Suppose that a certain catalyst is prepared by immersing a sheet of porous material in a solution that contains the catalytic agent, thereby depositing it in the pores. The resulting catalyst concentration $C_C(x)$ 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 laboratory 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 maintained 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 $)=(L$ be placed next to the impermeable barrier. For this arrangement, use a singular perturbation analysis to determine the steadystate 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 concentration 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$.446-450).$

Boundary Condition $1$: $C_A=C_{A0}.$ Boundary Condition $2$: $\frac{d{C}_A}{dx}=0.$ Governing equation is $\frac{d{C}_A}{dx}+J_A-R_A=0$ where $J_A=-D_A{C}_A$and $$ is the gradient. Therefore, the governing equation is $-D_A\frac{d^2{C}_A}{d{x}^2}+$k$C_C{C}_A=0.$ So in non$-$dimensionalized form, we have $=\frac{C_A}{C_{A0}}$ and n$=\frac{x}{L}$ where n is the greek letter eta. Therefore, $\frac{d^2}{d{n}^2}-$Da$[1-n^2]=0.$So the new BCs are $(0)=1$ and $\frac{d}{dn}(1)=0.$ Now E is the greek letter epsilon. So Esince Da$=ph{i}^2.$ Then E $\frac{d^2}{d{n}^2}-[1-n^2]=0.$

For part b$,$ only set up the problem until the stage where you get the Airy ODE $\frac{d^{2}f}{d{z}^2}=k^{2}z$f$,$ where k is a constant. You do not need to solve that equation, but you should be able to derive it up to that point.