An exothermic reaction, $A3B,$ takes place adiabatically in a stirred$-$tank reactor. This liquid reaction occurs at constant volume in a $20m^3$ reactor. The reaction can be considered to be first$-$order and irreversible

$r_A=k{C}_A$

with the rate constant given by

$k=2\mathrm{.}4{10}^{15}{e}^{-20000T}(min-1)$ where $TisinR.$

a$.$ Derive the mathematical model for the system relating exit temperature, $T$ and exit concentration $C_A$ to the inlet concentration $C_{Ai}$ and inlet temperature $T_i($Energy balance, component balance equations for $C_A$ and $C_B).$ State all assumptions you make.

b$.$ What is the degrees of freedom? Choose the variables whose functions must be specified at $t=0.$

c$.$ Linearize any non$-$linear terms in the model equations.

d$.$ Express model equations in terms of deviation variables.

e$.$ Specify the variables as inputs and outputs.

f$.$ If you are going to design control systems for the given problem, what will be your choices for the controlled and manipulated variables?

$($i$)$ Nominal steady state conditions are: $\frac{?}{\text{b}\text{a}\text{r}}(T)=75C\frac{\text{;}}{b}ar(C{)}_{Ai}=800\frac{\text{m}\text{o}\text{l}}{m^3}\frac{\text{;}}{b}ar(T{)}_i=25C\frac{?}{\text{b}\text{a}\text{r}}(q)=10\frac{m^3}{(min)}$ flow in and out of the reactor.

$($ii$)$ Physical property data: $C=0\mathrm{.}2$kca$\frac{l}{k}{g}_{-}^{0}C$;$=850k\frac{g}{m^3}$;$-H_R=500k\frac{J}{m}ol$