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

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

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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$).$

i$.$ Perform a Mass Balance for the reactant A under steady state conditions on an element

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

$[15$ Marks$]$