Consider the jacketed continuous stirred tank reactor (CSTR) sketched in Fig. PlO-5. The following information is obtained from testing the reactor and its control system. The transfer function of the reactor temperature to the jacket temperature is a first-order lag with a gain of 0.6”CPC and a time constant of 13 min. The transfer function of the jacket temperature to the coolant flow is a first-order lag with a gain of – 2.0°C/(kg/s) and a time constant of 2.5 min. The control valve is linear with constant pressure drop and is sized to pass 12 kg/s when fully opened. Its time constant is negligible. The reactor temperature transmitter is calibrated for a range of 50 to 100°C and has a time constant of 1 min. The jacket temperature transmitter is calibrated for a range of 0 to 100°C, and its time constant is negligible.
(a) Decide on the proper fail position of the control valve and the action of the controller for a simple feedback control loop with the reactor temperature controller manipulating the position of the coolant valve. Draw the block diagram showing all transfer functions, and write the closed-loop transfer function of the reactor temperature to its set point. Pay particular attention to the signs, which must correspond to the fail position of the valve and the controller action. (b) Write the characteristic equation for the single feedback loop and calculate its ultimate gain and period by direct substitution.
(c) Design a cascade control system for the reactor temperature with the jacket temperature as the intermediate process variable, specifying the action of both controllers. Draw the complete block diagram for the cascade control system showing all transfer functions and their signs.
(d) Assuming a proportional slave controller with a gain of 2%/%, write the transfer function for the jacket temperature loop and redraw the block diagram with the jacket temperature loop as a single block.
(e) Using the simplified block diagram from part
(d), write the characteristic equation of the reactor temperature loop in the cascade control system and calculate the ultimate gain and period of the loop by direct substitution