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CHAPTER
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CONTROLLERS AND FINAL CONTROL ELEMENTS
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In the previous chapter, the block-diagram representation of a simple control system (Fig. 9.2) was developed. This chapter will focus attention on the controller and final control element and discuss the dynamic characteristics of some of these components that are in common use. As shown in Fig. 9.2, the input signal to the controller is the error and the output signal of the controller is fed to the final control element. In many process control systems, this output signal is au air pressure and the final control element is a pneumatic valve that opens and closes as air pressure on the diaphragm changes. For the mathematical analysis of control systems, it is sufficient to regard the controller as a simple computer. For example, a proportional controller may be thought of as a device that receives the error signal and puts out a signal proportional to it. Similarly, the final control element may be regarded as a device that produces corrective action on the process. The corrective action is regarded as mathematically related to the output signal from the controller. However, it is desirable to have some appreciation of the actual physical mechanisms used to accomplish this. For this reason, we begin this chapter with a physical description of a pneumatic control valve and a simplified description of a proportional controller. Up to about 1960, most controllers were pneumatic. Although pneumatic controllers are still in use and function quite well in many installations, the controllers being installed today are electronic or computer-based instruments. For this reason, the proportional controller to be discussed in this chapter will be electronic
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or computer-based. The transfer functions that are presented in this chapter apply to either type of controller, and the discussion is in no way restrictive. After the introductory discussion, transfer functions will be presented for simplified or idealized versions of the control valve and the conventional controllers. These transfer functions, for practical purposes, will adequately represent the dynamic behavior of control valves and controllers. Hence, they will be used in subsequent chapters for mathematical analysis and design of control systems.
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The control valve shown in Fig. 10.1 contains a pneumatic device (valve motor) that moves the valve stem as the pressure on a spring-loaded diaphragm changes. The stem positions a plug in the orifice of the valve body. As the pressure increases, the plug moves downward and restricts the flow of fluid through the valve. This action is referred to as air-to-close. The valve may also be constructed to have air-to-open action. Valve motors are often constructed so that the valve stem position is proportional to the valve-top pressure. Most commercial valves move from fully open to fully closed as the valve-top pressure changes from 3 to 15 psig. In general, the flow rate of fluid through the valve depends upon the upstream and downstream fluid pressures and the size of the opening through the valve. The plug and seat (or orifice) can be shaped so that various relationships between stem position and size of opening (hence, flow rate) are obtained. In our example, we shall assume for simplicity that at steady state the flow (for fixed upstream and downstream fluid pressures) is proportional to the valve-top pneumatic pressure. A valve having this relation is called a linear valve. An extensive discussion of control valves is presented in Chap. 20.
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Controller
The control hardware required to control the temperature of a stream leaving a heat exchanger is shown in Fig. 10.2. This hardware, available from manufacturers of
FIGURE 10-1 Pneumatic control valve (air-to-close).
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Hot process stream
unit (thermocouple) FIGURE 10-2
Schematic diagram of control system.
such equipment, consists of the following components listed here along with their respective conversions: Transducer (temperature to current). Controller-recorder (current to current). Converter (current to pressure). Control valve (pressure to flow rate). Figure 10.2 shows that a thermocouple is used to measure the temperature; the signal from the thermocouple is sent to the transducer, which produces an output in the range of 4-20 ma, which is a linear function of the input. The output of the transducer enters the controller where it is compared to the set point to produce an error signal. The controller converts the error to an output in the range of 4-20 ma according to the control law stored in the memory of the computer. The only control law we have considered so far has been proportional. Later in this chapter other control laws will be described. The output of the controller enters the converter, which produces an output in the range of 3-15 psig, which is a linear function of the input. Finally, the output of the converter is sent to the top of the control valve, which adjusts the flow of cooling water to the heat exchanger. We shall assume that the valve is linear and is the pressure-to-
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