Page 706 - Bird R.B. Transport phenomena

686 Chapter 22 Interphase Transport in Nonisothermal Mixtures

Using the dimensional analysis discussion at the beginning of this section, predict the
form of the steady-state mass transfer coefficient correlation for creeping flow.
SOLUTION The dimensional analysis procedure in §19.5 may be used, with D as the characteristic length
p
and v the characteristic velocity. Then, from Eq. 19.5-11, we see that the dimensionless con-
Q
centration depends only on the product ReSc, in addition to the dimensionless position coor-
dinates and the geometry of the bed.
The most extensive data are for creeping flow at large Peclet numbers. Experimental data
on the dissolution of benzoic acid spheres in water 4 have yielded the result

Sh = ^ (ReSc) 1 / 3 ReSc > > 1 (22.3-43)
w
where E is the volume fraction of the bed occupied by the flowing fluid. Equation 22.3-43 is
reasonably consistent with the relation

Sh w = 2 + 0.991 (ReSc) 1 / 3 (22.3-44)
5
which incorporates the creeping flow solution for flow around an isolated sphere {e = 1) (see
§§22.2b). This suggests that the flow pattern around an isolated sphere is not much different
from that around a sphere surrounded by other spheres, particularly near the sphere surface
where most of the mass transport takes place.
No reliable data are available for the limiting behavior at very low values of ReSc, but
numerical calculations for a regular packing 6 predict that the Sherwood number asymptoti-
cally approaches a constant near 4.0 if based on a local difference between interfacial and bulk
compositions.
Behavior within the solid phase is far more complex, and no simple approximation is
7
wholly trustworthy. However, experiments to date show that where intraparticle mass trans-
port is described by Fick's second law, one can use the approximation

Sh m = - ^ * 10 (22.3-45)
where k cs is the effective mass transfer coefficient within the solid phase and ЯЬ is the diffu-
As
sivity of A in the solid phase. The equation is for "slow" changes in the solute concentration
bathing the particle. This is an asymptotic solution for a linear change of surface concentra-
8
tion with time, and has been justified 9 by calculations. For a Gaussian (bell-shaped) concen-
tration wave, "slow" means that the passage time (temporal standard deviation) of the wave
2
is long relative to the particle diffusional response time, which is of the order of D /64b .
p
As
Fick's second law must be solved with the detailed history of surface concentration when this
inequality is not satisfied.
In packed beds, as with tube flow, one must keep in mind the fact that there will be
nonuniformities in the concentration as a function of the radial coordinate. This was dis-
cussed in §14.5 and §20.3.

4
E. J. Wilson and С J. Geankopolis, Ind. Eng. Chem. Fundamentals, 5, 9-14 (1966). See also
J. R. Selman and C. W. Tobias, Advances in Chemical Engineering, 10, 212-318 (1978), for an extensive
summary of mass transfer coefficient correlations obtained by electrochemical measurements.
5
V. G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ. (1962), §14.
6
J. P. Stfrensen and W. E. Stewart, Chem. Eng. Sci., 29, 811-837 (1974).
A. M. Athalye, J. Gibbs, and E. N. Lightfoot, J. Chromatography, 589, 71-85 (1992).
7
H. S. Carslaw and ]. С Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press
8
(1959), §9.3, Eqs. 10 and 11.
9
J. F. Reis, E. N. Lightfoot, P. T. Noble, and A. S. Chiang, Sep. Sci. Tech., 14, 367-394 (1979).