Heat Transfer Handbook part 130. The Heat Transfer Handbook provides succinct hard data, formulas, and specifications for the critical aspects of heat transfer, offering a reliable, hands-on resource for solving day-to-day issues across a variety of applications. | THERMAL PROCESS CONTROL FOR MANUFACTURING 1287 1200 0 1000 800 o Ph o 600 S 400 200 0 50 100 150 200 250 300 350 400 Time s Figure Comparison of closed-loop step responses of a first-order thermal system with PI and thermostatic control. but dhcrhashs damping. Thd integral gain Ki improves thd tracking accuracy but decreases damping and system stability. Thd derivative gain Kd improves thd speed and damping but increases the sensitivity to measurement noise. For a linear first-order thermal system with gain K 10 K W and time constant t 600 s Fig. compares the closed-loop performance of a PID and a thermostatic controller with the differential gap during a step command to a desired temperature increase Td 1000 K. The PI controller has the gains Kp W K and Ki W K. The thermostatic controller switches on to c 600 W when e b 50 K and switches off to zero when e b -50 K. For the same power saturation level the PID controller gives smoother but slower asymptotic tracking the thermostatic controller may be tuned to the optimal actuator power level but its relay operation may accelerate wear. Software Implementation of SISO Controllers The controllers designed must be converted from the continuous to the discrete time domain to be programmed in software for sampled computer control. First the sampling period T can be selected by rules of thumb based on the step response of die ohen-loop thermal system T with ts 3t and t 3 3 s 4a where ts is the settling time d the delay and t the time constant of the thermal system which for conduction depends on the characteristic length L and diffusivity a. Next 1288 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING the controller Laplace variable 5 is approximated by the discrete z variable through a differential approximation such as 2 z -1 ew Q z s T EE M aAz EE Finally the discretized control relation of power Q k to the error e k for the kth sampling period is converted to the time domain via
