Dead-time plus capacity process

It is interesting to note the comparison between the controllability of this process and the two-capacity process. Taken on the basis of an equal ratio of secondary to primary element, the dead-time plus capacity process is TOO/ttIO or 8 times as difficult to control. Recall that the pure dead-time process was 12.5 times as difficult to control as the most difficult two-capacity process. 

Derivative is the inverse of integral action. In theory, it is characterized by a 90 phase lead, although because of physical limitations 45° is about all that can be expected. If perfect derivative (90 lead) were available, it could halve the period of the dead-time plus capacity loop by allowing the dcnd time to contribute all 180°. Remember that perfect derivative applied to the two-capacity process provided critical damping with zero proportional band. But Fig. 1.27 indicates that perfect derivative is limited to zero damping at a period of 2t[Pg.33]

If the same program is applied to a two-capacity process, the controlled variable will be more heavily damped than necessary. Therefore this program provides the minimum-time switching only for dead-time plus integrating processes. 

Between the most and least difficult elements lies a broad spectrum of moderately difficult processes. Although most of these processes are dynamically complex, their behavior can be modeled, to a large extent, by a combination of dead time plus single capacity. The proportional band required to critically damp a single-capacity process is zero. For a dead-time process. It Is Infinite. It would appear, then, that the proportional band requirement Is related to the dead time in a process, divided by Its time constant. Any proportional band, hence any process, would fit somewhere In this spectrum of processes. A discussion of multicapacity processes In Chap. 2 will reaffirm this point. 

Calculate the gain of a dead-time plus single-capacity process whose natural period under proportional control is 3.0 t-j. What is the ratio of r /ri Does this point fall on the curve of Fig. 1.26 ... 

From Fig. 2.2 it can be seen that the interacting multicapacity process differs from the dead time plus single-capacity process in the smooth upturn at the beginning of the step response. This curvature indicates that the dead time is not pure, but instead is the result of many small lags, and therefore the process will be somewhat easier to control. By the same token, derivative action will be of more value than it was in the case of dead time and a single capacity. Nonetheless, if we choose to estimate the necessary controller settings on the basis of a single-capacity plus dead-time representation we will err on the safe side. 

FIG 2.3. The step response of a multicapacity process can be reduced to dead time plus a single capacity. 

P.6 The process is essentially dead-time plus single-capacity, in that Tohi = 3.91. The dead time varies inversely with flow, which can very likely be compensated for by using an equal-percentage valve. 

Consider the heat exchanger as a single-capacity plus dead-time process where the dynamic gain of the capacity is expressed as... 

Because all the heat leaving the reactor flows through the walls and into the coolant, the capacities of reactants, walls, and coolant interact. But in view of the slight heat capacity of the bulb, its time constant does not significantly interact with the others. Basically the process is four-capacity plus dead-time. 

It is important to see over what range of processes complementary feedback has an advantage over two-mode control. A single-capacity plus dead-time process will respond to a step load change under complementary feedback as shown in Fig. 4.15. Without going into the derivation of the load response curve, it turns out that the integrated area per unit load change is... 

Again, processes do not fall into such neat classifications as two-capacity or single-capacity plus dead-time. The bulk of difficult processes lie between these limits. But the same control function described by Fig. 5.16 and Eq. (5.10) can be adjusted to accommodate dead time in addition to two capacities. Equation (5.11) indicates the required settings for optimal switching ... 

Both of these processes have similar characteristics, in that they are typically comprised of one large and many small capacities, i.e. valve actuator, transmitter, etc. The net result is a response indicative of a process with a dominant capacitance plus a dead time. Both of the above categories will be investigated and their specific differences and similarities will be identified. 

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