$4-15.$ Nonuniform Catalyst Distribution ${?}^{**}$

Suppose that a certain catalyst is prepared by immersing a sheet of porous material in a solu$-$

tion that contains the catalytic agent, thereby depositing it in the pores. The resulting catalyst con$-$

centration $C_C(x)$ is found to be nonuniform. Assume that it is given by

$C_C(x)=C_{C0}[1-(\frac{x}{L}{)}^2]$

where $L$ is the sheet thickness and $C_{C0}$ is a constant. The supported catalyst is to be used in a labo$-$

ratory experiment involving the irreversible reaction $AB,$ which follows the rate expression

$R_{VA}=-k{C}_C{C}_A.$

Entry of species A will occur only at one surface of the sheet, where the concentration will be main$-$

tained at $C_{A0}$; the opposite surface will be placed next to an impermeable barrier. The reaction is fast

enough that $Da=k{C}_{C0}\frac{L^2}{D_A}1.$

$($a$)$ It is intended that the side with depleted catalyst $(x=L)$ be placed next to the impermeable

barrier. For this arrangement, use a singular perturbation analysis to determine the steady$-$

state flux of A entering the sheet. Evaluate the first two terms in the expansion for the flux.

$($b$)$ Suppose that the sheet is accidentally reversed, so that the side with high catalyst concen$-$

tration is placed next to the barrier. Show that the first term in a perturbation expansion for

$C_A$ is governed by the Airy equation, which is

$\frac{d^{2}W}{d{z}^2}=zW.$

The solutions to this equation, called Airy functions, are given in Abramowitz and Stegun

$(1970,$ pp$.446-450).$