KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter 11 (11-23-04)

KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter 11 (11-23-04)

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1 Chapter 11 Dynamic Behavior and Stability of Closed-Loop Control Systems In this chapter we consider the dynamic behavior of processes that are operated using feedback control. This combination of the process, the feedback controller, and the instrumentation is referred to as a feedback control loop or a closed-loop system. Block Diagram Representation To illustrate the development of a block diagram, we return to a previous example, the stirred-tank blending process considered in earlier chapters.
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2 Chapter 11 Figure 11.1 Composition control system for a stirred-tank blending process.
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3 Chapter 11 Next, we develop a transfer function for each of the five elements in the feedback control loop. For the sake of simplicity, flow rate w 1 is assumed to be constant, and the system is initially operating at the nominal steady rate. Process In section 4.3 the approximate dynamic model of a stirred-tank blending system was developed: ( 29 ( 29 ( 29 1 2 1 2 (11-1) τ 1 τ 1 K K X s X s W s s s @ = + + + where 1 1 2 ρ 1 , , and (11-2) w V x K K w w w τ - = = =
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4 Chapter 11 Figure 11.2 Block diagram of the process.
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5 Chapter 11 Composition Sensor-Transmitter (Analyzer) We assume that the dynamic behavior of the composition sensor- transmitter can be approximated by a first-order transfer function: ( 29 ( 29 (11-3) τ 1 m m m X s K X s s e = + Controller Suppose that an electronic proportional plus integral controller is used. From Chapter 8, the controller transfer function is ( 29 ( 29 1 1 (11-4) τ c I P s K E s s e = + where and E ( s ) are the Laplace transforms of the controller output and the error signal e ( t ). Note that and e are electrical signals that have units of mA, while K c is dimensionless. The error signal is expressed as ( 29 P s e ( 29 p t e p e
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6 Chapter 11 ( 29 ( 29 ( 29 (11-5) sp m e t x t x t e = - % or after taking Laplace transforms, ( 29 ( 29 ( 29 (11-6) sp m E s X s X s e = - % The symbol denotes the internal set-point composition expressed as an equivalent electrical current signal. This signal is used internally by the controller. is related to the actual composition set point by the composition sensor- transmitter gain K m : ( 29 sp x t e % ( 29 sp x t e % ( 29 sp x t e ( 29 ( 29 (11-7) sp m sp x t K x t e = % Thus ( 29 ( 29 (11-8) sp m sp X s K X s e = %
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7 Chapter 11 Figure 11.3 Block diagram for the composition sensor- transmitter (analyzer).
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8 Chapter 11 Current-to-Pressure (I/P) Transducer Because transducers are usually designed to have linear characteristics and negligible (fast) dynamics, we assume that the transducer transfer function merely consists of a steady-state gain K IP : ( 29 ( 29 (11-9) t IP P s K P s e = Control Valve As discussed in Section 9.2, control valves are usually designed so that the flow rate through the valve is a nearly linear function of the signal to the valve actuator. Therefore, a first-order transfer function usually provides an adequate model for operation of an installed valve in the vicinity of a nominal steady state. Thus, we assume that the control valve can be modeled as ( 29 ( 29 2 (11-10) τ 1 v t v W s K P s s e = +
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This note was uploaded on 09/22/2009 for the course CHEMICAL CHE 401 taught by Professor Dr.muhammadal-arfaj during the Spring '09 term at King Fahd University of Petroleum & Minerals.

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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter 11 (11-23-04)

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