a process of grinding and thickening.
Particles are then very fine and the mixture is at a high concentration (50–60% by
weight). As the concentration of particles is increased (beyond 40% by weight for many
slurries), the mixture becomes more viscous and develops non-Newtonian properties.
Apart from rich concentrate slurries, drilling mud, sewage sludge, and fine limestone (cement
kiln feed slurry) behave as homogeneous flows. Typical particle sizes for homogeneous
mixtures are smaller than 40 �m to 70 �m (325–200 mesh), depending on the density
of the solids.
The presence of clays in certain circuits must not be ignored. If clay is not separated,
the slurry can be quite viscous. Pumps and pipes must be sized properly to handle the resultant
absolute (or dynamic) viscosity. Certain mines in Peru contain material called soft
high clay, which can increase the absolute (or dynamic) viscosity of the slurry up to 400
mPa at weight concentrations in excess of 45%. Dilution to lower concentration and
changes to recycling load are solutions to such a problem.
1-2-2 Heterogeneous Flows
In a heterogeneous flow, solids are not uniformly mixed in the horizontal plane. A gradient
of concentration exists in the vertical plane. Dunes or a sliding bed may form in
the pipe, with the heavier particles at the bottom and the lighter ones in suspension, particularly
at the critical deposition velocity. The different phases retain their properties
and the largest particles do not necessarily cause the biggest problems; it really depends
on the ratio that they are mixed with finer particles. Heterogeneous slurries are encountered
in many placer mining, phosphate rock mining, and dredging applications. Concentration
of particles remains low, typically less than 25% by weight in many dredging
applications and below 35% by weight in many tailing disposal applications. Heterogeneous
flows require a minimum carrier velocity. In some tailing applications of the
Taconite mines of Minnesota, the typical deposition velocity is in excess of 3.4–4 m/s
(11–13 ft/s).
Nature being complex, flows are encountered that have the characteristics of heterogeneous
or homogeneous flows. The concept of pseudohomogeneous flows is also used
when a large fraction of particles are fine but there remain a sufficient fraction of coarse
particles that may deposit as the flow speed is reduced below a minimum value.
1-2-3 Intermediate Flow Regimes
Intermediate regimes occur when some of the particles are homogeneously distributed
and others are heterogeneously distributed. Intermediate regime flows include tailings
from mineral processing plants and a wide range of industrial slurries.
1-2-4 Flows of Emulsions
Strictly speaking, an emulsion is not slurry. An emulsion is a mixture of two phases at
certain temperatures resulting in an essentially homogeneous flow. An example of an
emulsion is a mixture of bitumen at 70% by volume with water at 30% by volume. If sur-

factants are used, the bitumen remains well mixed with water in a certain temperature
range. Emulsions can become unstable under certain high shear rates or through very tight
clearances of pumps according to Nunez et al. (1996). In the 1990s, PDVSA-BITOR constructed
a 300 km (188 mi) pipeline in Venezuela with a diameter of 660–915 mm (26–36
in) for transporting their ORIMULSION fuel, a mixture of highly concentrated bitumen
and water. The fuel is a substitute for coal in thermal plants.
Emulsions do not encounter the deposition velocity of slurries, but as flows become
unstable, fine droplets of heavy oils or bitumen may coalesce into larger ones, causing
changes of flow.
1-2-5 Flows of Emulsions—Slurry Mixtures
A mixture of an emulsion and solids such as fine coal could be used to produce a fluid
with a high calorific value. Coal must be very fine to burn readily in combustion furnaces.
The flow is similar to a homogeneous flow with a high absolute (or dynamic) viscosity
component.
1-3 SINKING VELOCITY OF PARTICLES
AND CRITICAL VELOCITY OF FLOW
Various parameters of speed determine whether a mixture may separate or continue to
flow. In fact, the designer of a thickener or a mixer is often more interested in the sinking
velocity of particles. On the other hand, the designer of a pipeline has to pay attention to
the critical velocity of flow, settling speed, and whether the flow is vertical or horizontal,
particularly in the case of heterogeneous flows.
1-3-1 Sinking or Terminal Velocity of Particles
This is the minimum speed needed to maintain particles in suspension, particularly in a
process of mixing or thickening. This velocity is not identical with the critical velocity of
flow, and should not be confused with it.
Table 1-8 presents examples of sinking velocity of various soils. The designer of a
mixing system or a thickener is encouraged to conduct lab tests, since clays may be mixed
with sands in some areas, or the soil may be stratified, with layers of different materials.
1-3-2 Critical Velocity of Flows
GENERAL CONCEPTS OF SLURRY FLOWS
1.17
In Chapters 3, 4, and 5 the mechanics of solid suspensions are described in detail. An important
parameter to introduce in this chapter is the critical velocity of a slurry flow. Figure
1-8 plots the pressure loss per unit length on the y-axis, versus the velocity V of a slurry
flow on the x-axis. Five points are shown for flow at a constant volume concentration.
For this slurry of moderate viscosity, the flow is stationary and the solids clog the pipeline
below point 1. There is insufficient speed to move the particles. As the flow is accelerated,
the speed reaches point 1, which is called the deposition critical velocity V D, or minimum
speed to start the flow. Between points 1 and 2, the bed builds up, dunes form, and
the different phases are well separated. Between points 2 and 3 the flow is streaking but

Pressure drop per unit of length
1.18 CHAPTER ONE
TABLE 1-8 Sinking Velocity of Soil Particles (after Sulzer Pumps, 1998, with
permission of Elsevier)
Particle
Soil grain size identification
diameter, Mesh size, Sinking Sinking Grain size
micrometers US fine velocity, m/s velocity, ft/s by ASTM Grain size, international
0.2 3 × 10 –8 Clay Fine clay
0.6 2.8 × 10 –7 Coarse clay
1 7 × 10 –6
2 9.2 × 10 –6
5 17 × 10 –6 Silt Fine silt
6 25 × 10 –6
20 28 × 10 –5 Coarse silt
50 270 17 × 10 –4 Fine sand Intermediate silt to sand
60 230 25 × 10 –4
100 150 0.07 Fine sand
200 70 0.021 Medium coarse sand
250 60 0.026 Coarse sand
300 0.032
500 35 0.053 Coarse sand
600 30 0.063
1000 18 0.10 Very coarse sand
2000 10 0.17
1
slurry
2
3
water
Speed of flow
4
5
y/D
concentration
4 - 5
Pseudohomogeneous
3 - 4
Jumping and rolling
2 - 3
Streaking
1 - 2
Bedding
Below 1
Stationary and
clogging
FIGURE 1-8 Pressure drop versus velocity for water and for a slurry mixture. (After Sulzer
Pumps, 1998, with permission of Elsevier.)

GENERAL CONCEPTS OF SLURRY FLOWS
momentum is building up. Between points 3 and 4, there is sufficient speed to cause
jumping and rolling of the coarse particles. Above point 4, the speed is sufficiently high
to allow a pseudohomogeneous flow in which the fine particles act as a carrier for the
coarse particles. These stages are extensively reviewed in Chapter 4.
When absolute (or dynamic) viscosity is an important factor, such as in clayish slurries
or homogeneous flows, another parameter, the viscous transition critical velocity V T must
be determined. There are two regimes for flow of homogeneous mixtures. Flow at speeds
less than V T is associated with laminar flows, whereas flows above V T are characteristic of
turbulent flows.
Flow in the laminar regime is often characterized by a friction loss factor, which is
64/Re for the Darcy factor or 16/Re for the Fanning factor (this topic will be discussed in
more details in Chapter 2). As a result, the losses in the laminar regime appear to be a linear
function of speed, whereas in the turbulent regime they are proportional to the square
of the speed. As we will see in Chapter 5, researchers have struggled with special definitions
of a modified Reynolds number for non-Newtonian flows.
Emulsions have been pumped over long distances in laminar flow. Nunez et al. (1996)
demonstrates the existence of certain effects similar to comminution, i.e., breakup of
large droplets into finer ones, as well as coalescence of small particles into larger ones under
different flow regimes, shear rates, and constraints.
A question often asked is what the relationship between the sinking speed (as per
Table 1-8) and the deposition of critical velocity V D? This question comes up when the
rheology laboratory produces the results of thickening tests, and when there is not
enough money or time to conduct proper slurry loop tests. This point will be examined
in Chapter 4 and various approaches have been adopted over the years, from the simplest
assumption that the critical deposition velocity is 17 times as large as the terminal
or sinking velocity, to more complex mathematical formulae. Often in a lab test, the
coarse particles deposit rapidly while the fine particles are still in suspension. Proper
pump tests are often recommended, particularly when a multimillion dollar pipeline is
being designed. Tests can be conducted at a number of universities, provincial and state
research centers, or with the help of manufacturers of slurry pumps. Examples include
the Saskatchewan Science Research Center in Canada; the GIW research lab of KSB
Pumps (USA), described by Wilson et al. (1992); the Slurry Research Lab of Mazdak
International Inc. (U.S.A.); Texas A&M University (U.S.A.); and Melbourne University
(Australia), among others.
1-4 DENSITY OF A SLURRY MIXTURE
The density of a slurry mixture is a function of
� The density of the carrier fluid
� The density of the solid particles
� The concentration by volume of the solid phase
1.19
The density of the solid particles is determined carefully by various experimental
methods. Fine particles tend to entrap air, which the lab technician must remove by proper
agitation or by adding a small quantity of wetting agent.
Some materials exhibit a change of packing abilities and therefore density as a function
of particle size. If the solids are to be passed through a comminution process, a SAG
(semiautogeneous), or ball mill, they can occupy more volume per unit mass as they be-

1.20 CHAPTER ONE
come finer. The slurry engineer is therefore encouraged to measure the solid’s density at
the proposed size of particles to be transported in slurry form.
Certain errors can occur in evaluating the density of solids with heterogeneous mixtures.
If the heavier slurry particles settle out and a sample is taken, it may reflect a
greater density of finer particles. Due to these possible sources of error, the engineer is
encouraged to measure the density of the slurry mixture after proper mixing, and to use
the data on concentration by weight or by volume to work back to the density of the
solids.
The density of a slurry mixture is expressed as
100
�m = ���
(1-4)
Cw/�s + (100 – Cw)/�L where
Cw = concentration by weight
�m = density of the mixture phase
�l = density of the liquid phase
�s = density of the solid phase
Engineers use the term concentration by weight, as it is easier to convert back into the
total tonnage of solids to be transported through a pipeline or across an extractive metallurgy
plant. However, the characteristics of the mixture, the mechanics of flow, and the
resultant physical properties are more related to the concentration by volume.
The concentration by volume of solids in a mixture is expressed as
C w� m
Cv = � = ���
(1-5)
�s Cw/�s + (100 – Cw)/�L The concentration by weight of solids in a mixture is expressed as
Cv�s �
�m
100 C w/� s
Cv = = (1-6)
Example 1-A
A pipeline is designed to transport 140 metric ton (308,000 lb) of sand per hour. The specific
gravity of the sand particles is 2.65 (or the density is 2650 kg/m3 ��
Cv�s + (100 – Cv) ). The concentration
by weight is 30%. Determine the density of the mixture if the carrier fluid is water and determine
the resultant flow rate.
� Weight of sand in the mixture over a period of 1 hr is 140 metric ton (308,000 lb)
� Weight of equivalent volume of water equivalent to the sand content is 140,000
kg/2,650 kg/m 3 = 52,800
� Weight of water in the slurry mixture at a concentration of 30% by weight = 140,000
kg (100 – 30)/30 or 326,600 kg
� Total weight of slurry mixture transported in 1 hr is the sum of the weight of water and
sand or 140,000 + 326,600 = 466,600 kg/hr
� Total weight of equivalent volume of water is 326,000 + 52,800 = 378,800 kg or 378
m 3 of liquid, since density of water is 1000 kg/m 3
The density of the slurry mixture is therefore 466,600 kg/378 m 3 = 1,230 kg/m 3 . Alternatively,
the specific gravity of the slurry compared to water is 1.23. The flow rate, being
the volume per unit of time, is equivalent to 378 m 3 /hr.
C v� s

1-5 DYNAMIC VISCOSITY OF A
NEWTONIAN SLURRY MIXTURE
Although density is essentially a static property, absolute (or dynamic) viscosity is a dynamic
property and tends to reduce in magnitude as the shear rate in a pipeline increases.
Thus, engineers have had to define different forms of viscosity over the years, everything
from dynamic viscosity, to kinematic viscosity, to effective pipeline viscosity. The effective
pipeline viscosity will be discussed in detail in Chapters 3, 4, and 5. In this chapter,
the reader is introduced to basic concepts of the mixture of slurry in a stationary state.
This is effectively what the pump, or a mixer, might see at the start-up of a plant. As is often
the case, when the driver cannot deliver enough torque to overcome the absolute (or
dynamic) viscosity, the operator is forced to dilute the slurry mixture.
Plasticity as defined in Section 1-1-4 is an important parameter in determining overall
absolute (or dynamic) viscosity of a mixture of clay and water. There are, however, numerous
soils in nature, such as sand and water or gravel and water, in which the solids
contribute little to the overall absolute (or dynamic) viscosity, except in terms of their
concentration by volume.
1-5-1 Absolute (or Dynamic) Viscosity of Mixtures with Volume
Concentration Smaller Than 1%
For such solid–liquid mixtures in diluted form, Einstein developed the following formula
for a linear relationship between absolute (or dynamic) viscosity and volume concentration:
= 1 + 2.5� (1-7)
where
�m = absolute (or dynamic) viscosity of the slurry mixture
�L = absolute (or dynamic) viscosity of the carrying liquid
This is a very simple equation that is based on the following assumptions:
� Particles are fairly rigid
� The mixture is fairly dilute and there is no interaction between the particles
Such a flow is not encountered, except in laminar regimes of very dilute concentrations
(below a volume concentration of 1%).
1-5-2 Absolute (or Dynamic) Viscosity of Mixtures with Solids
with Volume Concentration Smaller than 20%
Thomas (1965) took the equation of Einstein further by calculating for higher volumetric
concentrations of Newtonian mixtures:
� m
� �L
where K 1, K 2, K 3, and K 4 are constants
GENERAL CONCEPTS OF SLURRY FLOWS
� m
� �L
1.21
= 1 + K 1� + K 2� 2 + K 3� 3 + K 4� 4 + . . . (1-8)

1.22 CHAPTER ONE
K 1 is the Einstein constant of 2.5 (from Equation 1-7), and K 2 has been found to be in
the range of 10.05–14.1 according to Guth and Simha (1936). It is difficult to extrapolate
the higher terms K 3 and K 4 in Equation 1-8. They are ignored with volumetric concentrations
smaller than 20%.
1-5-3 Absolute (or Dynamic) Viscosity of Mixtures with High
Volume Concentration of Solids
For higher concentrations, Thomas (1965) proposed the following equation with an exponential
function:
= 1 + K 1� + K 2� 2 + A exp(B�) (1-9)
where
K2 = 10.05
A = 0.00273
B = 16.6
Figure 1-9 is based on Equation 1-9 and is widely accepted in the slurry industry for heterogeneous
mixtures of a Newtonian rheology.
1-6 SPECIFIC HEAT
Thomas (1960) derived an equation for the specific heat of a mixture as a function of the
specific heat of the liquid and solid phases:
1-7 THERMAL CONDUCTIVITY AND
HEAT TRANSFER
CpmCws + CpmCwL Cpm= ��
(1-10)
100
Thermal conductivity is difficult to measure, as solids may settle during the test. Sometimes
it is recommended to apply a small quantity of gel to maintain the solids in suspension.
Orr and Dalla Valle (1954) derived the following equation for the thermal conductivity
of slurry mixtures:
where
k = thermal conductivity
and subscripts
l = liquid
m = mixture
s = solids.
� m
� �L
2kl + ks – 2�(kl – ks) ���
2kl + ks – 2�(kl + ks) k m = k l� � (1-11)

1.23
Ratio of viscosity of slurry mixture vs.
L
m
viscosity of carrier liquid
100
90
80
70
60
50
40
30
20
10
0
CV
Volumetric concentration of solids
0 10 20 30 40 50 60 70 80 90
C [%]
V m L
1 1.029
3 1.089
5 1.156
7 1.233
10 1.365
12 1.465
14 1.575
16 1.696
18 1.83
20 1.978
22 2.142
25 2.426
27 2.649
29 2.907
31 3.210
33 3.573
35 4.017
37 4.570
39 5.273
42 6.734
43 7.37
44 8.103
C [%]
V
m L
45 8.950
46 9.932
47 11.07
48 12.40
49 13.94
50 15.75
51 17.86
52 20.33
53 23.22
54 26.62
55 30.61
56 35.29
57 40.80
58 47.28
59 54.91
60 63.89
61 74.47
62 86.94
63 101.63
64 118.95
65 139.4
FIGURE 1-9 Ratio of viscosity of mixture versus viscosity of carrier in accordance with the Thomas equation for coarse slurries.

1.24 CHAPTER ONE
Metzner et al. (1959, 1960) published articles on heat transfer for slurries. The applications
for heat transfer problems have been confined to the nuclear industry, the processing
of tar sands, feeding slurry to autoclaves for thermal processing, or certain emulsionbased
slurries.
1-8 SLURRY CIRCUITS IN EXTRACTIVE
METALLURGY
It would be beyond this book to discuss the principles of extractive metallurgy. Slurry is a
very important component in the processing of ores to the final disposal of tailings and
shipping of concentrate. Chapter 7 is dedicated to equipment for slurry processing. There
are three main processes used for extractive metallurgy:
1. Hydrometallurgy, which implies processing the ore using a liquid medium
2. Electrometallurgy, which involves the application of electric and electro-chemical
processes to extract the ore
3. Pyrometallurgy, which involves the use of heat (roasting, smelting, etc.) for processing
the ore
The most common minerals of high metric tonnage (iron, aluminum, copper, titanium,
nickel, chromium, magnesium, zinc, etc.) are found in nature as oxides and sulfides and
as a combination of both. Ores are sometimes a mixture of rich metal composition and
poorer compositions called gangue. The gangue can be acidic or alkaline, and determines
the type of flux used for pyrometallurgy. Since ores come in all levels of complexity, various
methods of processing have been developed over the years.
The first process ore undergoes is called classification or ore dressing. The purpose
here is to separate the richer components of a mixture from the unuseful soils. Mineral
processing may be used to produce a single stream, as is the case with taconite circuits,
where iron extraction is the main activity. It may also create two streams, such as copper
concentrate and gold concentrate, when both minerals are found in the same ore
body .
Mineral processing is usually undertaken at the mine. Its purpose is to separate some
of the gangue before shipment of a concentrate. The concentrate is richer in the desired
mineral than the original soil.
1-8-1 Crushing
In Figure 1-10, a block diagram for crushing and grinding is presented. These are two of
the fundamental steps taken to start a slurry circuit. Large rocks are first crushed to an acceptable
size. Depending on the type of equipment used, crushing may be done in a single
step to feed a semiautogeneous mill, or in three steps (primary, secondary, and tertiary
crushing).
Rocks are transported from one crusher to another by conveying in a dry form. Their
initial size of 300–600 mm (1–2 ft) is reduced to 100–150 mm (4–6 in). Jaw, gyrator, and
cone crushers are commonly used during these stages. The crushed material is transported
by conveyors to a storage area called the stockpile. From the stockpile, crushed rocks are
transported to the grinding and milling circuit.

wells
Reclaim Tailings
Water Dam
Lake
River
stockpile
tank
water
30-40% solids
Tailings pipes
GENERAL CONCEPTS OF SLURRY FLOWS
Ore

1.26 CHAPTER ONE
which helped eliminate complicated gearboxes for SAG mills. Some of the largest SAG
mills are now built with a diameter of 12.2 m (40 ft).
The process of crushing is essentially a dry process but the process of milling is a wet
process in which slurry comes to play an important role. The use of water eliminates the
dangerous generation of dust associated with dry grinding.
It is not possible to undertake milling or wet grinding in a single step. The slurry is recirculated
as follows. Initially, the coarse and fine particles are separated through a coarse
screen or a special cyclone. Then the coarse particles are returned to the SAG mill and the
intermediate sized particles are sent to the ball mill. The fine sized particles are taken
from the cyclone overflow to a magnetic separation, electrostatic separation, or flotation
plant. It is therefore not uncommon to recirculate 250–350% of the feedstock through the
circuit of grinding and milling.
Attention should be paid to the presence of clays in the ore, and associated dynamic
viscosity. Flow from the ball mills can reach a concentration of 40–50% by weight and
certain non-Newtonian rheology may manifest itself. Over the years, a plant that started
in rocky ground may encounter more clay as it proceeds into deeper depths.
1-8-3 Classification
This is essentially a process to separate the particles according to their sinking rates in
water. Wet classifiers are used with grinding and may include rubber-lined or ceramic cyclones
(called hydrocyclones) and spiral mechanical classifiers.
The principle of the cyclone is to feed the slurry tangentially and force it to rotate. By
centrifugal forces, the coarser particles sink to the bottom of the cone while the finer particles
float to the top. Both streams separate. The underflow, which consists of coarse material,
and the overflow, which consists of fines, are then directed to other circuits. The
underflow is fed back to the ball mills and the overflow is directed to the flotation circuit
or other types of separators.
The spiral mechanical classifier is used in pools. The heavier particles are allowed to
settle in the pool while the finer particles float and flow out of the pool. The heavier particles
are then removed from the bottom of the pool with a spiral or mechanical device.
From cyclones or from the grinding circuit, the slurry may pass through different types
of separators such as flotation circuits, electrostatic separators, and magnetic separators.
Their purpose is to separate by chemical, electrical, or magnetic forces the minerals from
the gangue. These steps occur before further thickening prior to feeding the pipeline. The
gangue is diverted to tailings circuits (see Figure 1-11).
1-8-4 Concentration and Separation Circuits
After a considerable effort to reduce the sizes of particles, it is important to separate the
richer soils from the slimes or gangue. This step is achieved by using the properties of the
ore itself.
Gravity devices work on the principle that the ore (such as gold or diamonds) is heavier
than the gangue. These devices include shaking units, the classic miner’s pan, rocking
cradle devices, or more sophisticated gold concentrators or mineral sand concentrators.
The drawback of these systems is that they may not necessarily be able to treat the fine
particles produced by grinding and milling. In diamond extraction plants, X-ray machines
are used in conjunction with gravity separation to detect the diamonds.
Gravity devices are also used in a number of dry processes as well as slurry processes

1.27
wells
Lake
River
tank
Reclaim
Water
Tailings
Dam
Tailings pipes
30-40% solids
Crushing, grinding, milling (see fig 1- 10)
Flotation Electrostatic
Cells Separators
gangue
tailings
sump
reclaim
water
Magnetic
Separators
minerals
Gravity
Separators
tailings concentrate
thickener
thickener
slurry
pipeline
filtering
drying
smelting
burning
concentrate
at about 20%
mineral
FIGURE 1-11 Block fi 1diagram 11 for thickening and disposal of tailings, used as the basis for the design of slurry concentrate and
tailings pipelines.

1.28 CHAPTER ONE
for mineral sand and placer mining. In the case of mineral sands, the presence of a wide
range of heavy oxides allows the miner to separate the various components by using gravity
in conjunction with the magnetic or electrostatic properties.
Magnetic devices are especially useful, since iron is one of the most common minerals
and it is possible to separate iron oxides from gangue by applying a magnet. This is usually
done by introducing the slurry over a rotating drum, as shown in Figure 1-12; the drum
picks up the magnetic concentrate while the remaining soils are diverted away by the flow
of the remaining slurry. Magnetic devices are common in taconite processing plants, iron
ore plants, as well as in mineral sand plants.
Electrostatic devices were developed in Australia to process beach and mineral sands.
The sand ore is fed to a conducting and grounding rotor and is exposed to ionization. The
particles, which have certain electrostatic properties, are attracted by the electric charge
and are separated from the other particles. The nonconducting particles drop on the rotor
and are brushed away into a separate container (Figure 1-13).
Flotation devices use the principle of flotation to separate particles that are wettable
from other particles. This is a very common process with sulfides but is less efficient with
oxides. The cyclone overflow or the fine particles in the slurry after undergoing milling
and grinding are fed to a series of flotation tanks (or a flotation machine). An agitator provides
vigorous mixing. Air is introduced from a separate compressor line and chemical
reagents are added to create froth. The nonwettable minerals float on top of the froth and
are pumped away by froth-handling pumps or scraped away by mechanical devices. The
wettable particles, such as the gangue, do not float on top of the froth and sink to the bottom
of the tank. Special tailing pumps may then pump away the gangue, sometimes for
further grinding and processing (particularly gangue from the first flotation tank) and
sometimes to the final tailing box (see Figure 1.14).
The process of froth generation and flotation is more efficient when carried out in
steps. A series of up to six tanks may be constructed to drop gradually, with launders in
between.
Without reagents, only graphite or molybdenum would be nonwettable. Reagents have
been produced by the industry for different grades of sulfides, to depress or activate the
extraction of certain minerals, to control pH, etc.
Feed
Other soils
Magnetic
concentrate
rotating
magnet
FIGURE 1-12 A drum-type magnetic separator. The drum is sometimes replaced by a magnetic
belt on a special table.

GENERAL CONCEPTS OF SLURRY FLOWS
Feed
Nonconductor soils
Nonionizing
electrode
fine-wire
electrode
ionizing
Conductors
concentrate
FIGURE 1-13 An electrostatic separator.
1.29
The secondary grinding that is applied to the underflow of the circuits from flotation is
useful for extracting secondary ores, and has been applied successfully in copper–gold
ores for further extraction of the gold.
A mineral process plant includes gravity flows from hydrocyclones to ball mills and
pumped flows from SAG mills to hydrocyclones. A good plant layout must allow space
for repairs and long bends, and provide the ability to join and split flows. The use of
three dimensional computer modeling is a very useful tool for the design engineer to determine
the slope of launders and physical constraints to the layout of the plant (Figure
1-15)
froth bubbles with mineral concentrate
FIGURE 1-14 A flotation circuit.
aeration agitator
pump
secondary grinding

1.30 CHAPTER ONE
1-8-5 Piping the Concentrate
From these processes of classification, a concentrate is obtained. The slurry can be further
thickened or dewatered using thickeners to a concentration of 50–60% by weight. The
concentrate can then be pumped for hundreds of kilometers to a port from which it can be
shipped to a pyrometallurgy plant. At the port, a filtering plant can provide further dewatering.
Long pipelines are used to transport concentrate. At Cuajone, in Peru, an open launder
is used to transport tailings from an altitude of 3000 m (10,000 ft) down to sea level.
The potential energy drop is used to overcome the friction losses of the launder. In
Escondida, Chile, copper concentrate flows by gravity from an altitude in excess of
2500 m (8200 ft) above sea level over a distance in excess of 200 km (125 mi) to a port
at sea level.
Thus, in a typical copper extraction process the rocks are reduced to very fine particles
through vertimills, semiautogeneous mills, and ball mills. Further separation occurs
through flotation circuits and grinding.
1-8-6 Disposal of the Tailings
Once the concentrated ore has been extracted, the plant is left with the sands and slimes.
These are dewatered to an acceptable concentration by weight of 35–45%. Using thicken-
FIGURE 1-15 Three-dimensional computer representation of a grinding circuit with one
central SAG mill, a ball mill on each side, hydrocyclones at the top left, and pumps (at the floor
level). Courtesy of Hatch & Associates, Vancouver, Canada.

GENERAL CONCEPTS OF SLURRY FLOWS
ers, the flow separates into clarified water that is returned to the plant for use in the
milling and grinding circuits, and into an underflow of concentrated tailings. The underflow
from the tailings is then pumped away to a large disposal pond, called a tailings dam.
The sand sinks to the bottom of the tailings pond and the water from the top is returned
back to the plant for further use.
During the process of disposing of the tailings, spigots and other devices are used to
separate coarse from fine particles at the discharge point; the coarse particles are used to
build the beaches or the wall of the dam. Dams have been built up to a height of 200 m
(656 ft). The environmental engineer must make sure that no dangerous chemicals seep
through the ground. If the tailings contain dangerous substances, a plastic or clay (which
tends to create a seal) lining for the pond may be recommended to prevent seepage. Some
dangerous collapses of tailing dams have been reported over the years, with detrimental
consequences when they contained cyanide products, as in the case of certain gold mines.
The technology used in tailing pumps has improved. The casings can be designed to
withstand 6.9 MPa (1000 psi) of pressure. However, these are essentially single-stage
centrifugal pumps installed in series. Up to seven pumps have been installed in series in
mines such as National Steel in Minnesota and Kelian Gold in Indonesia.
1-9 CLOSED AND OPEN CHANNEL FLOWS,
PIPELINES VERSUS LAUNDERS
1.31
The reader will find that over the years the majority of references on slurry flows have focused
on pipelines because the interest in the field has concentrated on the ability to haul
coal, sand, and phosphate hydraulically.
Many mines, particularly in Chile and Peru, are located at very high altitudes. This demand
has increased interest in gravity flows. In the early 1970s, Southern Peru Copper installed
one of the first long, open launders to dispose of tailings to the sea. The launder
was of a concrete and fiberglass design, with a U-shaped cross section. Another example
of a long gravity pipeline for copper concentrate is the Escondida concentrate pipeline in
Chile, which is longer than 200 km (125 mi).
Despite the increasing importance of long gravity pipelines, equipment has not kept up
with the expanding need. Cave (1980) described tests on slurry turbines. A 350 mm × 300
mm (14� × 12�) was reported by Burgess and Abulnaga (1991).
Launders play a very important role in slurry flows of plants. Cyclone underflow is directed
to ball mills then to SAG mills by gravity. Flows in these circuits can cause
tremendous wear if provision is not made to control speeds. Launders in plants are typically
rubber lined. Long-distance pipelines are manufactured of rubber-lined steel or extra-thick,
high-density polyethylene (HDPE). Because of the importance of open launders,
gravity flows, and drop boxes, Chapter 6 is dedicated to these complex flows.
Obviously, not all mines are located on mountaintops, and slurry pipeline flow will
continue to be the main emphasis of researchers. In long pipelines, centrifugal pumps can
be installed at regular intervals; these require power to be brought in. For long-distance
pumping, positive displacement pumps compete well with centrifugal pumps. The positive
displacement pumps are of a diaphragm or hose design. They are extremely expensive.
A 17.3 MPa (2500 psi) pump may range in price between U.S. $600,000 and
$1,200,000 in year 2000 dollars. The higher capital investment required for positive displacement
pumps is offset by their higher efficiency. These pumps are built to much
smaller flow capacity than are large centrifugal pumps.

1.32 CHAPTER ONE
1-10 HISTORICAL DEVELOPMENT OF
SLURRY PIPELINES
One of the first large engineering projects that involved transportation of solids by liquid
was the dredging for the Suez Canal in the 1860s in Egypt. It was reported to have used
conduits to dispose of the sand–water mixture. Nora Blatch in 1906 was probably the first
person to conduct a systematic investigation of the flow of solid–water mixtures. She
used a 25 mm (1 in) horizontal pipe and measured the pressure gradients as a function of
flow, density, and solid concentration. As a result, between 1918 and 1924, a 200 mm (8
in) pipeline was installed in the Hammersmith power station, in London, England to
transport coal slurry over a distance of 660 yd.
In 1948, in France, the Institute of Research SOGREAH began a series of tests on
transporting sand and gravel in pipes with a diameter from 38–250 mm (1.5–10 in). These
extensive tests were the basis for the formulation by Durand of a number of equations that
will be reviewed in Chapter 4. These equations have been subject to further refinements
over the last 50 years.
In 1952, in the United Kingdom, the British Hydromechanic Research Association
(BHRA) started to study the hydraulic transport of lump coal, sand, gravel, and limestone.
Limestone pipelines were constructed in Trinidad and England in 1960. The Trinidad
pipeline had a diameter of 204 mm (8 in), a length of 9.6 km (6 mi), and was designed to
operate in a laminar flow regime. The limestone pipeline in England had a diameter of
250 mm (10 in) and was 112 km (70 mi) long.
In 1950, the Consolidated Coal Co. in the United States started to conduct research on
the hydrotransport of fine “nonsettling” slurries. Concentrated coal with a weight concentration
of 60% and particle size between minus 1168 �m (14 mesh) and minus 43 �m
(325 mesh) was transported. The pipeline transported 1.5 million tons of coal each year
between 1957 and 1964. The pipeline stretched 176 km (110 mi) from Cadiz, Ohio to
Eastlake in Cleveland, Ohio.
In 1957, the Colorado School of Mines collaborated with the American Gilsonite
Company and designed a pipeline with a diameter of 200 mm (8 in) to transport crushed
gilsonite. The pipeline was constructed between Bonanza, Utah and Grand Junction, Colorado.
The particle size was minus 4.7 mm (4 mesh) and solids were pumped at a weight
concentration of 48%. Two other pipelines were built in Georgia to transport kaolin in the
1960s.
In 1967, an iron ore concentrate slurry pipeline started to operate in Tasmania,
Australia. The pipeline had a diameter of 245 mm (9 5 – 8 in). Concentrate was transported
at a weight concentration of 60% with an average particle size of minus 149 �m (100
mesh) over a distance of 85 km (53 mi) through extremely rugged terrain (see Figure
1-16).
In 1970, the Black Mesa Pipeline, one of the longest pipelines ever built up to that
time, started operation between the Black Mesa Coal fields in Arizona and the Mohave
Power Plant in Nevada. Coal was ground to a particle size of minus 1168 �m (14 mesh),
and transported in a pipe with a diameter of 457 mm (18 in) over a distance of 437 km
(273 mi). Coal was dewatered at the end of the line through a mill before combustion with
preheated air.
Since the 1970s, a number of short and long slurry pipelines have been constructed.
Table 1-9 lists a number of such achievements. Now, at the beginning of the 21st century,
new complex, multiphase tar–sand pipelines are planned for northern Alberta,
Canada.

GENERAL CONCEPTS OF SLURRY FLOWS
1.11 SEDIMENTATION OF DAMS—
THE ROLE OF THE SLURRY ENGINEER
1.33
FIGURE 1-16 Long slurry pipelines must travel through isolated areas over long distances
and may involve pressures up to 2500 psi that require special positive displacement pumps.
Courtesy of Wirth Pumps, Germany.
In the last 150 years, world population has grown fast and our modern standards of living
depend on the production of electricity, and production of food for at least two seasons a
year. In an effort to meet these demands, engineers have built small as well as very large
dams. In certain areas, very large man-made lakes have been dug in the earth, such as behind
the Aswan High Dam in Egypt, the Ataturk Dam in Turkey, and new dams on the
Yellow River in China.
Some large rivers transport silt that tends to separate from water when the speed of the
flow is interrupted by a dam. This phenomenon is called siltation of dams. The problem

1.34 CHAPTER ONE
TABLE 1-9 Examples of Slurry Pipelines Built Since 1957
Site of pipeline,
Pipe
diameter
Pipeline length
Solids
transported,
million short Start-up
Ore or name of pipeline inch Mile km tons/yr date
Coal Consolidation, USA 10 108 175 1.3 1957
Black Mesa 18 273 440 4.8 1970
ETSI 38 1036 1675 25 1979
ALTON 24 180 112 10 1981
Belonovo–Novosibirsk, Siberia,
Russia
158 256 3.4 1985
Iron Savage River 9 53 86 2.25 1967
concentrate Waipipi (Iron Sands) 8,12 6 9.7 1.0 1971
Pena, Colorado 8 28 45 1.8 1974
Las Truchas, Mexico 10 16 27 1.5 1976
Sierra Grande, Argentina 8 20 32 2.1 1976
Samarco, Brazil 20 244 395 12 1977
Chongin, North Korea ? 61 98 4.5 1975
New Zealand Sands, NZ 12 5 8
Jian Shan, China 10 62 100
La Parla–Hercules, Mexico 8/14 52/182 85/295 4.5 1982
Copper ore Los Bronces 24 35 56
Copper Bougainville, PNG 6 20 32 1.0 1972
concentrate West Irian, Indonesia 4 69 111 0.3 1972
Pinto Valley 4 11 17 0.4 1974
OK Tedi, Papua New Guinea 6 96 155 1987
Escondida, Chile (gravity line) 9 102 165 ? 1994
Collahausi, Chile 7 125 203 1.0 1999
Freeport, Indonesia 5 71 115
Batu Hijau, Indonesia 6 11 18 1999
Alumbrera, Argentina 6 194 314 0.8 1998
Copper Bougainville, Papua NG 34 31 50
tailings Southern Peru Copper (gravity) 150 1972
Limestone Rugby 10 57 92 1.7 1964
Calaveras 7 17 27 1.5 1971
Michigan Limestone Tailings 20 1.2 2
Phosphate Chevron, Vernal, Wyoming 10 94 152 1.3–2.5 1986
ore concentrate Simplot 8 89 145
Wenglu 8 28 45
Dredging Dallas White Rock Lake, USA 24 33 21 11,000 gpm 1998

GENERAL CONCEPTS OF SLURRY FLOWS
1.35
of siltation of dams has not been well documented or studied. Chanson (1998) and Chanson
and James (1999) examined the siltation of Australian dams. Certain dams in Australia
became gradually fully silted between 1890 and 1960. They reported that the
Koorawatha dam in New South Wales (Figure 1-17) became fully silted with bed-load
material. The Cunningham Creek Dam in New South Wales, Australia was well studied
by Hellstrom (1941) according to Chanson. Sedimentation problems are more acute with
small dams than with medium-size and large reservoirs (Chanson and James,1999). Siltation
at Eildon in the State of Victoria occurred in 1940 after torrential rainfalls following
bushfires that had destroyed 50% of the catchment forest. The siltation at Eppalock in
Victoria followed extensive gold mining, tree clearing, and hydraulic mining between
1851 and 1890.
There were some extreme siltation cases. The Quipolly Reservoir No. 1, in Australia
underwent very rapid sedimentation between 1941 and 1943 at a rate in excess of
1143m 3 /km 2 /year (9600 ft 3 /mi 2 /y). The Korrumbyn Creek Dam sedimented in less than 7
years.
Since the 1950s, improvements in land management practices and a better understanding
of the problems of soil erosion have resulted in better approaches to the protection of
dams.
FIGURE 1-17 The Koorawatha Dam in Australia—fully silted. [From Chanson (1999).
Reprinted by permission of Butterworth-Heinemann.]

1.36 CHAPTER ONE
Chanson and James (1998) discussed the hazards of fully silted dams. The weight of
silt pressing against the concrete structure introduces a medium with a specific gravity
larger than the water for which the dam was designed. It is important in these cases to
monitor the structure.
In the 1970s and 1980s, the drought in Ethiopia and the Sudan reduced the flow of the
waters to the Nile to dramatically low levels. Egypt avoided famine by using its massive
man-made lake behind the Aswan High Dam (Lake Nasser). On the other hand, any specialist
who visits Egypt can feel that the fellahin (farmers) are speaking with nostalgia of
the “Tamye” or silt that used to come with the annual flood and enrich the land. This raw
material was the basis of natural nutrients, as well as mud for the construction of houses
and manufacture of bricks.
The Egyptian case is not unique. Certain dams in the United States are now the subject
of discussions on decommissioning and some were removed in the 1990s. The slurry engineer
can offer some much needed solutions. The twenty-first century will see slurry engineers
providing adequate solutions in terms of dredging the lakes that are sedimenting
and transporting the dredged silt to traditional lands, or to arid lands via special slurry
pipelines. A simple concept for such a solution is presented in Figure 1-18. It is proposed
that in certain areas, particularly where the accumulation of silt is likely to apply pressure
on the dam structure, submersible slurry pumps be installed on a permanent basis. Dredging
boats with dredging arms or submersible pumps and cutters would be used on the rest
of the lake. Where the capital investment does not justify it, small dredgers with submersible
pumps should be used. The slurry from these operations will be dilute, it would
be pumped to the shore through a floating plastic pipe. It may be pumped in pipelines and
diverted to canals for agricultural purposes. It may also be pumped to brick plants, where
it would be dewatered and the silt used as a raw material.
dredger
Agriculture
Submersible
pump
Dam
Brick
manufacture
Fi 118
FIGURE 1-18 Simplified flow sheet to remove silt behind dams. Silt would be dredged using
submersible pumps at predetermined locations or in association with boat dredgers. The silt
would be pumped in slurry form to agricultural farmlands or to special plants for the manufacture
of bricks.
silt

Slurry engineers can provide economic solutions to the siltation of dams. This effort
should be made in conjunction with environmental engineers, as new ecosystems often
form around large dams. The author hopes that this handbook will be useful to the decision
makers who have to deal with siltation of dams, while satisfying the concerns of environmental
engineers, as decommissioning is not always the solution.
1-12 CONCLUSION
In this first chapter, some of the basic properties of solids, which are important to the
composition of slurries, were reviewed. Their importance will be emphasized in the next
few chapters. They may lead to Newtonian as well as more complex non-Newtonian
flows that require special equations to determine the friction factors, velocity of flow,
pipe sizes, head, and efficiency losses in pumps.
Wear is an important cost to be paid for transporting solids by liquids. This will be discussed
later in the book when exploring slurry pumps and pipelines. The modern slurry
engineer can serve the mining and power industries by making possible the transportation
of minerals, coal, coal–crude oil mixtures over very long distances, and also play a major
role in dredging sediments behind dams to avoid dam failure and to provide arid lands
with much needed silt.
1-13 NOMENCLATURE
GENERAL CONCEPTS OF SLURRY FLOWS
A Constant
B Constant
Cc Coefficient of curvature
Cu Coefficient of uniformity
Cv Concentration by volume of the solid particles in percent
Cw Concentration by weight of the solid particles in percent
Cp Heat capacity
d10 Grain size at which 10% of the soil is finer
d30 Grain size at which 30% of the soil is finer
d50 Grain size at which 50% of the soil is finer
d60 Grain size at which 60% of the soil is finer
d80 Grain size at which 80% of the soil is finer
K1, K2, K3, K4 polynomial coefficients in Einstein’s equation for dynamic viscosity
k Thermal conductivity
LL Liquid limit of clay and silt soils
PI Plastic index of clay and silt soils
PL Plastic limit of clay and silt soils
Re Reynolds number
VD Deposition critical velocity
VT Viscous transition critical velocity
� Concentration by volume in decimal points
� Absolute (or dynamic) viscosity
� Density
1.37

1.38 CHAPTER ONE
Subscripts
l Liquid
m Mixture
s Solids
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1.39
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Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New
York: Elsevier Applied Sciences.
Further Reading:
Wilson, G. 1976. Construction of Solids-Handling Centrifugal Pumps. In Pump Handbook. Edited
by J. Karassik et al. New York: McGraw-Hill.

CHAPTER 12
FEASIBILITY STUDY
FOR A SLURRY PIPELINE
AND TAILINGS
DISPOSAL SYSTEM
12-0 INTRODUCTION
A consultant engineer has to convince his clients of the merits of a slurry pipeline over alternative
methods of transportation, whether it is for tailings disposal or concentrate shipping.
One very important step in the design of a slurry pipeline is to appreciate the economics
involved in the process. There is no question that this is the effort of a team of
engineers, geologists, and accountants.
It is therefore very important to appreciate the different and complex facets of a feasibility
study. The exercise of a feasibility study or basic engineering should go through a
number of steps, or follow a kind of checklist. In this chapter, the different steps are presented
for this purpose. A pipeline for the disposal of tailings may be a few kilometers or
miles long, whereas a slurry concentrate pipeline may be few hundred kilometers or miles
long. The role of the geologist or foundation engineer is critical to the successful construction
of a tailings dam. It would be beyond the scope of this book to discuss geophysics.
In recent years, there has been a trend toward disposing of tailings in the sea. Whether
tailings are disposed over land or in the sea, there are environmental concerns that must
be satisfied. The engineer should be aware of these issues. The presence of some corrosion
inhibitors, cyanide, or toxic materials in the tailings must be handled carefully. Tailings
dams are sometimes within reach of agricultural fields and seepage could have negative
effects on the quality of underground water. Environmental concerns may represent
hidden costs with particular repercussions on slurry projects.
This chapter presents an overview of basic engineering for a feasibility study. The
study consists of identifying the components of a pipeline (such as feeding station, main
and booster stations, emergency dump ponds, final disposal tailings pond, and area for
sub-sea disposal), the size of the pipeline based on the anticipated flow rate, and the material
of the pipe based on pressure, chemical attack, erosion, and corrosion. Other aspects
of the study outside the scope of the slurry engineer and which cannot be covered in this
book involve the geological survey, the cost of excavation, the cost of construction, power
lines, power stations, transformer stations, and SCADA or control systems. The specialists
involved in these areas rely on the slurry engineer for extensive information and
12.1

12.2 CHAPTER TWELVE
help in basic engineering by providing data on stability of soil, difficulty of the terrain,
cost of power transmission, etc. In turn, this collaborative information is fed to the estimators
and the project managers. The slurry engineer will be requested to make suggestions,
review the feasibility study, and help purchase the equipment.
12-1 PROJECT DEFINITION
At an early stage of the feasibility study, the project is defined in the following terms:
� Volume of slurry to be transported over the life of the project
� Annual pumped flow of slurry
� Starting point of the pipeline, such as a smelter or tailings dam, and a final point of the
pipeline, such as a port for export of the concentrate or power plant for burning coal
� Proposed contour of the pipeline
� Existing roads and need for new roads for access to the pipeline, tailings dam, or
booster station
� Proposed pressure rating of the pipeline and the number of main and booster stations
� Rheology of the slurry
� Environmental impact of the project
� Stability of soil along the contour of the pipeline and possibility of seismic problems or
landslides
� Need for a dewatering plant at the end of the pipeline
� Need for local generation of electricity for booster pump stations or for reclaim water
stations
� Need for a reclaim water pipeline to return water to the starting point of the slurry
pipeline
� Estimation of excavation costs if the pipeline is buried or if electric conduits are underground
� Estimation of costs for power poles to transmit electricity
� Protection of the pipelines from freezing in cold environments
� Allowance for water hammer and transients
� Allowance for thermal expansion in hot climates
� Required modifications to existing thickeners, or filtering and dewatering plants as
part of expansions of production and pumped flow rates
� Mitigation against erosion, abrasion, and corrosion
� Required purchase of land for pipeline contour, tailings dam, dewatering plant, and
booster stations
� Engineering costs
� Construction costs
A general schematic diagram for the tailings disposal pipeline (Figure 12-1) or the concentrate
pipeline (Figure 12-2) is made at an early stage to define the major components
of the pipeline.

Mineral
Process
Plant
slurry
thickener
Emergency
pond
isolation
valve
clarified
water
dilution
water
tailings
pipeline
feed
sump
reclaim water pipeline
pump station
(usually barge mounted)
corrosion
inhibitors
pipeline feed pump station
(from 1 to 9 pumps in series
up to 7.7 MPa (1100 psi))
tailings
pipeline
FIGURE 12-1 General schematic for tailings disposal pipeline.
spigot
cyclones
submerged
disposal
fines for
submerged
disposal
coarse
for banks
of pond
fines for
submerged
disposal
coarse
for banks
of pond
Tailings pond (single or multiple cells)
(or sometimes the sea is used instead of man made ponds)
Tailings Disposal Ponds (Dams)

Mineral
Process
Plant
slurry
Field pump test
Loop to adjust
concentration
thickener
Emergency
pond
isolation
valve
clarified
water
dilution
water
corrosion
inhibitors
concentrate
pipeline
feed & storage
agitator tank
pipeline feed pump station
centrifugal pumps up to 7.7 MPa,1100 psi
reciprocating up to 18 MPa (2600psi))
FIGURE 12-2 General schematic for concentrate pipeline.
concentrate
pipeline
filter/dewatering
plant

FEASIBILITY STUDY FOR A SLURRY PIPELINE
12-2 RHEOLOGY, THICKENER
PERFORMANCE, AND PIPELINE SIZING
12.5
Thickeners are often located at the starting point of the pipeline. Thickeners are installed
for a certain production capacity and can be modified for higher output through the use of
flocculants.
Certain slurries, particularly those rich in fines, silt, and clay, can prove troublesome
for the thickeners. At concentrations in excess of 50–55% by weight, the presence of such
fines could dramatically increase the viscosity and yield stress. This in turn could force
higher power to be needed for pipeline feed pumps, or could force the operator to dilute
the slurry.
Pilot plant tests are highly recommended. In fact, in large mines, a local pump test
loop is sometimes built at the location of the thickeners. The underflow or the concentrate
is pumped through the test loop in order to measure the viscosity and pressure drop. The
information from the test loop is then used to adjust the operation of the main pipeline
pumps and to feed information to the dewatering plant.
During the feasibility study, samples of the ore are sent to a rheology lab. Samples
should be taken from different boreholes. Some boreholes may yield coarser material at
the higher levels but finer materials at deeper depth. This information is used to predict
the performance of the pumps throughout the lifetime of the project. For example, in the
earlier years of the project, the slurry may be coarser and of heterogeneous flow. As the
life of the project progresses, finer material may be pumped at higher concentrations as
non-Newtonian flows.
Samples from different boreholes are also mixed for testing. The blended samples are
quite important as the thickeners may be handling soils from different excavation points,
such as a mixture of sulfides and oxides in different proportions.
From the rheology of the slurry and the optimum performance of the thickeners, the
slurry engineer decides the range of concentrations needed to pump the slurry. The speed
of operation is then decided on the basis of the ratio of coarse to fine particles, the velocity
of deposition, and the friction losses.
Example 12-1
Samples of tailings from a copper process plant are tested for viscosity and yield stress.
Results are plotted in Table 12-1. Determine the maximum concentration for designing
the thickness or pumping of slurry
It is obvious from the data that the viscosity and yield stress rise sharply above a
weight concentration of 55%. The slurry engineer would be wise to consider operations
above 55% as unstable.
Having decided that the maximum weight concentration is 55%, the information is
then given to the process engineer in charge of selecting the thickener. Upon review of the
TABLE 12-1 Combined Fine Tailings
Weight concentration Reduced viscosity (slurry/water) Yield stress (Pa)
40 4.95 1.5
44 7.45 2.7
49.9 12.5 6.2
55 26.4 12.4
60 45 25

12.6 CHAPTER TWELVE
data, the process engineer notices that the thickeners may perform well without flocculants
up to a maximum weight concentration of 50%. Because flocculants are expensive,
a trade-off study is conducted on the power consumption of pumping slurry at 55%. The
study reveals that there is an important increase in capital cost if investment in special
thickeners is made to thicken the slurry at 55%. The amount of expensive flocculants for
a weight concentration of 55% increases the operating cost and the slurry is more viscous
at 55% weight concentration. Despite the fact that there is an increase in the amount of
water pumped at 50% weight concentration, a good compromise is found between cost of
operation of the thickeners and the capital costs needed for the pipeline to handle the flow
for operation at a weight concentration of 53%. The thickeners, pipeline size, and pumps
are then sized to produce slurry at a 53% concentration by weight.
Thickeners (Figure 12-3) are considered the starting point of tailings and concentrate
slurry pipelines. For tailings pipelines, they feed directly into the tailings sump, but for
concentrate pipelines they feed special storage tanks with agitators.
The sump for the tailings pipeline (Figure 12-4) may be built of concrete or rubberlined
steel. A number of pipes are installed in the feed side such as:
� Dilution process water pipes
� slurry pipes
� pipes from emergency ponds
The concentrate storage tanks for slurry pipelines are essentially large tanks with agitators
(Figure 12-5). A small pump test loop near these tanks is used to test the concentrate before
feeding it to the pipeline pumps. Feed is essentially from the thickeners, but continuous
agitation in the tank and addition of viscosity control agents, corrosion inhibitors, and
even some dilution water are part of the process.
Not every slurry pipeline requires thickeners. Dredging pipelines and phosphate rock
pumping are both transport low-concentration slurries. These slurries are pumped over
shorter distances and use pipes and pumps that are physically relocated from one point to
another. Some pipelines operate totally as an open channel flow, such as the tailings
pipeline of Southern Peru Copper in Peru.
FIGURE 12-3 Thickeners. (Courtesy of Geho Pumps.)

FEASIBILITY STUDY FOR A SLURRY PIPELINE
FIGURE 12-4 Sump for tailings pipeline.
FIGURE 12-5 Concentrate storage tank.
12.7

12.8 CHAPTER TWELVE
12-3 RECLAIM WATER PIPELINE
Although slurry is pumped from the process plant to a tailings disposal site, reclaim water
is often pumped back from the tailings pond to the mine. A popular method of feeding the
reclaim water into a pipeline is by installing vertical turbine (mixed flow) pumps on a
barge or onshore near a pump station (Figure 12-6). The number of stages of these vertical
pumps is set by the total dynamic head and the possibility of installing booster pump
stations along the pipeline route. The pipeline material may be constructed of steel or
high-density polyethylene. The latter, however, is limited to a pressure rating of 1.4 MPa
(200 psi) on large pipe sizes (see Chapter 2 for more details on the pressure rating of
HDPE).
If the reclaim water pipeline is steel and the tailings have been neutralized for corrosion
using lime, the pipeline may gradually suffer from deposition of lime on the inside
walls. Over time, this increases the pipe’s roughness; friction losses increase and the
penalty could be higher power consumption. To prevent such a problem, polypig launching
and receiving stations are installed at the start and end of the pipeline. Polypigs are
sponge-filled bullets with brushes sized to the pipe diameter. As they move in the pipe,
they clean its surface.
The methods of sizing the reclaim pipeline for single-phase water were covered in
Chapter 2.
Floating pump stations are often designed as a catamaran for adequate stability. The
pumps are located in the middle of the barge. The catamaran is built with buoyancy
tanks on each side that can be filled. Some catamarans have a false bottom to protect the
suction of the pump. Reclaim water enters from the side pump inlet via a proper fish
screen. The fish screen prevents any fish or aquatic plants from being pumped back to
the mine.
FIGURE 12-6 Pump station.

12-4 EMERGENCY POND
FEASIBILITY STUDY FOR A SLURRY PIPELINE
12.9
An emergency pond should be carved out or built at the start of the pipeline near the
pump station or at the lowest point in the pipeline. The purpose of the emergency pond is
to provide a means of draining the slurry pipeline (Figure 12-7). The decision to dig an
emergency pond is often based on the ability to restart a pipeline after a shut down.
Restarting is often difficult with particularly coarse slurries, taconite, sand, and dredging
rocks. With finer slurries and clays, it may be possible to restart the pipeline without
draining it, provided that the maximum slope does not exceed the critical value (discussed
in Chapter 4).
An emergency pond is needed in cold areas to avoid freezing the pipeline after a shutdown.
Sometimes a special valve chamber is installed with a valve on a tee branch. This
valve automatically opens on power failure to divert slurry to the emergency pond.
An emergency pond needs its own pumping system. It can consist of a vertical slurry
pump floating on a pontoon or barge (Figure 12-8). The sump pump feeds a booster pump
that redirects the slurry back to the pipeline pump box or back to the thickeners.
Submersible pumps (Figure 12-9) with augers are also used for emergency ponds near
thickeners, particularly with concentrate pipelines. Special water sparges are installed
FIGURE 12-7 Emergency pond.

floats
pump
FIGURE 12-8 Emergency pond pumping system.
FIGURE 12-9 Submersible pump.
12.10
motor

around the emergency pond to dilute the slurry. Although fairly reliable, submersible
pumps require a special shop to rebuild them and replace the seals. In remote mines, they
tend to be less popular than vertical cantilever pumps.
It is also recommended to install emergency ponds near booster pump stations. These
should be connected to the booster station by a drainpipe. A pontoon on the pond with a
cantilever pump (Figure 12-8) is recommended to pump back the spill to the booster station
pump box.
If the tailings dam is to be located in a flood plain, the civil engineer may recommend
an emergency spillway. A decant pond may serve as an emergency spillway. Sometimes
it is more economical to provide adequate height of the walls of the dam to contain the
1:100 year flood, particularly when purchasing more land is an expensive proposition.
12-5 TAILINGS DAMS
FEASIBILITY STUDY FOR A SLURRY PIPELINE
Many pipelines are used for pumping tailings. Selecting a site for disposal of tailings is
based on many factors:
� The tailings dam must be able to be used for the life term of the mine (e.g., 10–20
years).
� The site bedrock or foundation must be stable to build the dam walls. These are typically
made of sand and coarse rejects and some are built at a rate of 4.6 m (15 ft) per
year.
� The site must not interfere with future expansion of the mine and must not be on an ore
deposit. For this reason, the tailings disposal system is sometimes a long distance away
from the mine or surrounding economic centers (towns, cities, and agricultural fields).
� The tailings disposal area must be designed to minimize contamination such as seepage
of liquids to surrounding areas.
� The volume of the tailings containment must be calculated to account for disposal volumes,
runoff of snow or rain, and the pumping out of reclaim water.
� The process of separation of slimes from coarse materials at the tailings dam must be
designed carefully. It can be as simple as a spigot when there is a considerable portion
of coarse materials, or as complex as a two-stage cyclone when there are a lot of fines
in the tailings.
� Accessibility to the site is important for repairs and for construction of the tailings
dam.
The guidelines for constructing a tailings dam have been established by the International
Commission on Large Dams (1982). These are reviewed briefly in the following
paragraphs. These are general principles that must be adapted to every site and condition.
It is important to be able to separate the coarse from the fine particles when building a
tailings dam. The coarse solids are used to build the dam walls, whereas the fines are used
to form the beaches (Figure 12-10).
12-5-1 Wall Building by Spigotting
12.11
One method of constructing the walls of a dam is to use the coarse material in the tailings.
The fines or slimes are allowed to sink to the bottom of the tailings pond or to form

12.12 CHAPTER TWELVE
toe
trench
Rock
Toe
slurry
(coarse material)
beaches between the water and the dam. When there is a high content of coarse particles,
as in the taconite mines of Minnesota (U.S.A.), it is sufficient to use a spigot to separate
coarse from fine particles. At the exit from the spigot, the coarse particles separate under
gravity and pressure while the finer particles are carried further away. A bulldozer relocates
and then compacts the material by rolling over it. The banks are gradually built this
way. This operation is difficult in the winter in Canada, Siberia, and northern United
States. Therefore, the actual construction of the dam is limited to the summer months.
Although the great majority of tailing dams are built on the concept of a single spigot,
some use the concept of multiple parallel spigots. In the single-spigot approach, all the
tailings are dispensed at one point. After a couple of days or so, the spigot is then moved
approximately 15 m (50 ft) away. At each location, the bulldozer is brought in to compact
the coarse material. The banks of the dam are thus gradually built.
In the multispigot system, the spigots are fixed in place. The diameter of the pipeline
is gradually reduced around the tailings dam. This method is particularly interesting in
very cold climates when construction of the dam is difficult in the six months of the year
when construction is not possible.
12-5-2 Deposition by Cycloning
cyclone
underflow
filter drain bed
cyclone overflow
is used to make
beaches of fines
beaches of fines
FIGURE 12-10 Using tailings to build dams and beaches.
decant water intake
A spigot may not be sufficient to separate coarse from fine particles. More pressure and
force may be needed. One particularly useful piece of equipment is the hydrocyclone
(which was presented in Chapter 7). The coarse material is diverted to the underflow and
the finer material to the overflow. In a certain ratio of coarse and fine particles, a single
cyclone is sufficient, but when the coarse material is less than 25% of the tailings, two cyclones
in series may be needed. It is strongly recommend that a cyclonability test be conducted
in a lab before deciding whether a single-stage or a two-stage cyclone is needed.
Example 12-2
Tailings from a mine were tested for cyclonability. The following particle size distribution
was obtained:
Particle size (microns) 152 110 74 53 44 37 29 25 22 17
Cumulative % passing 83.3 75.4 67 61 54 52 50 44 42 40
pond

It is clear that the fines (

12.14 CHAPTER TWELVE
Rock Toe
Toe
Trench
Coarse material
used for lifts
1
12-5-2-1 Mobile Cycloning by the Upstream Method
When there is not a sufficient percentage of coarse material in the tailings, it is very difficult
to build a high dam. The approach is then to move the crest of the dam progressively
upstream and to fill more and more surface area. This method is called the “upstream
technique” (Figure 12-11). In this technique, the delivery pipe is initially installed at the
toe wall of the dam. The cyclone underflow is placed inside the toe wall and therefore upstream
of the pipe and partially on top of the fines beach.
12-5-2-2 Mobile Cycloning by the Downstream Method
The slurry pipeline is placed on the internal dividing wall between the fines beach and the
coarse material. The cyclone underflow material is placed between the toe and starter
walls, or downstream from the delivery pipe. The starter wall acts initially as a storage
dam for fines until the coarse material develops an impoundment area to confine the fines.
The crest of the dam moves progressively downstream as the impoundment area is
raised. This provides a “full wedge” impoundment material of high strength. A bulldozer
is used to push the coarse material upward and toward the toe of the dam. As the dam rises,
the centerline of the impoundment area moves outward toward the outer toe of the
dam (Figure 12-12).
Rock
Toe
toe
trench
FIGURE 12-11 Upstream technique of dam building.
Coarse
material
FIGURE 12-12 Downstream technique of dam building.
2
Crest of the dam
2
Annual
lifts
Dividing
wall
beaches of fines
beaches of fines
1

FEASIBILITY STUDY FOR A SLURRY PIPELINE
FIGURE 12-13 Outside wall of tailings dam.
Figure 12-13 shows the outside wall of a tailings dam. The steps represent successive
lifts. Many tailing dams are constructed by annual lifts of 2.4 to 4.5 m (8 to 15 ft).
Sometimes cycloning is done alternatively upstream and downstream to suit the split
of coarse and fine materials. This technique is used when there is not sufficient coarse
material for the downstream method, but there is sufficient material to create a wedge of
coarse material that could be used more satisfactorily than a full upstream method (Figure
12-14).
12-5-2-3 Deposition by Centerline
In this method, the crest of the dam is fixed in plan with respect to the toe wall as the level
of the impoundment is raised.
12-5-2-4 Multicellular Construction
One method of constructing a dam consists of dividing the impoundment area into a number
of cells or small lakes and filling them one at a time. If one of them is used as a source
of reclaim water, it may be set at a lower elevation and therefore fed by the decant water.
This method is popular in hilly regions were it is difficult to find flat ground and the cost
of excavating a large pond would be prohibitive.
12-6 SUBMERGED DISPOSAL
12.15
When the material is simply too fine to build a dam, it may be recommended to submerge
the disposal point. When the volume of the tailings is also too low, as in certain gold

12.16
coarse
by
downstream
method
toe
drain
Rock
toe
FIGURE 12-14 Alternate upstream and downstream technique of dam building.
4
2
5
3
1
Starter
wall
coarse
by
upstream
method
beaches
of
slimes

FEASIBILITY STUDY FOR A SLURRY PIPELINE
mines where tailings may be pumped at a rate of 20 m 3 /hr (88 US gpm), submerged disposal
in a lake is an acceptable method.
12-6-1 Sub-Sea Deposition Techniques
The deposition of city wastewater in the sea has been done by large cities for a number of
decades. Since the late 1960s, more and more mines have turned to the sea for disposal of
tailings. In Peru, Southern Peru Copper has been disposing its tailings in the sea since the
early 1970s. In Chile, CMP has been disposing hematite tailings in the sea since the early
1990s. The disposal of tailings in the sea must take into account low and high tides, the
sea currents, and the redistribution of tailings. It must be environmentally friendly. Over
the years, new beaches may form from accumulated solids and new disposal points must
be selected.
12-7 TAILINGS DAM DESIGN
12.17
Having determined the location of the tailings dam, and having determined its geometry
and the rheology of the tailings, there are still a number of factors that affect the dam design.
Topsoil and unsuitable foundation materials may need to be removed within the footprints
of the dam. This material may have to be stockpiled for eventual closure and reclamation
at the end of the project. A high starter dam may have to be built initially from
compacted waste rock hauled from the mine.
It may be necessary to install a blanket drain of high-permeability material. A geotextile
filter blanket may have to be installed to separate the waste rock from cycloned sand.
To monitor the structural integrity of the dam, vibrating wire piezometers (VWPs) may be
installed beneath the foundation of the dam. The VWP should be placed in the blanket
drain, within the starter dam and within the cycloned sands.
A checklist for the design of a site tailings dam is presented in Table 12-2.
TABLE 12-2 Checklist for Site Tailings Dam Design Information
Crest elevation of starter dam
Final crest elevation of the dam
Crest width
Maximum crest length
Average upstream slope of starter dam
Average downstream slope of starter dam
Average upstream slope of cycloned sand
Average downstream slope of cycloned sand
Blanket drain

12.18 CHAPTER TWELVE
12-8 SEEPAGE ANALYSIS OF
TAILINGS DAMS
In order to determine the expected water level in the completed dam, seepage analysis is
conducted using the data from soils encountered at the site, as well as soils to be part of
the tailings or construction. Seepage from the blanket drain or cycloned sands should be
collected in a trench and sent to a sump where it can be measured.
Seepage analysis is not the responsibility of the slurry engineer and requires a good
geologist and complex computer programs. Seepage analysis is usually based on the permeability
of the soils as shown in Table 12-3. It is discussed here so that the slurry engineer
may design an appropriate monitoring sump.
12-9 STABILITY ANALYSIS FOR
TAILINGS DAMS
A stability analysis must be conducted to find the critical failure surface. This helps determine
the stable slope. Stability charts are available from the U.S. Naval FEC Soils Mechanics
Design Manual 7.01 (1986). Different commercial software packages are available.
They use data from bore hole and test pit logs.
The data from the stability analysis is critical to determine the final height or surface
area of the dam. Obviously, if the soils are weak and unstable, it will not be possible to
build a high dam. More surface area will be required with longer pipes.
The stability analysis includes determining the angle of friction for sands, the cohesion
strength for clays and silts, as well as the density of soils. These determine the strength of
the soils. Static and dynamic slope stability analyses, along with seismic analysis must
also be performed.
Settlement may occur during the construction phase of the dam or during its operation.
Alluvial soils under tailings dams are prone to settlement up to a depth of 3 m (9.85 ft).
An expert should be involved in the design of the walls.
The risk of liquefaction needs to be assessed, particularly when the tailings have a
very large percentage of fines. The plasticity index (see Chapter 1) needs to be measured
to assess the role of liquefaction.
When tailings dams are built on stable foundations, they can be built to a height of 60
m (200 ft). But if an accident happens, they may spill harmful liquids such as cyanide solutions,
with disastrous consequences.
TABLE 12-3 Examples of Materials Permeability for Seepage Analysis
Material Permeability (m/s)
Foundation clay soils 8.4 × 10 –9
Foundation silt 2.5 × 10 –7
Foundation gravel soil 10 –4
Foundation waste rock and sand for starter dams 5 × 10 –5
Foundation soils 4 × 10 –6
Impoundment tailings 6 × 10 –8
Drain sand 10 –4
Cycloned embankment sand 4 × 10 –6

12-10 EROSION AND CORROSION
Erosion and corrosion data are very important at an early stage of the design. There are a
number of recommended tests. One method is the ASTM G75 Slurry Abrasivity Determination
by the Miller number system. The Miller number is established as the relative rate
of mass or volume loss in 2 hours. Slurries with Miller numbers smaller than 50 are considered
relatively nonabrasive and can be pumped with piston reciprocating pumps. Slurries
with Miller numbers above 80 are considered relatively abrasive and require flushed
plunger or membrane pumps (see Section 10-6).
One method to reduce corrosion of steel pipes is to raise the pH by adding lime. The
engineer also has a number of choices for pipeline materials:
� For applications up to 1400 kPa (200 psi), high-density polyethylene pipes are available
for a number of low to medium abrasive slurries.
� For more abrasive slurries with particles up to 6 mm ( 1 – 4 in), rubber-lined pipes are
available for pressures up to 1200 psi.
� For coarser slurries and dredging applications, plain steel pipes are available.
Unlined plain steel pipes are sometimes installed with sacrificial thickness. This is
particularly the case with continuous welded underground pipelines. The reader should
refer to Chapters 9 and 11 for some of the particular approaches used to select materials
for different slurries. Some slurries such as laterite and taconite has been found to be very
abrasive with HDPE pipes.
The actual design and construction of the pipeline should comply with ANSI/ASME
B31.11 (1989 Edition) Slurry Transportation Piping Systems (discussed in the previous
chapter), as well as the standards of the American Water Works Association and local and
national regulations.
12-11 HYDRAULICS
FEASIBILITY STUDY FOR A SLURRY PIPELINE
To conduct a proper study on friction losses, it is a good idea to begin with practical historical
cases, as discussed in Chapter 11, as well as test data. The methods for hydraulic
friction loss estimation have been covered extensively in Chapters 2 to 5 for closed conduits
and in Chapter 6 for open channel flow. A pipeline may consist of both types of
flow. For example, slurry may be pumped to a tank situated at a high point, and then be
allowed to flow by gravity to the tailings pond.
12-12 PUMP STATION DESIGN
12.19
In Chapter 8, it was indicated that the maximum head from rubber-lined pumps is about
30 m (100 ft) and about 55 m (180 ft) for all-metal pumps. This rule of thumb quickly
helps the engineer determine the minimum number of pumps to be installed in series.
Example 12-3
A slurry pipeline is designed to handle flow at a pressure of 3.5 MPa (� 500 psi). The
slurry specific gravity is 1.5. Determine the number of pumps to be installed in the pump
station (assume a head factor of 0.83).

12.20 CHAPTER TWELVE
Solution
The total dynamic head is first calculated:
TDH = P/(�g) = 3.5 × 106 /(1500 × 9.81) � 231 m (or 758 ft)
Assuming a head ratio factor of 0.83 due to the concentration of slurry, the corrected head
is 231/0.83 = 279 m (or 915 ft).
Iteration 1
Assuming all-metal construction at 55 m/stage (180 ft), this would be closer to 5.1 stages.
The engineers must therefore install 6 stages:
279/6 = 46.5 m/stage
For the sake of process control, the engineer may install the variable frequency drive
in the last stage to accumulate fluctuation of (15% of head or ±7m (21 ft)—46.5 – 7 =
39.5 m (129.6 ft) on the lower side for the last stage and 46.5 + 7 = 53.5 m (176 ft) on the
upper side of the last stage. This leaves the opportunity to use fixed-speed motors on the
first 5 stages, and a variable frequency drive for the final stage.
If rubber-lined pumps are to be used, and assuming about 30 m/stage, 10 pumps are
required. To cut down on the number of rubber-lined pumps, polyurethane-lined pumps
may be used because they can operate up to a tip speed of 32 m/s (6300 ft/min). Considering
that the head is proportional to the square of the tip speed:
(32/28) 2 × 30 � 39 m/stage (128 ft/stage)
279 m/39 m � 7.2 m/stage or 8 pumps in a series
If rubber-lines pumps are to be used, and assuming about 30 m/stage, this results in an
installation requiring 10 pumps.
12-13 ELECTRIC POWER SYSTEM
It would be beyond the scope of this book to discuss electrical engineering. However,
some basic concepts need to be reviewed. The electric starters and transformers should be
installed above the 1:100 flood level or above the estimated slurry level in case of flooding
due to a pipeline rupture or back flow.
The estimated consumed power of reclaim water pumps and booster slurry pumps is
supplied to the electrical engineer. Power is brought to the mining site using 10 kV or
14.6 kV overhead or underground lines. Overhead transmission lines may be of an aluminum
conductor, steel reinforced (ACSR). Underground cables may be made of aluminum
or copper. On-site power generation is sometimes considered.
Overhead lines are less reliable than underground lines as they are subject to the following
risks:
� Trees add to the risks of overhead lines because the branches can sway or fall onto the
line, causing the line to fault.
� Lightening storms in some areas require the lines to have special lightening protection.
� Wind and ice formation can also cause momentary line-to-line shorts or even failure.
Although underground lines enjoy the protection of the earth overfill, they are not
problem-free. They are more expensive to install and suffer from longer mean time to repair
(MTTR) than overhead lines.

FEASIBILITY STUDY FOR A SLURRY PIPELINE
Poles for overhead lines may be constructed of steel, concrete, or wood. Concrete is
usually less expensive than steel and more resistant than wood.
The line size is optimized to minimize operating resistance and cost over the life of the
project.
12-14 TELECOMMUNICATIONS
Along the corridor of the power line, a system control and data acquisition (SCADA) or
instrumentation line is run back to the main control station. Telephone wires or fiber optics
are used to monitor remote start or shut pumps and valves. The computer system that
monitors the local control may be “programmable local controller” (PLC) based. The centralized
control room is usually called the distributed control system (DCS). SCADA controlled
systems are designed to include:
� Automatic controls to operate as required by the process
� Manual controls to override SCADA locally
� Remote control for remote start/stop or open/close as required
Motor protection devices should interface to the SCADA at least for status display.
Motor control circuits must be hardwired for motor start/stop control and run status.
“Four wire” types of motor control require one normally open contact to start the motor
and one normally closed to stop it. The advantage is that the motor will maintain current
operating state (running or stopped) in case of PLC failure or accidental shut-down.
Indoor enclosures for control systems in North America should be NEMA 12 with
locking handles. Outdoor enclosures must be NEMA 4X with locking handles.
It is recommended that each PLC have at least three communication ports—one for
the local operator interface, another for the host communications, and a third for the programming
terminal. It is recommended to use modular input/output cards. The system
should be designed with 20% spare capacity for input/output and with the capability for
100% expansion.
12-15 TAILINGS DAM MONITORING
12.21
Monitoring tailings dams can prevent dangerous accidents. A number of methods are recommended:
� Daily visual inspections. Records should be kept.
� Measurement of dry density and moisture contents to ASTM D 3017 standard every
800 cubic meters (28,250 cubic feet) of placed cycloned sand.
� Measurement of dry density and moisture to ASTM D 1556 every three months.
� Determining grain size distribution to ASTM D 422 every 5000 cubic meters (or
176,570 cubic feet) of placed cycloned sand, or after each shut-down of the cyclone
station, or after important changes to the ore.
� Measurement of specific gravity of the tailings in accordance with ASTM D 854, on a
monthly basis, after a shut-down of the cyclone station, or significant change to the
ore.

12.22 CHAPTER TWELVE
� Monthly measurements of the compaction of the deposited sands in accordance with
ASTM D 698 for relative density.
� Measurement of the shear strength characteristics to ASTM D 4767 every 100,000 cubic
meters (3.5 million cubic feet) of placed cycloned sand.
� Weekly measurement of pore-water pressure by the vibrating wire piezometer method.
� Monthly surface survey taken every 10 m (30 ft) along the downstream slope and starting
from the crest.
� Weekly monitoring of seepage by using a notch weir.
The results need to be compared with the original design calculations and assumptions to
make appropriate decisions.
12-16 CHOKE STATIONS AND IMPACTORS
Chokes and choke stations are used to maintain a full pipeline by applying an artificial
pressure loss at the end of the line or at a particular point. Chokes may be pinch valves or
cermet nozzles. Cermet nozzles are made from a composite of metal and ceramic.
Impactors (Figure 12-15) are essentially spring actuated relief valves that protect the
pipeline from surges and water-hammer-related damage. Impactors were developed for
use in the phosphate mines of Florida (U.S.A.), and are gaining acceptance in other fields
of mining. Special air relief valves (Figure 12-16) may need to be installed on tailings and
slurry pipelines at high points to protect the slurry systems. For slurry, the long body design,
familiar in sewage pipelines, is recommended.
12-17 ESTABLISHING AN APPROACH FOR
START-UP AND SHUTDOWN
It is very important to avoid blocking slurry lines. The profile of the pipeline must not
include any steep gradients in excess of the critical slope. Slurry pipelines should be
FIGURE 12-15 Impactor.

FEASIBILITY STUDY FOR A SLURRY PIPELINE
FIGURE 12-16 Air relief valve.
purged after they are shut down in order to avoid sanding the lines. It is not recommended
to start empty lines with slurry. It is preferable to first pump water or thickener
overflow into the line before gradually introducing slurry. Some slurry lines also require
minimum static head or pressure at start-up. Filling the line with water creates the
pressure needed for start-up, and is more effective than pumping against the air in an
empty line.
12-18 CLOSURE AND RECLAMATION PLAN
12.23
It is recommended even at the very start of a feasibility study or basic engineering exercise
to have a closure and reclamation plan. This is really the work of the environmental
engineer, civil engineer, and geophysicist. The hidden costs of abandoning a tailings dam
can be horrendous.
The closure plan may involve planting trees, adding topsoil on the impoundment, or
backfilling and grading with the use of tracked dozers. In one interesting site in South
America, a mine was disposing tailings in the sea. As the sand filled the shore, the seawater
retracted and farmers started moving in and reclaiming the land for agriculture. The
mine was far from reaching a point of closure, and had not established a reclamation plan.
Although this was a convenient closure plan for the mine, it was not an environmentally

12.24 CHAPTER TWELVE
safe solution. Tailings may contain chemicals that are particularly unhealthy to absorb, so
it is safer to grow vegetation such as timber for industrial use than it is to grow plants for
human or animal consumption. The slurry engineer must therefore work in close collaboration
with an environmental engineer and oceanographer.
12-19 ACCESS AND SERVICE ROADS
Access and service roads are an important cost to factor into the budget. It is of utmost
importance to be able to reach each pump station, valve chamber, choke station, and
emergency pond and to be able to drive around a tailings dam. In large mines, these
access roads are sometimes 20 m (60 ft) wide in order to accommodate heavy machinery.
12-20 COST ESTIMATES
Once the basic engineering scheme is completed, or even during its progression, data is
gathered for cost estimates. Written quotes for pumps, equipment, valves, and cyclones
are obtained from manufacturers. The contractors are requested to submit quotes for the
pipeline based on surveys and contour maps. Prices for earthworks are calculated on a
cost-efficient method or using excavation, load, haul, placement, and compaction fleet. It
is very useful to use local costs for similar projects because the labor costs, customs and
duties, and equipment leases change from country to country.
The specific soil investigation along the footprint of the dam and along the pipeline,
may yield large quantities of topsoil that must be stripped away and stockpiled. This information
should then be supplied to the contractor.
Contractor overhead, profit, and mobilization and demobilization costs are calculated
as an acceptable percentage of the lump sum of all installation and construction costs. Often,
the cost runs between 28–32% of the cost of the project. There may also be additional
import duties applied to equipment and materials from outside the country or even inside
the county. In some jurisdictions, these taxes are as high as 14–18%.
12-20-1 Capital Costs
Capital costs for a tailings project include the following (Table 12-4):
� Capital submission costs, such as all project construction and management costs
� Ongoing costs of studies, engineering, investigation, and new land purchases
� Sunk costs, such as costs for project evaluation and investigation of already purchased
land
� Sustaining capital or future capital costs to extend blanket drains, starter dams, and access
roads
� Exit costs to close the project, revegetate, and move back stockpiled topsoil
In some other applications, there may be a separate dewatering and filtering plant as
well as a ship loading facility for a concentrate. Construction of such facilities may repre-

FEASIBILITY STUDY FOR A SLURRY PIPELINE
TABLE 12-4 Checklist for Capital Costs Estimate for a Tailings Pipeline
Item Value in accepted currency
1) Capital submission
a) Access roads
b) Contingency
c) Thickeners modifications
d) Cycloning or spigotting
e) Impoundment
f) Project management (including commissioning)
g) Pipelines (slurry and water)
h) Pump stations
2) Ongoing costs
a) Land purchase
b) Project evaluation
3) Sunk costs
a) Previously purchased land
b) Project evaluation
Total capital submission
Total ongoing costs
Total ongoing costs
Sustaining capital costs
Exit capital costs
sent an important capital investment involving the purchase of land and equipment, construction
costs, power generation, mobilization, and demobilization.
12-20-2 Operation Cost Estimates
The annual cost of operating a slurry pipeline bears an important influence on the final
designs. In the 1950s and 1960s, it was common to see many booster stations of two or
three pumps in series. They were expensive to operate and maintain. With the advent of
high-pressure pumps up to 6.3 MPa (900 psi), the tendency is to centralize pump stations
and eliminate some overhead lines, transformers, and trucks to move spare parts, etc.
Annual operating costs for a tailings system may include the following (Table 12-5):
� Earthworks to grade and place sands
� Costs of moving spigots, discharge pipelines, and cyclone underflow pipes
12.25

12.26 CHAPTER TWELVE
TABLE 12-5 Checklist for Operation Costs
Activity Cost in approved currency
Equipment, fuel, earthworks and piping movement
Impoundment monitoring
Labor
Power requirement for slurry pumps
Power requirement for reclaim pumps
Cyclone data on power consumption
Flocculants for thickeners
Power transmission losses
Maintenance and spare parts
Maintenance contracts
Others
Other dewatering costs
Ship loading costs for concentrations
Contingency (10%)
� Salaries for employees to manage and monitor the tailings impoundment area
� Spare parts for pumps and cyclones and regular maintenance of mechanical and electrical
equipment
� Power for the tailings pumps
� Power for the reclaim pumps
� Cost of flocculants for the thickeners to maintain the concentration of the slurry
� Power transmission loss
� Cost of relining the tailing pipes with rubber-lined cases
� Cost of pipes and materials
� Contract services to maintain pumps and equipment
� Security and monitoring costs
� Travel and training costs
� Contingency costs (10%)
There may be more cost studies needed to conduct to take in account variations in
cost of material and inflation rate. These studies are better left to the experienced estimator.
Despite the most detailed financial studies, natural disasters can wreak havoc with the
most precise engineering estimates. For example, a landslide in Argentina buried a section
of the concentrate pipeline of Bajo Alumbera during the construction phase. It was
necessary to reroute the pipeline and add a booster pump station with a very expensive

price tag. The cost estimate is the result of teamwork between specialists of various
branches of engineering.
12-21 PROJECT IMPLEMENTATION PLAN
During the phase of basic engineering and feasibility studies, it becomes clear that there
may be long delays in the purchase of equipment, mobilization of the construction fleet,
obtaining permits approvals, securing financing, etc. A list of tasks is established with estimates
of required time to complete each step of construction and installation of the
pipeline and equipment. Such a list may include the following:
� Development of a tailings management strategy to support the mining plan
� Basic engineering designs and completion of prefeasibility studies
� Approval of the prefeasibility studies by the owner and financiers
� Detailed engineering designs and completion of feasibility study
� Further approvals and permits for construction
� Land purchase and acquisition of rights of way or lease agreements on land
� Detailed engineering and procurement
� Construction of access roads, site preparation, blanket drain, foundation for thickeners,
pump stations, and starter dams for tailings
� Excavation for buried pipelines, or installation of pipelines and their supports
� Installation of poles for overhead power lines or excavation for underground lines
� Construction of pump stations, MCC buildings, and installation of equipment
� SCADA (systems control and data acquisition) system installation
� Start-up and commissioning
� Demobilization of construction fleet
� Operation and monitoring
� Closure and reclamation
Based on such a task list, the project milestones are established with a precise date set
for start and completion of each phase of the project. To each milestone, a part of the
budget is allocated.
12-22 CONCLUSION
FEASIBILITY STUDY FOR A SLURRY PIPELINE
12.27
The different phases of the engineering of a slurry pipeline are presented in this chapter. It
is not always a straightforward process and may involve trade-off studies based on the
rheology of the slurry, the budget restrictions, the size and capabilities of the equipment,
and mining plans.
This effort is only accomplished through teamwork that involves engineers and designers
from different professions including an estimator, a purchasing officer, technicians
in test labs, geophysicists, and even specialists for reclamation of plants and vegeta-

12.28 CHAPTER TWELVE
tion. It is a very important step to sell the concept to decision makers and financing institutions,
and the expert in slurry systems should be consulted on the various options.
12-23 REFERENCES
The International Commission on Large Dams. 1982. Manual on Tailings Dams and Dumps, Vol 45.
Paris: Author.
U.S. Naval FEC Soils Mechanics Design Manual 7.01 (1986).
Further reading:
Aplin, C. L. and G. O. Argall. 1972. Tailing disposal today. In Proceedings of the First International
Tailing Symposium. Tucson, AZ. San Francisco: Miller Freeman.

APPENDIX B
UNITS OF MEASUREMENT
Acceleration
To convert from To Multiply by
foot/seconds2 (ft/sec2 ) meter/seconds2 (m/s2 ) 0.3048
meter/seconds2 (m/s2 ) foot/seconds2 (ft/sec2 Angular Momentum
) 3.28084
To convert from To Multiply by
kilogram-meter2 /second pound-foot2 /second 23.7304
(kg-m2/s)
Angular Speed
(lb-ft/sec)
To convert from To Multiply by
radian/second revolution per minute 9.5493
revolution per minute
Area
radian/second 0.10472
To convert from To Multiply by
square millimetre ( mm2 ) square inch (in2 ) 0.00155
square inch (in2 ) square millimetre ( mm2 ) 645.161
meter2 (m2 ) yard2 (yd2 ) 1.19599
yard2 (yd2 ) meter2 (m2 ) 0.83612
meter2 (m2 ) foot2 (ft2 ) 10.7639
foot2 (ft2 ) meter2 (m2 ) 0.09290
hectare (ha)
Density
acre 2.47105
To convert from To Multiply by
gram/centimeter3 (g/cm3 ) kilogram/meter3 (kg/m3 ) 1,000
lbm/in3 kilogram/meter3 (kg/m3 ) 27,679
lbm/ft3 kilogram/meter3 (kg/m3 ) 16.0185
kilogram/meter3 (kg/m3 ) lbm/ft3 0.062428
slug/foot3 kg/m3 515.379
lbm/ft3 slug/foot3 0.03106
B.1

B.2 APPENDIX B
Energy
To convert from To Multiply by
British Thermal Units ( BTU)
(ISO/TC 12)
Joules ( J) 1,055.06
Joule (J) foot-pound force (ft-lbf) 0.737562
calorie (mean) (ca) Joules (J) 4.19
megajoule (MJ) hour-horsepower (hph) 0.3725
megajoule (MJ)
Force
kilowatt-hour (kwh) 0.27778
To convert from To Multiply by
dyne (dy) Newton (N) 1 × 10 –5
pound-force (avoirdupois) (lbf) Newton (N) 4.4482
Newton (N) lbf 0.224809
kip
Heat coefficient and transfer
Newton 4,448.221
To convert from To Multiply by
Watt/meter2-degree Celsius British Thermal Unit/foot2- 0.17611
(W/m2 oC) hour-degree Fahrenheit
(BTU/ft2-hr o Length
F)
To convert from To Multiply by
meter (m) foot (ft) 3.28084
micrometer (�m) meter (m) 1 × 10 –6
micrometer (�m) inch (in) 39.37 × 10 –6
millimeter (mm) meter (m) 0.001
millimeter (mm) inch (in) 0.03937
fathom meter (m) 1.8288
foot (ft) meter (m) 0.3048
inch (in) meter (m) 0.0254
inch (in) millimeter (mm) 25.4
yard meter (m) 0.9144
kilometer statute mile (US) 0.621371
nautical mile meter 1,852
statute mile
Mass
meter 1,609.344
To convert from To Multiply by
carat (metric) gram (g) 0.2
grain gram (g) 0.0647989
gram kilogram (kg) 0.001
pound-mass (avoirdupois) (lbm) kilogram (kg) 0.453592
kilogram (kg) pound-mass (avoirdupois) (lbm) 2.20462
long ton kilogram (kg) 1,016.046
short ton ilogram (kg) 907.1847
metric ton kilogram (kg) 1,000
ounce mass (avoirdupois) kilogram (kg) 0.0228349

UNITS OF MEASUREMENT
ounce mass (troy or apothecary) kilogram (kg) 0.0311035
slug
Moment of inertia
kilogram (kg) 14.5939
To convert from To Multiply by
kilogram-meter2 (kg-m2 ) pound-foot2 (lb-ft2 Momentum
) 23.7304
To convert from To Multiply by
kilogram-meter/sec (kg-m/s)
Power
pound-foot/second (lb-ft/sec) 7.23301
To convert from To Multiply by
Btu (thermochemical)/second Watts (W) 1,054.3502
Watt foot-pound-force/second
(ft-lbf/sec)
0.737562
horsepower (hp) foot-pound-force/second
(ft-lbf/sec)
550
horsepower (hp) Watts (W) 745.6998
horsepower (metric)
or chevaux (ch)
Pressure
Watts (W) 735.499
To convert from To Multiply by
atmosphere Newton/meter2 (N/m2 ) 101,325
bar Newton/meter2 (N/m2 ) 100,000
centimeter of mercury (0oC) Newton/meter2 (N/m2 ) 1,333.22
centimeter of water (4oC) Newton/meter2 (N/m2 ) 98.0638
inch of mercury (32oF) Newton/meter2 (N/m2 ) 3,386.389
inch of mercury (60oF) Newton/meter2 (N/m2 ) 3,376.85
inch of water (60oF) Newton/meter2 (N/m2 ) 248.84
lbf/foot2 Newton/meter2 (N/m2 ) 47.8803
lbf/inch2 (psi) Newton/meter2 (N/m2 ) 6,894.757
Pascals (Pa) Newton/meter2 (N/m2 ) 1.0
Newton/meter2 (N/m2 ) lbf/inch2 Specific heat capacity
(psi) 0.00014504
To convert from To Multiply by
Joule/gram- British thernal unit/pound- 0.238846
degree Celsius (J/goC) degree Fahrenheit
(BTU/lb oF) kilojoule/kilogram- British thernal unit/pound- 0.238846
degree Celsius (kJ/kgoC) degree Fahrenheit
(BTU/lb o Speed
F)
To convert from To Multiply by
meter/second (m/s) feet/second (ft/sec) 3.28084
foot/second (ft/sec) meter/second (m/s) 0.3048
B.3

B.4 APPENDIX B
Stress
To convert from To Multiply by
Newton/millimeter2 (N/mm2 ) tonf/in2 0.064749
lbf/inch2 (psi) Newton/meter2 (N/m2 ) 6,894.757
ksi
Temperature
MPa 6.894757
To convert from To Multiply by
Celsius (T oC) Kelvin (T oK) (T oK) = (T oC) +273.15
Fahrenheit (T oF) Kelvin (T oK) (T oK)= 5/9
(T oF +
459.67)
Fahrenheit (T oF) Celsius (T oC) (T oC)= 5/9 (T
oF – 32)
Rankine (T oR) Kelvin (T oK) (T o Viscosity
K)= 5/9 (T
oR) To convert from To Multiply by
centistokes (cst) meter2 /second (m2 /s) 1 × 10 –6
stoke meter2 /second (m2 /s) 0.0001
foot2 /second (ft2 /s) meter2 /second (m2 /s) 0.092903
centipoise (cP) Newton-second/meter2 (N.s/m2 ) 0.001
centipoise (cP) Pascal-second (Pa·s ) 0.001
poise (P) Newton-second/meter2 (N·s/m2 ) 0.10
poise (P) Pascal-second (Pa·s ) 0.10
slug/foot-second Pascal-second (Pa·s ) 47.88025
lbm/foot-second Pascal-second (Pa·s ) 1.488216
lbf-second/foot2 Volume
Pascal-second (Pa·s ) 4.788
To convert from To Multiply by
barrel (petroleum) US gallon 42
gallon (Imperial liquid) liter (L) 4.54608
gallon (US liquid) liter (L) 3.78541
liter (L) meter3 (m3 ) 0.001
ounce (US fluid) meter3 (m3 ) 2.957352 ×
10 –5
foot3 meter3 (m3 ) 0.0283168
quart (US liquid) liter (L) 0.946353
Data from:
Mechanical Engineers Reference Book, 11th ed., A. Parrish (Ed.). London: Newnes-
Butterworths, 1978.
Pump Handbook, 2nd ed., J. Karassik, W. H. Ckrutzch, W. H. Fraser, and J. P.Messina
(Eds.). New York: McGraw Hill, 1986.