(Solved): Consider the liquid level control system shown in the disgram below. Height h of liquid: water (con ...

Consider the liquid level control system shown in the disgram below. Height h of liquid: water (constant density) in the tank (crose-sectional area $\mathrm{A}=0.25{\mathrm{m}}^2$ ) is defected by the transmitter (LT) and controlled by a PI controller (L.C). The transmitter transfer function ${\mathrm{K}}_{\mathrm{m}}=12\mathrm{mA}/\mathrm{m}$, controller paratneters are ${\mathrm{Ki}}_i=?5$ and ${\mathrm{tr}}_{\mathrm{i}}=2$ min. The flow rate ${\mathrm{q}}_{\mathrm{t}}$ is the disturbance variable and the flow rate $qy$ is the manipulated variable with a control valve with ${\mathrm{K}}_{\mathrm{v}}=3$ liters/min psi. There is a transducer betwoen the controller and the control valve that converts the $\mathrm{mA}$ signal from the controller to a pneumatic signal. The transducer transfer function ${\mathrm{Grt}}_{\mathrm{t}}=0.75\mathrm{psi}/\mathrm{mA}$. Figure 11.22 Liquid-level control system with pump in exit line. The system operates at steady state with $\mathrm{h}=0.5\mathrm{m},{\mathrm{q}}_1=10$ liters $/\min ,{\mathrm{q}}_2=10$ liters/min and $q_3=20$ liters $/\min$. The set point for $h$ is set at $0.5\mathrm{m}$. (a) Using deviation variables, obtain transfer function models for the uncontrolled process that show the variation of height $h$ with variation of flow rates $q1$ and $q3$. (b) Draw a closed-loop block diagram for the feedback control system. (c) Derive the transfer function ${\mathrm{H}}^{?}(\mathrm{s})/{\mathrm{Qi}}_1^{?}(\mathrm{s})$ for the feedback control loop. (d) If $q_1(\mathrm{t})$ is increased by a step increase from 10 liters/min to 15 liters/min, find an expression for the output response $h(t)$. Find the height of water in the tank $3\min$ after the step change. What is the new steady state level in the tank? (e) If $q_1(\mathrm{t})$ is ramped from 10 liters/min to 15 liters/min at a constant rate of 0.1 liters $/\min /\min$ (so that the change from 10 liters $/\min$ to 15 liters/min occurs over period of 50 minutes), find an expression for response of $h(t)$. What is the heigh water in the tank 30 minutes after the start of the ramp?