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Q$10$: 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

$AkB$

Where:

$r_A=-k{C}_A^2,[2^{nd}$ order reaction $]$

As represented in FIG. below, 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 $x=L.$

FIG. Representation of a cylindrical catalyst pore.

The mathematical model for this system can be expressed as follows:

Boundary Conditions:

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

$\{\begin{array}{cc}{C}_A=C_{A0},& x=0\end{array}$

$----------$

$\frac{d{C}_A}{dt}=0,x=L$

$\backslash$table$[[\backslash$table$[[$Rate constant$],[k(\frac{L}{g}$mols$)$