Process Controls & Dynamics
A CST tank of 5 m3 in volume is used to mix two streams of identical fluid at different temperatures. The level in tank is maintained constant. The first stream is at 70oC with a mass flow rate of 1.5 kg/s while the second is at 100oC with a mass flow rate of 4.2 kg/s. The density of the fluid is 750 kg/m3 with a cp value of 950 J/kg K. There is also heat loss to the surroundings according to Newton's law of cooling:
q = hA(T - Ti)
The value of the heat transfer coefficient is 5 W/m2 K while the surroundings temperature can be assumed to be constant at 25oC. The surface area of the tank is 6.3 m2. Since the tank wall material is a good conductor and relatively thin (unfortunately), the surface temperature of the tank can be assumed to be the same as the inside temperature.
a) Obtain the dynamic model for this process in terms of deviation variables if the temperature of the first inlet stream is the only input variable. Write your assumptions.
b) Obtain the transfer function of the process. What order is the process transfer function? What is the process time constant? How long would it take for the process to stabilize after a step change? Does the heat loss significantly contribute to the process lag?
c) What is the process gain? Find the steady state outlet temperature at 70oC. Using only the two values calculated in c), find the new steady state temperature if the inlet temperature of the first stream were to increase from 70 to 85oC.
d) What is/are the pole(s) of the transfer function? What dynamic behavior can we expect?
e) Confirm your finding in d) by obtaining the time domain model for a general a step input A (occurs at t = 0) from your transfer function. Replace the value of A with the value that corresponds to a step increase from 70 to 85oC and plot the function. Comment on the results.
f) Estimate all tuning parameters using the Cohen Coon technique for a PID controller.
g) Obtain the overall transfer function for the process if both the temperature sensor lag and actuator lag were to be included (they can both assumed to be first order). Use a sensor lag of 7 seconds with sensor gain equal to one and an actuator lag of 5 seconds with the actuator gain equal to one. What are the poles of the overall transfer function? Is it stable?
h) If the combined process-sensor-actuator model were to be combined with a PI controller with a controller gain, Kc, of 10 and an integral time, τI, of one minute, would the closed-loop be stable? Use the Routh array analysis for you proof.