By F.G. Shinskey
In almost every temperature and composition loop, the process we try to control has its dynamics distributed across space and distance from inlet to outlet. Attempting to lump its dynamics into dead time, plus lag, misrepresents its response and ignores information that can be useful in diagnosing behavior and estimating optimum controller settings. This is especially the case in heat transfer—common to all industries—where heat capacity and resistance are distributed across the entire process, and where the resulting dynamic properties are also subject to variation with flow. Temperature control, therefore, has its own behavior, requiring special attention for those auxiliaries that must provide heating and cooling over the full range of production rates.
To introduce the identification of distributed dynamics, let’s begin with a series of equal first-order lags, observing how they may interact with one another. A distillation column is a perfect example of a multiple-lag process, illustrating both interacting and noninteracting lags. Figure 1 below shows the flow of liquid reflux into tray 1 cascading downward from tray to tray, eventually leaving tray, n, counter to the flow of vapor upward. Feed, condenser and reboiler are omitted to simplify the presentation.
A column has interacting and noninteracting lags.
An increase in reflux flow entering tray 1 must raise the level above its overflow weir before outflow can begin to increase—a characteristic of a first-order lag. The same process is repeated from tray to tray, with the flow leaving tray n having the response of all the first-order lags in series.
Step responses of equal noninteracting lags.
Figure 2 describes the response of flow leaving tray n for n = 1, 5, 20 and 100, following a step increase in reflux. Time, t, is normalized by dividing by Στ, which in this case is simply the sum of all the lags: Στ = nτ, with τ being the hydraulic time constant of an individual tray, typically on the order of 5 seconds. Observe that the curve for n = 1 crosses t/Στ = 1 at 63.2%, a familiar property for a first-order lag. As n increases, the crossover point moves downward toward 50%, and the curves approach the response of pure dead time.
These are noninteracting lags. The flow entering a tray from above is unaffected by any level or flow on a lower tray. The flow of information is in the same direction as the liquid flow. The response is that expected from a series of equal lags uncoupled from one another, simulated electrically by a ladder network of equal resistors, R, and capacitors, C, isolated from one another by buffering amplifiers as shown in Figure 2; each individual time constant is τ = RC. The apparent dead time in this series of hydraulic lags will prevent us from controlling base level by manipulating reflux flow if there are more than a handful of trays. But surprisingly, the same trays behave as interacting lags when their composition response is examined.
Consider the response of the analyzer in the overhead vapor line of Figure 1 to a change in reflux flow. Increasing reflux will reduce the amount of heavy impurity in the overhead vapor in a matter of seconds. But the effect does not stop there. The same increase in reflux flow affects both the composition and flow of liquid entering tray 2, which, in turn, affect the composition of the vapor leaving it and entering tray 1. This produces a secondary effect on the composition of the vapor reaching the analyzer from tray 1. The compositions of both liquid and vapor leaving the trays continue to change all down the column and back upward for a long time after new flow rates have been established. The response can be simulated by the electrical ladder network of Figure 3, where isolating amplifiers are not included. The series of interacting lags differs from a series of noninteracting lags in two distinctive ways: the total time response Στ increases much faster with n, while the shape of the response curve hardly changes at all!
The total time response can be estimated from the understanding that the first capacitor, C, in the ladder is charged through a single resistor, R, while the last is charged through the entire series of n resistors. The total time response then varies with the sum of the series of integers 1 through n:
The sum of the series is readily calculated as (n2 + n) / 2. Time response can therefore be expected to vary roughly with the square of the number of stages. This has a profound effect on dynamic behavior: A column of 100 trays of 5 sec time constant each has a hydraulic response time of 500 sec or 8.3 min, but a composition response time of 25,250 sec or 7 hours!
Step responses of equal interacting lags.
The shape of the response curve is also surprising. Figure 3 shows normalized step-response curves for 1, 5 and 100 equal interacting lags. Note first that all curves cross t/Στ = 1 at 63.2% response. Then observe how little separates the curves as n increases. Raising n above 100 reveals no discernible change whatsoever, either to the eye or to the controller closing the loop around the process. This has an important corollary: simulation using as few as 20 interacting lags is quite sufficient to represent any number, including infinity, i.e., an infinite number of infinitesimally small lags. The latter is representative of a truly distributed process, comprised not of a series of discrete stages but of a continuum of capacity and resistance.