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Heat exchangers are more problematic in this respect. Their residence time and gain also vary inversely with flow for the same reasons as the stirred tank, but most exchangers have no internal recirculation as does a stirred tank. Their estimated dead time then also varies inversely with flow, making them much more difficult to control as flow decreases. The loop gain of a process is directly proportional to Kpτde/Στ. When all three parameters vary inversely with flow, the loop gain will vary, unless some form of compensation is applied. Suitable gain compensation is available in the form of the equal-percentage valve characteristic, whose gain varies directly with flow delivered. As a result, equal-percentage valves are recommended almost universally for temperature control. Integral and derivative settings should also be scheduled inversely with flow for compensation to be complete.
Another once-through process is the static mixer—a hollow pipe fitted with vanes or baffles providing radial mixing of the fed ingredients into a uniform blend at the exit. Longitudinal mixing is intentionally avoided, with the result that the static mixer’s dynamic response is dominated by dead time. In practice, it behaves much like a series of 20-50 noninteracting lags as described in Figure 2, but its parameters Kp and Στ vary inversely with flow like a heat exchanger. Here too, scheduling of controller settings as an inverse function of flow is necessary for complete compensation. A more effective solution is to install a circulating pump—the static mixer then behaves like a stirred tank.
Distributed processes require special tuning rules.
Rules for optimum tuning of PID controllers on distributed processes differ somewhat from those applied to first-order-plus-dead-time (FOPDT) models. Figure 5 compares the load response of a distributed process with an interacting PID controller tuned for minimum integrated absolute error (Min-IAE) against what would be Min-IAE tuning applied to a FOPDT model of the same process. In general, FOPDT tuning applies a wider proportional band along with shorter integral and derivative time settings than are optimal for distributed processes. Minimum-IAE settings are listed in Table 1.
The last column in the table is a measure of integrated error (IE) in the controlled variable per unit change in controller output Δm required to respond to a disturbance. IE can be related to operating cost such as excess energy applied or excess product giveaway, and is slightly less than IAE in a well-damped loop. IE/Δm is readily calculated as the product of the proportional band and integral time of the controller.
Minimum-IAE controller settings for distributed processes.
An example of an application of the tuning rules would be the temperature controller for the air-conditioned space describer earlier. The optimum proportional band for the PI controller would be 20Kp or 12%. If the band were depressed further to about 5%, a uniform oscillation would develop. Integral time could then be set at about 0.54 (1.55) or 0.84 times its period.
In conclusion, the distributed process—while undeniably complex—can be defined by only two parameters, and is relatively easy to control.
|About the Author|
F.G. Shinskey is a process control consultant based in North Sandwich, N.H.
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