One of the most important variables that must be controlled in industrial and chemical systems is temperature. A simple representation of a thermal control system is shown in Figure P4.3 [14]. The temperature T of the process is controlled by the heater with a resistance R. An approximate representation of the dynamic linearly relates the heat loss from the process to the temperature difference T - Te. This relation holds if the temperature difference is relatively small and the energy storage of the heater and the vessel walls is negligible. Also, it is assumed that the voltage eh applied to the heater is proportional to edesired or eh = kEb = kaEbe(t), where ka is the constant of the actuator. Then the linearized open-loop response of the system is

where
Ñ‚ = MC/(pA),
M = mass in tank,
A = surface area of tank,
p = heat transfer constant,
C = specific heat constant,
K1 = a dimensionality constant, and
eth = output voltage of thermocouple.
Determine and compare the open-loop and closed-loop systems for
(a) Sensitivity to changes in the constant K = k1kaEb;
(b) The ability to reduce the effects of a step disturbance in the environmental temperature ΔTe(s); and
(c) The steady-state error of the temperature controller for a step change in the input, edesired.
Figure P4.3
Temperature control system.

Transcribed Image Text:

5.(s) g (s) = rs + TS+ Heater Envirunmenl E, Process Thermocouple e(r) e pn t4 Kikw Edlesired