Problem $(3)$: Consider a cylindrical shaped porous catalyst particle of radius $R$ and length $L$ as shown. This particle is in a catalytic reactor, where it is submerged in a gas stream containing the reactant A and the product $B.$ In the neighborhood of the surface of the particle under consideration, we presume that the concentration is $C_{AR}$ moles of A per unit volume. Species A diffuses through the tortuous passages in the catalyst and is converted to $B$ on the catalytic surfaces according to a first$-$order chemical reaction $R_A=-k_1^{\text{'}\text{'}}a{c}_A,$ where $a$ is the available catalytic surface per unit volume $($of solids $+$ voids$).$ Show that the molar flow $W_{AR}$ at the surface $r=R$ is given by:

$W_{AR}=-(2RL)D_A{c}_{AR}m\frac{I_1(mR)}{I_0(mR)}$

where $m=\sqrt[\phantom{2}]{\frac{k_1^{\text{'}\text{'}}a}{D_A}}$ and $D_A$ is the "effective diffusivity".