Problem $3(20/100)$: Gas$-$phase catalytic conversion is carried out in a membrane reactor with no

pressure drop at $800K.$ Two parallel reactions follow elementary rate laws $($D is desired product$)$:

$A{k}_{1}D$

$A{k}_{2}U$

$-1,-A,-D-v$

As shown in the schematic above, the reactant $A$ is supplied to the reaction compartment through the

membrane walls. The rate of permeation is relatively low and it can be assumed that the concentration

of $A$ in the membrane interior $(C_{AM})$ is constant. Reactions occur in the surrounding shell filled with a

catalyst. Products are diluted by the inert carrier gas flow $(F_{\text{I}})$ so that it can be assumed that there is no

change in total volumetric flow rate in the reaction compartment $(v_0=25\frac{L}{m}in).$

The known parameters are reaction rate constants, $k_1=1mi{n}^{-1}$ and $k_2=2mi{n}^{-1}$ and the membrane

transport coefficient, $k_c=0\mathrm{.}05mi{n}^{-1}.$ The membrane interior is kept at a constant pressure of $5$ bar and

the catalytic bed is kept at $1$ bar. The reaction compartment $($catalytic bed$)$ volume is $10L.$

a$)$ Calculate the outlet molar flow rate of $A(F_A)$ : Derive equations required and evaluate the

numerical value of $F_A.$

b$)$ Calculate the rate of production of $D$ in terms of its outlet molar flow rate $(F_D)$ : Derive

equations required and evaluate the numerical value of $F_D.$