D$2.$ You are working for a company that is trying to use membrane GP to remove carbon dioxide from natural gas

$($methane$).$ Carbon dioxide has a higher permeability than methane in polymer membranes. Unfortunately, none

of the commercially available membranes have a high enough selectivity $\backslash$alpha CO$2-$CH$4$ at an acceptable value of the

CO$2$ permeance, $($PCO$2$

$/$tms$).$ A small membrane research company claims they can design a polymer membrane

that is selective for CO$2$ and will have a high enough CO$2$ permeance. Your company$$s design group wants to

separate feed that is $40$ mol$\%$ carbon dioxide and $60$ mol$\%$ methane. The goal is to produce $84$ mol$\%$ methane

retentate and recover $70\%$ of the methane in the retentate using a single perfectly mixed membrane module with

a retentate pressure of $6\mathrm{.}2$ atm and a permeate pressure of $1\mathrm{.}04$ atm. The membrane area required for $1\mathrm{.}0$ kmol$/$s

of feed should be $70,000$ m$2$

or less. Find the cut $\backslash$theta $,$ selectivity $\backslash$alpha CO$2-$CH$4$

$,$ and permeance PCO$2$

$/$tms

required.