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 $(H_2),$ 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$)$

Note: the theoretical separation factoer for knudsen eq$.$ Is $3\mathrm{.}72.$

The actual separation factor id $0\mathrm{.}6957.$

Please answer question $2$