Below is a mixing process between hot and cold streams of water. A feedback control system was installed to control the output temperature.

$f_1$

$T_1$

Data:

Steady state values: $\frac{?}{\text{b}\text{a}\text{r}}(f_2)=0\mathrm{.}3\frac{m^3}{s}\frac{,}{b}ar(T_1)=25C.$

Temperature Sensor$/$Transmitter: Input $20-100$C$,$ Output $4-20$mA$,$ Time constant $1$ secnod$.$

Control Valve: linear characteristics, Time Constant $3$ second, Air to open, $\frac{?}{\text{b}\text{a}\text{r}}(P)=1$ atm.

Assume the volume of liquid in the tank is constant.

$1-$ Size the control valve assumes $1\mathrm{.}5$ overcapacity factor.

$2-$ What should be the action of the controller?

$3-$ Develop a mathematical model for the process.

$4-$ Put the model in deviation form and take the Laplace transform for the model.

$5-$ If P$-$controller only is used, calculate the ultimate gain of the controller.

$6-$ What does the ultimate gain means?