7_high_order - Spring 2006 Process Dynamics, Operations,...

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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 7: High Order Overdamped Processes 7.0 context and direction Chemical processing plants are characterized by large time constants and time delays. For control engineering, we can often approximate these high-order systems by the FODT (first-order-dead-time) model. Dead time in a process increases the difficulty of controlling it. DYNAMIC SYSTEM BEHAVIOR 7.1 big and slow - high-order overdamped systems We began our study of process control by considering a mixed tank. Applying a material or energy balance to a well-mixed tank produces a first-order lag system. We subsequently combined two balances to produce a second-order system. In one case, two material balances described storage of material in two tanks. In another case, a single tank stored both material and energy. Energy and material balances show that the tank causes a dynamic lag between input and output, because it takes time to adjust the amount of mass or energy distributed throughout the tank. We might thus expect that more storage elements would lead to higher-order behavior, and require higher-order equations to describe them. The classic illustration of a high-order system is a set of n tanks in series: each tank feeds the next, and a change in the inlet stream composition C A0 must propagate through multiple tanks to be felt at the output C An . The individual tank models are ' Ai ' 1 Ai ' Ai i FC FC C V dt d = (7.1-1) They are combined by eliminating the interior stream variables to produce a single transfer function between input and output. () 1 s 1 s 1 s 1 ) s ( C ) s ( C n 2 1 ' 0 A ' An + τ + τ + τ = " (7.1-2) Let us illustrate high-order behavior and (7.1-2) by first imagining a single well-mixed overflow tank of time constant τ . If we introduce a step increase in the inlet concentration, we will (by the well-mixed assumption) immediately detect a rise in the outlet stream – the familiar first-order lag response. If we have instead two tanks in series, each half the volume of the original, we will detect a second-order, sigmoid response at the outlet. revised 2006 Mar 29 1
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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 7: High Order Overdamped Processes Each tank has a smaller individual time constant, and their sum is the time constant τ of the original tank. If we continue to increase the number of tanks in the series, always maintaining the total volume, we observe a slower initial response with a faster rise around the time constant. This behavior is shown in Figure 7.1-1. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time/tau step response 1 2 6 12 Figure 7.1-1. Step response for tanks in series; equal time constants revised 2006 Mar 29 2
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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 7: High Order Overdamped Processes The step response shows that high-order systems have a longer start-up period before rising toward the final value.
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7_high_order - Spring 2006 Process Dynamics, Operations,...

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