98 ◾  Distillation Control, Optimization, and Tuning
The self-regulating process gain, K p , can be measured from the
setpoint change and controller output (valve position) change
(Equation 9.2).
(set point change, % of span)
K p =
(9.2)
(controller output change, % at steady state)
For example, suppose the step change in the controller was
from 70°C to 76°C with a 200°C span, and the steady-state
valve position increased from 50% to 52% open. The process
gain, K p , would be [(76 − 70)/200]/[(52 − 50)/100] = 1.5.
Tuning a control loop for minimum IAE after a step
change in setpoint led to the discovery of the 7/1 overshoot/
undershoot ratio for finding the optimum proportional gain,
K c . However, the integral time constant is too long; that is,
the reset is too slow, for good response to load changes. An
integral time constant of 26 min did not give a good response
to a load change, as shown in Figure 9.7.
The optimum integral (reset) time constant for excellent
response to a load change is about 1.0 times the response
time, that is, 13 min. Reducing the integral time constant,
T i , to 6.5 created instability in Figure 9.7. A correlation for
many combinations of self-regulating control loops is shown
in Equation 9.3. In other words, with K c = 4.08 and K p = 1.0,
the optimum integral time constant for excellent response to
load change was calculated as 0.88 times the response time,
or 11 min.
T i = 0.55 (K c K p ) 0.33 response time
(9.3)
where:
T i = integral time constant (in minutes) for excellent
response to load changes