$(30$ marks$)$ Biodiesel can be produced by conversion of triglycerides $($TG$)$ into fatty acid methyl esters

$($ME$)$ in the presence of methanol and sodium hydroxide $($catalyst$).$ The reaction occurs in $3$ steps,

producing diglycerides $($DG$)$ and monoglycerides $($MG$)$ before ultimately yielding glycerol $($G$)$ as a

co$-$product. DG and MG act as surfactants, and are thus undesirable. Each reaction is first order, and

reversible. The relevant reactions are listed below:

Let $k_1,k_2$ and $k_3$ represent the forward reaction rate constants and $k_{-1},k_{-2}$ and $k_{-3}$ represent the reverse

reaction rate constants for the first, second and third reactions, respectively.

i$)$ Considering the goal is to maximize production of methyl esters $($ME$),$ identify two

strategies to maximize ME yield. Explain and justify your answer

ii$)$ For a batch reactor, write all of the mass balance equations that would be needed to track

the moles of all components in this system

iii$)$ Develop$/$write the corresponding energy balance for the system, assuming adiabatic

operation