Exercise $6\mathrm{.}2$: Consider the problem of diffusion and reaction in a cylindrical pore $($e$.$g$.,$ in a solid catalyst$)$ where component A reacts at the walls of the cylinder according to

${?}^{r}B$

where

$r=k{C}_A^2(second-$order reaction$)$

In this system, component A diffuses into the pore due to lower concentration of A inside the pore than at the pore mouth. Since $B$ is produced by the reaction, the concentration of $B$ inside the pore is larger than at the inlet, causing diffusion of $B$ out of the pore. At the inlet of the pore $)=(0,$ the concentration is $C_{A0}.$ The end of the pore $)=(L$ is assumed to be sealed, so there is no flux of $A$ at $x=L.$ The mathematical model for this system can be expressed as follows:

$D_A\frac{d^2{C}_A}{d{x}^2}=k{C}_A^2$

$C_A(0)=C_{A0}$

$\frac{d{C}_A(L)}{dx}=0$

The second boundary condition is the $"$no flux at $x=L"$ condition. This is a split BVP that can be solved using the shooting method. Here are some data for the problem:

$k=0\mathrm{.}01\frac{L}{\text{g}\text{m}\text{o}\text{l}\text{s}}(rate$ constant$)$

$C_{A0}=1\mathrm{.}0$gmo$\frac{l}{L}(inlet$ concentration$)$

$D_A=1{10}^{-3}c\frac{m^2}{s}(diffusivity)$

$L=1cm(lengthof$ pore$)$

Prepare an Excel spreadsheet to solve this problem using the Euler method. Use Goal Seek to implement the shooting method. Be sure to get a reasonable value of the unknown boundary condition $($by experimentation$)$ before invoking Goal Seek.