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FIGURE 9-8 Block diagram of control system.
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error E) is used to maintain control. A set of equations generally does not clearly indicate the relationships shown by the block diagram. In the next several chapters, tools will be developed that will enable us to reduce a block diagram such as the one in Fig. 9.8 to a single block that relates T (s) to Ti or TA. We shall then obtain the transient response of the control system shown in Fig. 9.8 to some specific changes in Td and T(. However, we shall first pause in Chap. 10 to look more carefully at the controller and control element blocks, which have been skimmed over in the present chapter.
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9.1. The two-tank heating process shown in Fig. P9.1 consists of two identical, wellstirred tanks in series. A flow of heat can enter tank 2. At time t = 0, the flow rate of heat to tank 2 suddenly increases according to a step function to 1000 Btu/min, and the temperature of the inlet water Ti drops from 60 F to 52 F according to a step function. These changes in heat flow and inlet water temperature occur simultaneously. (a) Develop a block diagram that relates the outlet temperature of tank 2 to the inlet temperature to tank 1 and the flow of heat to tank 2. (b) Obtain an expression for T;(s) where T{ is the deviation in the temperature of tank 2. This expression should contain numerical values of the parameters. (c) Determine TX(~) and TV. (d) Sketch the response T;(t) versus t. Initially, Ti = Tt = 7 ~ = 60 F and q = 0. The following data apply: w = 250 Ib/min holdup volume of each tank = 5 ft3 density of fluid = 50 lb/ft3 heat capacity of fluid = 1 Btu/(lb) (OF) 4
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Tank1
Tank2
FIGURE I !b1
9.2. The two-tank heating process shown in Fig. P9.2 consists of two identical, wellstirred tanks in series. At steady state, T, = Tt, = 60 F. At time t = 0, the temperature of each stream, entering the tanks changes according to a step function, i.e., Ti = lOu(t), TL = 20u(t) where Lrd and TL are deviation variables. (a) Develop the block diagram that relates Z ;, the deviation in temperature in tank 2, to TA and Td. (b) Obtain an expression for T;(s).
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Tank 1
Tank 2
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WJ = w1+
FIGURE P9-2
(c) Determine 2 2(2). The following data apply: WI = w2 = 250 lb/min Holdup volume of each tank = 10 ft3 Density of fluid = 50 lb/ft3 Heat capacity of fluid = 1 Btu/(lb)( F) 9.3. The heat transfer equipment shown in Fig. P9.3 consists of two tanks, one nested inside the other. Heat is transferred by convection through the wall of the inner tank. The contents of each tank are well mixed. The following data and information apply: 1. The holdup volume of the inner tank is 1 ft3. The holdup of the outer tank is 1 ft3. 2. The cross-sectional area for heat transfer between the tanks is 1 ft2. 3. The overall heat transfer coefficient for the flow of heat between the tanks is 10 Btu/(hr)(ft*)( F). 4. The heat capacity of fluid in each tank is 1 Btu/(lb)( F). The density of each fluid is 50 lb/ft3. Initially the temperatures of the feed stream to the outer tank and the contents of the outer tank are equal to 100% The contents of the inner tank are initially at lOOoF. At time zero, the flow of heat to the inner tank (Q) is changed according to a step change from 0 to 500 Btu/hr. (a) Obtain an expression for the Laplace transform of the temperature of the inner tank, T(s). (b) Invert T(s) and obtain T for time = 0, 5 hr, 10 hr, and 00. Q
1 0 Ib/hr -1 1 , Inner tank /
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