$(35)$

$3.$ Direct conversion of $C{O}_2$ to methanol $(C{H}_{3}OH)$ has potential to be an environmentally friendly method for reducing $C{O}_2$ emissions while producing an important feedstock chemical.

$C{O}_2+3H_{2}harrC{H}_{3}OH+H_{2}O$

A simplified schematic of a pilot scale process is shown below. The fresh feed $($Stream $1)$ contains $50.mo\frac{l}{m}in$ of $C{O}_2,10\%$ excess $H_2,$ and $3mo\frac{l}{m}in$ of a $N_2$ impurity. The single pass conversion of $C{O}_2$ across the reactor is $25\%$ and the overall conversion is $80\%.$ The liquid product stream from the condenser $($Stream $4)$ contains all of the $H_{2}O$ and $95\%$ of the methanol leaving the system. The vapor stream from the condenser $($Stream $5)$ is divided into a purge $($Stream $6)$ and a recycle $($Stream $7).$

a$)$ Why is a purge stream needed for this process? $(2$ points$)$

b$)$ What is the molar flow rate of $H_{2}O$ in Stream $4?(($:$6pts$