Chemical Reaction Analysis and Desing, Apuntes de Ingeniería Química

¡Descarga Chemical Reaction Analysis and Desing y más Apuntes en PDF de Ingeniería Química solo en Docsity! Chemical Reactor Analysis and Design Gilbert F. Froment Rijksuniversiteit Gent, Belgium Kenneth B. Bischoff University of Delaware John Wiley 8 Sons New York Chichester Brisbane Toronto Copyrrght @ 1979 by John Wlley & Sons. Inc All rights reserved. Published simultaneousiy tn Cdna Reproduction or translatron of any pan of thrs work beyond that permrtted by Sections 107 and 108 of the 1976 United States Copyright Act without the permrssion of the copyrrght owner is unlawful. Requests for permtssron or further tnformation should be addressed to the Permrsstons Department. John Wiley & Sons. Library of Congress Cataloging in Publication Data Frornent. Gilbert F. Chemical reactor analysis and design. Includes index. 1. Chemical reactors. 2. Chemical reacttons. 3. Chemical engineering. I . Bischoti. Kenneth B.. joint author. 11. Title. Printed in the United States of America 1 0 9 8 7 6 5 4 3 2 Rijksuniversiteit Gent, at the University of Maryland, Cornell University, and the University of Delaware. Some chapters were taught by G.F.F. at the University of Houston in 1973, at the Centre de Perfectionnement des Industries Chimiques at Nancy, France, from 1973 onwards and at the Dow Chemical Company, Terneuzen, The Netherlands in 1978. K.B.B. used the text in courses taught at Exxon and Union Carbide and also at the Katholieke Universiteit Leuven, Belgium, in 1976. Substantial parts were presented by both of us at a NATO- sponsored Advanced Study Institute on "Analysis of Fluid-Solidcatalytic Systems" held at the Laboratorium voor Petrochemische Techniek, Rijksuniversiteit, Gent, in August 1974. We thank the following persons for helpful discussions, ideas, and critiques: among these are dr. ir. L. Hosten, dr. ir. F. Dumez, dr. ir. J. Lerou, ir. J. De Geyter and ir. J. Beeckman, all from the Laboratorium voor Petrochemische Techniek of Rijksuniversiteit Gent; Prof. Dan Luss of the University of Houston and Professor W. D. Smith of the University of Rochester. Gilbert F. Froment Kenneth B. Biihoff . . . vlll PREFACE Contents Notation Greek Symbols Subscripts Superscripts xvii xxxiii xxxix xxxix Part One-Chemical Engineering Kinetics 1 Elements of Reaction Kinetics 1 .I Reaction Rate 1.2 Conversion and Extent of Reaction 1.3 Order of Reaction E,uample 1.3-1 The Rare ofan Autocaralytic Reacrion, 13 1.4 Complex Reactions Esumple 1.4-1 Comp1e.r Reaction Nertt~orks, 19 E.rattipk I .J-2 Cu~a l~ t i c Cracking of Gusoil, 24 E.uumple 1.4-3 Rate Determinin,g Step und S~eudv-Sture Appro.uimution, 27 E.uample 1.4-4 Classicul Unimoleculur Rure Theory. 30 E.rample 1.4-5 Thermal Cracking of Efhune, 35 Example 1.4-6 Free Radical Addition Polymeri~ation Kinetics, 38 15 Influence of Temperature 42 E-\ample 1.5-1 Determination of the Actiration Enery?: 43 E.uample 1.5-2 Acticurion Energy for Comp1e.u Reuctions. 44 1.6 Determination of Kinetic Parameters 46 1.6-1 Simple Reactions 46 1.6-2 Complex Reactions 47 E.rump1e 1.6.2-1 Rure Constunr Deiermination by file Himmelblau-Jones- Bischoj'method. 50 Example 1.6.2-2 Olejin Codimerization Kinetics, 53 E.rample 1.6.2-3 Thermal Cracking of Propane, 57 1.7 Thermodynamicaily Nonideal Conditions 60 E.uumple 1.7-1 Reaction of Dilure Strong Electro!vres, 63 E-~umple 1.7-2 Pressure Eficts in Gus-Phase Reactions, 64 2 Kinetics of Heterogeneous Catalytic Reactions 2.1 Introduction 2.2 Rate Equations Exumple 2.2-1 Cnmpetitir-e Hydrogenation Reocrions. 94 E.xumple 2.2-2 Kinetics of Erhyiene O.ridur~on on a Supporred Silver Carafvsr, 101 2.3 Model Discrimination and Parameter Estimation 2.3.a Experimental Reactors 2.3.b The Differental Method for Kinetic Analysis 2.3.c The Integral Method of Kinetic Analysis 2.3.d Sequential Methods for Optimal Design of Experiments 2.3.d-1 Optimal Sequential Discrimination Exurnpie 2.3.d.l-i Model Discrimination in rhe Dehydrogeno~ion of f-Burene inro Buradiene, 121 E.rumple 2.3.d.l-I Ethanol Deh.vdrogenarion. Seqrientiul Discnminarion Using rhe Inregra! Method of'Kineric Anall~is, 125 2.3.d-2 Sequential Design Procedure for Optimal Parameter Estimation E.xumple 2.3.d.2-I Sequentiuf Descqn of Experimenrs /or Optimuf Puramerer Esrimution in n-Penfane Isomeriiur!on. Integral 'Method oJ'Kinrrlc Analysis. 129 3 Transport Processes with Fluid-Solid Heterogeneous Reactions Part I Interfacial Gradient Effects 3.1 Surface Reaction Between a Solid and a Fluid 3.2 Mass and Heat Tramfer Resistances 3.2.a Mass Transfer Coefficients 3.2.b Heat Transfer Coefficients 3.2.~ Multicomponent Diffusion in a Fluid E.~umple 3.2.c-1 Use of Mean Efectice Binarv Drffusic~ry, 149 3 3 Concentration or Partial Pressure and Temperature Differences Between Bulk Fluid and Surface of a Catalyst Particle E.rampfe 3.3-1 Interfaciui Gradienrs rn Erhunol Dehydrogenarion Expertments, 15 1 Part 11 Intraparticle Gradient Effects 3.4 Catalyst Internal Structure 3 5 Pore Diusion 3.5.a Definitions and Experimental Observations E.rcunpIe 3.5.0-1 Effect of Pore D~jiusion in the Cracking ofdlkanes on Zeolites, 164 3.5.b General Quantitative Description of Pore Diffusion x CONTENTS 7. l .c The Momentum Equation 7.2 The Fundamental Equations 7.2.a The Continuity Equations 7.2.b Simplified Forms of the "General" Continuity Equation 7.2.c The Energy Equation 7.2.d Simplified Forms of the "General" Energy Equation 8 The Batch Reactor 8.1 The Isothermal Batch Reactor Exmple 8.1-1 Example of Derivurion of a Kinetic Equation by Means oj Butch Data, 364 8.2 The Nonisothermal Batch Reactor Example 8.2-1 Hydrolysis of Acetyluted Cusror Oil Ester, 370 83 Optimal Operation Policies and Control Strategies 8.3.a Optimal Batch Operation Time Example 8.3.0-1 Optimum Conversion und iWu.~irnum Profit for u Firs!-Order Reuction, 376 8.3.b Optimal Temperature Policies E.rumple 8.3.6-1 Optimal Temperarure Trujec!orres for Firsi-Order Rerrrsible Reucrions, 378 E.uumple 8.3.b-2 Oprimum Temperature Policiestor Conseczrtice und Purullel Reuct~ons, 383 9 The Plug Flow Reactor 9.1 The Continuity, Energy, and Momentum Equations E.xump1e 9.1-1 Dericurion of u Kineric Equution from E.t-prrimenrs in un Isoihermul Tubulur Reuctor wiih Plug Flotr,. Thermul Cracking of Propune. 397 9.2 Kinetic Analysis of Nonisothermal Data Esumple 9.2-1 Dericarion o fu Rare Equurionfor rhe Thermul Crucking of Acerone from Nonisorhermul Dora, 402 9 3 Design of Tubular Reactors with Plug Flow E.uumple 9.3-1 An Adiubur~c Reuctor with Plug Flow Conditions, 408 E.rumple 9.3-2 Design of u Nonisothermai Reucror for Tl~ermoi Cracking of Ethane, 410 10 The Perfectly Mixed Flow Reactor 10.1 Introduction 10.2 Mass and Energy Balances 10.2.a Basic Equations 10.2.b Steady-State Reactor Design E.xumple IO.2.b-I Single Irrecersible Reaction in u Srirred Flow Reoctor, 424 CONTENTS xiii 10.3 Design for Optimum Selectivity in Complex Reactions 10.3.a General Considerations 10.3.b Polymerization Reactions 10.4 Stability of Operation and Transient Behavior 10.4.a Stability of Operation E.rample 10.4.0-I Mulripiicity and Sfabiiity in un Adiabatic Stirred Tunk Reactor, 446 10.4.b Transient Behavior Exumple 10.4.b-I Temperalure Osciliariom in u Mixed Reactor for ihe Vapor Phase Chlormarion of Merhyl Chloride, 452 11 Fixed Bed Catalytic Reactors Part I Introduction 1 1.1 The Importance and Scale of Fixed Bed Catalytic Processes 11.2 Factors of Progress: Technological Innovations and Increased Fundamental Insight 11.3 Factors Involved in the Preliminary Design of Fixed Bed Reactors 11.4 Modeling of Fixed Bed Reactors Part 11 Pseudo-Homogeneous Models 11.5 The Basic OneDimensional models 11.5.a Model Equations E.rumple 11 .5 .~- I Culcu/anon of Pressure Drop m Packed Beds, 48 1 1 1.S.b Design of a Fixed Bed Reactor According to the One-Dimensional pseudo-Homogeneous Model 1 1 .5 .~ Runaway Criteria E.rump1e 11.5.~- 1 Application ofthe Firsr Runaway Criterion of Van Wel~rnaere and Fromenr, 490 11.5.d The Multibed Adiabatic Reactor 11.5.e Fixed Bed Reactors with Heat Evchange between the Feed and Effluent or between the Feed and Reacting Gases. "Autothemic Operation" I 1.5.f Non-Steady-State Behavior of Fixed Bed Catalytic Reactors Due to Catalyst Deactivation 11.6 One-Dimensional Model with Axial Mixing 11.7 Two-Dimensional Pseudo-Homogeneous Models 1 1.7.a The Effective Transport Concept 11.7.b Continu~ty and Energy Equations I I .7.c Design of a Fixed Bed Reactor for Catalytic Hydrocarbon Oxidation Part 111 Heterogeneous Models 11.8 One-Dimensional Model Accounting for Interfacial Gradients 1 f .8.a Model Equations xiv CON^ 11.8.b Simulation of the Transient Behavior of a Reactor 549 E.~umple 1 I .8.b-1 .4 Gus-Solid Reaction in u Fixed Bed Reactor, 551 11.9 One-Dimensional %Ide l Accounting for Interfacial and Intraparticle Gradients 11.9.a Model Equations Exumple 11.9.~-1 Stmulur ion of u Fuuser-!Monrecaf~ni Reactor for High-Pressure Methunoi Synthesis. 562 E.~ample 11.9.~-2 Simulurion of an Industrial Reactor for I-Bu~ene Dehydrogenation into Butudiene, 571 11.10 Two-Dimensional Heterogeneous &lodeis 12 Nonideal Flow Patterns and Population Balance Models 592 12.1 Introduction 12.2 Age-Distribution Functions Example 12.2-1 RTD of a Perfect/y ibfixed Vessel. 595 Example 12.2-2 Determination of RTDfrom Experimenrol Tracer Cur~ve. 596 E,~ampie 12.2-3 Calculutron of Age-Disrriburion Funcrionsfrom E.rperimento/ Dufa, 598 ' 12.3 Interpretation of Flow Patterns from Age-Distribution Fulctions 12.3.a Measures of the Spectrum of Fluid Residence Times E.rurnple 12.3-1 Aye-Distriburion Func~iom for a Series ofn-Stirred Tanks, 603 Exumple 12.3-2 RTDfor Combinations oj~Noninteracting Regions, 605 12.3.b Detection of Regions of Fluid Stagnancy from Characteristics of Age Distributions 12.4 Application of Age-Distribution Functions Example 12.4-1 Mean Vulue of'Rute Constant in a Well-Mixed Reactor, 609 E.rumple 12.4-2 Second-Order Reaction in a Stirred Tank. 61 1 Exumple 12.4-3 Reactions in Series Plug Flow and Perfecfly Mired Reucrors. 61 2 12.5 Flow Models 12.5.a Basic Models Example 12.5.~-I Axial Dispersion ~Lfodelfor kiminar Flow in Round Tubes, 620 12.5.b Combined Models Example 12.5.b-I Transient .Mass Tramfer in a Packed Column, 631 Example 12.5.b-2 Recycle Model for Large-Scale S4ixing Egects, 634 12.5.c Flow Model Parameter Estimation 12.6 Population Balance Models Example 12.6-1 Population Balonce Modei for Micromixing, 646 Example 12.6-2 Surfae Reaction-Induced Changes m Pore-Size Distribution, 653 13 Fluidized Bed Reactors 13.1 Introduction 13.2 Fluid Catalytic Cracking CONTENTS xv Engineer~ng units S.I. unlts c,. c, c, C.4br C8b . . . heat exchange surface for a packed bed on the side of the heat transfer medium gas-liquid interfacial area per unit liquid volume interfacial area per unit tray surface frequency factor absorption factor, L'!mF gas-liquid interfac~al area per unlt gas + liquid volume stoichiometric coefficient parameters (Sec. 8.3.b) surface to volume ratio of a particle external particle surface area per unit catalyst mass external particle surface area per unit reactor volume order of reactlon with respect to A order of reaction with respect to '4, gas-liquid interfacial area per unit packed volume liquid-solid interfacial area per unit packed volume reaction component fictitious component vector of fictitious components stoichiometric coefficient order of reaction with respect to B molar concentration of species A. B, j molar concentrations of species A. B . . . in the bulk fluid molar concentrations of adsorbed A, B . . . drag coefficient for spheres mpl:mp3 mp2,'mp3 mPZ'kg cat. mP2;kg cat. m,z!m,' mpZ mp3 kmol/m3 kmolirn,' kmol,kg cat. kmolikg cat. xviii - NOTATION S.I. units molar concentration of reacting component S of solid coke content of catalyst molar concentration of vacant active sites of kg cokelkg cat. kmolikg cat. kg cokeikg cat. kmolikg cat. catalyst total molar concentration kmolllg cat. kmolikg cat. of active sites inlet concentration vector of concentrations molar concentration of d at eqcilibrium molar concentration of .4 in front of the interface molar concentration of fluid ieactanc inside the solid molar concentration of sorbed poison inside catalyst, with respect to core boundary equilibrium molar concentration of sorbed poison inside catalyst reactanr molar concentration at centerline of particle (Chapter 3) Laplace transform of C, molar concentration of fluid reactant in front of the solid surface molar concentration of A inside completely reacted zone of solid specific heat of fluid specific heat of solid Damkahler number for poisoning, k,, RID., molecular diffusivities of A, B in liquid film molecular diffusivity for A in a binary mixture of A and B xix NOTATION Engineeriog vnits S.L units Dx Knudsen diffusivity mira hr mms Der Dear Des effective diffusivities for m/m hr or mms or transport in a (pseudo-) m/m, br m/m, s continuum, or (Chapter 13) in emulsion phase Deo gas phase effective main, hr m0 ms diffasivity in axíal direction in a multiphase packed bed De liquid phase effective m¿Sm, hr mim, s diffusivity in axial direction in a multiphase packed bed Doo effective pore diffusivity m/m cat. hr m/m cats for poison Dias De effective diffusivities in m/m, hr mim, $ axial, respectively radial directions in a packed bed Des effective diflusivity for mm, hr m/m, s transport of A through a grain (Sec. 4:4) De effective diflusivity for transport of A in the pores between the grains (Sec. 4.4) Den measure of divergence between rival models for the nth experiment in the ith grid point Dia eddy diffusivity for species m/m hr mms j ín the 1 disection D, eddy diffustvity in the + m/m br m/m s direction D; effective diffusivity for m/m, hr mim, s transport through completely reacted solid (Chapter 4) Dim effective molecular my? ¿m hr m/m s diffusiviry of jin a multicomponent mixture d wall thickness m m de bubble diameter m m de coil diameter m m de particle diameter m m de reactor diameter m m xx NOTATION Engineering units S.1. units k rate coefficient with respect to unit solid mass for a reaction with order n with respect to fluid reactant A and order m with respect to solid component S coking rate coefficient gas phase mass transfer coefficient referred to unit interfacial area liquid phase mass transfer coefficient referred to unit inierfacial area mass transfer coefficient (including interfacial area) between flowing and stagnant liquid in a multiphase reactor ki-I, k72 mass transfer coefficient (including interfacial area) beween regions I and 2 of flow model (Chapter 12) kc rate coefficient based on concentrations k g gas phase mass transfer coefficient; when based on concentrations; when based on mole fractions ; when based on partial pressures; in a fluidized bed interfacial mass transfer coeficient for catalyst poison mass transfer coefficient between liquid and catalyst surface, referred to unit interfacial area kp reaction rate coefficient based on partial pressures kw rate coefficient for propagation reaction in addition polymerization NOTATION mf3"(kmol A)'-" mf3"(krnol .A)' -" (kmol S)-" (krnol s)-" m:'"- " hr- mP3(m- "s" kg cokeikg cat. hr kg coke!kg cat atm or hr-' s(N;m2) or s- ' mG3 m: hr ~ n ~ ' / . m , ~ s hr- '(kmoli s-'(kmolf m3',1-ta +b' . . . l m3; I - W - W ..j m,3/mp' hr; mfJ, rn; S; kmol/mp2 hr: kmol!mP2 s; kmol/mpz hr atm kmollmpL s (Nim'); mf3;m,%r m /','m," s xxiii Engineering units S.I. units k, k , ~ , k,e krp k , k,. kt, k,. kE k , li., k , . k 2 . . . k; k ; k ; k ; (k6c)b ( k d b reaction rate coefficient (Chapter 3) rate coefficient for catalytic reaction subject to poisoning rate coefficient for first-order poisoning reaction at core boundary surface-based rate coefficient for catalytic reaction (Chapter 5) rate coefficients for termination reactions volume-based rate coefficient for catalytic reaction during poisoning, resp. in absence of poison rate coefficient based on mole fractions slutriation rate coefficient (Chapter 13) reaction rate coefficients rate coefficient of catalytic reaction in absence of coke mass transfer coefficient in case of equirnolar counterdiKusion, k,yJl mass transfer coefficient between stagnant liquid and catalyst surface in a multiphase reactor surface based reaction rate coefficient for gas-solid reaction mass transfer-coefficient from bubble to interchange zone. referred to unit bubble volume overall mass transfer coefficient from bubble to emulsion, referred to unit bubble volume m,'/m2 cat. hr mJ3!m2 cat. s ml J,'m2 cat. hr m,'/mz cat. s m13/m2 cat. hr m13/mZ cat. s m131m2 cat. hr mf3.!m2 cat. s m3/kmol hr or hr- ' mJ/kmol s or s-' mJ3/m3 cat. hr m,31m3 cat. s see k,, k, . k , depending on rate dimensions see k , kmol A m13/mb3 hr mJ3/m,' s xxiv N O T A T I O ~ Engineering units S.I. units (kce)b mass transfer coeficient from interchange zone to emulsion, referred to unit bubble volume ( k t A mass transfer coefficient from bubble + interchange. zone to emulsion, referred to unit bubble + interchange zone volume L volumetric liquid flow rate also distance from center to surface of catalyst pellet (Chapter 3) also distance between pores in a solid particle (Sec. 4.5) and thickness of a slab (Sec. 4.6) total height of fluidized bed height of a fluidized bed at minimum fluidization molar liquid flow rate modified Lewis number. .I,./P,c,, D, vacant active site ratio of initial concentrations CewiC,, molecular weight of species j mean molecular weight monomer (Sec. 1.4-6) Henry's coefficient based on mole fractions. also order of reaction mt total mass m total mass flow rate mi mass flow rate of component j N stirrer revolution speed; also runaway number, 2f f /R ,pc ,k , (Sec. 11.5.~) 'VA molar rate of absorption per unit gas-liquid interfacial area kgi kmol kg/kmoi NOTATION xx Engineering units S.I. units Sh' rate of reaction of A at interface radius of bend of coil radial position of unpoisoned or unreacted core in a sphere reaction rate per unit pellet volume mean pore radius reaction component also dimensionless group, f i (Chapter 11) Schmidt number, p/pD internal surface area per unit mass of catalyst external surface area of a pellet modified Sherwood number for liquid film. kuA,.D,, modified Sherwood number, k, L, D, (Chapter 3 ) modified Sherwood number for poisoning, k,,R:D,, stoichiometric coefficient also parameter in Danckwerts' age distribution function also Laplace transform variable experimental error variance of model i order of reaction with respect to S pooled estimate of variance temperature bed temperature at radius R, critical temperature maximum temperature temperature of surroundings temperature instde solid, resp. at solid surface m2cat.,'kg cat. m'cat., kg cat. m2 m ' xxviii NOTATION clock time also age of surface element (Chapter 6) reference time reduced time time required for complete conversion (Chapter 4) contact time . 4 transfer function of flow model (Chapter 12) overall heat transfer coefficient linear velocity bubble rising velocity, absolute bubble rising velocity, with respect to emulsion phase emulsion gas velocity, interstitial interstitial velocity L(,L interstitial velocity of gas, resp. liquid fluid velocity in direction 1 superficial velocity superficial gas velocity terminal velocity of particle reactor volume or volume of considered "point " volume of a particle equivalent reactor volume. that is, reactor volume reduced to isothermality bubble volume crit~cal volume also volume of bubble + interchange zone volume of interchange zone product molar volume bubble volume corrected for the wake corrected volume of bubble + interchange zone Engineering units S.I. units hr s hr s m, hr m s m,':rn,' hr m ' m,' s mc3:m,' hr mi'&.i s rn, hr m,! s m.' mr3 NOTATION xxix Engineering units S.I. units w,. w,. w,. w:, volume of interchange zone, m3 m3 corrected for wake total catalyst mass kg cat. kg cat. mass of amount of catalyst kg kg with diameter d, increase in value of reacting $ f mixture Weber number, p,L2 d;Q2ur amount of catalyst in bed j kg kg of a multibed adiabatic reactor cost of reactor idle time, reactor charging time. Sihr % is reactor discharging time and of reaction time weighting factor in objective function (Sec. 1.6-2) price per kmole of chemical species A j fractional conversion fractional conversion of A. B. j ... fractional conversion of A at equilibrium conversion of acetone into ketene (Chapter 9) total,conversion of acetone (Chapter 9) mole fraction in liquid S, kmol $, kmol phase on plate n eigenvector of rate coefficient matrix K ( E x . 1.4.1-1) .conversion of 4. B . . . kmol conversion of A, 8 . .. for kmol/m' constant density radius of grain in grain m model of Sohn and Szekely (Chapter 4) kmol kmol/m3 xxx NOTATION Greek Symbols Engineering units S.I. units convective heat transfer coefficient also profit resulting from the conversion of 1 kmole of .4 into desired product (Sec. 11.S.d) also weighung factor in objective function (Sec. 2.3.~-2) vector of flow model parameters (Chapter 12) deactivation constants convective heat transfer coefficient, packed bed side stoichiometric coeficient of component j in a single. with respect to the ith, reaction convective heat transfer coefficient on the side of the reaction mixture convective heat transfer coefficient on the side of the heat rransfer medium convective heat transfer coefficient for a packed bed on the side of the heat transfer medium convective heat transfer coefficient in the vicinity of the wall wall heat transfer coefficient for solid phase wall heat transfer coefficient for fluid kg cat..;kg coke or hr- ' kcal:m2 hr 'C kg cat. kg coke or s - ' kJ,m's K xxxii i Engineering units S.I. units radical involved in a bimolecular propagation step; also weighting factor in objective function (Sec. 2.3.c); stoichiometric coefficient (Chapter 5); cast of 1 kg of catalyst (Chapter 11); dimensionless adiabatic temperature rise, x, - &/To (Sec. 11.5.c) also Prater number = (- AH)DtCi/Lc (Chapter 3) locus of equilibrium conditions in x - T diagram '. locus of the points in x - T diagram where the rate is maximum locus of maximum rate along adiabatic reaction paths in r - T diagram Hatla number, also dimensionless activation energy, EiRT (Section 11.5.c and Chapter 3) also weighting factor in objective function (Section 2.3.c) d molar ratio steam/ hydrocarbon 6.4 expansion per mole of reference component A, (q + s - a - b)/a E void fraction of packing m13/mr3 mj3/m,' &A expansion factor, yA,6, EG gas hold up mG3/mr3 mG3/m,3 EL liquid holdup mL3/m,' mL3/m.' xxxiv NOTATION VG 'lb llquid holdup in flowing fluid zone in packed bed void fraction of cloud, that is, bubble + interchange zone pore volume of macropores void fraction at minimum fluidization internal void fraction or porosity pore volume of micropores dynamic holdup factor used in pressure drop equation for the bends; also correction factor in (Sec. 4.5- 1 ) quantlty of fictitious component effectiveness factor for solid particle effectiveness factor for reaction in an unpoisoned catalyst utiilzation factor, liquid side global utilization factor effectiveness factor for particle + film fractional coverage of catalyst surface; also dimensionless time. D,I/L' (Chapter 3), ak'C,r (Chapter 4); residence time reactor chang~ng time reaction time reactor discharging time reaction time corresponding to final conversion reactor idle time angle described by bend of coil matrix of eigenvalues Engineering units S.I. units mL3, mr3 mL3;m,' S rad Subscripts with respect to A, B . . . coke gas; also global (Chapters 6 and 14) or regenerator (Chapter 13) liquid poison reactor (Chapter 13) at actual temperature a reference temperature adsorption; also in axial direction adiabat~c bulk ; also bubble phase bubble + interchange zone; also critical value: based on concentration desorption emulsion phase; also effective or exit stream from reactor at chemical equilibrium fluid; also film; aiso at final conversion average; also grain or pas interface; aiso ith reaction with respect to jth component liquid: also in 1 direct~on maximum; also measurement point (Chapter I?) tray number pellet, particle; also based on partial pressures reactor dimension; also surroundings also in radial direct~on inside solid; also surface based or superficial velocity surface reaction total: also tube volume based at the wall based on mole fractions initial or inlet condition; also overall value Superscripts T transpose d stagnant fraction of Ruid f flowing fraction of fluid s condition at external surface 0 in absence of poison or coke radical calculated or estimated value xxxix ELEMENTS OF REACTION KINETICS We begin the study of chemical reactor behavior by considering only "local" regions. By this we mean a "point" in the reactor in much the same way as is customary in physical transport phenomena, that is, a representative volume element. After we develop quantitative relations for the local rate of change of the amount of the various species involved in the reaction, they can be "added together" (mathematically integrated) to described an entire reactor. In actual experiments, such local phenomena cannot always be unambiguously observed, but in principle they can be discussed. The real-life complications will then be added later in the book. 1.1 Reaction Rate The rate of a homogeneous reaction is determined by the composition of the reaction mixture, the temperature, and the pressure. The pressure can be deter- mined from an equation of state together with the temperature and composition; thus we focus on the influence of the latter factors. Consider the reaction It can be stated that A and B react at rates and Q and S are formed at rates where N j represents the molar amount of one of the chemical species in the reaction, and is expressed in what follows in kmol, and t represents time. The following equalities exist between the different rates: Each term of these equalities may be considered as the rate of the reaction. This can be generalized to the case of N chemical species participating in M independent' chemical reactions, with the convention that the stoichiometric coefficients, a,,, are taken positive for products and negative for reactants. A comparisori'with Eq. (1.1-1) would give A, = A, r , = -a (for only one reaction the subscript, i, is redundant, and z i j - + z j ) , A 2 = B , r 2 = - b , A 3 = Q , z 3 = q , A 4 = S , a , = s . The rate of reaction is generally expressed on an intensive basis. say reaction volume, so that when V represents the volume occupied by the reaction mixture: For the simpler case: where CA represents the molar concentration of A (kmolim3). When the density remains constant, that is, when the reaction volume does not vary, Eq. (1.1-5) re- duces to In this case, it suffices to measure the change in concentration to obtain the rate of reaction. ' By independent is meant that no one of the stoichiometric equations can be derived from the others by a linear combination. Discussions of this are giver. by Denbigh [I], Prigoglne and Defay 121, and Aris [3]. Actually, some of the definitions and manipulations are true for any set of reactions, but it is convenient to work with the minimum, independent set. 4 CHEMICAL ENGINEERING KINETICS 1.2 Conversion and Extent of Reaction Conversions are often used in the rate expressions rather than concentrations, as follows: x ; = N A o - N A x b = N B o - N B (1.2-1) For constant density, x i = c4, - c, .xi = CEO - C , (1.2-2) Most frequently, fractional conversions are used: which show immediately how far the reaction has progressed. One must be very careful when using the literature because it is not always clearly defined which kind of conversion is meant. The following relations may be derived easily from Eq. (1.2-1) to (1.2-3): y'. = N . y . J 1 0 - 1 (1.2-4) An alternate, but related, concept to the conversion is the extent or degree of advancement of the general reaction Eq. (1.1-3), which is defined as a quantity that is the same for any species. Also where N j , is the initial amount of A , present in the reaction mixture. For multiple reactions, Equations 1.2-3 and 1.2-7 can be combined to give ELEMENTS OF REACTION KINETICS 5 When the rate coefficient, k(hr-I), is known, Eq. 1.3-4 permits the calculation of the rate, r , , for any concentration of the reacting component. Conversely, when the change in concentration is known as a function of time, Eq. (1.3-4) permits the calculation of the rate coefficient. This method for obtaining k is known as the "differential" method: further discussion will be presented later. Integration of Eq. (1.3-4) leads to Thus, a semilog plot of C,/C,, versus t permits one to find k. A more thorough treatment will be given in Sec. 1.6. The integrated forms of several other simple-order kinetic expressions, ob- tained under the assumption of constant density, are listed in Table 1.3-1. Table 1.3-1 Integrated forms of simple kinetic expres- sions (constant density) Zero order kt = CA, - CA k t = C,,.u, First order A - Q Second order Z A - Q + S Caddell and Hurt [8] presented Fig. 1.3-1, which graphically represents the various simple integrated kinetic equations of Table 1.3-1. Note that for a second- order reaction with a large ratio of feed components, the order degenerates to a pseudo first order. 8 CHEMICAL E N G I N E E R I N G KINETICS Figure 1.3-1 Graphical representation of uariolts simple integratt kinetic equations ( f ram Caddell and Hurt [a]). IENTS OF REACTION KINETICS All reactions are, in principle, reversible, although the equilibrium can be sufficiently far toward the products to consider the reaction irreversible for simplicity. The above considerations can be used for the reverse reaction and lead to similar results. For example, if we consider the simple reversible first-order reaction: From the stoichiornetry, C, + C, = C,, + C, and, = (kt + k2K.4 - kz(C.4, + C,,) The solution to this simple differential equation is The equilibrium concentration of A is given by, k2 C.4.q = --- k 1 + k, (CA. + CQ,) In terms of this, the equation can be written, (CA - CAcq) = (CAo - CArq)e-(kl+kl)r or Note that the last equation can be written in terms of conversions t o give the result : 10 CHEMICAL ENGINEERING KINETICS Example 1.3-1 The Rate of an Autocatalytic Reaction An autocatalytic reaction has the form 1 A + Q - Q + Q 2 Here, -- dC* - - klCACQ + k, CQ2 dt Thus, d - ( C , + C,) = 0 dt or C , + CQ = constant = C , , + CQo =-- C , In this case it is most convenient to solve for CQ: and C, would be found from C,(t) = co - CQ(t) Note that initially some Q must be present for any reaction to occur, but A could be formed by the reverse reaction. For the irreversible case, k2 = Si, Here, both A and Q must be present initially for the reaction to proceed. These kinetic results can also be deduced from physical reasoning. A plot of CQ(t) gives an "S-shaped" curve, starting at CQ(0) = CQo and ending at C Q ( m ) =Co = C,, + Coo; this is sometimes called a "growth curve" since it represents a buildup and then finally depletion of the reacting species. Figures 1 and 2 illustrate this. ELEMENTS OF REACTION KINETICS 13 Autocatalvtic reaction A + Q - + Q + Q 4.0 IkvCoti Figure 1 CQjC,, versus dimensionless time. C Q G O Figure 2 Dimensionless rate cersus C$CQ, Autocatalytic reactions can occur in homogeneously catalytic and enzyme systems, although usually with different specific kinetics. For the general reaction (1.1-3), the following treatment is used (see Aris [3] for more details): N 2 a jA j = 0 j = 1 In most cases the forward reaction depends only on the reactants and so the a; corresponding to those j with positive z j are zero. Similarly, the reverse reaction usually depends only on the products. Aris [6] has given the relations for these for the case of simple reactions where the stoichiometric equation also represents the molecular steps: There are cases, however, where this is not true, as in product inhibition or autocatalytic reactions. In the former, increasing product concentration decreases the rate, and so the a; are negative when they correspond to positive a,; thus. for all these situations: aja; < 0 and x jq j 2 0 (1.3-8b) which is useful in deducing certain mathematical features of the kinetics. The only exceptions are autocatalytic reactions where the aja; > 0 for the species inducing the autocatalytic behavior. Also note that the rate can -be etpressed in terms of only the extent (and other variables such as temperature, of course) and the initial composition. This is seen by substituting for the concentrations in Eq. 1.3-7, Thus again we see that the progress of a reaction can be completely described by the single variable of extent/degree of advancement or conversion. ELEMENTS OF REACTION KINETICS 15 For consecutive reactions: 1 7 A - Q - S R, = k,CAa' (1.4-4) RQ = klCAa' - kzCQq' (1.4-5) R, = k2CQq' (1.4-6) Equations 1.4-4 to 1.4-6 can also be easily integrated for first-order reactions: and These results are illustrated in Fig. 1.4-2. If experimental data of C,, CQ are given as functions of time, the values of k , and k , can, in principle, be found by comparing the computed curves, as in Fig. 1.4-2, with the data. However, it is often more effective to use an analog computer to quickly generate many solutions as a function of (k,, k,), and compare the outputs with the data. The maximum in the Q curve can be found by differentiating the equation for CQ and setting this equal to zero in the usual manner with the following result: Again, it is often simpler to find the selectivity directly from the rate equations. Dividing gives 18 CHEMICAL ENGINEERING KINETICS Figure 1.4-2 Consecutivefirsz-order reactions. Concentrations versus time for various ratios k J k , . which has the solution where Example I .4-I Complex Reaction Networks Many special cases are given in Rodigin and Rodigina [12]. The situation of general first-order reaction networks has been considered by Wei and Prater [ I 31 in a particularly elegant and now classical treatment. Boudart [5] also has a more abbreviated discussion. ELEMENTS OF REACTION KINETICS 19 The set of rate equations for first-order reversible reactions between the N components of a mixture can be written where the y, are, say, mole fractions, kji is the rate coefficient of the reaction Ai -r A, and In matrix form: It is simplest to consider a three-component system, where the changes in com- position with time-the reaction paths-can be followed on a triangular diagram. Figure 1 shows these for butene isomerization data from the work of Haag, Pines, and Lago (see Wei and Prater) [I 33. We observe that the reaction paths all converge to the equilibrium value in a tangent fashion, and also that certain ones (in fact, two) are straight lines. This has important implications for the behavior of such reaction networks. It is known from matrix algebra that a square matrix possesses Neigenvalues, the negatives of which are found from where I is a unit matrix and 5, 2 0 for the rate coefficient matrix. Also, N-eigen- vectors, x,, can then be found from Kx, = -I,x, (e) 20 CHEMICAL ENGINEERING KINETICS represents all the contributions that make up the reaction paths. Special initial conditions of, say, (,(0) # 0, <,, *(0) = 0 leave only one term on the right-hand side, and this one direction thus is that of the special straight-line reaction paths. Thus, knowing the rate constants, k j i , a series of matrix computations will permit one to determine the proper (real) starting compositions for straight-line reaction paths, which are (,(t). The above figure shows an experimental determination of these paths. Wei and Prater also show that only IV - 1 such paths need to be found, and the last can be computed from matrix manipulations. In addition to illustrating many features of monomolecular reaction networks, Wei and Prater illustrated how these results, especially the straight line reaction paths, could be helpful in planning experiments for and the determination of rate constants, and this will be discussed later. Also, these same methods have been used in the "stochastic" theory of reaction rates, which consider the question of how simple macroscopic kinetic relations (e.g., the mass action law) can result from the millions of underlying molecular collisions-see Widom for comprehen- sive reviews [14]. w not her common form of mixed consecutive-parallel reactions is the following: Q + B - S Successive chlorinations of benzene, for example, fall into this category. The main feature is the common second reactant B, so that in a sense the reactions are also parallel. The rate expressions are There is no simple solution of these differential equations as a function of time. However, the selectivities can again be found by dividing the equations: This is precisely the same as for the simpler first-order case considered above, and so would result in the same final results. Thus, the common reactant, B, has no effect on the selectivity, but will cause a different behavior with time. An important consequence of this is that a "selectivity diagram" or a plot of Cp, Cs, . . . versus C,, or conversion, x,, is often rather insensitive to details of the reaction network ELEMENTS OF REACTION KINETICS 23 other than the concentrations of the main chemical species. This concept is often used in complicated industrial process kinetics of catalytic cracking, for example, to develop good correlations of product distributions as a function of conversion. Example 1.4-2 Catalytic Cracking of Gasoif An overall kinetic model for the cracking of gasoils to gasoline products was developed by Nace, Voltz, and Weekman 1151. The actual situation was a catalytic reaction and the data were from specific reactor types, but mass-action type rate expressions were used and illustrate the methods of this section. The overall reaction is as follows: 1 A - Q \A S where A represents gasoil, Q gasoline, and S other products ( C , - C,, coke). For the conditions considered, the gasoil cracking reaction can be taken to be approximately second order and the gasoline cracking reaction to be first order (see Weekman for justification of this common approximation for the com- plicated cracking reaction) 116, 171. Then, the kinetic equations are (where y represents weight fractions): This parallel-consecutive kinetic scheme can be integrated, but an expression for the important gasoline selectivity can also be found directly by formally dividing Eqs. a and b: Integrating gives 24 CHEMICAL ENG~NEERING KINETICS Space velocity, wt/(wtilhrl Figure I Comparison of experimental conversions with model predictions for different charge stocks. Catalyst residence rime: 1.25 min. (Nace, Voltz, and Weekman [I 51). where Ei(x) = exponential integral (tabulated function) Figure 1 shows the conversion versus (reciprocal) time behavior for four different feedstocks, and a catalyst residence time of 1.25 min in the fluidized bed reactor ELEMENTS OF REACTION KINETICS 25 The rate equations are: This is the same case solved earlier, and is illustrated in Fig. 1.4-2. There are some interesting and useful features of this simple system that will iilustrate the important concept of the rate determining step. Note from Fig. 1.4-2 that when k, % k , , the two reactions are almost separate in time, and the overall rate of product formation is dominated by the slow reaction 2. Alge- braically, from the integrated rate equations given above, after a certain time in- terval : % = k , ~ , k - (- %) for 2 + 1 k , For the opposite case of k , %= k, , the integrated rate equations in a different rearrangement give: again after a certain time interval. Thus, the overall rate of product formation is dominated by the slow reaction 1. This shows that the overall rate is always dominated by any slow steps in the reaction sequence;' this concept of a "rate limiting step" will be used many times in the ensuing discussions. One of the most useful applications pertains to the notion of a stationary or steady state of the intermediate. Ifa stationary state between the main reactant and ' This material was adapted from Kondrat'ev 1251. 28 CHEMICAL ENGINEERING KINETICS product is to exist for this simple case, the rate of disappearance of A must be approximately equal to the rate of production of P. This would make a plot of Cp(t) the mirror image of CA(t). From Fig. 1.4-2, or from Eq. e, it is seen that this is almost true for large k , / k , > 10 + a. Physically, a large value of k , , relative to k , , means that as soon as any I is formed from reaction of A, it is immediately trans- formed into P, and so the product formation closely follows the reactant loss. Thus, the intermediate is very short lived, and has a very low concentration; this can also be seen in Figure 1.4-2. The sum of Eq. a, b, and c gives If the stationary state exists, and the reactant loss and product formation are approximately equivalent, and so which is the usual statement. Then, from Eq. b which is indeed small for finite C, and ( k , / k , ) $= 1. Also, and the exact details of the intermediate need not be known. Rigorous justification of the steady-state approximation has naturally been of interest for many years, and Bowen, Acrivos, and Oppenheim [26] have resolved the conditions under which it can be properly used. The mathematical question concerns the correctness of ignoring the derivatives in some of a set of differential equations (i.e., changing some to algebraic equations), which is analogous to ignoring the highest derivatives in a single differential equation. These questions are answered by the rather complicated theory of singular perturbations, discus- sion of which is given in the cited article. Predictions from the steady-state approximation have been found to agree with experimental results, where it is appropriate. This should be checked by using relations such as Eq. h to be sure that the intermediate species concentrations are, in fact, much smaller than those of the main reactants and products in the reac- tion. When valid, it permits kinetic analysis of systems that are too complicated ELEMENTS OF REACTION KINETICS 29 to conveniently handle directly, and also permits very useful overall kinetic rela- tionships to be obtained, as is seen in Ex. 1.4-4 to 1.4-6. Example 1.4-4 Classical Unimolecular Rate Theory Another interesting example of complex reactions is in describing the chemical mechanism that may be the basis of a given overall observed kinetics. A question of importance in unimolecular decompositions (e.g.. cyclohexane, nitrous oxide, 320 methane-see Benson [27])-is how a single molecule becomes sufficiently energetic by itself to cause it to react. The theory of Lindemann [28] explains this by postulating that actually bimolecular collisions generate extraenergetic molecules, which then decompose: A* A Q + . . . ( slow) Then, the rate of product formation observed is To find .4,* its kinetics are given by: To solve this differential equation in conjunction with a similar one for species A would be very difficult, and recourse is usually made to the "steady-state ap- proximation." This assumes that dCA./dt - 0, or that the right-hand side of Eq. d is in a pseudo-equilibrium or stationary state. Justification for this was provided in the last example. With this approximation, Eq. d is easily solved: Then, Now, at high concentrations (pressure), k,CA % k , (recall reaction 3 is pre- sumably slow), and so, 30 CHEMICAL ENGINEERING KINETICS In reality, the reaction might proceed by the following steps: kr A, - 2R; Initiation (1.4-12) k Hydrogen R; + A , --L R I H + R; abstraction (1.4-13) Propagation k Radical R; -L A, + R; decomposition (1.4-14) k R; + R ; --% A , Termination (1.4-15) R; and R; are radicals (e.g., when hydrocarbons are cracked CH;, C2H;, H'). The rate of consumption of A , may be written: The rate of initiation is generallymuch smaller than the rate of propagation so that in Eq. (1.4-16) the term k,CA, may be neglected.Theproblem is now to express C,,, which are difficult to measure, as a function of the concentrations of species which are readily measurable. For this purpose, use is made of the hypothesis of the steady-state approximation in which rates of change of the concentrations of the intermediates are assumed to be approximately zero, so that or, in detail, These conditions must be fulfilled simultaneously. By elimination of CR, one obtains a quadratic equation for C,, : the solution of which is, ELEMENTS OF REACTION KINETICS 33 Since k , is very small. this reduces to so that Eq. (1.4- 16) becomes: which means that the reaction is essentially first order. There are other possibilities for termination. Suppose that not (1.4-15) but the following is the fastest termination step: It can be shown by a procedure completely analogous to the one given above that the rate is given by which means that the reaction is of order 3/2. Goldfinger, Letort, and Niclause [31] (see Laidler [IO]) have organized resultsof this type based on defining two types of radicals: p-a radical involved as a reactant in a unimolecular propagation step. 8-a radical involved as a reactant in a bimolecular propagation step. Usually the p radical is larger than the B radical, so that (termination rate constant magnitude)@p) < (pp) < (pp) (1.4-20) This leads to the results shown in Table 1.4-1. Table 1.4-1 Overall Orders for Free Radical Mechanisms First-Order Initiation S e d O r d e r Initiation Shnple Simple Overall Termination Third Body Termination Third Body Order - - - - - -- - - BB 2 BB PP BBM 4 BP BBM F ~ P BPM I PI' &M PPM + W M 0 34 CHEMICAL ENGINEERING KINETICS Note that in the above example a first-order initiation step was assumed, and with a termination step involving both R;(B) and R;(p) , an overall first-order reaction was derived, in agreement with Table 1.4-1. The alternate R ; + R ; termination was of the (BB) type, leading to a three-half-order reaction. Franklin [32] and Benson 129) have summarized methods for predicting the rates of chemical reactions involving free radicals and Gavalas [33] has shown how the steady-state approximation and use of the chain propagation reactions alone (long-chain approximation) leads to reasonably simple calculation of the relative concentrations of the nonintermediate species. Also see Benson [34]. Example 1.4-5 Thermal Cracking of Ethane The overall reaction is C2H6 = C2H4 + HZ and can be considered to proceed by the following mechanism: Initiation: Eq. 1.4-12: C2H6 A 2CH; (A 1 ) (R i ) Hydrogen abstraction: Eq. 1.4-13: CH; + C2H, A CH, + C2H; (R;) (A,) (RlH) (Ri) and: k4 H ' + C2H6 - Hz + C2H; (R;) (A,) (R,H) (R;) Radical decomposition: Eq. 1.4-14: C,H; A C2H, + H' (Ri) ('42) ( R ; ) Termination: ELEMENTS OF REACTION KINETICS 35 Thus, the ethyl radical is both and p, although the slowness of its decomposi- tion reaction tends to make the former more important. Thus, with first-order initiation and approximate (Bp) behavior, the overall order is again approximately unity. Further details are given in Steacie [37] and Laidler [lo] and Benson [27] among others. This rather involved example illustrated the large amount of information that can be obtained from the general free radical reaction concepts. Example 1.44 Free Radical Addition Polymerization Kinetics Many olefinic addition polymerization reactions, such as that of ethylene or styrene polymerization, occur by free radical mechanisms. The initiation step can be activated thermally o r by bond breaking additives such as peroxides. The general reaction scheme is: k aM, +bI - P , Initiation (a) PI + M , Propagation (b) k Pa-, + M, A P, k, P, + P,,, - M , + , Termination (c) where M, is the monomer, 1 is any initiator, P, is active polymer, and iU ,+ , is inactive. Note that all the propagation steps are assumed to have the same rate constant, k,, , which seems to be reasonable in practice. Also, a or b can be zero, depending on the mode of initiation. The rates of the reactions are dM1 -= - ari - k,, IM , 1 P, dt where ri is the initiation rate of formation of radicals. Aris 131 has shown how these equations may be analytically integrated to give the various species as a function of time for an initiation step first order in the monomer, M,, and a simple termina- tion step of an extension of Eq. (b), P, + M , -. M E + , .The more general case is most easily handled by use of the steady-state approximation, whereby dP Jdt = 0, 38 CHEMICAL ENGINEERING KINETICS as discussed above. Then each of equations e to fare equal to zero and, when added together, give 0 = ri - k,(C P,)' (!3) which states that under thesteady-state assumption, the initiation and termination rates are equal. Thus, Eq. d is changed to for initiation independent of monomer, a = 0 in Eq. a, or for small magnitude of monomer used in the initiation step relative to the propagation or polymerization steps (usually the case). There are several possibilities for initiation, as mentioned above: second order in monomer (thermal), first order in each monomer and initiator catalyst, I, or first order in I. For the latter, the initiation rate of formation of radicals is given by, ri = k , l (j) so that The rate of monomer disappearance is, then, This expression for the overall polymerization rate is found to be generally true for such practical examples of free radical addition polymerization as poly- ethylene, and others. Even further useful relations can be found by use of the above methods. Con- sider the case of reactions in the presence of "chain transfer" substances as treated by Alfrey in Rutgers [38] and Boudart [ S ] . This means a chemical species, S, that reacts with any active chain, P,, to form an inactive chain but an active species, S ' : This active species can then start a new chain by the reaction ELEMENTS OF REACTION KINETICS 39 Thus, S acts as a termination agent as far as the chain length of P,, but does propagate a free radical S' to continue the reaction. In other words, the average chain length is modified but not the overall rate of reaction. These effects are most easily described by the number average degree of poly- merization, P,, which is the average number of monomer units in the polymer chains. This can be found as follows. For no chain transfer: rate of monomer molecules polymerized (PN), = rate of new chains started - kpr M I a, 1 I l l With a chain transfer agent present, this is changed to and shows the decrease in average chain length with increasing S. Further details about the molecular weight distribution of the polymer chains can be obtained by simple probability arguments. If the probability of adding another monomer unit to a chain is p, the probability of a chain length P (number distribution) with random addition is N ( P ) = ( 1 - p ) p P - l (starting with the monomer) (q) which is termed the "most probable" or "Schultz-Flory" distribution. Note that C,"=, N ( P ) = 1 , a normalized distribution. The number average chain length is, then. 40 CHEMICAL E N G I N E E R I N G KINETICS for in detail. deviations from the straight line may be experienced in the Arrhenius plot for the overall rate. If there is an influence of transport phenomena on the measured rate, deviations from the Arrhenius law may also be observed; this will be illustrated in Chapter 3. From the practical standpoint, the Arrhenius equation is of great importance for interpolating and extrapolating the rate coefficient to temperatures that have not been investigated. With extrapolation, take care that the mechanism is the same as in the range investigated. Examples of this are given later. Example 1.5-1 Determination of the Activation Energy For a first-order reaction, the following rate coefficients were found: Temperature ("C) k(hr- ') 48.5 0.044 70.4 0.534 90.0 3.708 These values are plotted in Fig. 1, and it follows that: L. 105 r Figure 1 Determination of activation energy. ELEMENTS OF REACTION KINETICS 43 Example 1.5-2 Activation Energy for Complex Reactions The overall rate equation based on a complex mechanism often has an overall rate constant made up of the several individual constants for the set of reactions. The observed activation energy is then made up of those of the individual reactions and may be able to be predicted, or used as a consistency check of the mechanism. For example, the Rice-Henfeld mechanism for hydrocarbon pyrolysis has overall rate expressions such as Eq. 1.4-18: Thus, and Equating the temperature coefficients: d In ko (R.H.S.) dfllT)=d(llT) gives the relationship: Eo = +(El + E2 + E 3 - E 4 ) An order of magnitude estimate of the overall activation energy is given by using typical values for the initiation. hydrogen abstraction, radical decomposition, and termination steps: This is the size of overall activation energy that is observed. Note that it is much lower than the very high value for the difficult initiation step, and is thus less than the nominal values for breaking carbon-carbon bonds. For the specific Example 1.4-5 of ethane pyrolysis, Eq. i of that example shows that the overall rate constant is: 44 CHEMICAL ENGINEERING KINETICS and Eo = &El + E3 + E4 - E 5 ) Values from Benson [27), p. 3.54, give Benson states that observed overall values range from 69.8 to 77 kcal/mol(291.8 to 321.9 kJ/mol) and so Eq. e provides a reasonable estimate. Laidler and Wojciechowski 140) present another table of values, which lead to Eo = 65.6 kcaI/mol (274.2 kJ/mol). Both estimates are somewhat low, as mentioned in EX. 1.4-5. For the second-order initiation mechanism, the rate constant is and Using Laidler and Wojciechowski's values This seems to be a more reasonable value. The exponential temperature dependency of the rate coefficient can cause enormous variations in its magnitude over reasonable temperature ranges. Table 1.5-1 gives the magnitude of the rate coefficient for small values of RT!E. It follows then that the "rule" that a chemical reaction rate doubles for a 10 K Table 1.51 Variation offate coefficient with temperature RTIE EIRT k/Ao ELEMENTS OF REACTION KINETICS 45 Unfortunately, the method is not an automatic panacea to all problems of complext first-order kinetics. The only directly measured quantities are the y j . The i, are found by a matrix transformation using the k j i . However, we don't yet know these since, in fact, this is what we are trying to find. Thus, a trial-and- error procedure is required, which makes the utilization of the method somewhat more complicated. Wei and Prater suggest an experimental trial-and-error scheme that is easily illustrated by a simple example and some sketches. The three-species problem to be considered is (e.g., butene isomerization): The compositions can be plotted on a triangular graph as shown in Fig. 1.6.2-1. The arrows indicate the course of the composition change in time and the point "em is the equilibrium position. Thus, an experiment a starts with pure A , and proceeds to equilibrium along the indicated curve. Now the above scheme for three components will give three A,,,, one of which is zero. It can be shown that the other two &--each corresponding to a (,--will give a straight line reaction path on the above diagram, lines @) @ and @ @. The first experiment didn't give a straight line and so one of the (, is not pure A , . Thus, a second experiment is done, @I which again probably won't give a straight line. Finally, a t experiment a, a straight line is found and possibly confirmed by experiment @with the indicated initial composition (mixture of -5 parts A, and 1 part A,) . The com- positions for experiment a or @are plotted as In[(, - i,,] versus time and the slope will be 1,. The other straight-line path @ @ can be found from matrix calculations, and then confirmed experimentally. For larger numbers of reacting 48 C H E M I C A L E N G I N E E R ~ N G KINETICS species, more (N - 1) of the straight line paths must be found experimentally by the iterative technique. Then the kji are found. Obviously, this is a rather laborious procedure and is most realistically done with bench scale studies. However, as Wei and Prater strongly pointed out, extensive data must be taken if one really wants to find out about the kinetics of the process. Finally, the entire procedure is only good for first-order reactions, which is another restriction. However, many industrial reactions are assumed first order in any event, and so the method can have many applications. For example, see Chapter 10 in Boudart [5] . Gavalas 1461 provides another technique for first-order systems that again estimates values for the eigenvalues of the rate coefficient matrix. Another method that can be used is to take the Cj measured as a function of time, and from them compute the various slopes, dCJdt. The general form of kinetic expressions can then be written, for M reactions, as: where C is the N-vector of concentrations. Then, since all the k's appear in a linear fashion, at any one temperature, standard linear regression techniques can be used, even with the arbitrary rate forms rj,, to determine the rate constants. Un- fortunately, however, this differential method can only be used with very precise data in order to successfully compute accurate values for the slopes, dCj/dt. An alternate procedure was devised by Himmelblau, Jones, and Bischoff [47]. This was to take the basic equations (1.6.2-2) for the C j and directly integrate (not formally solve) them: which leads to M 11 cJ{ti) - cJ< t~ ) = 2 kp rjp(c(t)Mt (1.6.24) - p = 1 to - Directly Integrals of measured measured data Notice again that the k's occur linearly no matter what the functions r j p are, and so standard linear regression methods, including various weighting, and so on, ELEMENTS OF REACTION KINETICS 49 can be used. Also, only integration of experimental data is necessary (not dif- ferentiation), which is a smoothing operation. Thus, it seems that the advantages of linear regression are retained without the problems arising with data differentia- tion. Equation 1.6.2-4 can now be abbreviated as A# The standard least squares method would minimize the following relation: where n = number of time data points xj = C,(ti) - Cdt,), experimental value of dependent variable M x j = 1 k p X i j , , calculated value of dependent variable p = 1 wij = any desired weighting function for the deviations Standard routines can perform the computations for Eq. 1.6.2-6 and will not be further discussed here. The result would be least squares fit values for the kinetic parameters, k,. This latter technique of Himmelblau, Jones, and Bischoff (H-J-B) has proved to be efficient in various practical situations with few, scattered, data available for complex reaction kinetic schemes (see Ex. 1.6.2-1). Recent extensions of the basic ideas are given by Eakman, Tang, and Gay [48,49, SO]. It should be pointed out, however, that the problem has been cast into one of linear regression at the expense of statistical rigor. The 'independent variables", X i j p , do not fulfill one of the basic requirements of linear regression: that the X i , have to be free of experimental error. In fact, the Xi jp ate functions of the-dependent variables CAti) and this may lead to estimates for the parameters that are erroneous. This p;oblem will be discussed further in Chapter 2, when the estimation of parameters in rate equations for catalytic reactions will be treated. Finally, all of the methods have been phrased in terms of batch reactor data, but it should be recognized that the same formulas apply to plug Bow and constant volume systems, as will be shown later in this book. Example 1 A.2-1 Rate Constant Determination by the Himmelblau- Jones-Bischofl Method To illustrate the operation of the H-J-B method described above, as well as gain some idea of its effectiveness, several reaction schemes were selected, rate constants 50 CHEMICAL ENGINEERING KINETICS error was introduced, except for the value of k , in run 5. For the more complex model in Table 2, even without introducing random error, the values of k , and k3 deviated as much as 10 percent from the original values. After analyzing all of the computer results, including trials not shown, it was concluded that most of the error inherent in the method originates because of the sensitivity of the rate coefficients to the values obtained in the numerical integration step. If the concentration-time curves changed rapidly during the initial time increments, and if large concentration changes occurred,. significant errors resulted in the calculated rate parameters. It has been found that data- smoothing techniques before the numerical integration step help to remedy this problem. Another source of error is that errors in the beginning integrals tend to throw off all the predicted values of the dependent variables because the predicted values are obtained by summing the integrals up to the time of interest. Thus, it would seem that the use of unequal time intervals with more data at short times is im- portant in obtaining good precision. Example 1.6.2-2 Kinetics of O w n Codimerization Paynter and Schuette [51] have utilized the above technique for the complex industrial process of the codimerization of propylene and butenes to hexene, heptene, octene, and some higher carbon number products of lesser interest. Not only are there a variety of products, but also many possible feed compositions. This is actually a catalytic process, but the mass-action kinetics used can serve to illustrate the principles of this section, as well as previous parts of this chapter. The most straightforward reaction scheme to represent the main features of this system are: 5 2C4-, - CB where the concentrations are: C3-propylene; C, - ,-butene-1 : C, - ,-butene-2 (both cis and trans); C,-hexene; C7-heptenes; C,-octenes. ELEMENTS OF REACTION KlNETlCS 53 The C,' compounds are not of primary interest, and so an approximate overall reaction was used to account for their formation: 6 ( C 3 + C 4 - , f C4-2)+(C6 f C 7 + C s ) -----) Cgf To obtain the proper initial selectivity, a further overall reaction was introduced: 8 3C3 - cg+ Finally, the butene isomerization reaction was also accounted for: C 4 - , & C 4 - , , with equilibrium constant K 1: 12 7' The straightforward mass action rate equations then are -- dC3 - -2klC3' - k2C,C4-, - k,C,C4- , - k,C,(C6 + C7 + C,) - 3k8C3' dt (a) dC4 - , -- dt - -k2C3C4- , - 2k4C4-1' - k6C4- , (C6 + C7 + C8) Certain aspects of these rate equations are obviously empirical, and illustrate the compromises often necessary in the analysis of complex practical industrial reacting systems. 54 CHEMICAL ENGINEERING KINETICS Paynter and Schuette found that with a "practical" amount of data, the direct determination of the eight rate constants by the H-J-B method (or presumably by others) could adequately fit the data, but the constants were not consistent in all ways. Thus, several other types of data were also utilized to independently relate certain of the rate constants, and these concepts are considered here. The initial selectivities of C,/C, and C,/C, are found by taking the ratios of Eqs. d, e, or f under initial conditions: For pure butene-1 feed this reduces to which again reduces, for pure butene-1 feed, to Thus, with pure butene-1 feed, a plot of C , versus C, has an initial slope of (kl/k2)(C3/C4 - Eq. (i), and knowing the feed composition yields (k l /k2); see Fig. 1 . Similarly, for a given ratio of C4-2 and C4- ,, plus (C,/C4), Eq. (h) yields Figure I Hexenes versus heptenes, T = 240°F. (Paynter and Schuette [Sl]). ELEMENTS OF REACTION KINETICS 55 Table I Molecular scheme for the thermal cracking ofpropane Reaction Rate Rate equation is assumed for these. The equilibrium constants Kc , , Kc, and Kc, are obtained from thermodynamic data (F. Rossini et al.) [54]. It follows that the total rate of disappearance of propane RclH, is given by while the net rate of formation of propylene is given by The experimental study of Froment et al. (loc. cit) was carried out in a tubular reactor with plug flow. The data were obtained as follows: total conversion of propane versus a measure of the residence time, VR/(FC,,,),; conversion of propane into propylene versus VR/(FclH,)o and so on. V, is the reactor volume reduced to isothermal and isobaric conditions, as explained in Chapter 9 on tubular reactors and (F,3,,)o is the propane feed rate. It will be shown in Chapter 9 that a mass balance on propane over an isothermal differential volume element of a tubular reactor with plug flow may be written 58 CHEMICAL ENGINEERING KINETICS In Eq. (a) a more general notation is used. aij is the stoichiometric coefficient of the ith component in the ith reaction. After integration over the total volume of an isothermal reactor, Eq. a yields the various flow rates F j at the exit of the reactor, for which V,j(F,,,,),, has a certain value, depending on the propane feed rate of the experiment. If Eq. a is integrated with the correct set of values of the rate coefficients k , . . . k , the ex- perimental values of Fj should be matched. Conversely, from a comparison of experimental and calculated pj the best set of values of the rate coefficients may be obtained. The fit of the experimental F . b means of the calculated ones, F j , I . can be expressed quantitatively by comput~ng the sum of squares of deviations between experimental and calculated exit flow rates, for example. These may eventually be weighted to account for differences in degrees in accuracies between the various F j so that the quantity to be minimized may be written, for n experi- ments: Sundaram and Froment [Ioc. cit] systematized this estimation by applying non- linear regression. The results a t 80OoC are given in Table 2. The estimation was repeated at other temperatures so that activation energies and frequency factors could be determined. Figure 1 compares experimental and calculated yields for various components as a function of propane conversion at 800°C. Table 2 Values for the rate coefficients of the molecularscheme forpropane crackingaf80O0C Rate coefficient Value (s-' or : m3 kmol-' s-') ELEMENTS OF REACTION KINETICS 59 1.7 Thermodynamically Nonideal Conditions 34 I I 1 I 30 - - - camput& Experimental 0 - C1H4 =C3Y 0 -CH4 - X = H a X - - - 10 I I I I 50 60 70 80 90 1W It was mentioned in Sec. 1.3 that the rate "constant" defined there is actually only constant for thermodynamically ideal systems, and that in general it may vary with composition. Also, the classical form of the mass action law gives for the reac- tion 2.2 2.0 1.8 t I" - 1.8 * 'i - 0 ; 1.4 1.2 1.0 60 CHEMICAL ENGINEERING KINETICS Propane conversion. % --t Figure I Comparison o j experimental and calculated yields jor various components as afirnction of propane conversion at 800°C.