Page 644 - Bird R.B. Transport phenomena

624 Chapter 20 Concentration Distributions with More Than One Independent Variable

Consider the steady, two-dimensional flow of a binary fluid around a submerged ob-
ject, such as that in Fig. 4.4-1. In the vicinity of the solid surface, the equations of change
given in §§18.2 and 3 may be simplified as follows, provided that p, \л, к, C pf and %b are
AB
essentially constant (except in the pg term), and that viscous dissipation can be neglected:
dV x #V V
Continuity: -^ + — = 0 (20.2-1)
2
/ dv dv\ dv d v
x
x
Motion: p\ v ——I- v -г— = pv ——(- a —r- 2
V x dx xl •' dy) c dx dy
— Too) + J)g x£(co A — OJ AX) (20.2-2)
Energy: pC t\ v x % + Py Ц) = к ^ - (jf- - jf )r (20.2-3)
A
д(х) А д(х) /
Y
p
Continuity of A: P\v -^ + v - ^ - 4 = ^ -jf + A (20.2-4)
x y A B
The equation of continuity is the same as Eq. 12.4-1. The equation of motion, obtained
from Eq. \Ъ.Ъ-Ъ, differs from Eq. 12.4-2 by the addition of the binary buoyant force term
Р£т£(й>л ~~ <*>л°о)- The energy equation, obtained from Eq. (F) of Table 19.2-4, differs from
Eq. 12.4-3 by the addition of the chemical heat-source term -\(H /M ) - (H /M )]r .
A
A
B
A
B
Equation 20.2-4 is obtained from Eq. 19.1-16 by setting a> = <o {x, y) and neglecting the
A
A
diffusion in the x direction. More complete equations, valid for high-velocity, variable-
property boundary layers, are available elsewhere. 1
The usual boundary conditions on v x are that v x = 0 at the solid surface, and v x =
v (x) at the outer edge of the velocity boundary layer. The usual boundary conditions on Г
e
in Eq. 20.2-3 are that T = T (x) at the solid surface, and T = T at the outer edge of the
0 x
thermal boundary layer. The corresponding boundary conditions on ш А in Eq. 20.2-4 are
that o) = o) (x) at the surface and co = co at the outer edge of the diffusional boundary
A A0 A Ays
layer. Thus there are now three boundary layers to consider, each with its own thickness.
In fluids with constant physical properties and large Prandtl and Schmidt numbers, the
thermal and diffusional boundary layers usually lie within the velocity boundary layer,
whereas for Pr < 1 and Sc < 1 they may extend beyond it.
For mass transfer systems the velocity v ]f at the surface is usually not zero, but de-
pends on x. Hence we set v y = v (x) at у = 0. This boundary condition is appropriate when-
o
ever there is a net mass flux between the surface and the stream, as in melting, drying,
sublimation, combustion of the wall, or transpiration of the fluid through a porous wall.
Clearly, some of these processes are possible with pure fluids, but for simplicity we have
deferred their consideration to this chapter (see also §§18.3 and 22.8 for related analyses).
With the help of the equation of continuity, Eqs. 20.2-1 to 4 can be formally inte-
grated, with the boundary conditions just given, to obtain the following set of boundary
layer balances:
Continuity + motion:
dv d Г , .dv Г*
/л -г— = — pv (v — v )dy + — p(v — v )dy
e
x
ы
dy y= 0 ax Jo x ' e x ' cix Jo c x
- | pg p(T - TJdy - I pg ((o) - (o )dy + pi) v (20.2-5)
J о x J о x A Ax o c
Continuity 4- energy:
= j - П pz>,C,,(T, - T) dy - \ ' (§i - ^ \ d y - pv C (T x - T ) (20.2-6)
o
A
p
0
=0 UXJo Jo \M M /
y A B
1
See, for example, W. H. Dorrance, Viscous Hypersonic Flow, McGraw-Hill, New York (1962), and
K. Stewartson, The Theory of Laminar Boundary Layers in Compressible Fluids, Oxford University Press (1964).