Q$1.$

The following reactions are carried out in an isothermal continuous stirred tank

reactor.

$AB,-r_A=k_1{C}_A,k_1=1\mathrm{.}1mi{n}^{-1}$

$BC,-r_B=k_2{C}_B^2,k_2=0\mathrm{.}1\frac{m^3}{k}$molmin

$B$ is desired product and concentration of $B$ is controlled using PI controller. The feed

stream of $1\frac{m^3}{m}in$ containing $0\mathrm{.}8$kmo$\frac{l}{m^3}$ at steady state operation passes through

the reactor having a volume of $2m^3.$ If the volume of reaction mixture does not change

with time.

a$.$ Derive the transfer function between inlet concentration of A and outlet

concentration of $B,$ and find the ultimate value of outlet concentration of $B$ when

inlet concentration is increased to $0\mathrm{.}9$kmo$\frac{l}{m^3}.$

b$.$ The transfer functions of measuring device and control valve are given below. Using

Routh Hurwitz criterion, determine the ultimate gain of controller if integral time

is $0\mathrm{.}1.$

$G_m(s)=G_f(s)=1$

Please write your answer by step step and clearly. I will remember to evaluate your answer.