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42 MATHEMATICAL MODELS OF CHEMICAL ENOINEERING SYSTRMS reactant A in each tank are (with units of kg - mol of A/min) v dCA1 - = F(cAO - CAlI - VlklCA1 ’ dt I y dCAz - = F(C,, ’ dt - cA2) - v, k2 cA2 (3.3) v dCA3 - = F(cA2 3 dt - CA3) - v3 k,CA3 The specific reaction rates k, are given by the Arrhenius equation k, = CIe-E/RTm n = 1, 2, 3 (3.4) If the temperatures in the reactors are different, the k’s are different. The n refers to the stage number. The volumes V, can be pulled out of the time derivatives because they are constant (see Sec. 3.3). The flows are all equal to F but can vary with time. An energy equation is not required because we have assumed isothermal operation. Any heat addition or heat removal required to keep the reactors at constant temperatures could be calculated from a steadystate energy balance (zero time derivatives of temperature). The three first-order nonlinear ordinary differential equations given in Eqs. (3.3) are the mathematical model of the system. The parameters that must be known are Vi, V, , V, , kl, k2, and k, . The variables that must be specified before these equations can be solved are F and CA,. “Specified” does not mean that they must be constant. They can be time-varying, but they must be known or given functions of time. They are theforcingfunctions. The initial conditions of the three concentrations (their values at time equal zero) must also be known. Let us now check the degrees of freedom of the system. There are three equations and, with the parameters and forcing functions specified, there are only three unknowns or dependent variables: CA1, CA2, and cA3. Consequently a solution should be possible, as we will demonstrate in Chap. 5. We will use this simple system in many subsequent parts of this book. When we use it for controller design and stability analysis, we will use an even simpler version. If the throughput F is constant and the holdups and temperatures are the same in all three tanks, Eqs. (3.3) become ~ dt & dt (3.5) dCA3 dt
EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENOINEERING SYSTEMS 43 where r = V/F with units of minutes. There is only one forcing function or input variable, CA0 . 3.3 CSTRs WITH VARIABLE HOLDUPS If the previous example is modified slightly to permit the volumes in each reactor to vary with time, both total and component continuity equations are required for each reactor. To show the effects of higher-order kinetics, assume the reaction is now nth-order in reactant A. Reactor 1: dV’=F -F dt ’ ’ f (v,c,,) = Fo CAO - FICAI - V,kl(CAl) (3.6) Reactor 2: dVZ=F -F dt ’ 2 f (v, cA2) = FICA, - F, CAZ - v, k,(C,,) (3.7) Reactor 3 : dV3 - = F, - F3 dt f 05 cA3) = F2 cA2 - F, CAS - v, k,(CA,) (3.8) Our mathematical model now contains six first-order nonlinear ordinary differential equations. Parameters that must be known are k,, k,, k,, and n. Initial conditions for all the dependent variables that are to be integrated must be given: CA1, cA2, CA3, VI, V2, and V, . The forcing functions CAo(r) and Focr, must also be given. Let us now check the degrees of freedom of this system. There are six equations. But there are nine unknowns: C*i, CA,, C,, , VI, V,, V,, F,, F,, and F,. Clearly this system is not sufficiently specified and a solution could not be obtained. What have we missed in our modeling? A good plant operator could take one look at the system and see what the problem is. We have not specified how the flows out of the tanks are to be set. Physically there would probably be control valves in the outlet lines to regulate the flows. How are these control valves to be set? A common configuration is to have the level in the tank controlled by the outflow, i.e., a level controller opens the control valve on the exit
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I WILLIAM WIlllAM 1. LUYBEN PROCESS
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The Series Bailey and OUii: Biochem
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PROCESS MODELING, SIMULATION, AND C
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ABOUT THE AUTHOR William L. Luyben
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CONTENTS Preface Xxi 1 1.1 1.2 1.3
Page 11 and 12: CONTENTS .. . xlu 5.7 Batch Reactor
Page 13 and 14: CONTENTS xv 10.3 10.4 Performance S
Page 15 and 16: comwrs xvii 151.3 Transpose of a Ma
Page 17 and 18: CONTENTS Xix 20.4 Sampled-Data Cont
Page 19 and 20: . Xxii PREFACE Another significant
Page 21 and 22: PROCESS MODELING, SIMULATION, AND C
Page 23 and 24: 2 PROCESS MODELING, SIMULATION. AND
Page 25 and 26: 4 PROCESS MODELING, SIMULATION, AND
Page 27 and 28: 6 PROCESS MODELING. SIMULATION, AND
Page 29 and 30: 8 PROCESS MODELING, SIMULATION, AND
Page 31 and 32: 10 PROCESS MODELING, SIMULATION, AN
Page 33 and 34: 12 PROCESS MODELING, SIMULATION, AN
Page 35 and 36: CHAPTER 2 FUNDAMENTALS 2.1 INTRODUC
Page 37 and 38: FUNDAMENTALS 17 big, complex system
Page 39 and 40: FUNDAMENTALS 19 z=o I z + dz z=L FI
Page 41 and 42: FUNDAMENTALS 21 The minus sign come
Page 43 and 44: FUNDAMENTALS 23 Molar flow of A lea
Page 45 and 46: FUNDAMENTALS 25 For liquids the PP
Page 47 and 48: FUNDAMENTALS 27 Flow of energy (ent
Page 49 and 50: FUNDAMENTALS 29 The units of this f
Page 51 and 52: FUNDAMENTALS 31 FIGURE 2.9 Laminar
Page 53 and 54: FUNDAMENTALS 33 Then Eq. (2.48) bec
Page 55 and 56: FUNDAMENTALS 35 Wenzel in Chemical
Page 57 and 58: FUNDAMENTALS .37 This exponential t
Page 59 and 60: FUNDAMENTALS 39 a direction 0 = 45
Page 61: EXAMPLES OF MATHEMATICAL MODELS OF
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Page 107 and 108: 88 COMPUTER SIMULATION Our discussi
Page 109 and 110: 90 COMPUTER SIMULATION lation packa
Page 111 and 112: 92 COMPUTER SIMULATION For this rea
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94 COMPUTER SIMULATION TABLE 4.1 Ex
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96 COMPUTER SIMULATION Clearly, the
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98 COMPUTER SIMULATION TABLE 4.2 Ex
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I()() COMPUTER SIMULATION Settingfl
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ANALYSIS OF MULTNARIABLE SYSTEMS 57
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ANALYSIS OF MULTIVARIABLE SYSTEMS 5
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ANALYSIS OF MULTIYARIABLE SYSTEMS 5
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FIGURE 16.5 Multivariable INA (- 1,
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ANALYSIS OF MULTIYARIABLE SYSTEMS 5
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DESIGN OF CONTROLLERS FOR MULTIVARI
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DESIGN Ok CONTROLLERS FOR MULTIVARI
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