Page 585 - Bird R.B. Transport phenomena

§18.7 Diffusion and Chemical Reaction Inside a Porous Catalyst 565

We now define an "effective diffusivity" for species A in the porous medium by
dc,
дг . = -Q) - (18.7-4)
Л; A
in which c A is the concentration of the gas A contained within the pores. The effective
diffusivity %b must be measured experimentally. It depends generally on pressure and
A
temperature and also on the catalyst pore structure. The actual mechanism for diffusion
in pores is complex, since the pore dimensions may be smaller than the mean free path
of the diffusing molecules. We do not belabor the question of mechanism here but as-
sume only that Eq. 18.7-4 can adequately represent the diffusion process (see §24.6).
When the preceding expression is inserted into Eq. 18.7-3, we get, for constant
diffusivity

(18.7-5)
dr
We now consider the situation where species A disappears according to a first-order
chemical reaction on the catalytic surfaces that form all or part of the "walls" of the
winding passages. Let a be the available catalytic surface per unit volume (of solids +
voids). Then R = -k"ac , and Eq. 18.7-5 becomes (see Eq. C.l-6)
A A
1 d (ДЛСА (18.7-6)
dr
This equation is to be solved with the boundary conditions that c A = c AR at r = R, and
that c is finite at r = 0.
A
2
2
Equations containing the operator (l/r )(d/dr)[r (d/dr)] can frequently be solved by
using a "standard trick"—namely, a change of variable c /c = (l/r)/(r). The equation
A AR
for/(r) is then
(18.7-7)
This is a standard second-order differential equation, which can be solved in terms of ex-
ponentials or hyperbolic functions. When it is solved and the result divided by r we get
the following solution of Eq. 18.7-6 in terms of hyperbolic functions (see §C5):

I*- (18.7-8)

Application of the boundary conditions gives finally

(18.7-9)

In studies on chemical kinetics and catalysis one is frequently interested in the molar
flux N AR or the molar flow W AR at the surface r — R:

2
W fiR = 4irR N AR = - (18.7-10)
When Eq. 18.7-9 is used in this expression, we get
(18.7-11)

This result gives the rate of conversion (in moles/sec) of Л to В in a single catalyst particle
of radius R in terms of the parameters describing the diffusion and reaction processes.