The CSTR shown in the figure was operated at a steady state with a feed concentration $(C_{AO})$ of $0\mathrm{.}5\frac{\text{m}\text{o}\text{l}}{m^3}$ and the feed concentration was suddenly increased to $1\mathrm{.}0\frac{\text{m}\text{o}\text{l}}{m^3}.$

All other process variables $($T$,$ V and F$)$ are constant. For this system:

a$.$ Develop a dynamic model for the concentration in the reactor $(C_A).$ The reaction rate term is defined as $r_A=\frac{-k_1{C}_A}{(1+k_2{C}_A)}$

b$.$ Obtain a transfer function $(G(s))$ showing the relationship between $C_A$ and $C_{A_0}.$

c$.$ Obtain the dynamic response of $C_A(t)$ for the step change in $C_A.$

Find the new steady state value of $C_A$ using the following data

$V=2\mathrm{.}0m^3,F=1\mathrm{.}0\frac{m^3}{h},k_1=1\mathrm{.}0h^{-1},k_2=1\mathrm{.}0\frac{m^3}{\text{m}\text{o}\text{l}}$