Page 643 - Bird R.B. Transport phenomena

§20.2 Steady-State Transport in Binary Boundary Layers 623

The total number of moles of A that have crossed the interface at time t through the surface
S(t) can be obtained from integration of Eq. 20.1-71 as follows:

M (t) = [ Г c dz dy dx = S(t)c Г (1 - erf(z/S)Wz
f
A J J Jo A A0 JQ
2
= Slt)c -$= Г Г ex V(-£ )dCdz
A0

2
Г [S(t)/S(t)] dt
= S(t)c A0^-j4® AB
2
-^]lS(t)] dt (20.1-73)
An equivalent expression can be obtained by integrating Eq. 20.1-72:

= f SU)N Ct)d~t
AzO

(20.1-74)
2
lS(t)/S(t)] dt
Both Eq. 21.1-73 and Eq. 21.1-74 can be checked by verifying that dM /dt = N (t)S(t).
A
AzO
If S(t) = at", where я is a constant, the above results simplify to
(20.1-75)

For the diffusion into the surrounding liquid from a gas bubble whose volume is increasing
linearly with time, n = \ and 2n + 1 = \. This is of course an approximate result, in which cur-
vature has been neglected, and is therefore valid only for short contact times. Related results
have been obtained for interfaces of arbitrary shapes, 212 and experimentally verified for sev-
eral laminar and turbulent systems. 213

§20.2 STEADY-STATE TRANSPORT IN
BINARY BOUNDARY LAYERS
In §12.4 we discussed the application of boundary layer analysis to nonisothermal flow
of pure fluids. The equations of continuity, motion, and energy were presented in
boundary layer form and were solved for some simple situations. In this section we ex-
tend the set of boundary layer equations to binary reacting mixtures, adding the equa-
tion of continuity for species A so that the concentration profiles can be evaluated. Then
we analyze three examples for the flat-plate geometry: one on forced convection with a
homogeneous reaction, one on rapid mass transfer, and one on analogies for small mass-
transfer rates.

12
J. B. Angelo 7 E. N. Lightfoot, and D. W. Howard, AIChE Journal 12, 751-760 (1966).
n W. E. Stewart, in Physicochemical Hydrodynamics (D. B. Spalding, ed.), Advance Publications Ltd.,
London, Vol. 1 (1977), pp. 22-63.