Process dynamic characteristics can be classified into three broad categories:
• Self-regulating
• Non-self-regulating
• Open-loop unstable
Figure 3-7. Installed Characteristics for a Linear Valve for a Range of Pressure Drop Ratios:
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characteristics to a step change in input is shown in Figure 3-9.
Self-regulating processes are those that, if all inputs are fixed, will seek their own equilibrium.
Figure 3-8a shows a tank that has a fixed restriction in the outflow line. If the flow into the ves-sel is fixed, the level will reach equilibrium when the hydrostatic pressure at the base of the vessel causes the outflow to exactly equal the inflow. This is the analogy of many physical pro-cesses. For instance, in a thermal process if there is a certain rate of heat input, the temperature will rise or fall until the heat lost, either to the environment or when carried away by effluent streams, is exactly equal to the rate of heat input. Most processes fall into this category.
Non-self-regulating processes can be depicted by the hydraulic analogy of Figure 3-8b. Here, there is a fixed flow rate out of the tank that is controlled by the flow controller. The outflow does not depend upon the level in the tank. Unless the inflow is precisely the same as the out-flow, the level will continue to rise or fall until either the tank overflows or becomes empty.
A mathematical expression for a process of this type is given by the following integral equa-tion:
(3-6) where: h = height of fluid in tank (i.e., level)
A = cross-sectional area fin, fout = volumetric flow rates
Figure 3-8. Hydraulic Analogies of Common Types of Process Characteristics
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For this reason, non-self-regulating processes are often called integrating processes. In actual practice, liquid-level control can usually be represented as an integrating process. Another example of an integrating process is pressure control for a liquid-vapor system. If the heat input exceeds the heat removal rate, the pressure will continue to rise.
The number of processes that are self-regulating is much greater than the number that are non-self-regulating.
A few processes are unstable in the open loop (i.e., a loop without feedback control). These are called “runaway” processes. A typical example is a jacketed exothermic reactor. As the reactor temperature is increased, the reaction rate increases. But at the higher reaction rate, more heat is generated; also more heat is removed due to a higher differential temperature between the reactants and the jacket. If there is insufficient heat transfer, then excess heat will be generated, causing the temperature to rise even further.
Figure 3-10 illustrates heat flow versus temperature curves for an exothermic reactor. The curved line represents heat generation; the straight lines represent heat removal. The solid straight line represents one heat removal relation, determined by the heat transfer surface area.
Figure 3-9. Step Input Response of Common Types of Process Characteristics
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The intersection of the heat generation and heat removal curves represents an equilibrium point; at that point as much heat is being removed as is being generated. If the temperature is above (below) that point, and if the heat generation curve lies above (below) the heat removal curve, then excess heat is being generated (removed) which will cause the temperature to move away from the equilibrium point. On the other hand, if the heat generation curve is below (above) the heat removal curve, then the temperature will move toward the equilibrium point. The solid line for heat removal represents an open-loop unstable process, whereas the dashed line represents an open-loop stable process.
In chapter 4, we will mention the use of the derivative mode in a controller to provide a stabi-lizing effect in control loops containing open-loop unstable processes.
Types of Dynamic Response
The simplest type of self-regulating process is one in which there is a single location for mass or energy storage. An example of such a process is shown in Figure 3-11. Here, a constantly flowing stream is heated in a well-mixed vessel. If the heat storage is negligible in both the walls of the vessel and in the heating coil itself, then the only point of heat storage is in the fluid itself. On a step increase in the heat input, the temperature will respond as shown in Fig-ure 2-4. This is the typical response of a first-order lag; hence the process can be described by a two-parameter process model that consists of a process gain and a time constant.
Another elementary type of response is pure dead time. Usually associated with the physical Figure 3-10. Heat Flow versus Temperature Curves for Exothermic Reactor
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well-insulated flowing pipeline, where the temperature is measured at two points separated by a considerable distance. The temperatures recorded from the two measuring points would be identical except they would be separated by the time required for the fluid to move from the upstream to the downstream point of measurement. This process can be described by a single parameter model that represents the dead time, as shown in Figure 2-5.
First-order lags and dead times can be considered as “elements” that, combined in a myriad ways, comprise the dynamic characteristics of real processes. Most self-regulating processes are not as simple as those just described; rather, they consist of multiple locations for mass or energy storage plus, perhaps, the time for the movement of the material. We will describe sev-eral elementary combinations, giving a physical example and a hydraulic analogy of each. Our purpose is to give the reader an intuitive understanding of why processes behave as they do.
Two different hydraulic analogies can be given for two locations for mass storage. In Figure 3-12a, the level in tank 2 has no influence on the flow of fluid between the tanks, whereas in Fig-ure 3-12b, the driving force for flow is the difference in level between the tanks. Both of these situations represent two first-order lags, but in Figure 3-12a the lags are said to be uncoupled, whereas in Figure 3-12b, the lags are said to be coupled.
In both cases, the response to a step change in input will be an “S-shaped curve” rather than the idealized first-order lag response of Figure 2-4. A more detailed discussion of the shape of the S-shaped curve follows later in this section.
We note, however, that coupled lags are representative of many physical processes. For exam-ple, heat transfer from the hot to the cold side of a heat exchanger is dependent upon the differ-ential temperature between the two sides.
Figure 3-11. Process with Single Point of Energy Storage 7
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It is interesting to consider the effect of many lags in series, both coupled and uncoupled lags.
Continuing with our hydraulic analogies, assume that we start with one tank whose time con-stant is some value τ. The response to a step change in inflow is the familiar first-order lag graph, which is designated by “1” in both Figures 3-13 and 3-14.
Now suppose we replace that tank with two smaller tanks, each of which is half the size of the original tank. It is reasonable to assume that the time constant of each of the smaller tanks is τ / 2. These tanks can either be configured as “uncoupled” lags or as “coupled” lags, as shown in Figures 3-13 and 3-14. Note that the mass is now distributed between the two tanks. The response to a step input change is shown by the graphs labeled “2” in Figures 3-13 and 3-14.
Figure 3-12. Hydraulic Analogies for Two Points of Mass Storage IL[HG
Now replace the two tanks with three smaller tanks, each of which is a third the size of the original tanks and has a time constant of τ/ 3. The step response is shown by the graphs labeled “3” in Figures 3-13 and 3-14. Continue in this manner, using N tanks, each holding 1/N volumetric units and having a time constant of τ/ N.
For larger and larger values of N, the responses will continue to be S-shaped curves. With the uncoupled first-order lags, the graphs will be more severely curved. In the limit, as N approaches , the S-shaped response will be converted into a pure dead time, with a time delay of τ minutes. (Chemical engineers will recognize this thought process as the passage from a stirred-tank reactor to a plug flow reactor.)
With coupled first-order lags, the apparent dead time does not increase as drastically, but the system becomes increasingly sluggish. The process appears to have a very long dominant time constant, much longer than if all the mass were concentrated at one location (a single first-order lag).
The physical import of this discussion is to demonstrate that the more widely the mass or energy is distributed, the more the process assumes the characteristics of dead time, particu-larly with uncoupled first-order lags. With coupled first-order lags, the process takes on the characteristics of a moderate amount of dead time plus a long dominant time constant.
Figure 3-13. The Response of Multiple Uncoupled First-order Lags in Series
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Let us consider a real example, rather than a hydraulic analogy. Suppose we have a liquid-filled, well-mixed tank that is heated by an internal steam coil, as shown in Figure 3-15. This is the same process shown in Figure 3-11, except that now we will consider the heat stored in the heating coil, the walls of the coil, and in the temperature-sensing bulb itself.
Figure 3-14. The Response of Multiple Coupled First-order Lags in Series
Figure 3-15. Example of Coupled First-order Lags
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Suppose that initially the process is stable, with the steam valve in an intermediate position.
There will be some amount of pressure drop across the steam valve. Assuming that there is a constant steam supply pressure, this will determine the pressure inside the coil. Since the steam is condensing within the coil, it will be in a saturated condition. Hence, its temperature is indirectly determined by the position of the valve.
From this stable condition, let us make a step increase in the valve position. This will increase the steam flow, causing the pressure in the coil to rise. The steam within the coil represents a point of (thermal) energy storage. The steam pressure, and hence the temperature in the coil, will not rise instantaneously. It will rise only as the mass of steam within the coil increases.
This will be similar to the response of a first-order lag, with a very short time constant.
As the temperature of the steam within the coil rises, the temperature difference between the steam and the metal in the coil walls increases. This causes increased heat flow into the metal walls of the coil, which represents another point of energy storage. The temperature of the coil walls increases, which increases the temperature difference between the coil and the fluid itself. This in turn causes an increase in heat flow into the fluid, which causes the fluid temper-ature to rise. Finally, as the fluid tempertemper-ature rises, the tempertemper-ature difference between the fluid and the thermal bulb increases. This causes an increase in heat flow, and consequently an increase in the temperature sensed by the thermal bulb.
This somewhat tedious discussion shows that this process can be represented by the hydraulic analogy, much as in Figure 3-14 but with four tanks in series, representing a series of four cou-pled first-order lags. Hence, the expected response to a step change in valve position would be an S-shaped curve with a long response, something like those shown in Figure 3-14.
Let us consider another type of response. Start with the process represented by Figure 3-11, but now suppose that the vessel has a metal tank wall of significant thickness, as shown in Fig-ure 3-16. This represents a point of energy storage. This is a first-order lag that is not in series with the main signal flow, but is at the side. A step change in heat into the tank must furnish heat not only to raise the fluid temperature, but also to raise the temperature of the metal wall of the vessel. A hydraulic analogy of a side lag is given in Figure 3-17.
If the ratio of the side lag time constant to the basic time constant is small (e.g., a very thin ves-sel wall and a high heat-transfer coefficient from the fluid to the wall), then the response will be essentially that of a first-order lag. On the other hand, suppose that the ratio of the side lag time constant to the basic time constant is large (e.g., a very thick vessel wall and a low heat-transfer coefficient from the fluid to the wall). In this case, the output response to a step input will be a relatively rapid rise to partial equilibrium, followed by a slow change the rest of the way to equilibrium. Figure 3-18 shows the output response for a range of ratios of the side lag to basic time constants. This phenomenon can explain such observed physical behavior as an initial rapid change in the variable followed by a lengthy period of slowly drifting change.
These examples have illustrated the types of response that may be expected if the points of mass or energy storage (first-order lags) are in parallel rather than in series, or at the side. In
Figure 3-16. Example of Side Lag
Figure 3-17. Hydraulic Analogy of Side Lag
Figure 3-18. Response of Basic Lag Plus Side Lag 7
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real processes, infinite combinations of these elementary examples are possible. We can have points of energy storage, some in parallel, some in series, and some at the side. Thus, depend-ing upon the physical parameter values, we might expect an infinite number of responses for a step input change. But the responses will usually have the form of an S-shaped curve. Other forms of response that we occasionally encounter will be covered later.
We have purposely avoided giving rigorous forms of models, such as explicit transfer func-tions, for these examples. Doing so would not be meaningful. In a real industrial process appli-cation, we probably will not know the precise structure of the system—how many points of mass or energy storage there are in series, parallel, and so on. It would therefore be futile to try to construct an exact mathematical model. This situation differs from that of electro-mechani-cal systems (flight control, robotics, etc.), which are often comprised of discrete components and hence are amenable to precise mathematical modeling. For industrial process control, we have done well to recognize the potential forms of response and some of the factors that give rise to them.
In summary, we have concluded that the step response of most (but not all) self-regulating pro-cesses will be an S-shaped curve. We cannot formulate a precise process model, but we can approximate the response with a simplified form. The form we will choose is first-order plus dead time (FOPDT). This requires three parameters: process gain, pseudo-dead time, and pseudo-time constant. We will leave for chapter 6 a discussion of the task of obtaining actual numerical values for these parameters.
A self-regulating process can exhibit other forms of response, for example, the underdamped (decaying oscillation) form shown in Figure 3-19. While this type of response is often seen in closed-loop systems (when the controller is in the automatic mode), it is very unusual to see this as an open-loop response (when the controller is in the manual mode). The reason for this is that feedback is present in the closed loop.
Figure 3-19. Open-Loop Response: Damped Oscillatory Behavior
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as to what might cause oscillation in the open loop. Suppose that in a chemical process there is a point of control action (i.e., a control valve) and that downstream from this there is a sensor, perhaps a composition analyzer. Suppose further that between the point of control and the sen-sor there is a chemical recycle loop. This is a form of feedback that could result in oscillation.
As another example, suppose we have a cascade control system. (See chapter 9 for a full dis-cussion of cascade control.) Normally, the outer, or primary, loop should be significantly slower than the inner, or secondary, loop. But if this is not the case, then when the secondary controller is in the automatic mode and the primary is in manual, a step change in the output of the primary controller may cause the primary measurement to oscillate because feedback is present in the secondary loop.
These examples are hypotheses of how oscillation could occur in the open loop. While actual situations resembling these may occur, the situation is sufficiently unusual in industrial pro-cess control systems that it need not be considered further here.2
A more important consideration, although still somewhat rare, is that of an inverse response.
With an inverse response, when the valve undergoes a step change, the measurement tends to initially go the “wrong” way. It then reverses and comes to an equilibrium in the predicted direction, as shown in Figure 3-20a. This situation is also applicable to some integrating pro-cesses, whose response is shown in Figure 3-20b.
Figure 3-20. Inverse Open-loop Response
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An oft-cited example of inverse response is the shrink-and-swell phenomena that occurs in the feedwater drum level of large steam generators. The drum level will remain constant if there is a constant feedwater rate, if there are no water losses and no blow-down within the boiler, and if the steam take-off rate, in mass units per hour, exactly matches the feedwater rate.
Suppose, however, that from this equilibrium situation the steam demand valve is opened slightly, increasing the steam flow. If the feedwater rate is unchanged, what will the drum level do? The first and obvious thought is that the drum level will go down, since more steam is being withdrawn than is being replaced with feedwater. However, the boiler drum does not respond that way because of a secondary phenomenon. Within the drum, as well as within the water tubes inside the boiler, there is water at its flash-point temperature. There are com-pressed bubbles of vapor, and when the steam demand is increased, the steam pressure falls slightly. This causes some of the water to flash into steam; it also causes the bubbles to expand.
The expansion of vapor bubbles in the drum, together with the water forced from the tubes into the drum as a result of the expansion of vapor in the tubes themselves, causes the water level in the drum to rise momentarily.
The expansion of vapor bubbles in the drum, together with the water forced from the tubes into the drum as a result of the expansion of vapor in the tubes themselves, causes the water level in the drum to rise momentarily.