$(1)$ Consider a temperature controlled, continuous flow stirred$-$tank reactor with a single first order exothermic chemical reaction. The tank is temperature controlled with a cooling coil.

Use the following symbols:

The flow of reactant solution into the tank is $q$ and flow rate of product solution out of the tank is also $q$;

The tank volume is $V$;

The overall heat transfer coefficient is $U_o$;

The heat transfer area is $A$;

The specific heat of the reactant solution is $C_p$;

The initial concentration of $A$ is $C_{AO}$ and $B$ is $0$ ;

The solution density is $$;

The heat is reaction is $H_R$;

The coolant rate is $w.$

The reaction is

$AB$

$r=k{C}_A$ where $k=k_0{e}^{-\frac{E}{R}T}$

$($a$)$ Develop a material balance equation for the tank of volume $V$;

$($b$)$ Develop an energy balance in terms of $U_o$ with a heat transfer area A and specific heat $C_p$;

$($c$)$ Write the two equations from part $($a$)$ and $($b$)$ in deviation variable form;

$($d$)$ Write the coefficients of the two equations from part $($c$)$ in terms of $a_{11},a_{12},$ etc;

$($e$)$ Write the two equations from part $($c$),$ as defined in part $($d$),$ in state$-$space form;

$($f$)$ Solve for the transfer function,

$\frac{C_A^{\text{'}}}{T_C^{\text{'}}}$

$($g$)$ Find the poles $($eigenvalues$)$ for this system, in terms of your coefficients from part $($f$).$