Mass Transfer
429
Whitman (1923) took this a step further in his two-film theory, in which it was assumed that mass
transfer from a gas bubble could be described by molecular diffusion through the two stagnant
films. From
Pick's first law
with a constant concentration gradient through the film, we get for the
mass transfer through the liquid film:
where
Da
is the diffusion coefficient for component A and ^ is the thickness of the liquid film. The
film theory gives a simple relation between the diffusivity and the mass transfer coefficient.
However, it is not possible to calculate
k,
from
DA,
since the film thickness is normally not known.
In fact, there is, strictly speaking, no stagnant film surrounding the bubbles, although the liquid
velocity relative to the surface is low very close to the bubble. Furthermore, the assumption of a
constant concentration gradient may be wrong, e.g. if the diffusing species A is consumed by a
chemical reaction within the film. Despite these shortcomings, the film model is probably the most
widely used model to illustrate the concept of mass transfer.
Not all models for mass transfer make use of a stagnant film. In the so-called surface renewal
theory (Danckwerts, 1970), discrete liquid elements close to the gas-liquid interface are thought to
be interchanged with a well-mixed bulk liquid. Each element stays close to the surface for a certain
time, during which it is considered to be stagnant. The transfer of compound A from the gas phase
to the element is determined by the exposure time of the liquid element by solving the unsteady-
state form of Pick's law of diffusion (sometimes called Fick’s second law o f diffusion). The mean
residence time at the surface,
zren
is introduced in this model, and it can be shown that
k,
is related to
Da
through:
It is seen from Eq. (10.10) that a slightly different relation between
ks
and
DA
than predicted from
the film theory is obtained. The dependence of
kt
on
DA
is experimentally found to be between that
predicted by the film theory and that predicted by the surface-renewal theory. Again, a calculation
of
k,
is not possible from the model, since it is difficult to find a value for the mean residence time
at the surface.
A more rigorous approach is to connect the mass transfer coefficient to a solution o f the flow field
close to a surface. This can be done using boundary-layer theory (see e.g Cussler, 1997). The mass
transfer coefficient, averaged over a length L (m), for transfer of a compound present in the solid
phase of a sharp-edged plate, is given by:
(10.9)
ki = 4 D aI t ™
(
10
.
10
)
(
10
.
11
)
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