fluid_mechanics

8.4 Dimensional Analysis of Pipe Flow 411
y
R = D/2
δ s
Velocity
profile, u = u(y)
Viscous sublayer
x
or
∋
Rough wall
δ s
Smooth wall
∋
F I G U R E 8.19 Flow in the
viscous sublayer near rough and smooth
walls.
of the density than it is of viscosity. Second, we have introduced two additional dimensionless
parameters, the Reynolds number, Re rVDm, and the relative roughness, eD, which are not
present in the laminar formulation because the two parameters r and e are not important in fully
developed laminar pipe flow.
As was done for laminar flow, the functional representation can be simplified by imposing
the reasonable assumption that the pressure drop should be proportional to the pipe length. 1Such
a step is not within the realm of dimensional analysis. It is merely a logical assumption supported
by experiments.2 The only way that this can be true is if the /D dependence is factored out as
¢p
1
2rV / 2 D f aRe, e D b
As was discussed in Section 8.2.3, the quantity ¢pD1/rV 2 22 is termed the friction factor, f. Thus,
for a horizontal pipe
where
¢p f / rV 2
D 2
f f aRe, e D b
(8.33)
The major head
loss in pipe flow is
given in terms of
the friction factor.
For laminar fully developed flow, the value of f is simply f 64Re, independent of eD. For turbulent
flow, the functional dependence of the friction factor on the Reynolds number and the relative
roughness, f f1Re, eD2, is a rather complex one that cannot, as yet, be obtained from a theoretical
analysis. The results are obtained from an exhaustive set of experiments and usually presented
in terms of a curve-fitting formula or the equivalent graphical form.
From Eq. 5.89 the energy equation for steady incompressible flow is
p 1
g a V 1 2
1
2g z 1 p 2
g a V 2 2
2
2g z 2 h L
where h L is the head loss between sections 112 and 122. With the assumption of a constant diameter
1D 1 D 2 so that V 1 V 2 2, horizontal 1z 1 z 2 2 pipe with fully developed flow 1a 1 a 2 2, this
becomes ¢p p 1 p 2 gh L , which can be combined with Eq. 8.33 to give
h L major f / D
V 2
2g
(8.34)