Please answer each of the following independent questions.

a$)(10$ marks$)$ Methanol is produced by partial oxidation of methane; but methanol can undergo further oxidation to produce $C{O}_2$ and $H_{2}O.$ This series reaction can be represented by $A+\frac{1}{2}{O}_{2}B+\frac{3}{2}{O}_{2}C+2D.$ You're your goal is to maximize the yield of methanol $($B$),$ and limit production of $C{O}_2(C).$ What type of reactor $($or reactor combination$)$ would you choose to maximize the yield of methanol, with $(-r_A)=k_A{A}_A^2$ and $(r_C)=k_B{c}_B,$ where $k_A=0\mathrm{.}025Lmo{l}^{-1}{s}^{-1},k_B=0\mathrm{.}25s^{-1}.$ Would you prefer high or low conversion of methane $($A$)?$ Explain and justify your answer $($calculations are not required$).$

b$)(10$ marks$)$ Assuming that the selected reactor for the methanol synthesis process in $($a$)$ is a pseudo$-$homogeneous PFR operating adiabatically, write the energy balance and mass balance equations that account for both reactions and all five components in the system.

c$)(10$ marks$)$ You have been asked to design a chemical reactor system for an autocatalytic reaction $AC,$ with $(-r_A)=k_A{c}_A{c}_C.$ There is no $C$ in the feed, although a one$-$time addition of $C$ at start$-$up is possible. The liquid phase reaction takes place isothermally and isobarically. What would be your two best design options, if the primary objective is to minimize reactor volume and capital cost. PROVIDE DETAILED REASONING TO SUPPORT YOUR ANSWER.