Page 222 - Bird R.B. Transport phenomena

206 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

which shows that it is the integral of the local rate of viscous dissipation over the volume
of the entire flow system.
We now want to examine E v from the point of view of dimensional analysis. The
2
quantity Ф,, is a sum of squares of velocity gradients; hence it has dimensions of (v /l ) ,
Q
0
where v 0 and / are a characteristic velocity and length, respectively. We can therefore
0
write
E v = {pvlll){ix/kv p)№ v dV (7.5-2)
Q
2<
where Ф о = (l /v ) & v and dV = l^dV are dimensionless quantities. If we make use of
o
o
the dimensional arguments of §§3.7 and 6.2, we see that the integral in Eq. 7.5-2 depends
only on the various dimensionless groups in the equations of change and on various
geometrical factors that enter into the boundary conditions. Hence, if the only significant
dimensionless group is a Reynolds number, Re = l v p/iJL, then Eq. 7.5-2 must have the
o o
general form
^ / зт2ч ч. (& dimensionless function of Re\
(7.5-3)
E v v = (OVQIQ) X , . i. • i t - /r7 r o 4
u u
H
\and various geometrical ratios/
In steady-state flow we prefer to work with the quantity E v = E /w, in which w = p(v)S is
v
the mass rate of flow passing through any cross section of the flow system. If we select
the reference velocity v 0 to be (v) and the reference length to be Vs, then
/
0
2
E = \(v) e (7.5-4)
v v
in which e , the friction loss factor, is a function of a Reynolds number and relevant di-
v
mensionless geometrical ratios. The factor \ has been introduced in keeping with the
form of several related equations. We now want to summarize what is known about the
friction loss factor for the various parts of a piping system.
For a straight conduit the friction loss factor is closely related to the friction factor.
We consider only the steady flow of a fluid of constant density in a straight conduit of
arbitrary, but constant, cross section S and length L. If the fluid is flowing in the z direc-
tion under the influence of a pressure gradient and gravity, then Eqs. 7.2-2 and 7.4-7
become
(z-momentum) - (Pi ~ p )S + (pSL)g z (7.5-5)
2
(mechanical energy) E v ~ Pi) + Чг (7.5-6)
Multiplication of the second of these by pS and subtracting gives

Ev = 7s (7.5-7)

If, in addition, the flow is turbulent then the expression for F Hs in terms of the mean hy-
draulic radius R may be used (see Eqs. 6.2-16 to 18) so that
h

E = \{vf^f (7.5-8)
v
K h
in which / is the friction factor discussed in Chapter 6. Since this equation is of the form
of Eq. 7.5-4, we get a simple relation between the friction loss factor and the friction
factor

for turbulent flow in sections of straight pipe with uniform cross section. For a similar
treatment for conduits of variable cross section, see Problem 7B.2.