$8\mathrm{.}1.$ Determine the analytical expression for a step set point change in the following processes under P$-$only and PI feedback control. You should select values for the tuning constant that give acceptable performance.

$($a$)$ Example $3\mathrm{.}1$ with $C_A$ as the controlled variable, $C_{A0}$ as the manipulated variable, and $\frac{?}{{?}_{\frac{?}{?}}SP}=0\mathrm{.}1$mol$\frac{e}{m^3}.$

EXAMPLE $3\mathrm{.}1.$

Information. The system is the liquid in the tank. The tank has been designed

well, with baffling and impeller size, shape, and speed such that the concentration

should be uniform in the liquid $($Foust et al$.,1980).$

Assumptions.

Well$-$mixed vessel

Density the same for A and solvent

Constant flow in

Data.

$F_0=0\mathrm{.}085\frac{m^3}{m}in$;$V=2\mathrm{.}1m^3$;$C_{\text{A}\text{i}\text{n}\text{i}\text{t}}=0\mathrm{.}925$mol$\frac{e}{m^3}$;$C_{A0}=0\mathrm{.}925$mol$\frac{e}{m^3}$;

thus, $C_{A0}=1\mathrm{.}85$mol$\frac{e}{m^3}$ after the step

The system is initially at steady state at $\{$:$t=0)$

Note that the inlet concentration, $C_{A0},$ remains constant after the step change has

been introduced to this two$-$component system.