process control

Starting from the mathematical model in the time domain of the system shown in Figure 1, use
Laplace transforms to determine all transfer functions that describe this process in deviation
state in the open loop. Determine initially any first order dependencies between variables and
calculate their respective gains and time constants. Furthermore, combine these first order
transfer functions to obtain the transfer functions describing the effect of temperatures 1
′
()
and 2
′
() to changes to the temperatures 1
′ (), 2
′ () and
′
(). Clearly show all steps taken
and clearly indicate the units of all involved variables.
[Marks: 9/22]
b) Assuming a generic P controller, derive the closed-loop response transfer function and comment
on the order of the transfer functions to changes to the load and setpoint, relating this to the
speed of the response, for:
i. The case that the manipulated variable is 1() and 2 is constant
ii. The case that the manipulated variable is 2() and 1 is constant.
[Marks: 3/22]
F2, T2(t)
Tank 2
F, Ti(t)
F1, T1(t)
Tank 1
Ts1(t)
Ts2(t)
F3, T2(t)
c) Based on the obtained transfer functions construct the block diagram of the closed loop process
and build it in Simulink. Name all blocks and streams in a descriptive way, so that they are
identifiable in relation to the actual process. Provide a screenshot of the final Simulink diagram.
Consider further that initially the system is at steady state and the following cases:
 At time = 0 there is a step change in the inlet temperature () equal to 20 o
C and the
system used is that of case b) i. above.
 At time = 0 there is a step change in the inlet temperature () equal to 20 o
C and the
system used is that of case b) ii. above.
i. Using a P controller, simulate in Simulink the process response, under the above-described
step changes, for the following values: 1, 10, 100 psi/o
C. Consider an adequate
simulation time to reach a new steady state. Plot the responses of 1
′
() and 2
′
() with
time (one plot per variable including the responses for all values). Discuss the following:
presence or not and trend, if any, of an oscillatory behaviour, stability or not of the
response. Use numeric data to support your arguments.
[Marks: 3/22]
ii. Starting from the general equation describing the closed-loop response of the feedbackcontrolled system and using the Final Value Theorem, calculate the offset for the various
values considered in part i. Compare the calculated values with those predicted by
Simulink and discuss trends in relation to theory.
[Marks: 2/22]
d) Consider that again the system is at steady state and the following cases:
 At time = 0 there is a step change of 20 o
C in the set-point of the process and the system
used is that of case b) i. above.
 At time = 0 there is a step change of 20 o
C in the set-point of the process and the system
used is that of case b) ii. above.
i. Using a PI controller, simulate in Simulink the process response, under the above-described
cases, for a constant value of 100 psi/o
C and for a value of 50 s. Consider an adequate
simulation time to reach a new steady state. Plot the response of the controlled variable
with time (one plot for both cases). Discuss the following: Presence or not and trend, if any,
of an offset.
[Marks: 2/22]
ii. Starting from the characteristic equation of this closed-loop system, investigate its
oscillatory and stability behaviour for the cases considered in part i. Compare the findings
with those predicted by Simulink and discuss trends in relation to theory.
[Marks: 3/22]
Numerical values (assumed constant):
 Volume of fluid in the tank 1: 1 = 0.5 m3
 Volume of fluid in the tank 2: 2 = 0.4 m3
 Overall heat transfer coefficient between heating medium and fluid in tank 1: 1 = 400 W/m2
-K
 Overall heat transfer coefficient between heating medium and fluid in tank 2: 2 = 450 W/m2
-K
 Heat transfer area between heating medium and fluid in the tank 1: 1 = 4 m2
 Heat transfer area between heating medium and fluid in the tank 2: 2 = 5 m2
 Inlet flow rate of the fluid in tank 1: = 1 × 10−3 m3
/s
 Density of the fluid in the inlet feed stream and in the tanks: = 1000 kg/m3
 Specific heat of the fluid in the inlet feed stream and in the tanks: = 3500 J/kg-K
 Transfer function for the final control element: () = , where = 1 o
C/psi
 Transfer function for the measuring device: () = , where = 1 o
C / o
C