Page 583 - Bird R.B. Transport phenomena

§18.7 Diffusion and Chemical Reaction Inside a Porous Catalyst 563

cules of A penetrate only slightly into the film, they cannot get far enough to "see" the
outer boundary of the film, and hence they cannot distinguish between the true bound-
ary condition and the approximate boundary condition that we use. The same kind of
reasoning was encountered in Example 12.2-2 and Problem 12B.4.
The form of the boundary conditions in Eqs. 18.6-3 to 5 suggests the method of com-
[/3
bination of variables. Therefore we try c /c AQ = /(17), where 77 = y(a/94b z) . This com-
AB
A
bination of the independent variables can be shown to be dimensionless, and the factor
of "9" is included to make the solution look neater.
When this change of variable is made, the partial differential equation in Eq. 18.6-2
reduces to an ordinary differential equation
(18.6-6)

with the boundary conditions/(0) = 1 and/(00) = 0.
This second-order equation, which is of the form of Eq. C.I-9, has the solution

/=Q Г exp(- (18.6-7)
V
The constants of integration can then be evaluated using the boundary conditions, and
one obtains finally
p _ _ f 00 _ _
3
3
exp(-r] ) drj I exp(-r) ) drj
С A J V J it (18.6-8)

for the concentration profiles, in which Г(§) = 0.8930 .. is the gamma function of f. Next
.
the local mass flux at the wall can be obtained as follows

4 Ay\y=0 c
y=0 уаг\ \ A0/ dyj y=0
= + (-*- " (18.6-9)

Then the molar flow of A across the entire mass transfer surface at у = 0 is

\l/3
(18.6-10)
where Г ф = I ф = 1.1907....
Г
The problem discussed in §18.5 and the one discussed here are examples of two types
of asymptotic solutions that are discussed further in §20.2 and §20.3 and again in Chapter
22. It is therefore important that these two problems be thoroughly understood. Note that
2/3
in §18.5, W^ ос (2) и 1/2 , whereas in this section W A ос (ЯЬ Ь) . The differences in the ex-
АВ
лв
ponents reflect the nature of the velocity gradient at the mass transfer interface: in §18.5,
the velocity gradient was zero, whereas in this section, the velocity gradient is nonzero.
§18.7 DIFFUSION AND CHEMICAL REACTION
INSIDE A POROUS CATALYST
Up to this point we have discussed diffusion in gases and liquids in systems of simple
geometry. We now wish to apply the shell mass balance method and Fick's first law to
describe diffusion within a porous catalyst pellet. We make no attempt to describe the