The ethylene epoxydation is to be carried out using a cesium$-$doped silver catalyst in a packed$-$bed

reactor.

$(1)C_2{H}_4+\frac{1}{2}{O}_2{C}_2{H}_{4}O,-r_{IE}=\frac{k_{1E}{P}_E{P}_O^{0\mathrm{.}58}}{(1+K_{IE}{P}_E{)}^2}$

Along with the desired reaction, the complete combustion of ethylene also occurs

$(2)C_2{H}_4+3O_{2}2C{O}_2+2H_{2}O,-r_{2E}=\frac{k_{2E}{P}_E{P}_O^{0\mathrm{.}3}}{(1+K_{2E}{P}_E{)}^2}$

$[$M$.$ Al$-$Juaied, D$.$ Lafarga, and A$.$ Varma, Chem. Eng. Sci. $56,395(2001)].$

It is proposed to replace the conventional PBR with a membrane reactor in order to improve the selec$-$

tivity. As a rule of thumb, a $1\%$ increase in the selectivity to ethylene oxide translates to an increase in

profit of about $$2$ million $\frac{?}{yr}.$ The feed consists of $12\%($mole$)$ oxygen, $6\%$ ethylene, and the remainder

nitrogen at a temperature of $250C$ and a pressure of $2$atm. The total molar flow rate is $0\mathrm{.}0093mo\frac{l}{s}$ to

a reactor containing $2kg$ of catalyst.

Additional information:

$\{\begin{array}{c}\text{:}{K}_{1E}=6\mathrm{.}50at{m}^{-1}\\ K_{2E}=4\mathrm{.}33at{m}^{-1}\end{array}$

$($a$)$ What conversion and selectivity of ethylene epoxide to $C{O}_2$ are expected in a conventional PBR$?$

$($b$)$ What would be the conversion and selectivity if the total molar flow rate were divided and the

$12\%$ oxygen stream $($no ethylene$)$ were uniformly fed through the sides of the membrane reactor,

and $6\%$ ethylene $($no oxygen$)$ were fed at the entrance?

$($c$)$ Repeat $($b$)$ for a case when ethylene is fed uniformly through the sides and oxygen is fed at the

entrance. Compare with parts $($a$)$ and $($b$).$ Describe what you find.