Considered the well stirred non$-$isothermal reactor in the figure below. An exothermic liquid phase $1^{st}$ order reaction in the form $AB$ takes place. The density is assumed to be constant. A coil in which heat transfer oil at a temperature of $T_c$ is immersed in the reactor.

a$.$ Develop the model that described the dynamic behavior $($unsteady state operation$)$ of the system.

b$.$ Perform degrees of freedom analysis.

$F_j$ : volumetric flow rate of stream $j(d\frac{m^3}{m}in)$

$C_{Aj}$: molar concentration of $A$ in $j^{th}$ stream $(mo\frac{l}{d}{m}^3)$