Page 637 - Bird R.B. Transport phenomena

§20.1 Time Dependent Diffusion 617

the heat of reaction removed through the solid. The concentration profile is a generalization
of that in Eq. 20.1-16:
X
x A ~ AO erf(Z - erf <p (20.1-23)
~ X АО 1 + erf <p
The dimensionless flux <p is given by
1 - x )(N Az0 + N ) dU (20.1-24)
Bz0
Ax
2 N Az0 - x (N AzQ + N ) dZ z=o
Bz0
A0
The relation between the interfacial fluxes and the terminal compositions is
(x M - x J(N AZO + у (20.1-25)
A
~ x
A
Equations 20.1-16,10, and 18 are included as special cases of the last three equations. The last
one is a key result for mass transfer calculations.

EXAMPLE 20.1-2 Gas A is absorbed by a stationary liquid solvent S, the latter containing solute B. Species A re-
acts with В in an instantaneous irreversible reaction according to the equation aA + bB —>
Gas Absorption with Products. It may be assumed that Fick's second law adequately describes the diffusion
Rapid Reaction ' processes, since A, B, and the reaction products are present in S in low concentrations. Obtain
3 4
expressions for the concentration profiles.
SOLUTION Because of the instantaneous reaction of A and B, there will be a plane parallel to the
liquid-vapor interface at a distance z R from it, which separates the region containing no A
from that containing no B. The distance z R is a function of f, since the boundary between A
and В retreats as В is used up in the chemical reaction.
The differential equations for c A and c are then
B
dc
A t ^ A for 0 < z < z (t) (20.1-26)
R
dc 2
R
= 3), d c B for Z (t) < Z < (20.1-27)
R
These are to be solved with the following initial and boundary conditions:
I.C.: at t = 0, C B — С йоо for z > 0 (20.1-28)
B.C.I: at z = 0, C A = C A0 (20.1-29)
B.C. 2, 3: at z = z (t), (20.1-30)
R
B.C. 4: at z = z (t), (20.1-31)
R
B.C. 5: atz = oo = c (20.1-32)
r
B
Here c A0 is the interfacial concentration of A, and c Boo is the original concentration of B. The
fourth boundary condition is the stoichiometric requirement that a moles of A consume b
moles of В (see Problem 20B.2).

' T. K. Sherwood, R. L. Pigford, and C. R. Wilke, Absorption and Extraction, 3rd edition, McGraw-
Hill, New York (1975), Chapter 8. See also G. Astarita, Mass Transfer with Chemical Reaction, Elsevier,
Amsterdam (1967), Chapter 5.
For related problems with moving boundaries associated with phase changes, see H. S. Carslaw
4
and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press (1959). See also S. G.
Bankoff, Advances in Chemical Engineering, Academic Press, New York (1964), Vol. 5, pp. 76-150; J. Crank,
Free and Moving Boundary Problems, Oxford University Press (1984).