Single-capacity process

Since the phase lag can never exceed 90°, the first-order lag cannot oscillate under proportional control. This was also true of the integrating process. Therefore we can make a general statement that a singlecapacity process can be controlled without oscillation at zero proportional band. This means that the valve will be driven fully open or fully closed on an infinitesimal error, so that the loop is operating at top speed all the time. Since the proportional band is zero, no offset can develop. A single-capacity process must therefore be categorized as the easiest to control. 

Examples of pure single-capacity processes are rare. The most common one is a tank being filled through a valve which is rigidly coupled to a float. The level is prevented from overshooting the set point because the rigid coupling eliminates any delay in feedback action. 

Having established the ease with which a single-capacity process may be controlled, the complications involved in adding a second capacity may be evaluated. Since each capacity contributes a phase lag approaching 90 , the total phase lag in the loop can only approach 180 . As a result, the loop can oscillate only at zero period. This is exactly like a first-order lag with an integrating controller. 

Occurrences of either pure dead-time or ideal single-capacity processes are rare. The reasons for this are twofold ... 

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. 

Anyone who has tried to control composition in a stirred tank knows that it is not a single-capacity process. It would only be single-capacity if the contents of the vessel were perfectly mixed. But no mixer can move material from the inlet pipe to the exit pipe in zero time-it is impossible. Consequently some dead time must exist, i.e., that time required for the agitator to transport a particle of fluid from inlet to outlet. The presence of any dead time changes the control situation entirely, for now the process is capable of oscillating in a closed loop, which places a limitation on both controller gain and speed of response. 

The question often arises whether proportional, reset, and derivative are really the best control modes for every application. For the easier-to-control processes, their use can be justified. A single-capacity process and some two-capacity processes need only narrow-band proportional action. Derivative is of great value in processes with two or three capacities. But for the more difficult processes, it has been found that reset action is essential. 

FIG 4.13. A proportional-plus-reset controller is the complement of a single-capacity process. 

A single-capacity process can tolerate zero proportional band and zero reset time. Compared to a dead time process, it can be concluded that the easier the process is to control, the less critical are its mode adjustments. 

FIG 5.9. In the presence of differential gap, even a single-capacity process will limit-rycle. 

In this equation, the substituent parameters and reflect the incremental resonance interaction with electron-demanding and electron-releasing reaction centers, respectively. The variables and r are established for a reaction series by regression analysis and are measures of the extent of the extra resonance contribution. The larger the value of r, the greater is the extra resonance contribution. Because both donor and acceptor capacity will not contribute in a single reaction process, either or r would be expected to be zero. 

We can consider a simple model for the glass transition that consists of only a single relaxation process. For the response to the modulation, it has been shown that the reversing (real) and kinetic (imaginary) heat capacities (CpR and CpK) may be approximated as follows (11,30,34) ... 

There are several exceptions to Table 3.5 which would permit glass or plastic containers of no more than 1 gallon capacity to be used for class lA and class IB liquids (a) if a metal container would be corroded by the liquid (b) if contact with the metal would render the liquid unfit for the intended purpose (c) if the application required the use of more than one pint of a class lA liquid or more than one quart of a class IB liquid (d) an amount of an analytical standard of a quality not available in standard sizes needed to be maintained for a single control process in excess of 1/16 the capacity of the container sizes allowed by the table and (e) if the containers are intended for export outside the United States. 

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