Page 142 - Separation process principles 2

3.7 Two-Film Theory and Overall Mass-Transfer Coefficients 107

at different rates. Equations (3-202) and (3-203) become, the Marangoni effect, is discussed in some detail by Bird,
Stewart, and Lightfoot [28], who cite additional references.
112
NA,~ = ~(cA, - CA,) = (CA, - CA,)(SDAB) The effect can occur at both vapor-liquid and liquid-liquid
- - interfaces, with the latter receiving the most attention. By
adding surfactants, which tend to concentrate at the inter-
face, the Marangoni effect may be reduced because of stabi-
lization of the interface, even to the extent that an interfacial
mass-transfer resistance may result, causing the overall
mass-transfer coefficient to be reduced. In this book, unless
otherwise indicated, the Marangoni effect will be ignored
and phase equilibrium will always be assumed at the phase
interface.

Gas-Liquid Case
In the limit, for a high rate of surface renewal, sS~/DA~,
(3-204) reduces to the surface-renewal theory, (3-200). For Consider the steady-state mass transfer of A from a gas
low rates of renewal, (3-205) reduces to the film theory, phase, across an interface, into a liquid phase. It could be
(3- 188). At conditions in between, kc is proportional to Dig, postulated, as shown in Figure 3.21a, that a thin gas film ex-
where n is in the range of 0.5-1.0. The application of the ists on one side of the interface and a thin liquid film exists
film-penetration theory is difficult because of lack of data for on the other side, with the controlling factors being molecu-
6 and s, but the predicted effect of molecular diffusivity lar diffusion through each film. However, this postulation is
brackets experimental data. not necessary, because instead of writing

3.7 TWO-FILM THEORY AND OVERALL
-
MASS-TRANSFER COEFFICIENTS (3-206)
Separation processes that involve contacting two fluid we can express the rate of mass transfer in terms of mass-
phases require consideration of mass-transfer resistances in transfer coefficients determined from any suitable theory,
both phases. In 1923, Whitman [63] suggested an extension with the concentration gradients visualized more realisti-
of the film theory to two fluid films in series. Each film pre- cally as in Figure 3.21b. In addition, we can use any number
sents a resistance to mass transfer, but concentrations in the of different mass-transfer coefficients, depending on the se-
two fluids at the interface are assumed to be in phase equi- lection of the driving force for mass transfer. For the gas
librium. That is, there is no additional interfacial resistance phase, under dilute or equimolar counter diffusion (EMD)
to mass transfer. This concept has found extensive applica- conditions, we write the mass-transfer rate in terms of partial
tion in modeling of steady-state, gas-liquid, and liquid- pressures:
liquid separation processes.
The assumption of phase equilibrium at the phase inter-
face, while widely used, may not be valid when gradients of where kp is a gas-phase mass-transfer coefficient based on a
interfacial tension are established during mass transfer be-
partial-pressure driving force.
tween two fluids. These gradients give rise to interfacial tur-
For the liquid phase, we use molar concentrations:
bulence resulting, most often, in considerably increased
mass-transfer coefficients. This phenomenon, referred to as

Gas 1 Gas Liquid I Liquid
phase I film film 1 phase
PA6 I Liquid
I phase
PA, 1 PAL
I I
I C~,
I
I I CA6
1- c
Transport
Transport
Figure 3.21 Concentration
gradients for two-resistance
theory: (a) film theory; (b) more
(a) (b) realistic gradients.