ES 391b Chemical Process Control

Final Exam Dec 1997

3hrs. Calculators Allowed. One Page of Notes Table of Laplace transforms and Matlab texts allowed.

1.Two reactant streams A and B flow into a well mixed CSTR and react exothermally releasing a fixed amount of heat Q=(-H) =1.0 BTU per lb/hr of mass hold up in the CSTR. Water is circulated through a cooling coil in the CSTR at a fast rate, so that the temperature rise in the coil is minimal. The flow rates of A and B are both m=1000 lb/hr and the hold up in the CSTR is w=5000 lb. The area for heat transfer in the cooling coil is A=30 ft2 and its overall heat transfer coefficient is

U=600 BTU/(hr. ft2. F). The inlet feed streams A and B are both at a temperature of TA=TB= 80 F, whereas the inlet temperature of the water in the cooling coil is TC=70 F. Heat Capacity, Cp =1.0 BTU/(lb.F), for all streams. Exit temp of CSTR is T ( F)

a) Set up the differential equations which describe the dynamic behavior of the exit temperature in the CSTR.

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b) Calculate the Steady State temperature Ts of the reactant exit stream from the CSTR

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c) Determine the transfer function relating a change in the cooling coil temperature TC to a change in the CSTR temperature T

and hence find the value of T, 5 minutes after TC suddenly drops to 60 F

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2.

Consider the feed back loop shown in the figure above. The controller Gc is a PI controller with tuning parameters Kc and Ti .

The process transfer function Gp= and the measurement transfer function

In order to tune this feedback loop an appropriate Bode plot is constructed and the result is shown in the AR and Phase plot attached.

a) Mark the Bode diagram and show your calculations to determine the value of the two tuning parameters using Zeigler Nichols rules.

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b) If you had to construct the AR vs ω curve, what expression would you use to calculate the appropriate AR as a function of frequency ω in rad/s

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Attachment: Bode Plot

3.A block diagram of a control system is shown below. For what values of the gains Kc and Kf is the system stable? L is a load variable, R is the set point and C is the controlled variable

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4. Write a Matlab function tune.m which accepts the gain, K, the time constant, tau and the dead time, theta, of a First order plus dead time (FO+DT)model. It should also accept a vector of tuning constants tn, which contains the P, I, and D tuning values for a PID controller.

The function should form a set point, closed loop feed back control system with the (FO+DT) model above, using a 2nd order Pade approximation for the deadtime.

The function should then plot two step responses to a set point step of magnitude 2, in two separate subplots. The first subplot should be the step response for a PI controller and the second subplot should be for a PID controller. The plots should be labeled.

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Attachment for Question 2.