It is desired to control the exit temperature T2 of the beat exchanger shown In Figure by adjusting the steam flow rate w3, unmeasured disturbances occur in inlet temperature T1. The dynamic behavior of the heat exchanger can be approximated by the transfer functions:

where the time constants and time delays have units of seconds. The control valve has the following steady-state characteristics: w_s = 0.6√p – 4 where p is the controller output expressed in milliamps. At the nominal operating condition, p = 12mA. After a sudden change inthe controller output, w_s reaches a new steady-state value in 20s (assumed to take five time constants). The temperature transmitter has negligible dynamics and is designed so that its output signal varies linearly from 4 to 20mA as T₂ varies from 120 to 160^oF.

(a) If a proportional-only feedback controller is used what is K_cm? What is the frequency of the resulting oscillation when K_c = K_cm?

(b) Estimate K_cm using the Routh criterion and a 1/1 Padé approximation for the time-delay term. Does this analysis provide a satisfactory approximation?

2.5es Ti(s) "F [=) -1b/s W:(s) 10s + + 1 Ib/s T(s)_ 0.9e-21 5s + 1 [=] dimensionless T{(s)