fluid_mechanics

428 Chapter 8 ■ Viscous Flow in Pipes
follows. For a smooth pipe 1eD h 02 the Colebrook equation,
Eq. 8.35a, becomes
where from Eq. 3,
1
1f 2.0 log ae D h
3.7 2.51
Re h 1f b
2.0 log a 2.51
Re h 1f b
f 0.269 a 5
If we combine Eqs. 4, 5, and 6 and simplify, Eq. 7 is obtained for a.
1.928 a 5 2 2 log12.62 10 4 a 3 2 2
(5)
(6)
(7)
By using a root-finding technique on a computer or calculator, the
solution to Eq. 7 is determined to be a 0.614 ft, in agreement
(given the accuracy of reading the Moody chart) with that obtained
by the trial and error method given above.
Note that the length of the side of the equivalent square duct
is aD 7.348 0.918, or approximately 92% of the diameter
of the equivalent duct. It can be shown that this value, 92%, is a
very good approximation for any pipe flow—laminar or turbulent.
The cross-sectional area of the duct 1A a 2 53.9 in. 2 2 is
greater than that of the round pipe 1A pD 2 4 50.3 in. 2 2. Also,
it takes less material to form the round pipe 1perimeter pD
25.1 in.2 than the square duct 1perimeter 4a 29.4 in.2. Circles
are very efficient shapes.
8.5 Pipe Flow Examples
Pipe systems may
contain a single
pipe with components
or multiple
interconnected
pipes.
In the previous sections of this chapter, we discussed concepts concerning flow in pipes and ducts.
The purpose of this section is to apply these ideas to the solutions of various practical problems.
The application of the pertinent equations is straightforward, with rather simple calculations that
give answers to problems of engineering importance. The main idea involved is to apply the energy
equation between appropriate locations within the flow system, with the head loss written in
terms of the friction factor and the minor loss coefficients. We will consider two classes of pipe
systems: those containing a single pipe 1whose length may be interrupted by various components2,
and those containing multiple pipes in parallel, series, or network configurations.
F l u i d s i n t h e N e w s
New hi-tech fountains Ancient Egyptians used fountains in
their palaces for decorative and cooling purposes. Current use of
fountains continues, but with a hi-tech flair. Although the basic
fountain still consists of a typical pipe system (i.e., pump, pipe,
regulating valve, nozzle, filter, and basin), recent use of computercontrolled
devices has led to the design of innovative fountains
with special effects. For example, by using several rows of multiple
nozzles, it is possible to program and activate control valves to
produce water jets that resemble symbols, letters, or the time of
day. Other fountains use specially designed nozzles to produce
coherent, laminar streams of water that look like glass rods flying
through the air. By using fast-acting control valves in a synchronized
manner it is possible to produce mesmerizing three-dimensional
patterns of water droplets. The possibilities are nearly limitless.
With the initial artistic design of the fountain established, the initial
engineering design (i.e., the capacity and pressure requirements
of the nozzles and the size of the pipes and pumps) can be
carried out. It is often necessary to modify the artistic and/or engineering
aspects of the design in order to obtain a functional,
pleasing fountain. (See Problem 8.64.)
II: Q = ?
III: D = ?
I: Δp = ?
8.5.1 Single Pipes
The nature of the solution process for pipe flow problems can depend strongly on which of the various
parameters are independent parameters 1the “given”2 and which is the dependent parameter 1the
“determine”2. The three most common types of problems are shown in Table 8.4 in terms of the parameters
involved. We assume the pipe system is defined in terms of the length of pipe sections used
and the number of elbows, bends, and valves needed to convey the fluid between the desired locations.
In all instances we assume the fluid properties are given.
In a Type I problem we specify the desired flowrate or average velocity and determine the
necessary pressure difference or head loss. For example, if a flowrate of 2.0 galmin is required
for a dishwasher that is connected to the water heater by a given pipe system as shown by the figure
in the margin, what pressure is needed in the water heater?
In a Type II problem we specify the applied driving pressure 1or, alternatively, the head loss2
and determine the flowrate. For example, how many galmin of hot water are supplied to the dishwasher
if the pressure within the water heater is 60 psi and the pipe system details 1length, diameter,
roughness of the pipe; number of elbows; etc.2 are specified?