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 $A2$ 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 $G_C=K_c,G_v=\frac{1}{3s+1},G_p=,G_d=\frac{1}{4s+1},G_m=\frac{1}{2s+1},$

Consider the following standard first order plus time delay $($FOPTD$)$ model for the same

bio$-$lubricant production process

tilde$(G{)}_{(s)}=\frac{7e^{-8s}}{3s+1}$

a$.$ Derive an expression for a controller with time delay compensation using the

direct synthesis approach with FOPTD desired response. Discuss the nature of the resulting controller.

$[14$ marks$]$

b$.$ Design an internal model controller $($IMC$)$ for the process using the Pade

approximation model. Discuss the nature of the resulting controller. Given:

$e^{-s}$~$=\frac{1-\frac{}{4}s}{1+\frac{}{4}s}$