Page 641 - Bird R.B. Transport phenomena

§20.1 Time Dependent Diffusion 621

Fig. 20.1-3. Time-dependent diffusion from a
soluble wall of A into a semi-infinite column of
liquid B.
=
c A
at

Liquid В

C A ~ A0
c
Solid A

EXAMPLE 20.1-4 Figure 20.1-3 shows schematically the concentration profiles for the diffusion of A from a
slightly soluble wall into a semi-infinite body of liquid above it. If the density and diffusivity
Influence of Changing are constants, then this problem is the mass transfer analog of the problems discussed in §§4.1
Interfacial Area on and 12.1. The diffusion is described by the one-dimensional version of Fick's second law, Eq.
Mass Transfer at an 19.1-18,
Interface '
10 11
(20.1-62)
at
along with the initial condition that c A = 0 throughout the liquid, and the boundary condi-
tions that c A = c A0 at the solid-liquid interface and c A = 0 infinitely far from the interface. The
solution to this problem is

(20.1-63)

from which we can get the interfacial flux

(20.1-64)

Equation 20.1-63 is the mass transfer analog of Eqs. 4.1-15 and 12.1-8.
In Fig. 20.1-4 we depict a similar problem in which the interfacial area is changing with
time as the liquid spreads out in the x and у directions, so that the interfacial area is a function
of time, S(t). The initial and boundary conditions for the concentration are kept the same. We
wish to know the function c (z, t) for this system.
A

SOLUTION The velocity distribution for this varying interfacial area problem is v x = +{ax, v v = +\ay,
v z = -az, where a = d In S/dt. Then the diffusion equation for this system is

dc
1
J H ! " 5 dz dz 1 (20.1-65)

10
D. Ilkovic, Collec. Czechoslov. Chem. Comm., в, 498-513 (1934). The final result in this section was
obtained by Ilkovic in connection with his work on the dropping-mercury electrode.
"V. G. Levich, Physicochemical Hydrodynamics, 2nd edition (English translation), Prentice-Hall,
Englewood Cliffs N.J. (1962), §108. This book contains a wealth of theoretical and experimental results on
diffusion and flow phenomena in liquids and two-phase systems.