A start$-$up process plant planned to produce a bio$-$lubricant $($B$)$ which is a blend of two

components $A1$ and $A2$ using a stirred tank reactor. It is required that components $A1$

and $A2$ are thoroughly blended to obtain a homogeneous mixture of product $B.$ Based

on the product specifications, it is desired that the mole of A$2$ in the product B should

not exceed $100wt\%.$ As the Process Control Engineer, you are required to:

a$.$ Propose a feedback control system for the production of the bio$-$lubricant with a

well$-$defined objective and appropriate illustration. Develop a transfer functions

and block diagrams for the proposed bio$-$lubricant production for step$-$point

changes considering the individual components of the feedback control system.

$[17$ marks$]$

b$.$ Propose the value of controller gain $(K_c)$ that will make the bio$-$lubricant

production process stable using Routh Stability Criterion if the following

parameters are defined for the developed feedback control system.

Given that $G_C=K_c,G_v=\frac{1}{3s+1},G_p=G_d=\frac{1}{4s+1},G_m=\frac{1}{2s+1},$