Linearization

and $10$ Marks$]$

A CSTR with constant volume $V,$ has dynamic

$V\frac{d{C}_A}{dt}=F_{in}{C}_{A_{in}}-F_{\text{o}\text{u}\text{t}}{C}_A-C_A{k}_0{e}^{-{}_0\frac{E_0}{RT}},$

$V{c}_p\frac{dT}{dt}=c_p{F}_{in}{T}_{F_{in}}-c_p{F}_{\text{o}\text{u}\text{t}}T-HV{C}_A{k}_0{e}^{-n_0\frac{E_p}{RT}}-UA(T-T_c),$

Where density $,$ heat capacity $c_p,$ coolant temperature $T_c,$ activation energy $E_l,$ inlet concentration $C_{A_{in}}$ and Boltzman's $R$ are constant.

$F_{in}$ is the input variable, $C_A$ and $T$ are state variables. Take ${}_0$ the last number of your student ID $($if last number $0,$ take $2$ last numbers $($example $($:$2112340=21123{}_0{}_0\})..$ Linearize the CSTR equation at $\frac{?}{\text{b}\text{a}\text{r}}(F{)}_{in}\frac{,}{b}ar(C{)}_A$ and $\frac{?}{\text{b}\text{a}\text{r}}(T).$ Present the linearized model in state$-$space equation which is in matrix form.