Page 568 - Bird R.B. Transport phenomena

548 Chapter 18 Concentration Distributions in Solids and in Laminar Flow

Fig. 18.2-2. Film model for mass transfer; component A
Slowly Main fluid stream
-moving-*-]- in turbulent flow is diffusing from the surface into the gas stream through
gas film a hypothetical stagnant gas film.

^ A l
^surface
at edge of film
x A2

A Direction
of flow

The rate of mass transfer at the liquid-gas interface—that is, the rate of evapora-
tion—may be obtained from Eq. 18.2-1 as follows:
\dx
c9> AB B ^ (18.2-14)
1 - x dz = +- ' dz
M
By combining Eqs. 18.2-13 and 14 we get finally

x ) (18.2-15)
fe " A2
2
This expression gives the evaporation rate in terms of the characteristic driving force
X X
A\ ~~ A2-
By expanding the solution in Eq. 18.2-15 in a Taylor series, we can get (see §C2 and
Problem 18B.18)

N, x ) x x x ) (18.2-16)
2
(z 2 - A1 M A2 A2
The expression in front of the bracketed expansion is the result that one would get if the
convection term were entirely omitted in Eq. 18.0-1. The bracketed expansion then gives
the correction resulting from including the convection term. Another way of interpreting
this expression is that the simple result corresponds to joining the end points of the x A
curve in Fig. 18.2-1 by a straight line, and the complete result corresponds to using the
curve of x A versus z. If the terminal mole fractions are small, the correction term in brack-
ets in Eq. 18.2-16 is only slightly greater than unity.
The results of this section have been used for experimental determinations of gas
diffusivities. 2 Furthermore, these results find use in the "film models" of mass transfer.
In Fig. 18.2-2 a solid or liquid surface is shown along which a gas is flowing. Near the
surface is a slowly moving film through which A diffuses. This film is bounded by the
surfaces z = z } and z = z . In this "model" it is assumed that there is a sharp transition
2
from a stagnant film to a well-mixed fluid in which the concentration gradients are negli-
gible. Although this model is physically unrealistic, it has nevertheless proven useful as
a simplified picture for correlating mass transfer coefficients.

2
С Y. Lee and C. R. Wilke, lnd. Eng. Chem., 46, 2381-2387 (1954).