It is desired to control the exit temperature T₂of the heat exchanger shown in Fig. E11.18 by adjusting the steam flow rate w_s. 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 characteristic:

where p is the controller output expressed in mA. At the nominal operating condition, p = 12 mA. After a sudden change in the controller output, ws reaches a new steady-state value in 20 s (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 20 mA as T₂ varies from 120 to 160^ˆ˜F.

(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 direct substitution and a 1/1 Pade approximation for the time-delay term. Does this analysis provide a satisfactory approximation?

T,(s) F 2.5e-s W;(s) 10s +1 T(s) 0.9e-2s T{(s) [=] Ib/s -[=] dimensionless 5s + 1 Ws w, = 0.6Vp 4