fluid_mechanics

412 Chapter 8 ■ Viscous Flow in Pipes
0.08
f
0.06
0.04
0.02
0
0.00001
Completely
turbulent flow
0.0001
0.001
e
D
0.01
0.1
Equation 8.34, called the Darcy–Weisbach equation, is valid for any fully developed, steady, incompressible
pipe flow—whether the pipe is horizontal or on a hill. On the other hand, Eq. 8.33
is valid only for horizontal pipes. In general, with V 1 V 2 the energy equation gives
p 1 p 2 g1z 2 z 1 2 gh L g1z 2 z 1 2 f / rV 2
D 2
Part of the pressure change is due to the elevation change and part is due to the head loss associated
with frictional effects, which are given in terms of the friction factor, f.
It is not easy to determine the functional dependence of the friction factor on the Reynolds
number and relative roughness. Much of this information is a result of experiments conducted by
J. Nikuradse in 1933 1Ref. 62 and amplified by many others since then. One difficulty lies in the
determination of the roughness of the pipe. Nikuradse used artificially roughened pipes produced
by gluing sand grains of known size onto pipe walls to produce pipes with sandpaper-type surfaces.
The pressure drop needed to produce a desired flowrate was measured and the data were
converted into the friction factor for the corresponding Reynolds number and relative roughness.
The tests were repeated numerous times for a wide range of Re and eD to determine the
f f1Re, eD2 dependence.
In commercially available pipes the roughness is not as uniform and well defined as in the
artificially roughened pipes used by Nikuradse. However, it is possible to obtain a measure of the
effective relative roughness of typical pipes and thus to obtain the friction factor. Typical roughness
values for various pipe surfaces are given in Table 8.1. Figure 8.20 shows the functional dependence
of f on Re and eD and is called the Moody chart in honor of L. F. Moody, who, along
with C. F. Colebrook, correlated the original data of Nikuradse in terms of the relative roughness
of commercially available pipe materials. It should be noted that the values of eD do not necessarily
correspond to the actual values obtained by a microscopic determination of the average
height of the roughness of the surface. They do, however, provide the correct correlation for
f f1Re, eD2.
It is important to observe that the values of relative roughness given pertain to new, clean
pipes. After considerable use, most pipes 1because of a buildup of corrosion or scale2 may have a
relative roughness that is considerably larger 1perhaps by an order of magnitude2 than that given.
As shown by the figure in the margin, very old pipes may have enough scale buildup to not only
alter the value of e but also to change their effective diameter by a considerable amount.
The following characteristics are observed from the data of Fig. 8.20. For laminar flow,
f 64Re, which is independent of relative roughness. For turbulent flows with very large Reynolds
numbers, f f1eD2, which, as shown by the figure in the margin, is independent of the Reynolds
number. For such flows, commonly termed completely turbulent flow 1or wholly turbulent flow2, the
laminar sublayer is so thin 1its thickness decreases with increasing Re2 that the surface roughness
completely dominates the character of the flow near the wall. Hence, the pressure drop required is a
TABLE 8.1
Equivalent Roughness for New Pipes [From Moody
(Ref. 7) and Colebrook (Ref. 8)]
Equivalent Roughness, E
Pipe Feet Millimeters
Riveted steel 0.003–0.03 0.9–9.0
Concrete 0.001–0.01 0.3–3.0
Wood stave 0.0006–0.003 0.18–0.9
Cast iron 0.00085 0.26
Galvanized iron 0.0005 0.15
Commercial steel
or wrought iron 0.00015 0.045
Drawn tubing 0.000005 0.0015
Plastic, glass 0.0 1smooth2 0.0 1smooth2