## Modelling and Design of Feedback Controllers with SIMULINK

Problem 1: Cooling system

Consider the well-mixed cooling system shown below: Cooling system design and operating conditions:

Jacket heattransfercoefficient:                  U = 1.225×106 J/hr.m2.C Area ofjacket/tankinterface:                                 A = 31m2

Fluiddensity:                                                 r= 993Kg/m3

Fluidheatcapacity:                                       Cp = 4190 J/KgC

Mass of fluidintank:                                     M = 18,200 Kg Steady-stateinlettemperature:                                 T1 = 82C

Steady-stateoutlettemperature:                 T2 = 38 C (setpoint)

You may assume the fluid volume in the tank is constant and the fluid heat exchanger properties are constant but T1, T2, Tw, and F may change.

# Questions:

• Derive the transfer functions that relate T2 to T1, Tw, and F. Calculate the values for time constant andgains
• Sketch the instrumentation diagram for the process (transmitter, controller, controlvalve)
• With Simulink determine the PID tuning parameters using the closed- loop (constant oscillation) tuning method. Set point is constant. T1 changes –5 oC and F changes –0.4m3/hr

Additional information: Process delay = 0.5 hr

Transmitter output range = 4 – 20 mA Transmitter calibration range = 26 – 42 oC Transmitter time constant = 0

Valve gain = 1

# Problem 2: Heat exchanger

Consider the process within a heat exchanger as given below: Acetone enters at 120°C, heating up the incoming decane stream from 80°C to 95°C (at steady state). The outlet temperature of decane stream is maintained by manipulation of acetone inlet valve. At steady state operation, the flow rate of decane is 390 kmol/hr.

At time, t = 0, the decane flow rate is suddenly decreased to 250 kmol/hr, affecting the outlet decane temperature as shown in the response table shown below. For the second step testing, the acetone valve is suddenly opened from 20 to 30 %, giving a temperature response also tabulated in the table below.

Outlet Decane Temperature Responses to Step Tests

 Time (min) Decane Temp 0 90.916 0.3 90.916 0.6 90.916 0.9 92.2342 1.2 93.4804 1.5 94.6042 1.8 95.5946 2.1 96.4604 2.4 97.2085 2.7 97.8498 3 98.4003 3.3 98.8711 3.6 99.2718 3.9 99.6125 4.2 99.9024 4.5 100.149 4.8 100.36 5.1 100.539 5.4 100.692 5.7 100.823 6 100.934 6.3 101.03 6.6 101.111 6.9 101.181 7.2 101.241 7.5 101.292 7.8 101.335 8.1 101.372 8.4 101.404
 Time (min) Decane Temp 0 85.258 0.3 85.258 0.6 85.9871 0.9 87.1483 1.2 88.3826 1.5 89.5295 1.8 90.5287 2.1 91.3793 2.4 92.0932 2.7 92.6878 3 93.1855 3.3 93.5976 3.6 93.9385 3.9 94.2203 4.2 94.4535 4.5 94.6464 4.8 94.8059 5.1 94.938 5.4 95.0472 5.7 95.1376 6 95.2124 6.3 95.2742 6.6 95.3253 6.9 95.3676 7.2 95.4022 7.5 95.4309 7.8 95.4546 8.1 95.4742 8.4 95.4903

Decane 390 to250kmol/hr                                 Acetone Valve 20% to30%

Valve gain is between 1 and 2 with zero time constant.

Transmitter has unity gain with negligible dead time and time constant

• Tune PI and PID controllers using the open loopequations
• Provide some conclusions

Solution

Problem 1.  Matlab Simulation

• Find the constant oscillation

This is my simulation block. First I set PID coefficients Kp=  20, Ki = 0, Kd = 0

The result is The oscillation is descending, so should increase Kp

When Kp = 40 The ouput is over oscillating. I need to decrease Kp

When Kp = 33 The output is constantly oscillating so from this value we can find the PID coefficients.

Kp = 16.5 / 2 * 1.7 = 14

Ti = 0.50 * Tk = 1.4

Td = 0.125 * Tk = 0.35

Kp = 14 Ki = 10 Kd = 4.9

Using this coefficient we get the following result. Problem 2.  Heat Exchanger

Using the data output of step input, we can plot the graph using MATLAB  This is the change of the acetone temperature versus the step input of the flow rate of decane.

With linear approximation we can get linear model.

The following MATLAB graph shows the output of the temperature when step input the acetone valve change. From this graph we can get PID controller coefficients. 