Consider a CSTR with a heater where heat $Q_H$ is transferred from a heater coil to the tank. There is a second$-$order reaction $r=k{c}_A^2$ with the heat of reaction $H_R$ and reaction rate constant $k=k_0{e}^{-\frac{E}{R}T}$ where $E$ is an activation energy and $R$ and $k_0$ are constants. The tank volume $V,$ solution density $,$ and heat capacity $c_P$ are kept constant. There is no $B$ in the feed stream.

a$)$ What is the entire set of transfer functions for this MIMO system?

b$)$ Draw a block diagram for this process, as well as plausible feedback control schemes if the controlled variables are $c_B$ and $T$ and the manipulated variables are $Q_H$ and $c_{A,i}.$