Page 707 - Bird R.B. Transport phenomena

§22.4 Definition of Transfer Coefficients in Two Phases 687

EXAMPLE 22.3-4 In both gas-liquid 10 and liquid-liquid 11 contactors, sprays of liquid drops or clouds of bubbles
are frequently encountered. Contrast their mass transfer behavior with that of solid spheres.
Mass Transfer to Drops
and Bubbles
SOLUTION
Many different types of behavior are encountered, and surface forces can play a very impor-
tant role. We discuss surface forces in some detail in §22.7. Here we consider only some limit-
ing cases and refer readers to the above-cited references.
Very small drops and bubbles behave like solid spheres and can be treated by the corre-
lations in Example 22.3-3 and in Chapter 14. However, if both adjacent phases are free of sur-
factants and small particulate contaminants, the interior phase circulates and carries the
adjacent regions of the exterior phase along. This stress-driven "Hadamard-Rybczinski circu-
lation //12 increases the mass transfer rates markedly, often by almost an order of magnitude,
and the rates can be estimated from extensions 1316 of the "penetration model" discussed in
§18.5. Thus, for a spherical bubble of gas A with diameter D rising through a clean liquid B,
the Sherwood number on the liquid side lies in the range 16

(22.3-46)

where v t is the terminal velocity (see Eqs. 18.5-19 and 20).
The size at which the transition from the solid-like behavior to circulation occurs de-
pends on degree of surface contamination and is not easily predicted.
Very large drops or bubbles oscillate, 13 and both phases follow a modified penetration
model,
Sh. (22.3-47)

with angular frequency of oscillation 1
192<7
(22.3-48)
3
D (3p D + 2p c )
where a is the interfacial tension, and p D and p are the densities of the drops and the continu-
c
ous medium.
The success of this model implies that the boundary layer is refreshed once every oscilla-
tion, but there is also a small effect of periodic stretching of the surface.

§22.4 DEFINITION OF TRANSFER COEFFICIENTS
IN TWO PHASES

Recall that in §10.6 we introduced the concept of an overall heat transfer coefficient, U, to
describe the heat transfer between two streams separated from each other by a wall. This
overall coefficient accounted for the thermal resistance of the wall itself, as well as the
thermal resistance in the fluids on either side of the wall.

10
J. Stichlmair and J. F. Fair, Distillation Principles and Practice, Wiley, New York (1998).
11
J. С Godfrey and M. M. Slater, Liquid-Liquid Extraction Equipment, Wiley, New York (1994).
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, The Hague (1983).
12
J. B. Angelo, E. N. Lightfoot, and D. W. Howard, AIChE Journal, 12, 751-760 (1966).
13
14
J. B. Angelo and E. N. Lightfoot, AIChE Journal, 14, 531-540 (1968).
W. E. Stewart, J. B. Angelo, and E. N. Lightfoot, AIChE Journal, 16, 771-786 (1970).
15
R. Higbie, Trans. AIChE, 31, 365-389 (1935).
16
17
R. R. Schroeder and R. C. Kintner, AIChE Journal, 11,5-8 (1965).