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

Below is a mixing process between hot and cold streams of water. A feedback control system was installed to control the output temperature.
Data:
Steady state values: ?bar(f2)=0.3m3s,bar(T1)=25C.
Temperature Sensor/Transmitter: Input 20-100C, Output 4-20mA, Time constant 1 secnod.
Control Valve: linear characteristics, Time Constant 3 second, Air to open, ?bar(P)=1atm.
Assume the volume of liquid in the tank is constant.
1- Size the control valve assumes 1.5 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?
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