$12\mathrm{.}4.$ Consider the liquid$-$level control system shown in Fig. P$12-4.$ The tanks are noninteract$-$

ing. The following information is known:

The resistances on the tanks are linear. These resistances were tested separately, and

it was found that if the steady$-$state flow rate $qcfm$ is plotted against steady$-$state tank

level $hft,$ the slope of the line $d\frac{q}{d}h$ is $2f\frac{t^2}{m}in.$

The cross$-$sectional area of each tank is $2f{t}^2.$

The control valve was tested separately, and it was found that a change of $1$ in pressure

to the valve produced a change in flow of $0\mathrm{.}1cfm.$

There is no dynamic lag in the valve or the measuring element.

$($a$)$ Draw a block diagram of this control system, and in each block give the transfer func$-$

tion, with numerical values of the parameters.

$($b$)$ Determine the controller gain $K_c$ for a critically damped response.

$($c$)$ If the tanks were connected so that they were interacting, what is the value of $K_c$ needed

for critical damping?

$($d$)$ Using $1\mathrm{.}5$ times the value of $K_c$ determined in part $($c$),$ determine the response of the

level in tank $2$ to a step change in set point of $1$ in of level.