Consider the following stirred tank heating system used to heat liquid water:

The volume of the water, $V,$ in the tank is constant equal to $500$ liters, and the mass flow rate of water in and out, $w,$ is constant equal to $100k\frac{g}{m}in.$ Following properties of liquid water may be assumed constant: density $=996k\frac{g}{m^3},$ specific heat $C=4\mathrm{.}18k\frac{J}{k}gK.$ Initially, the process is operating at steady state with $T_{i,ss}=10C,$ and $T_{ss}=50C$ which is also set as the set point. Calculate the rate of heat input $Q_{ss}$ at steady state. In terms of these variables the process transfer function is given as

$T^{\text{'}}\frac{s}{Q^{\text{'}}}(s)=\frac{K}{s+1}$

where $K=(\frac{1}{w}C),=\frac{V}{w}$

A closed loop feedback control system is designed to control the temperature in the tank $(T)$ with heat supplied by the heater $(Q)$ as the manipulated variable and temperature of the inlet stream $(T_i)$ as the disturbance variable. There is a transducer between the controller and heater that converts the controller output signal $p^{\text{'}}(t)$ to heat input signal $Q^{\text{'}}(t).$ The transfer function of the transducer $G_{Tr}=Q^{\text{'}}\frac{s}{P^{\text{'}}}(s)=wC.$ TT is the temperature transmitter with transfer function $G_{TT}=1.$

$($a$)$ Draw a closed$-$loop block diagram for the feedback control system.

$($b$)$ Use the direct synthesis method with no time delay and two different values of ${}_c=2$ minutes and $5$ minutes. In each case determine the controller parameters $K_c$ and ${}_{\text{I}}.$

$($c$)$ For the feedback control loop, obtain the transfer function $T^{\text{'}}\frac{s}{T_{sp}}\text{'}($s$)$

$($d$)$ The set point $T_{sp}$ is changed by a step increase to $70C.$ Obtain expressions for response of temperature $T$ for the two controller parameters. Comment on which value of ${}_c$ provides a better control of temperature $T.$