fluid_mechanics

396 Chapter 8 ■ Viscous Flow in Pipes
D V
(1) (2)
μ
Δp = p 1
– p 2 = F(V, , D, )
Dimensional analysis
can be used to
put pipe flow parameters
into dimensionless
form.
8.2.3 From Dimensional Analysis
Although fully developed laminar pipe flow is simple enough to allow the rather straightforward
solutions discussed in the previous two sections, it may be worthwhile to consider this
flow from a dimensional analysis standpoint. Thus, we assume that the pressure drop in the horizontal
pipe, ¢p, is a function of the average velocity of the fluid in the pipe, V, the length of
the pipe, /, the pipe diameter, D, and the viscosity of the fluid, m, as shown by the figure in the
margin. We have not included the density or the specific weight of the fluid as parameters because
for such flows they are not important parameters. There is neither mass 1density2 times
acceleration nor a component of weight 1specific weight times volume2 in the flow direction involved.
Thus,
There are five variables that can be described in terms of three reference dimensions 1M, L, T2.
According to the results of dimensional analysis 1Chapter 72, this flow can be described in terms
of k r 5 3 2 dimensionless groups. One such representation is
(8.17)
where f1/D2 is an unknown function of the length to diameter ratio of the pipe.
Although this is as far as dimensional analysis can take us, it seems reasonable to impose a
further assumption that the pressure drop is directly proportional to the pipe length. That is, it takes
twice the pressure drop to force fluid through a pipe if its length is doubled. The only way that
this can be true is if f1/D2 C/D, where C is a constant. Thus, Eq. 8.17 becomes
which can be rewritten as
or
(8.18)
The basic functional dependence for laminar pipe flow given by Eq. 8.18 is the same as that
obtained by the analysis of the two previous sections. The value of C must be determined by
theory 1as done in the previous two sections2 or experiment. For a round pipe, C 32. For ducts
of other cross-sectional shapes, the value of C is different 1see Section 8.4.32.
It is usually advantageous to describe a process in terms of dimensionless quantities. To this end
we rewrite the pressure drop equation for laminar horizontal pipe flow, Eq. 8.8, as ¢p 32m/VD 2
and divide both sides by the dynamic pressure, rV 2 2, to obtain the dimensionless form as
This is often written as
where the dimensionless quantity
¢p F1V, /, D, m2
D ¢p
mV
f a / D b
D ¢p
mV C/
D
¢p
/ Cm V
D 2
Q AV 1p 4C2 ¢pD 4
m/
¢p
1
2 rV 132m/V D 2 2
64 a m
2 1
2 rV 2 rVD b a / D b 64
Re a / D b
¢p f / rV 2
D 2
f ¢p1D/21rV 2 22
is termed the friction factor, or sometimes the Darcy friction factor [H. P. G. Darcy
(1803–1858)]. 1This parameter should not be confused with the less-used Fanning friction