Page 580 - Bird R.B. Transport phenomena

560 Chapter 18 Concentration Distributions in Solids and in Laminar Flow

An exactly analogous problem occurred in Example 4.1-1, which was solved by the
method of combination of variables. It is therefore possible to take over the solution to
that problem just by changing the notation. The solution is 3

r, 9 f x/V4Z AB z/v m ^
(18.5-15)

or
= = erfc - (18.5-16)
440

In these expressions "erf x" and "erfc x" are the "error function" and the "complemen-
tary error function" of x, respectively. They are discussed in §C6 and tabulated in stan-
dard reference works. 4
Once the concentration profiles are known, the local mass flux at the gas-liquid in-
terface may be found as follows:

(18.5-17)
dx x=0
Then the total molar flow of A across the surface at x = 0 (i.e., being absorbed by a liquid
film of length L and width W) is

TIL (18.5-18)
The same result is obtained by integrating the product v c over the flow cross section
max A
at z = L (see Problem 18C.3).
Equation 18.5-18 shows that the mass transfer rate is directly proportional to the
square root of the diffusivity and inversely proportional to the square root of the "expo-
sure time," t exp = L/v . This approach for studying gas absorption was apparently first
max
proposed by Higbie. 5
The problem discussed in this section illustrates the "penetration model" of mass
transfer. This model is discussed further in Chapters 20 and 22.

EXAMPLE 18o5-l Estimate the rate at which gas bubbles of A are absorbed by liquid В as the gas bubbles rise at
v
their terminal velocity through a clean quiescent liquid.
Gas Absorption from t
Rising Bubbles

3
The solution is worked out in detail by the method of combination of variables in Example 4.1-1.
4
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 9th printing
(1973),pp.310etseq.
5
R. Higbie, Trans. AIChE, 31, 365-389 (1935). Ralph Wilmarth Higbie (1908-1941), a graduate of the
University of Michigan, provided the basis for the "penetration model" of mass transfer. He worked at
E. I. du Pont de Nemours & Co., Inc., and also at Eagle-Picher Lead Co.; then he taught at the University
of Arkansas and the University of North Dakota.