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FIGURE 11-2 Block diagram for a chemical-reactor control system.
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BLOCK DIAGRAM OF A CHEMICAL-REACTOR CONTROL SYSTEM
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ri= 2, r,= 1, rd= 0.5 K,= &
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Equivalent block diagram for a chemical-reactor control system (CR is now in concentration units).
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based controller, the block for the controller in Fig. 11.2 would be replaced by two blocks; one for the electronic controller and one for the converter, which converts the controller output (ma) to the pneumatic signal (psig). An equivalent diagram is shown in Fig. 11.3 in which some of the blocks have been combined. Numerical quantities for the parameters in the transfer functions are given in Fig. 11.3. It should be emphasized that the block diagram is written for deviation variables. The true steady-state values, which are not given by the diagram, must be obtained from the analysis of the problem. The example analyzed in this chapter will be used later in discussion of control system design. The design problem will be to select a value of Kc that gives satisfactory control of the composition C2 despite the rather long transportation lag involved in getting information to the controller. In addition, we shall want to consider possible use of other modes of control for the system.
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11.1. In the process shown in Fig. Pll. 1, the concentration of salt leaving the second
tank is controlled using a proportional controller by adding concentrated solution through a control valve. The following data apply:
1. The controlled concentration is to be 0.1 lb salt/ft3 solution. The inlet concen-
tration ci is always less than 0.1 lb/ft3. 2. The concentration of concentrated salt solution is 30 lb salt/ft3 solution. 3. Transducer: the output of the transducer varies linearly from 3 to 15 psig as the concentration varies from 0.05 to 0.15 lb/ft3. 4. Controller: the controller is a pneumatic, direct-acting, proportional controller. 5. Control valve: as valve-top pressure varies from 3 to 15 psig; the flow through the control valve varies linearly from 0 to 0.005 cfm.
LINEAR
CLOSED-LOOP SYSTEMS
Concentrated
J _ -z ~--= ---
solution
Controller + Set point
Salt solution 1 ft3/mi n ci
FIGURE Pll-1
6. It takes 30 set for the solution leaving the second tank to reach the transducer at the end of the pipe. Draw a block diagram of the control system. Place in each block the appropriate transfer function. Calculate all the constants and give the units.
CHAPTER
CLOSED-LOOP TRANSFER FUNCTIONS
Standard Block-Diagram Symbols
In Chap. 9, a block diagram was developed for the control of a stirred-tank heater (Fig. 9.2). In Fig. 12.1, the block diagram has been redrawn and incorporates some standard symbols for the variables and transfer functions, which are widely used in the control literature. These symbols are defined as follows:
R = set point or desired value C = controlled variable E = error B = variable produced by measuring element M = manipulated variable U = load variable or disturbance G, = transfer function of controller Gt = transfer function of final control element G2 = transfer function of process H = transfer function of measuring element
In some cases, the blocks labeled G, and G t will be lumped together into a single block as was done in Chap. 9. The series of blocks between the comparator and 143
LINEAR CLDSEJMBOP
SYSTEMS
FIGURE 12-1
Standard control system nomenclature.
the controlled variable, which consist of G,, Gi, and G2, is referred to as the forwurd path. The block H between the controlled variable and the comparator is called the feedback path. The use of G for a transfer function in the forward path and H for one in the feedback path is a common convention. The product GH, which is the product of all transfer functions (G,GlGzH) in the loop, is called the open-loop transfer function. We call GH the open-loop transfer function because it relates the measured variable B to the set point R if the feedback loop (of Fig. 12.1) is disconnected (i.e., opened) from the comparator. The subject of this chapter is the closed-loop transfer function, which relates two variables when the loop of Fig. 12.1 is closed. In more complex systems, the block diagram may contain several feedback paths and several loads. An example of a multiloop system, which is shown in Fig. 12.2, is cascade control. Several multiloop systems of industrial importance are presented in Chap. 18.
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