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62 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS The equations of state for steam (or the steam tables) can be used to calculate temperature TJ and pressure P, from density pJ . For example, if the perfectgas law and a simple vapor-pressure equation can be used, where M = molecular weight of steam = 18 A, and B, = vapor-pressure constants for water (3.70) Equation (3.70) can be solved (iteratively) for TJ if pJ is known [from Eq. (3.67)]. Once TJ is known, P, can be calculated from the vapor-pressure equation. It is usually necessary to know PJ in order to calculate the flow rate of steam through the inlet valve since the rate depends on the pressure drop over the valve (unless the flow through the valve is “critical”). If the mass of the metal surrounding the jacket is significant, an energy equation is required for it. We will assume it negligible. In most jacketed reactors or steam-heated reboilers the volume occupied by the steam is quite small compared to the volumetric flow rate of the steam vapor. Therefore the dynamic response of the jacket is usually very fast, and simple algebraic mass and energy balances can often be used. Steam flow rate is set equal to condensate flow rate, which is calculated by iteratively solving the heattransfer relationship (Q = UA AT) and the valve flow equation for the pressure in the jacket and the condensate flow rate. B. COOLING PHASE. During the period when cooling water is flowing through the jacket, only one energy equation for the jacket is required if we assume the jacket is perfectly mixed. PJ vJ cJ z = Fw CJ PATJO - TJ) + ho 4Ui.t - q) (3.7 1) where TJ = temperature of cooling water in jacket pJ = density of water C, = heat capacity of water TJ, = inlet cooling-water temperature Checking the degrees of freedom of the system during the heating stage, we have seven variables (C, , CB , T, TM, TJ , p J, and W,) and seven equations [Eqs. (3.61), (3.62), (3.64), (3.65), (3.67), (3.69), and (3.70)]. During the cooling stage we use Eq. (3.71) instead of Eqs. (3.67), (3.69), and (3.70), but we have only TJ instead OfT,,p,,and WC. 3.10 REACTOR WITH MASS TRANSFER As indicated in our earlier discussions about kinetics in Chap. 2, chemical reactors sometimes have mass-transfer limitations as well as chemical reaction-rate limitations. Mass transfer can become limiting when components must be moved
EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 63 FL Liquid product Enlarged view of bubble surface 0 FB ’ 0 0 “. 0 ’ D 1 L, ,kmLiquid\ pB 0 0 0 0 c BO oooc * Liquid feed oo”o I Gas feed FA PA cnc .-..>m*r ‘V’y FIGURE 3.11 Gas-liquid bubble reactor. from one phase into another phase, before or after reaction. As an example of the phenomenon, let us consider the gas-liquid bubble reactor sketched in Fig. 3.11. Reactant A is fed as a gas through a distributor into the bottom of the liquid-filled reactor. A chemical reaction occurs between A and B in the liquid phase to form a liquid product C. Reactant A must dissolve into the liquid before it can react. A+B:C If this rate of mass transfer of the gas A to the liquid is slow, the concentration of A in the liquid will be low since it is used up by the reaction as fast as it arrives. Thus the reactor is mass-transfer limited. If the rate of mass transfer of the gas to the liquid is fast, the reactant A concentration will build up to some value as dictated by the steadystate reaction conditions and the equilibrium solubility of A in the liquid. The reactor is chemical-rate limited. Notice that in the mass-transfer-limited region increasing or reducing the concentration of reactant IS will make little difference in the reaction rate (or the reactor productivity) because the concentration of A in the liquid is so small. Likewise, increasing the reactor temperature will not give an exponential increase in reaction rate. The reaction rate may actually decrease with increasing temperature because of a decrease in the equilibrium solubility of A at the gas-liquid interface.
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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
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CONTENTS xv 10.3 10.4 Performance S
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comwrs xvii 151.3 Transpose of a Ma
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CONTENTS Xix 20.4 Sampled-Data Cont
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. Xxii PREFACE Another significant
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PROCESS MODELING, SIMULATION, AND C
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2 PROCESS MODELING, SIMULATION. AND
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4 PROCESS MODELING, SIMULATION, AND
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6 PROCESS MODELING. SIMULATION, AND
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8 PROCESS MODELING, SIMULATION, AND
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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 75 and 76: EXAMPLES OF MA~EMATICAL MODELS OF C
Page 81: 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
Page 115 and 116: 96 COMPUTER SIMULATION Clearly, the
Page 117 and 118: 98 COMPUTER SIMULATION TABLE 4.2 Ex
Page 119: 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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