The system shown in Fig below is controlled by a proportional controller. The concentration of salt in solution leaving the tank is controlled by adding a concentrated solution through a control valve.

The following data apply:

Concentration of concentrated salt solution $C_1=25$lbsal$\frac{t}{f}{t}^3$ solution.

Controlled concentration $C=0\mathrm{.}1lb$ salt $\frac{?}{f{t}^3}$ solution.

$3.$Control valve: The flow through the control valve varies from $0\mathrm{.}002$ to $0\mathrm{.}0006f\frac{t^3}{m}in$ with a change of valve$-$top pressure from $3$ to $15.$ This relationship is linear.

Distance velocity lag: It takes $1$min for the solution leaving the tank to reach the concentration$-$measuring element at the end of the pipe.

Neglect lags in the valve.

$($a$)$ Draw a block diagram of the control system. Place in each block the appropriate transfer function. Calculate all the constants and give the units.

$($b$)$ Using a frequency$-$response diagram and the Ziegler$-$Nichols rules, determine the settings of the controller.

$($c$)$ Using the controller settings of part $($b$),$ calculate the offset when the set point is changed by $0\mathrm{.}02$ unit of concentration.