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44 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS line to increase the outflow if the level in the tank increases. Thus there must be a relationship between tank holdup and flow. Fl =A”,) F2 =.& F, =-I&j) (3.9) The f functions will describe the level controller and the control valve. These three equations reduce the degrees of freedom to zero. It might be worth noting that we could have considered the flow from the third tank F, as the forcing function. Then the level in tank 3 would probably be maintained by the flow into the tank, F, . The level in tank 2 would be controlled by F,, and tank 1 level by F, . We would still have three equations. The reactors shown in Fig. 3.1 would operate at atmospheric pressure if they were open to the atmosphere as sketched. .If the reactors are not vented and if no inert blanketing is assumed, they would run at the bubblepoint pressure for the specified temperature and varying composition. Therefore the pressures could be different in each reactor, and they would vary with time, even though temperatures are assumed constant, as the C,‘s change. 3.4 TWO HEATED TANKS As our next fairly simple system let us consider a process in which two energy balances are needed to model the system. The flow rate F of oil passing through two perfectly mixed tanks in series is constant at 90 ft3/min. The density p of the oil is constant at 40 lb,,,/ft3, and its heat capacity C, is 0.6 Btu/lb,“F. The volume of the first tank V, is constant at 450 ft3, and the volume of the second tank V, is constant at 90 ft3. The temperature of the oil entering the first tank is T, and is 150°F at the initial steadystate. The temperatures in the two tanks are T1 and T2. They are both equal to 250°F at the initial steadystate. A heating coil in the first tank uses steam to heat the oil. Let Q1 be the heat addition rate in the first tank. There is one energy balance for each tank, and each will be similar to Eq. (2.26) except there is no reaction involved in this process. Energy balance for tank 1: d(pcfitvLT1’ = pC,(Fo To - FIT,) + Q1 Energy balance for tank 2: OC, v, T,) dt = PC,(FIT, - F, W (3.11) Since the throughput is constant F, = F, = F, = F. Since volumes, densities, and heat capacities are all constant, Eqs. (3.10) and (3.11) can be simplified. PC, v, % = &,FCG - T) + QI pC, V, 2 = PC, F(T, - T,)
EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 6 Let’s check the degrees of freedom of this system. The parameter values that are known are p, C,, Vi, V, , and F. The heat input to the first tank Q1 would be set by the position of the control valve in the steam line. In later chapters we will use this example and have a temperature controller send a signal to the steam valve to position it. Thus we are left with two dependent variables, T1 and T2, and we have two equations. So the system is correctly specified. 3.5 GAS-PHASE, PRESSURIZED CSTR Suppose a mixture of gases is fed into the reactor sketched in Fig. 3.2. The reactor is filled with reacting gases which are perfectly mixed. A reversible reaction occurs : / 2A+ B- The forward reaction is 1.5th-order in A; the reverse reaction is first-order in B. Note that the stoichiometric coefficient for A and the order of the reaction are not the same. The mole fraction of reactant A in the reactor is y. The pressure inside the vessel is P (absolute). Both P and y can vary with time. The volume of the reactor I’ is constant. riction (control valve) into another vessel which is held at a constant pressure P, (absolute). The outflow will vary with the pressure and the composition of the reactor. Flows through control valves are discussed in more detail in Part III; here let us use the formula C, is the valve-sizing coefficient. Density varies with pressure and composition. (3.14) where M = average molecular weight MA = molecular weight of reactant A MB = molecular weight of product B
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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
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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 and 62: EXAMPLES OF MATHEMATICAL MODELS OF
Page 63: EXAMPLES OF MATHEMATICAL MODELS OF
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Page 81 and 82: EXAMPLES OF MATHEMATICAL MODELS OE
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
Page 113 and 114: 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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