Q$.1(20$ pts$)$ Consider an isothermal CSTR with constant holdup, in which a chemical reaction A $$ B takes

place $(r_A=k{C}_A^{0\mathrm{.}5}).$

Species A enter the reactor with a concentration of $C_{A0}$ and volumetric flow rate of $v^{}.$

The input concentration $C_{A0}$ may vary but the flowrate $v^{}$ is kept constant.

The mass balance gives the following differential equation:

$C_{A0}(t)-C_A(t)-\frac{V}{(v^{})}k{C}_A^{0\mathrm{.}5}(t)=\frac{V}{(v^{})}\frac{d{C}_A(t)}{dt}$

Starting with the given mass balance equation,

a$)$ Obtain a transfer function that relates the output $(C_A(t))$ to the input $(C_{A0}(t)).$

b$)$ Find the initial and final steady$-$state values of input and output for a step change

of $+M.$

c$)$ Plot the input $(C_{A0}(t))$ and output $(C_A(t))$ for a step change of $+M.$

DATA: $V_{rxr}=20m^3,k=0\mathrm{.}1\frac{m^2}{(mol).\text{m}\text{i}\text{n}},v_{in}^{}=1\frac{m^3}{\text{m}\text{i}\text{n}}\frac{,}{b}ar(C{)}_{A0}=35\frac{\text{m}\text{o}\text{l}}{m^3},M=10\frac{\text{m}\text{o}\text{l}}{m^3}$