Question $2$:

$[25$ marks$]$

Consider a chemical transformation of a reactant A in a heterogeneous catalytic packed bed reactor under isothermal linear kinetics with axial diffusion.

The reaction is the following

$AB,r_A=-k{C}_A,$

Pag e $|3$

Diffusion along the $z-$axis is controlled by a Fickian$-$like expression, so that in parallel with transport by convection due to the superficial velocity of the feed $v_0,($that is $A_n{v}_0{C}_A)$ there is also a diffusion$-$like flux given by the Fickian relation:

$J_A=-d_E\frac{d{C}_A}{dz},(\frac{\text{m}\text{o}\text{l}\text{e}\text{s}}{\text{A}\text{r}\text{e}\text{a}\text{t}\text{i}\text{m}\text{e}})$

acting along the longitudinal axis $($z$-$coordinate$).$

$(Z)$ on the tube with constant cross$-$sectional area, $A_n,$ and obtain a second order ODE for