$14\mathrm{.}11.$ A stirred$-$tank chemical reactor is shown in Figure Q$14\mathrm{.}11$ with the following reaction.

$AB,H_{rxn}=0$

$r_A=-k_0{e}^{-\frac{E}{R}T}\frac{C_A}{1+k{C}_x}$

The available sensors and control valves are shown in the figure, and no changes to these are allowed. The goal is to control the reactor concentration of A tightly by single$-$loop or cascade control, whichever is better. For each of the following disturbances, design the best control system and explain

your design: $(a)$ the heating medium pressure $(P_1),(b)$ the solvent feed temperature $(T_s),(c)$ the reactant feed pressure $(P_2),$ and $(d)$ the inhibitor concentration $(C_x)$ which enters in the solvent. $($To assist in the analysis, prepare a block diagram for each case showing the appropriate single$-$loop or cascade control system.$)$