A fresh feed stream at $2\mathrm{.}5C$ and $55$atm $($abs$.)$ containing $CO$ and $H_2$ joins a recycle stream, and the combined stream is fed to a reactor. The reactor outlet stream contains $20\mathrm{.}0$mol$\%H_2$ and $39\mathrm{.}9$mol$\%CO.$ The single$-$pass conversion of $H_2($the limiting reactant$)$ is $0\mathrm{.}80.$ This stream enters a condenser operating at $80$atm$(\frac{?}{abs})$ and $38C,$ where most $($not all$)$ of the methanol is condensed and removed as product at a rate of $100mo\frac{l}{m}in$; none of the other components are condensed. The gas stream leaving the condenser contains $CO,H_2,$ and uncondensed methanol vapor that is in equilibrium with the liquid methanol product stream. This gas stream is divided into a recycle stream $($which mixes with the fresh feed$)$ and a purge stream.

$($a$)(15$ points$)$ Copy the flowchart onto your answer sheet and fully label the flowchart. Circle all your unknowns.

Label the output streams from the condenser with the overall flow rate and mole fractions.

$($b$)(20$ points$)$ Show that the following subsystems have the following degrees$-$of$-$freedom. You have to use the specified method to do the DOF analysis for each subsystem.

$($i$)$ Overall system $($extent of reaction DOF$)=3$

$($ii$)$ Reactor $($atomic species DOF$)=1($note: $C$ and $O$ balances are NOT independent$)$

$($iii$)$ Condenser $=0$

$($iv$)$ Splitting point $=4$

$($v$)$ Mixing point $=5$

Circle your unknowns clearly on your diagram so you do not have to list them in your DOF.

$($c$)25$ points$)$ Solve the condenser system to calculate

$($i$)$ the molar flow rate of gas leaving the reactor, and

$($ii$)$ the composition of the vapor stream exiting the condenser.

$($d$)(25$ points$)$ The goal of this problem is to calculate the overall conversion of $H_2.$ Write out $($but do not solve$)$ all the equations in the order needed to solve this problem.

Be specific: circle the unknowns you are solving for in each equation, indicate equations that need to be solved simultaneously, etc.

$($e$)(15$ points$)$ Upon solving the problem, the fresh feed was found to contain $230\mathrm{.}0mo\frac{l}{m}in$ of $H_2$ and $158,8mo\frac{l}{m}in$ of $CO.$ By using the compressibility factor equation of state, calculate the volumetric flow rate of this stream in $\frac{m^3}{m}in.$

$($Note: remember to add $8K$ and $8$atm respectively to the $T_c$ and $P_c$ for $H_2!)$