The following liquid$-$phase reactions happen simultaneously:

$AB,r_1=k_1{C}_A$

$BC,r_2=k_2$

$AD,r_3=k_3{C}_A$

The reactions are carried out isothermally with a feed of pure A having an initial concentration $C_{40}=3mo\frac{l}{L},$ and a feed rate of $20\frac{L}{s}($assume no density change on reaction$).$ The fractional conversion of $A(x_A)$ at the exit of the reactor is $0\mathrm{.}80.$ The current reactor used is a Continuously Stirred Tank Reactor $($CSTR$)$ because of

simplicity.

a$)$ What should be the volume of CSTR used?

b$)$ Find $C{A}_A,C{B}_B$ and yield of $B(Y_B).$

c$)$ Derive the expression for maximum $C_B$ at an optimum residence time? What is the maximum yield of $B($i$.$e$.$ yield at maximum conversion of $B)?$

Other data:

$B$ is the desired product. $C$ and $D$ are undesired side products.

$k_1={10}^{-2}{s}^{-1}$

$k_2={10}^{-3}mo\frac{l}{L}-s$

$k_3={10}^{-2}$s$^(-1)$

$C_{B0}=0$

C$\_($C$\_0)=0$

$C_{D0}=0$