PID Controllers, 2nd Edition

Chapter 7.9 - Control Paradigms: System Structuring

In this section we illustrate how complex control systems can be built from simple components by using the paradigms we have discussed. The problem is quite complex. It involves selection of measured variables and control variables, and it requires significant physical understanding of the process.

The Process

The process to consider is a chemical reactor. A schematic diagram is shown in Figure 7.36. Two substances A and b are mixed in the reactor. They react to form a product. The reaction is exothermic, which means that it will generate heat. The heat is dissipated through water that is circulating in cooling pipes in the reactor. The reaction is very fast; equilibrium is achieved after a time that is much shorter than the residence time of the reactor. The flow qA of substance A is considerably larger than qB. Efficiency of the reaction and the heat generation is essentially proportional to the flow qB.

A static process model is useful in order to understand the control problem. Figure 7.37 shows the efficiency and the heat generation as a function of temperature. In the figure we have drawn a straight line that corresponds to the cooling power. There are equilibria where the power generated by the reaction is equal to the cooling power represented at points P and Q in the figure. The point P corresponds

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Figure 7.36 Schematic diagram of a chemical reactor.

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Figure 7.37 Static process model for the exothermic reactor.

to an unstable equilibrium. It follows from Figure 7.37 that if the temperature is increased above P the power generated by the reaction is larger than the cooling power. Temperature will thus increase. The catalyst in the reactor may be damaged if the temperature becomes too high. Similarly if the temperature decreases below point P it will continue to decrease and the reaction stops. This phenomena is called "freezing." Freezing starts at the surface of the cooling tube and will spread rapidly through the reactor. If this happens the reactor must be switched off and restarted again.

Design Requirements

There are considerable risks in running an exothermic reactor. The reactor can explode if the temperature is too high. To reduce the risk of explosion, the reactors are placed in special buildings far away from the operator. Because of the risk of explosion, it is not feasible to experiment with controller tuning. Consequently, it is necessary to compute controller setting beforehand and verify that the settings are correct before starting the reactor. Safety is the overriding requirement of the control system. It is important to guarantee that the reaction temperature will not be too high. It is also important to make sure that process upsets do not lead to loss of coolant flow, and that stirring does not lead to an explosion. It is also desirable to operate the reactor efficiently. This means that freezing must be avoided. Besides it is desirable to keep the efficiency as high as possible. Because of the risks, it is also necessary to automate start and stop as well as normal operation. It is desirable to avoid having to run the reactor under manual control. In this particular case the operator can set two variables, the reactor temperature and the ratio between the flows qA and qB. The reaction efficiency and the product quality can be influenced by these two variables.

 

7.9.2 Controller Structure

The reactor has five valves. Two of them, V1 and V2, influence the coolant temperature. The flow of the reactor is controlled by V3 and V4, and the product flow is controlled by the valve V5. In this particular application the valve V5 is controlled by process steps downstream. (Compare this with the discussion of surge tanks in Section 7.5.)

There are five measured signals: the reactor temperature Tr, the level in the reactor tank L, the cooling temperature Tv, and the flows qA and qB. The physical properties of the process gives a natural structuring of the control system. A mass balance for the material in the reactor tank shows that the level is essentially influenced by the flow qA and the demanded production. It follows from the stochiometry of the reaction that the ratio of the flows qA and qB should be kept constant for an efficient reaction. The reactor temperature is strongly influenced by the water temperature, by the temperature of the coolant flow and the flows qA and qB . Coolant temperature is influenced by the valve V1 that controls the amount of flow and by the steam valve V2.

This simple physical discussion leads to the diagram shown in Figure 7.38, which shows the causality of the variables in the process. The valve V5 can be regarded as a disturbance because it is set by downstream process units. Figure 7.38 suggests that there are three natural control loops:

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Figure 7.38 Causality diagram for the process variable.

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Figure 7.39 Block diagram for the level control through valve V3.

  1. Level control: Controlling the tank level with valve V3.
  2. Temperature control: Control of the reactor temperature with valves V1 and V2.
  3. Flow ratio control: Control of ratio qB/qA with valve V4. These control loops are discussed in detail.

7.9.3 Level Control

The block diagram for the level control is shown in Figure 7.39. The primary function is a proportional feedback from the level to the flow qA, which is controlled by the valve V3. The reactor is also used as a surge tank to smooth out the difference between actual production and commanded production. The level in the tank will vary during normal operations. Reasonable limits are that the level should be between 50% and 100%. If the proportional band of the controller is chosen as 50%, the control variable will be fully closed when the tank is full and half-open when the tank is half-full. It is important that the reactor temperature remains within given bounds. The flow qA is constrained, therefore, by two selectors based on measurements of the temperature in the reactor tank (Tr) and the coolant temperature (Tv). When starting the reactor the level is kept at the lower limit until the coolant temperature becomes sufficiently high. This is achieved by combination of limiters, multipliers, and selectors, as shown in Figure 7.39.

7.9.4 Temperature Control

Figure 7.40 gives a block diagram for controlling the reactor temperature. Since the chemical reaction is fast compared to temperature and flow dynamics, the reactor can be viewed as a heat exchanger from the control point of view. During normal conditions the temperature is controlled by adjusting the coolant flow through the valve V1. The primary control function is a feedback from temperature to the valves V1 and V2. The setpoint in this control loop can be adjusted manually. The parameters of this control loop can be determined as follows. The transfer function from coolant flow to the reactor temperature is approximately given by

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where the time constant typically has values T1 = 300 s and T2 = 50 s. The following rough calculation gives approximate values of the controller parameter. A proportional controller with gain K gives the loop transfer function

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The characteristic equation of the closed loop becomes

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Figure 7.40 Block diagram showing temperature control through valves V1 and V2.

The closed system is thus of second order. The relative damping ? and the the undamped natural frequency ? are given by

07_9-9.jpg

The approximation in the first expression is motivated by T1 >> T2. With a relative damping ? = 0.5 the Equation (7.16) then gives ? ˜ 1/T2. Furthermore it follows from Equation (7.17) that

07_9-10.jpg

The loop gain is thus essentially determined by the ratio of the time constants. The controller gain becomes

07_9-11.jpg

and the closed-loop system has the undamped natural frequency.

07_9-12.jpg

If PI control is chosen instead, it is reasonable to choose a value of the integration time.

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Control can be improved by using derivative action. The achievable improvement depends on the time constant of the temperature sensor. In typical cases this time constant is between 10 s and 40 s. If it is as low as 10 s it is indeed possible to obtain improved control by introducing a derivative action in the controller. The derivative time can be chosen to eliminate the time constant T2. We then obtain a system with the time constants 300 s and 10 s. The gain can then be increased so that

07_9-14.jpg

and the undamped natural frequency of the system then becomes ? ˜ 0.1 rad/s. If the time constant of the temperature sensor is around 40 s, the derivative action gives only marginal improvements.

The heat generated by the chemical reaction is proportional to the flow qA. To make sure that variations in qA are compensated rapidly we have also introduced a feedforward from the flow qA. This feedforward will only operate when the tank level is larger than 50% in order to avoid freezing when the reactor is started.

To start the reaction the reactor must be heated so that the temperature in the reaction vessel is larger than Tc (compare with Figure 7.37). This is done by using the steam valve V2. Split range control is used for the steam and water valves (compare Section 7.5). The water valve is open for low signals (3-9 PSI) and the steam valve is open for large pressures (9-15 PSI).

To avoid having the reactor freez, it is necessary to make sure that the reaction temperature is always larger than Tc. This is the reason for the extra feedback from water temperature to Tv through a maximum selector. This feedback makes sure that the steam valve opens if the temperature in the coolant flow becomes too low. Cascade control would be an alternative to this arrangement.

7.9.5 Flow Ratio Control

The ratio of the flows qA and qB must be kept constant. Figure 7.41 shows how the efficiency of the reaction depends on qB when qA is kept constant. The flow qB is controlled with a ratio control system (as shown in Figure 7.42), which is the primary control function. The reaction rate depends strongly on qB. To diminish the risk of explosion, there is a nonlinearity in the feedback that increases the gain when qB/qA is large. The flow loop has several selectors. At startup it is desirable that substance B is not added until the water temperature has reached the critical value Tc and the reactor tank is half-full. To achieve this the feedback from the water temperature and tank level have been introduced through limiters and a minimum selector. There are also limiters and a selector that closes valve V4 rapidly if flow qA is lost. There is also a direct feedback from qA through limiters and selectors and a feedback from the reactor temperature that closes valve V4, if the reactor temperature becomes too high.

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Figure 7.41 Reaction yield as a function of qB at constant qA.

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Figure 7.42 Block diagram for controlling the mixing ratio qB /qA through valve V4.

7.9.6 Override Control of the Outlet Valve

The flow out of the reactor is determined by valve V5. This valve is normally controlled by process steps downstream. The control of the reactor can be improved by introducing an override, which depends on the state of the reactor. When starting the reactor, it is desirable to have the outlet valve closed until the reactor tank is half-full and the reaction has started. This is achieved by introducing the tank level and the tank temperature to the setpoint of the valve controller via limiters and minimum selectors as is shown in Figure 7.43. The valve V5 is normally controlled by qsp. The minimum selector overrides the command qsp when the level L or the temperature Tr are too low.

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Figure 7.43 Block diagram for controlling the outflow of the reactor through valve V5.

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