Page 657 - Bird R.B. Transport phenomena

§20.4 Boundary Layer Mass Transport with Complex Interfacial Motion 637

(b) To get the surface-averaged value of the mass flux, we integrate the above expression
over all в and ф and divide by the sphere surface:
Г f "N R sin в dO с1ф
JO JO A
2 С" sin 3 Ode
2TTR C A0 3^ AB V X
3
Vcos в - 3 cos в + 2
2
- u )du

(20.3-28)

In going from the second to the third line, we made the change of variable cos в = и, and
to get the fourth line, we factored out (1 — u) from the numerator and denominator. Equa-
tion 20.3-28 was cited in Eq. 18.5-20 in connection with absorption from gas bubbles. 8 This
equation is referred to again in Chapter 22 in connection with the subject of mass transfer
coefficients.

§20.4 BOUNDARY LAYER MASS TRANSPORT WITH
COMPLEX INTERFACIAL MOTION " 3
1
Time-dependent interfacial motions and turbulence are common in fluid-fluid transfer
operations. Boundary layer theory gives useful insight and asymptotic relations for these
systems, utilizing the thinness of the concentration boundary layers for small ЯЬ (as in
АВ
liquids) or for flows with frequent boundary layer separation (as at rippling or oscillating
interfaces). Mass transfer with simple interfacial motions has been discussed in §18.5 for a
laminar falling film and a circulating bubble, and in Example 20.1-4 for a uniformly ex-
panding interface. Here we consider mass transfer with more general interfacial motions.
Consider the time-dependent transport of species Л between two fluid phases, with
initially uniform but different compositions. We start with the binary continuity equa-
tion for constant p and ЯЬ (Eq. 19.1-16, divided by p):
АВ
^ А А + \г А (20.4-1)

We now want to reduce this to boundary layer form for small 4b , and then present so-
AB
lutions for various forced-convection problems with controlling resistance in one phase.
We use the following boundary layer approximations:
(i) that the diffusive mass flux is collinear with the unit vector n normal to the
nearest interfacial element. (This approximation is used throughout the bound-
ary layer sections of this book. Higher-order approximations, 4 not treated here,
are appropriate for describing boundary layer diffusion near edges, wakes, and
separation loci.)

8
V. G. Levich, Physkochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ. (1962), p. 408,
Eq. 72.9.
1
J. B. Angelo, E. N. Lightfoot, and D. W. Howard, AIChE Journal, 12, 751-760 (1966).
2
W. E. Stewart, J. B. Angelo, and E. N. Lightfoot, AIChE Journal, 16, 771-786 (1970).
3
W. E. Stewart, AIChE Journal, 33, 2008-2016 (1987); 34,1030 (1988).
J. Newman, Electroanal. Chem. and Interfacial Electrochem., 6,187-352 (1973).
4