Page 196 - Bird R.B. Transport phenomena

180 Chapter 6 Interphase Transport in Isothermal Systems

Pressure
PL

--D-

z = 0 z = L

Fig. 6.2-1. Section of a circular pipe from z = 0 to
z = L for the discussion of dimensional analysis.

The system is either in steady laminar flow or steadily driven turbulent flow (i.e.,
turbulent flow with a steady total throughput). In either case the force in the z direction
of the fluid on the inner wall of the test section is

RdOdz (6.2-1)
r=R
In turbulent flow the force may be a function of time, not only because of the turbulent
fluctuations, but also because of occasional ripping off of the boundary layer from the
wall, which results in some distances with long time scales. In laminar flow it is under-
stood that the force will be independent of time.
Equating Eqs. 6.2-1 and 6.1-2, we get the following expression for the friction factor:

RdOdz
Jo Jo V dr I r=R
f(t) = (6.2-2)

Next we introduce the dimensionless quantities from §3.7: f = r/D, z = z/D, v z = v /(v ),
z
z
2
t = (v )t/D, Ф = (9> - tyo)/p(v ) , and Re = D(v )p/fi. Then Eq. 6.2-2 may be rewritten as
z
z
z
L/D 27r
1 D 1 f f f dv \\ (6.2-3)
7
-" L Re J Jo V <W|> =i/2
o
This relation is valid for laminar or turbulent flow in circular tubes. We see that for
flow systems in which the drag depends on viscous forces alone (i.e., no "form drag")
the product of /Re is essentially a dimensionless velocity gradient averaged over the
surface.
Recall now that, in principle, dvjdf can be evaluated from Eqs. 3.7-8 and 9 along
with the boundary conditions 1
B.C. 1: at f = \, for 2 > 0 (6.2-4)
B.C. 2: at z = 0, (6.2-5)
B.C. 3: at f = 0 and z = 0, (6.2-6)

1 2 2
Here we follow the customary practice of neglecting the (d /dz )v terms of Eq. 3.7-9, on the basis
of order-of-magnitude arguments such as those given in §4.4. With those terms suppressed, we do not
need an outlet boundary condition on v.