Please Solve Question $2!$

A sol$-$gel derived $-$alumina membrane has a porosity $=0\mathrm{.}5,$ thickness $=3m,$ average pore

diameter $d_p=4nm($assuming circular opening$),$ and pore channel tortuosity $=1\mathrm{.}5.$ When the

membrane was tested for separation of a binary gas mixture containing $80$mol$\%$ carbon

monoxide $($CO$)$ and $20$mol$\%$ hydrogen $),$ the fluxes for $CO$ and $H_2$ were $3\mathrm{.}45$mol$*m^{-2}*s^{-1}$ and

$2\mathrm{.}4$mol$*m^{-2}*s^{-1},$ respectively. The feed flow rate was very large so that the feed side composition

and pressure changes caused by permeation were negligible. Following are operation conditions

under which gas permeation through the membrane is expected to exhibit Knudsen diffusion

behavior:

Temperature $T=300K$; total feed pressure $P_f=\frac{1}{b}ar$ ; permeate pressure $P_p$~~$0$ bar $($under

vacuum$).$

$(1)$ Calculate the Knudsen $($theoretical$)$ separation factor $($i$.$e$.$ permeation selectivity$)$ for $H_2$ over

$CO$ and the actual separation factor obtained in the above experiment.

$(2)$ If the difference between the theoretical and actual separation factors was caused by leaking

through nonselective macropores $($defects$)$ in the membrane, estimate the permeance of CO

and $H_2$ through the $4-$nm membrane pores and the defects, respectively. Based on the

calculation discuss the impact of the defects to the permeation of $H_2$ and $CO,$ respectively.

$($NOTE: NOT ALL INFORMATION PROVIDED ARE NEEDED FOR SOLVING THE PROBLEMS$)$