Page 673 - Bird R.B. Transport phenomena

Problems 653

(e) Show that the terms of the two leading orders in Eq. 20C.1-3 give
2
2
2
d«*A , \ _ У . Зу ] dr s дь> А _ \д ш А <
2
(20C.1-4)
the second-order terms being designated by dashed underlines.
4
(e) This equation has been analyzed thoroughly in the electrochemical literature. The results
for n A0 are further considered in Problem 20C.2.
20C.2. Effect of surface curvature on absorption from a growing bubble (Fig. 20A.2). Pure gas A is
flowing from a small capillary into a large reservoir of initially pure liquid В at a constant
molar flow rate W . The interfacial molar flux of A into the liquid is predictable from the
A
Levich-Koutecky-Nezvman equation
(20C.2-1)
— C A0
in which
ко (3w y*
= = A (20C.2-2)
t l 4TTCJ
for purely radial motion and a spherical bubble. Equation 20C.2-1 is a consequence of Eq.
20C.1-4.
(a) Give an expression for the number of moles of A absorbed over a bubble lifetime .
t
0
(b) Use Eq. 20C.2-1 to obtain more accurate results for the absorption rates in Problem 20A.2.
20C.3. Absorption with chemical reaction in a semi-inh'nite medium. A semi-infinite medium of
material В extends from the plane boundary x = 0 to x = oo. At time t = 0 substance A is
brought into contact with this medium at the plane x = 0, the surface concentration being c A0
(for absorption of gas A by liquid B, for example, c A0 would be the saturation concentration).
Substances A and В react to produce С according to the irreversible first-order reaction A + В
—» С. It is assumed that A is present in such a small concentration that the equation describing
the diffusion plus chemical reaction process is

(20C.3-1)

in which k"' is the first-order rate constant. This equation has been solved for the initial condi-
tion that c A = 0 at t = 0, and the boundary conditions that c A = c A0 at x = 0, and c A = 0 at x =
oo. The solution is 3

(20C.3-2)

(a) Verify that Eq. 20C.3-2 satisfies the differential equation and the boundary conditions.
(b) Show that the molar flux at the interface x = 0 is

N A (20C.3-3)

4 J. Koutecky, Czech. ]. Phys., 2, 50-55 (1953). See also V. Levich, Physicochemical Hydrodynamics, 2nd
edition, Prentice-Hall, Englewood Cliffs, N.J. (1962). The right sides of Levich's Eqs. 108.17 and 108.18
should be multiplied by t . See also J. S. Newman, Electrochemical Systems, 2nd edition, Prentice-Hall,
2n
Englewood Cliffs, N.J. (1991).
P. V. Danckwerts, Trans. Faraday Soc, 46, 300-304 (1950).
5