Fundamentals of ChemicalReactionEngineeringFundal11entalsofChel11icalReactionEngineeringMarkE. DavisCalifornia Institute of TechnologyRobertJ. DavisUniversity of VirginiaBoston Burr Ridge, IL Dubuque, IA Madison,WI New York SanFrancisco St. LouisBangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico CityMilan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei TorontoMcGraw-Hill Higher Education 'ZZADivision of TheMGraw-Hill CompaniesFUNDAMENTALS OF CHEMICAL REACTION ENGINEERINGPublished by McGraw-Hili, a business unit of The McGraw-Hili Companies, Inc., 1221 Avenue of theAmericas, New York, NY10020. Copyright 2003 by The McGraw-Hili Companies, Inc. All rights reserved.No part of this publication may be reproduced or distributed in any form or by any means, or stored in adatabase or retrieval system, without the prior written consent of The McGraw-Hili Companies, Inc., including,but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.Some ancillaries, including electronic and print components, may not be available to customers outside theUnited States.This book is printed on acid-free paper.InternationalDomestic1234567890DOCroOC0987654321234567890DOCroOC098765432ISBN 0-07-245007-XISBN 0-07-119260-3(ISE)Publisher: Elizabeth A. JonesSponsoring editor: Suzanne JeansDevelopmental editor: Maja LorkovicMarketing manager: Sarah MartinProject manager: Jane MohrProduction supervisor: Sherry L. KaneSenior media project manager: Tammy JuranCoordinator of freelancedesign: Rick D. NoelCover designer: Maureen McCutcheonCompositor:TECHBOOKSTypeface: 10/12 Times RomanPrinter: R. R. Donnelley/Crawfordsville, INCover image: Adaptedfromartwork provided courtesy of Professor Ahmed Zewail's group at Caltech.In 1999, Professor Zewail received the Nobel Prize in Chemistry for studies on the transition states ofchemical reactions using femtosecond spectroscopy.Library of Congress Cataloging-in-Publication DataDavis, Mark E.Fundamentals of chemical reaction engineering / Mark E. Davis, Robert J. Davis. - 1st ed.p. em. - (McGraw-Hili chemical engineering series)Includes index.ISBN 0-07-245007-X (acid-free paper) - ISBN 0-07-119260-3 (acid-free paper: ISE)I. Chemical processes. I. Davis, Robert J. II. Title. III. Series.TP155.7.D38 2003660'.28-dc21 2002025525CIPINTERNATIONAL EDITIONISBN 0-07-119260-3Copyright 2003. Exclusive rights by The McGraw-Hill Companies, Inc., for manufacture and export.This book cannot be re-exported fromthe country to which it is sold by McGraw-HilI. The InternationalEdition is not available in North America.www.mhhe.comMcGraw.HiliChemical EngineeringSeriesEditorial Advisory BoardEduardoD.Glandt, Dean, School of Engineering and Applied Science, University of PennsylvaniaMichaelT. Klein, Dean, School of Engineering, RutgersUniversityThomasF. Edgar, Professor of Chemical Engineering, University of Texas at AustinBailey and OllisBiochemical Engineering FundamentalsBennett and MyersMomentum, Heat and Mass TransferCoughanowrProcess Systems Analysis and Controlde NeversAir Pollution Control Engineeringde NeversFluid Mechanics for Chemical EngineersDouglasConceptual Design of Chemical ProcessesEdgar and HimmelblauOptimization of Chemical ProcessesGates, Katzer, and SchuitChemistry of Catalytic ProcessesKingSeparation ProcessesLuybenProcess Modeling, Simulation, and Control forChemical EngineersMarlinProcess Control:Designing Processes and ControlSystems for Dynamic PerformanceMcCabe, Smith, and HarriottUnit Operations of Chemical EngineeringMiddleman and HochbergProcess Engineering Analysis in SemiconductorDevice FabricationPerry and GreenPerry's Chemical Engineers' HandbookPeters and TimmerhausPlant Design and Economics for Chemical EngineersReid, Prausnitz, and PolingProperties of Gases and LiquidsSmith, Van Ness, and AbbottIntroduction to Chemical Engineering ThermodynamicsTreybalMass Transfer OperationsToMary, Kathleen,and our parents Ruth andTed._________________C.Ott:rEHl:SPreface xiNomenclature xiiChapter1TheBasicsofReactionKineticsforChemical ReactionEngineering 11.1 The Scope of Chemical ReactionEngineering I1.2 The Extent of Reaction 81.3 The Rate of Reaction 161.4 General Properties of the Rate Function foraSingle Reaction 191.5 Examples of Reaction Rates 24Chapter 2RateConstantsofElementaryReactions 532.1 Elementary Reactions 532.2 Arrhenius Temperature Dependence of theRateConstant 542.3 Transition-State Theory 56Chapter 3ReactorsforMeasuringReactionRates 643.1 Ideal Reactors 643.2 Batch and Semibatch Reactors 653.3 Stirred-Flow Reactors 703.4 Ideal Tubular Reactors 763.5 Measurement of Reaction Rates 823.5.1 Batch Reactors 843.5.2 Flow Reactors 87Chapter 4TheSteady-StateApproximation:Catalysis 1004.1 Single Reactions 1004.2 The Steady-State Approximation 1054.3 Relaxation Methods 124Chapter 5HeterogeneousCatalysis 1335.1 Introduction 1335.2 Kineticsof Elementary Steps: Adsorption,Desorption, and Surface Reaction 1405.3 Kinetics of OverallReactions 1575.4 Evaluation of KineticParameters 171Chapter 6EffectsofTransport LimitationsonRatesofSolid-CatalyzedReactions 1846.1 Introduction 1846.2 External Transport Effects 1856.3 Internal Transport Effects 1906.4 Combined Internal and External TransportEffects 2186.5 Analysis of Rate Data 228Chapter 7MicrokineticAnalysisofCatalyticReactions 2407.1 Introduction 2407.2 AsymmetricHydrogenation of ProchiralOlefins 240ixx Contents7.3 Ammonia Synthesis on Transition MetalCatalysts 2467.4 Ethylene Hydrogenation on TransitionMetals 2527.5 Concluding Remarks 257Chapter 8Nonideal FlowinReactors 2608.1 Introduction 2608.2 Residence Time Distribution (RTD) 2628.3 Application of RTD Functions to thePrediction of Reactor Conversion 269804 Dispersion Models forNonidealReactors 2728.5 Prediction of Conversion with an Axially-Dispersed PFR 2778.6 Radial Dispersion 2828.7 Dispersion Models forNonideal Flowin Reactors 282Chapter 9Nonisothermal Reactors 2869.1 The Nature of the Problem 2869.2 Energy Balances 2869.3 Nonisothermal Batch Reactor 2889.4 Nonisothermal Plug Flow Reactor 2979.5 Temperature Effects in a CSTR 3039.6 Stability and Sensitivity of ReactorsAccomplishing Exothermic Reactions 305Chapter 10ReactorsAccomplishingHeterogeneousReactions 31510.1 Homogeneous VersusHeterogeneousReactionsin Tubular Reactors 31510.2One-Dimensional Models for Fixed-BedReactors 31710.3Two-Dimensional Models for Fixed-BedReactors 325lOAReactor Configurations 32810.5Fluidized Beds with Recirculating Solids 331Appendix AReview of Chemical Equilibria 339A.1 Basic Criteria for Chemical Equilibrium ofReacting Systems 339A.2 Determination of EquilibriumCompositions 341Appendix BRegressionAnalysis 343B.1 Method of Least Squares 343B.2 Linear Correlation Coefficient 344B.3 Correlation Probability with a ZeroY-Intercept 345BA Nonlinear Regression 347Appendix CTransportinPorousMedia 349C.1 Derivation of Flux RelationshipsinOne-Dimension 349C.2 Flux Relationshipsin Porous Media 351Index 355This book is an introduction to the quantitative treatment of chemical reaction en-gineering. Thelevelof thepresentationiswhat weconsider appropriateforaone-semester course. The text provides a balanced approach to the understandingof:(1) both homogeneous and heterogeneous reacting systems and (2) both chemicalreaction engineering and chemical reactor engineering. We have emulated the teach-ings of Prof. Michel Boudart in numerous sections of this text. For example, much ofChapters1 and 4are modeled after hissuperb text that is nowout of print (Kineticsa/Chemical Processes), but they have been expanded and updated. Each chapter con-tainsnumerousworked problemsand vignettes. Weusethevignettestoprovide thereaderwithdiscussionson real, commercial processesand/or usesof themoleculesand/or analyses described in the text. Thus, the vignettes relate the material presentedto what happens in the world around us so that the reader gains appreciation for howchemical reaction engineering and its principles affect everyday life. Many problemsinthistext requirenumerical solution. Thereadershouldseekappropriatesoftwarefor proper solution of these problems. Since this software is abundant and continuallyimproving, the reader should be able to easily find the necessary software. This exer-cise is useful for students since they will need to do this upon leaving their academicinstitutions. Completion of the entire text will give the reader a good introduction tothe fundamentals of chemical reaction engineering and provide a basis for extensionsintoother nontraditional usesof theseanalyses, forexample, behavior of biologicalsystems, processing of electronic materials, and prediction of global atmospheric phe-nomena. We believe that the emphasis on chemical reaction engineeringasopposedto chemicalreactor engineeringis the appropriate context fortraining futurechemi-cal engineers who will confront issues in diverse sectors of employment.We gratefully acknowledge Prof. Michel Boudart who encouraged us to write thistext and who has provided intellectual guidance to both of us. MED also thanks MarthaHepworth for her effortsin convertinga pile of handwritten notesintoa final prod-uct. In addition, Stacey Siporin, John Murphy, and Kyle Bishop are acknowledged fortheir excellent assistancein compilingthesolutionsmanual. Thecover artwork wasprovided courtesy of Professor Ahmed Zewail'sgroupat Caitech, and we gratefullythank them for their contribution. We acknowledge with appreciation the people whoreviewedour project,especially A. Brad Antonof CornellUniversity, whoprovidedextensive comments on content and accuracy. Finally, we thank and apologize to themanystudents who suffered through the early drafts as course notes.We dedicate this book to our wives and to our parents for their constant support.Mark E. DavisPasadena, CARobert J. DavisCharlottesville. VANomenclaturexiiCi or[Ai]CBCiSCpCpdedpdtDaDeDijDKiDrDTADaDaEEDE(t)EItactivity of species iexternal catalyst particle surface area per unit reactorvolumerepresentation of species icrosssectional area of tubular reactorcross sectional area of a poreheat transfer areapre-exponential factordimensionless group analogousto theaxial Peclet numberfor theenergy balanceconcentration of speciesiconcentration of species i in the bulk fluidconcentration of speciesi at the solid surfaceheat capacity per moleheat capacity per unit masseffective diameterparticle diameterdiameter of tubeaxial dispersion coefficienteffective diffusivitymolecular diffusion coefficientKnudsen diffusivity of species iradial dispersion coefficienttransition diffusivity fromtheBosanquet equationDamkohler numberdimensionlessgroupactivation energyactivation energy for diffusionE(t)-curve; residence time distributiontotal energy in closed systemfrictionfactor in Ergun equation and modifiedErgun equationfractional conversion based on speciesifractionalconversion at equilibriumhihtHt:.Ht:.HrHwHwIIIikkkcKaKcKpKxKLm,.MiMMSniNomeoclatllrefugacityof speciesifugacityat standard state of purespeciesifrictional forcemolar flowrate of species igravitational accelerationgravitational potential energy per unit massgravitational constantmassof catalystchange in Gibbsfunction("free energy")Planck's constantenthalpy per mass of stream iheat transfer coefficiententhalpychange in enthalpyenthalpy of the reaction (often called heat of reaction)dimensionless groupdimensionlessgroupionicstrengthColburn I factorfluxof speciesi with respect to a coordinate systemrateconstantBoltzmann's constantmasstransfer coefficientequilibrium constant expressed in termsof activitiesportion of equilibrium constant involving concentrationportion of equilibrium constant involvingtotal pressureportion of equilibrium constant involving mole fractionsportion of equilibrium constant involving activitycoefficientslength of tubular reactorlength of microcavityin Vignette 6.4.2generalized length parameterlength in a catalyst particlemass of stream imass flowrate of stream imolecular weight of species iratio of concentrations or moles of twospeciestotal massof systemnumber of molesof species ixiiixiv Nomenclatl J[eNiNCOMPNRXNPPeaPerPPqQQrLlSScSiSpSSAScSESh(t)tTTBTsTBufluxof speciesinumber of componentsnumber of independent reactionspressureaxial Peelet numberradial Peelet numberprobabilityheat fluxheat transferredrate of heat transferreaction rateturnover frequencyor rate of turnoverradial coordinateradius of tubular reactorrecyele ratiouniversal gas constantradius of pelletradius of poredimensionless radial coordinate in tubular reactorcorrelation coefficientReynolds numberinstantaneous selectivitytospecies ichange in entropysticking coefficientoverallselectivity tospeciesisurface area of catalyst particlenumber of activesiteson catalystsurface areaSchmidt numberstandard error on parametersSherwood numbertimemean residence timestudent t-test valuetemperaturetemperature of bulk fluidtemperature of solid surfacethird bodyin a collision processlinear fluidvelocity(superficial velocity)vViVpVRVtotalWeX-Zz"Iirr8(t)8-eNomeoclatl irelaminar flowvelocity profileoverall heat transfer coefficientinternal energyvolumetric flow ratevolumemean velocity of gas-phase speciesivolume of catalyst particlevolume of reactoraverage velocity of all gas-phase specieswidth of microcavity in Vignette 6.4.2length variablehalf thethickness of a slab catalyst particlemole fraction of speciesidefined by Equation (B.1.5)dimensionless concentrationyield of species iaxialcoordinateheight abovea reference pointdimensionlessaxial coordinatecharge of speciesiwhen usedasa superscript istheorder of reaction withrespect tospeciesicoefficients;fromlinear regressionanalysis,fromintegration, etc.parameter groupingsin Section 9.6parameter groupingsin Section 9.6Prater numberdimensionless groupdimensionlessgroupsArrhenius numberactivity coefficient of speciesidimensionlesstemperaturein catalyst particledimensionlesstemperatureDirac delta functionthicknessof boundary layermolar expansion factor based on speciesideviation of concentration fromsteady-state valueporosity of bedporosity of catalyst pelletxvxvi Nomenclat! j[eYJoYJep.,~PPBPpT-TViwintraphase effectiveness factoroverall effectivenessfactorinterphase effectiveness factordimensionless timefractional surface coverage of species idimensionlesstemperatureuniversal frequencyfactoreffective thermal conductivity in catalyst particleparameter groupingsin Section 9.6effective thermal conductivity in the radial directionchemical potential of species iviscositynumber of moles of species reacteddensity(either mass or mole basis)bed densitydensity of catalyst pelletstandard deviationstoichiometric number of elementary stepispace timetortuositystoichiometric coefficient of species iThiele modulusThiele modulus based on generalized length parameterfugacity coefficient of speciesiextent of reactiondimensionless length variable in catalyst particledimensionlessconcentration in catalyst particle forirreversible reactiondimensionlessconcentration in catalyst particle forreversible reactiondimensionless concentrationdimensionlessdistancein catalyst particleNotationused forstoichiometricreactionsandelementarystepsIrreversible(one-way)Reversible(two-way)EquilibratedRate-determining__~ _ 1 ~TheBasicsofReactionKineticsforChemicalReactionEngineering1.1I TheScopeof ChemicalReactionEngineeringThe subject ofchemical reactionengineeringinitiatedandevolvedprimarilytoaccomplishthetask of describinghowtochoose, size, anddeterminetheoptimaloperating conditions for a reactor whose purpose is to produce a given set of chem-icals in a petrochemical application.However,the principles developed for chemi-cal reactorscan beappliedto most if not allchemically reactingsystems(e.g., at-mospheric chemistry, metabolic processes in living organisms, etc.). In this text, theprinciples ofchemical reactionengineeringarepresentedinsuchrigor tomakepossiblea comprehensiveunderstanding of thesubject. Masteryof theseconceptswill allowfor generalizationstoreactingsystemsindependentof theiroriginandwill furnishstrategies for attackingsuch problems.The two questions that must be answered for a chemically reacting system are:(1) what changes are expected to occur and (2) how fast will they occur? The initialtask in approaching the description of a chemically reacting system is to understandtheanswer tothefirstquestion by elucidating thethermodynamics of theprocess.For example, dinitrogen (N2) and dihydrogen (H2) are reacted over an iron catalystto produce ammonia (NH3):N2 + 3H2 = 2NH3, - b.H, =109 kllmol (at 773 K)whereb.H, is the enthalpy of the reaction (normally referred to as the heat of reac-tion). This reaction proceeds in an industrial ammonia synthesis reactor such that atthereactor exit approximately50 percent of thedinitrogenisconvertedtoammo-nia. At first glance, one might expect to make dramatic improvements ontheproduction of ammoniaif, forexample, a newcatalyst(asubstancethat increases2 CHAPTER1 TheBasicsof ReactionKinetics forChemical ReactionEngineeringthe rate of reaction without being consumed) could be developed. However, a quickinspection of thethermodynamicsof thisprocessrevealsthatsignificant enhance-mentsintheproductionof ammoniaarenotpossibleunlessthetemperatureandpressure of the reactionarealtered.Thus,the constraints placed on a reactingsys-tem bythermodynamics should always be identified first.EXAMPLE1.1.1 IInorder toobtainareasonablelevel of conversionata commerciallyacceptablerate, am-monia synthesis reactors operateat pressures of 150 to 300 atm andtemperatures of 700 to750 K. Calculate the equilibrium mole fractionof dinitrogen at 300 atm and 723K startingfroman initial composition of XN2= 0.25, XHz= 0.75(Xi is the mole fractionof speciesi).At 300 atm and 723 K,the equilibrium constant, Ka, is 6.6X10-3. (K. Denbigh, The Prin-ciples of Chemical Equilibrium, Cambridge Press, 1971, p. 153). Answer(See Appendix A for a brief overviewof equilibria involving chemical reactions):CHAPTER1 TheBasics of Rear.tionKineticsforChemical ReactionEngineeringThe definition of theactivity of species i is:fugacityat thestandard state, that is, 1 atm for gasesand thus3K= ]a fI/2 f3/2 (t'O )N, H, JNH]Use of the Lewisand Randall rulegives:[fNH; ] [ ]]J;2]J;2 I atm/; = Xj cPj P, cPj = fugacitycoefficient of pure component i at T and Pof systemthen[XNH; ][ cPNH;] IK =KK-K = --- P- 1 atma X (p P XlI2 X3/2 -:1,1(2-:1,3/2 [ ] [ ]N, H, 'VN, 'VH,Upon obtaining eachcPjfromcorrelationsor tables of data (available in numerousref-erencesthat contain thermodynamic information):If a basis of 100 molisused (g isthe number of moles of N2reacted):thenN2HzNH3total2575o100(2g)(100- 2g)------- =2.64(25- g)l/2(75- 3g)3/2Thus, g=13.1 andXN, (25- 13.1)/(100 26.2)=0.16. At 300atm, theequilibriummole fractionof ammonia is 0.36 whileat 100 atm it fallsto approximately 0.16. Thus, theequilibrium amount of ammonia increases with the total pressure of the system at a constanttemperature.4 CHAPTER1 The Basics of ReactionKinetics for Chemical ReactionEngineeringThenext taskindescribingachemicallyreactingsystemistheidentifica-tion of thereactionsandtheir arrangementinanetwork. The kineticanalysisofthenetworkisthennecessaryforobtaininginformationontheratesof individ-ual reactionsandansweringthequestionof howfast thechemical conversionsoccur.Each reaction of the network is stoichiometrically simple in thesense thatit can be described by thesingle parameter called the extent of reaction (see Sec-tion1.2). Here,a stoichiometrically simple reaction will just be called a reactionfor short. Theexpression"simple reaction"shouldbeavoidedsinceastoichio-metrically simple reaction does not occur in a simple manner. In fact, most chem-ical reactions proceed through complicated sequences of steps involving reactiveintermediates that do not appear in the stoichiometries of the reactions. The iden-tification of theseintermediates and thesequence of steps are the core problemsof thekineticanalysis.If a step of thesequence can be written as it proceeds at the molecular level,itis denoted as an elementary step (or an elementary reaction), and it represents an ir-reducible molecular event. Here, elementary steps will be called steps for short. Thehydrogenation of dibromine isan example of a stoichiometrically simple reaction:If this reaction would occur by Hz interacting directly with Brz to yield two mole-cules of HBr, thestep would be elementary. However, it does not proceed aswrit-ten.It isknownthatthehydrogenation of dibrominetakesplace inasequence oftwostepsinvolvinghydrogenandbromineatomsthat donot appearinthestoi-chiometry of the reaction but exist in the reactingsystem inverysmall concentra-tionsasshown below(an initiator is necessary tostart the reaction, for example, aphoton: Brz + light -+ 2Br, and the reaction is terminated by Br + Br +TB -+ Brzwhere TBisa third body that isinvolved in the recombination process-see belowfor further examples):Br +Hz-+HBr + HH + Brz-+HBr + BrIn this text,stoichiometric reactions and elementary steps are distinguished bythe notation provided in Table1.1.1.Table1.1.1I Notationusedforstoichiometricreactionsandelementarysteps.Irreversible (one-way)Reversible(two-way)EquilibratedRate-determiningCHAPTER1 TheBasics of ReactionKinetics forChemical ReactionEnginAering 5Indiscussions on chemical kinetics, the terms mechanismor model fre-quentlyappearandareusedtomeananassumedreactionnetworkor aplausi-blesequenceof stepsfora givenreaction. Sincethelevelsof detail ininvesti-gating reaction networks, sequences and steps are so different, the wordsmechanismandmodel havetodatelargelyacquiredbadconnotationsbecausetheyhavebeen associatedwith muchspeculation. Thus, theywill beusedcare-fullyinthistext.As achemicallyreactingsystemproceeds fromreactants toproducts, anumber of species calledintermediatesappear, reachacertainconcentration,andultimatelyvanish. Threedifferenttypesof intermediatescanbeidentifiedthat correspondtothedistinctionamongnetworks, reactions, andsteps. Thefirst typeof intermediateshasreactivity, concentration, andlifetimecompara-bletothose of stablereactantsandproducts. Theseintermediatesaretheonesthatappearinthereactionsof thenetwork. For example, considerthefollow-ingproposal for howtheoxidationofmethaneat conditionsnear700Kandatmospheric pressure may proceed (see Scheme l.l.l). The reacting system mayevolve fromtwostable reactants, CH4and 2, totwostable products, CO2 andH20, through a network of four reactions.The intermediates areformaldehyde,CH20; hydrogenperoxide, H20 2;andcarbonmonoxide, CO. Thesecondtypeof intermediateappearsinthesequenceof stepsfor anindividual reactionofthe network. These species (e.g., free radicals in the gas phase) are usually pres-ent inverysmall concentrations andhave short lifetimes whencomparedtothoseof reactantsand products. Theseintermediateswill be called reactivein-termediatestodistinguishthemfromthemorestablespeciesthataretheonesthatappear inthereactionsof thenetwork. ReferringtoScheme1.1.1, fortheoxidationof CH20 togiveCOandH20 2,thereactionmayproceedthroughapostulatedsequenceof twostepsthatinvolve two reactiveintermediates, CHOand H02 . Thethirdtypeof intermediateiscalledatransitionstate, whichbydefinitioncannot beisolatedandisconsideredaspecies intransit. Eachele-mentarystepproceeds fromreactants toproducts througha transitionstate.Thus, foreachof thetwoelementarystepsintheoxidationof CH20, thereisa transitionstate. Althoughthe nature of thetransitionstate for the elementarystep involving CHO, 02' CO, and H02is unknown, other elementary steps havetransitionstates that havebeenelucidatedingreaterdetail. For example, theconfiguration shown in Fig. 1.1.1is reached for an instant in the transition stateof thestep:Thestudyof elementarysteps focusesontransitionstates, andthekineticsof these steps represent the foundation of chemical kinetics and the highest levelof understandingof chemicalreactivity. Infact, theuseof lasersthatcangen-eratefemtosecondpulseshasnowallowedfor the"viewing"ofthereal-timetransition from reactants through the transition-state to products (A. Zewail, The6CHAPTER1 TheBasicsof ReactionKinetics forChemical ReactionEngineeringCHAPTER1 TheBasicsof ReactionKineticsfor Chemical ReactionEngineeringBr7BrICH/ I "'CHH 3~ O W)..HOHFigure1.1.1IThe transitionstate (trigonalbipyramid)of the elementarystep:OH- +C2HsBr ~ HOC2Hs +Br-The nucleophilicsubstituent OH-displacesthe leavinggroup Br-.Br-)HIH '" C/ CH3IOHJ8 CHAPTER1 TheBasicsof ReactionKinetics forChemical ReactionEngineeringChemical Bond:Structureand Dynamics, AcademicPress, 1992). However, inthe vast majority of cases, chemically reacting systems are investigated in muchless detail.The level of sophistication that is conducted isnormally dictated bythepurpose of theworkandthestateof development of thesystem.1.2 I TheExtent of ReactionThechangesina chemically reactingsystem can frequently, butnotalways(e.g.,complex fermentation reactions), be characterized by a stoichiometric equation. Thestoichiometric equation for a simple reaction can be written as:NCOMP0= L:viA;i=1(1.2.1)where NCOMPisthenumber of components, A;, of thesystem. Thestoichiomet-ric coefficients, Vi' are positive for products, negative for reactants, and zero for inertcomponentsthat donotparticipateinthereaction. For example, manygas-phaseoxidation reactions use air as the oxidant and the dinitrogen in the air does not par-ticipate in the reaction (serves only as a diluent).In the case of ammonia synthesisthe stoichiometric relationshipis:Application of Equation(1.2.1)totheammonia synthesis, stoichiometric relation-ship gives:For stoichiometric relationships, the coefficients can be ratioed differently, e.g., therelationship:can bewritten alsoas:sincetheyare justmolebalances. However, for anelementaryreaction, thestoi-chiometry iswrittenasthereactionshould proceed. Therefore, anelementary re-action such as:2NO +O2-+2N02CANNOT bewritten as:(correct)(not correct)CHAPTER1 TheBasicsof ReactionKinetics for Chemical ReactionEngineering 9EXAMPLE1.2.1 IIf there are several simultaneous reactions taking place, generalize Equation (1.2.1) to a sys-tem of NRXN different reactions. For the methane oxidation network shown in Scheme1.1.1,write out the relationships fromthe generalized equation. AnswerIf there are NRXN reactions and NCOMP species in the system, the generalized form of Equa-tion(1.2.1)is:NCOMPo= 2: vi,jAi, j 1; ",NRXNi(1.2.2)For the methane oxidation network shown in Scheme1.1.1:0=OCOz +IHzO lOz + OCO + OHzOz +lCHzO- ICH40=OCOz + OHp- lOz +lCO +1HzOz - lCHzO + OCH4o= ICOz + ORzO ! Oz- ICO + OHzOz + OCHzO + OCH40=OCOz +IHzO +! Oz + OCO- I HzOz + OCHp + OCH4or in matrix form:I-Iooo1o-1o1-1o1ooI- o ~ lHzOzCHpCH4Notethatthesumof thecoefficients of a columnin thematrixiszeroif thecomponent isan intermediate.Consider a closed system, that is, a system that exchanges no mass with its sur-roundings. Initially,there aren?moles of component Ai present in thesystem. If asingle reaction takes place that can be described by a relationship defined by Equa-tion(1.2.1), then the number of moles of component Ai at any time t will be givenbythe equation:ni (t) = n? +Vi CH4 + CO~ /dCAdtdPA-=-kPdt A(constant V)[constant V:Ci= P;j(RgT) ](1.5.5)(1.5.6)EXAMPLE1 .5.1 IThus, for first-order systems, therate, r, is proportional (viak) totheamountpresent, ni' inthesystemat anyparticulartime. Althoughat first glance, first-order reaction rates mayappear toosimple to describe real reactions, such is notthe case (see Table 1.5.1). Additionally, first-order processes are many times usedtoapproximatecomplexsystems, forexample, lumpinggroups of hydrocarbonsinto a generic hypothetical component so that phenomenological behavior can bedescribed.In this text, concentrationswill be written in either of two notations. The nota-tions Ci and[AJare equivalent in terms of representing the concentration of speciesi or Ai, respectively. These notations are used widely and the reader should becomecomfortablewith both.The natural abundance of 235U in uranium is 0.79 atom %. If a sample of uranium is enrichedto3at. %andthenisstoredinsalt minesunder theground, howlongwill it take thesam-ple to reach the natural abundance level of 235U (assuming no other processes form235U; thisisnot thecaseif238Uispresent sinceitcandecaytoform235U)? Thehalf-lifeof235Uis7.13 X108years. AnswerRadioactivedecaycanbedescribedasafirst-orderprocess. Thus, foranyfirst-orderdecayprocess,theamount of material present declinesinan exponential fashionwithtime. Thisiseasytoseebyintegrating Equation(1.5.3)to give:11; 11? exp( - kt), where11?istheamount ofl1ipresent att O.CHAPTER1 The Basics of ReactionKinetics for Chemical ReactionEngineering 27The half-life, tj, is defined as the time necessary to reduce the amount of material in half. Fora first-orderprocesstj canbeobtained asfollows:! n?= n? exp( -ktj)orGiventj, avalue of kcanbecalculated. Thus, fortheradioactivedecayof235U, thefirst-order rate constant is:In(2)k= -- 9.7X10-10years-It12To calculate the time required to have 3 at. % 235U decay to 0.79 at. %, the first-order expression:can be used. Thus,n, = exp( -kt) or tn?(nO)In ~ ikEXAMPLE1.5.2 Ior averylong time.In(_3)0.79t = =1.4X109years9.7X10-10NzOs decomposes into NOz and N03 with a rate constant of 1.96X1014exp [ -10,660/T]s-l.At t = 0, pure NzOsisadmitted into a constant temperature and volume reactor withaninitialpressure of 2 atm. After 1 min, what is the total pressure of the reactor? T= 273 K . AnswerLet n bethe number of moles of NzOssuch that:dn-=-kndtSince n nO (1 - I):dfdtk(l - f),f = O@ t = 0Integration of thisfirst-order, initial-value problemyields:In(ft) kt f o r t ~ O28 CHAPTER1 TheBasics of ReactionKinetics for Chemical ReactionEngineeringorf I - exp( -kt) for t2:: 0At 273K, k= 2.16X10-3S-I. After reaction for Imin:f=I exp[ -(60)(2.16X 10-3)) =0.12EXAMPLE1.5.3 IFrom the ideal gaslaw at constant T andV:P n n(1+ sf)pO nO nOFor this decomposition reaction:Thus,p=pO(1+f) 2(1+ 0.12) =2.24 atmOftenisomerizationreactionsarehighlytwo-way(reversible). For example, theisomeriza-tion of I-butene to isobuteneisan important stepin theproduction of methyltertiarybutylether (MTBE), a common oxygenatedadditivein gasoline usedto lower emissions.MTBEis produced by reacting isobutenewith methanol:In order tomake isobutene, n-butane (an abundant, cheap C4 hydrocarbon) can be dehydro-genated toI-butene then isomerized to isobutene. Derive an expression for the concentrationof isobutene formedas a functionof time by the isomerization of I-butene:k,CHCH2CH3( ) CH3CCH3k2 IICH2 AnswerLetisobutenebedenotedascomponent J andI-buteneasB. If thesystemisatconstantTandV, then:dfB1, JdtCHAPTER1 TheBasicsof ReactionKinetics for Chemical ReactionEngineeringSince[B] = [BJO(l - Is):= [1]0 +[B]Ols [B]O(M +Is), ;W = [I]/[BJO*0Thus,A'l'b' dis 0t eqUl 1 flum - = , so:dt29[B]O(M +nq)[B J0(1 - IEq)Insertion of the equilibrium relationship into the rate expression yields:disdtor after rearrangement:dis kl(M +1)dt=(M +nq) (tEq- Is), Is=0 @t =0Integration of this equation gives:In[_ll= [kl(M +1)]t, M *01 _ Is M +IEqnqor{ [ ( kl(M+ 1))]}Is= IEq1 - exp - M+IEqtUsing thisexpression for Is:Consider thebimolecular reaction:A + B --+ products (1.5,7)Using the Guldberg-Waage form of the reaction rate to describe this reaction gives:(1.5.8)30 CH APTER1 TheBasicsQfReactionKinetics forChemical ReactiQnEngineeringFrom Equations(1.3.4)and(1.5.8):1 dnir=vYdtor(variable V)(constant V)dnAV-= -knAnBdtdCA- = -kCACBdtdPAkdt = - R r PAPB [constant V:Ci=g(1.5.9)(1.5.10)(1.5.11)Forsecond-order kineticprocesses, thelimitingreactantisalwaystheappropriatespecies tofollow(let species denotedas Abe the limitingreactant). Equations(1.5.9-1.5.11)cannot beintegrated unlessCsisrelated toCA' Clearly,thiscan bedone via Equation(1.2.5)or Equation (1.2.6).Thus,or if the volumeis constant:- c2If M= - = - = - then:c1nB= nA +- 1)CB= CA +- 1)PB = PA+- 1)(variable V)}(constant V)(constant V)(1.5.12)Inserting Equation (1.5.12)into Equations(1.5.9-1.5.11)gives:(variable V) (1.5.13)(1.5.14)(1.5.15)If Visnot constant, thenV =VO (1+SAJA) byusingEquation(1.2.15) andtheidealgaslaw. Substitution of this expression into Equation (1.5.13)gives:CHAPTER1 The-.BBS.icsof ReactionKinetics fOLChemical ReactionEngineeringk ( ~ ) (1- JA)[M - JA](1+SAJA)EXAMPLE1.5.4 I31(1.5.16)Equal volumesof 0.2Mtrimethylamineand0.2Mn-propylbromine(bothinbenzene)were mixed, sealed in glasstubes, and placed into a constant temperature bathat 412 K.After varioustimes, thetubeswere removedand quickly cooled to roomtemperature tostopthereaction:+N(CH3)3 + C3H7Br =? C3H7 N(CH3)3Br-The quaternization of a tertiary amine gives a quaternary ammoniumsalt that is not solublein nonpolar solventssuch asbenzene. Thus, thesalt can easily be filteredfromtheremain-ing reactants and the benzene. From the amount of salt collected, the conversion can be cal-culated and the data are:51325344559801001204.911.220.425.631.636.745.350.755.2Are these data consistent with a first-or second-order reaction rate? AnswerThereactionoccursintheliquid phaseandtheconcentrationsaredilute. Thus, a goodas-sumption is that thevolume of thesystemis constant. Since C ~ = C ~ :(first-order) In[_1]=kt1 - fA(second-order)fA=k C ~ tI fAInordertotest thefirst-ordermodel, theIn[ 1 - isplottedversus t whilefor thesecond-order model, [,)is plotted versust (see Figures1.5.1and1.5.2). Notice thatbothmodelsconformtotheequationy ex I t + az. Thus, thedata can befittedvialinear32 CHAPTER1 TheBasicsof ReactionKineticsforChemical ReactionEngineering1.0 -,----------------....,0.80.60.40.20.0o 20 40 60 80 100 120 140Figure1.5.1 IReaction rate data forfirst-order kineticmodel.regressiontobothmodels (seeAppendixB). Fromvisual inspectionof Figures 1.5.1and1.5.2, the second-order modelappearsto give a better fit. However, the resultsfromthe lin-ear regression are (5 is the standard error):5(aIl =2:51 X 10-45(a2) 1.63X 10-2first-order (Xl =6.54X10-3a2=5.55X10-2Ree0.995second-order (Xl =1.03X10-25(al) =8.81 X 10-5(X2 -5.18X 10-35(a2) 5.74X 10-3Ree0.999Both models give high correlation coefficients (Reel, and this problemshows howthecorrelationcoefficient maynotbeuseful indetermining"goodnessof fit." Anappropriatewaytodetermine"goodnessof fit"is toseeif themodelsgivea2that isnot statisticallydifferent fromzero. This is the reason for manipulating the rate expressionsinto formsthathavezerointercepts(i.e., a knownpoint fromwhichtocheckstatistical significance). If astudent1*-test isusedtotestsignificance(see Appendix B), then:(X2 015(a2)CHAPTER1 TheBasicsof ReactionKineticsfor Chemical ReactionEngineering 331.41.21.00.80.60.40.20.0o 20 40 60 80 100 120 140t ~ =Figure1.5.2 IReaction rate data forsecond-order kinetic model.The valuesof t*for the first-and second-order models are:15.55X10-2- 01 = 3.391.63X10-2-5.18X 10-3- 01t; = --------:-- = 0.965.74XFor 95 percent confidence with 9 data points or 7 degrees of freedom(from table of studentt*values):expected deviationt;xp= = 1.895standard errorM . minSincet ~ >t;xpandt; '>,()"~ :c0.2 40q.Sat)"tl:lci:0.1 200 0T i m e ~42 CHAPTER1 TheBasicsof ReactionKineticsfor Chemical ReactionEngineeringConsiderthereactionnetworkof twoirreversible(one-way), first-orderreac-tionsin parallel:yDPA~ S P (1.5.33)Again,like theseries network shown in Equation (1.5.22), the parallel network ofEquation (1.5.33)can represent a variety of important reactions. For example, de-hydrogenationof alkanescanadheretothis reactionnetworkwherethedesiredproduct DPisthealkeneandtheundesiredside-product SPisahydrogenolysis(C- C bond-breaking reaction)product:~ CHz=CHz + HzCH3~ CH4 + carbonaceous residue on catalystUsingtheGuldberg-WaageformofthereactionratestodescribethenetworkinEquation (1.5.33)gives for constant volume:(1.5.34)withc2 + c2p + cgp= CO= CA + CDP +CSPIntegration of the differential equation for CAwith CA= c2at t =0 gives:(1.5.35)Substitution of Equation (1.5.35)into the differential equation for CDP yields:The solution to thisdifferential equationwithCDP= cgpat t = 0 is:(1.5.36)CHAPTER1 TheBasicsQf ReactiQnKinetics forChemical ReactionEngineeringLikewise, the equation for Csp can be obtained and it is:The percent selectivity and yield of DP forthis reaction networkare:43(1.5.37)(1.5.38)andCD?y=-C ~EXAMPLE1.5.8 IThe followingreactions are observed when an olefin is epoxidized with dioxygen:alkene+O2===>epoxideepoxide+O2===>CO2+H20alkene+O2===>CO2+H20(1.5.39)Derive the rate expression for this mixed-parallel series-reaction network and the expressionforthe percent selectivity to the epoxide. AnswerThe reaction network is assumed tobe:k,A+O ~ EP+O CDCDwhere A: alkene,0:dioxygen, EP:epoxide, andCD:carbon dioxide. Therate expressionsfor thisnetwork are:dt44 CHAPTER1 TheBasicsof ReactionKinetics forChemical ReactionEngineeringThe percent selectivity to EP is:EXAMPLE1.5.9 IEXAMPLE1.5.10In Example1.5.6, the expression for the maximum concentration in a series reaction networkwasillustrated. Example1.5.8showed howtodeterminetheselectivityina mixed-parallelseries-reaction network. Calculatethemaximumepoxideselectivityattained fromthereac-tion network illustrated in Example1.5.8 assuming an excess of dioxygen. AnswerIf thereisan excess of dioxygenthen Co can be held constant. Therefore,From this expression it is clear that the selectivity for any CA will decline as k2CEP increases.Thus, the maximum selectivity willbe:and that thiswould occur at t = 0, aswas illustrated in Example1.5.7. Here, the maximumselectivityis not 100 percent at t = but rather the fractionkI/(k2 + k3) duetothe parallelportion of the network.Find the maximum yield of the epoxide using the conditions listed for Example1.5.9. AnswerThe maximumyield,will occur at CEPx. If kI= kICO, k2= k2CO'k3= k3CO'y Cand x= the rate expressionsfor this network can be written as:dydtdxdtNote theanalogyto Equation (1.5.23). Solving thedifferentialequation for yand substitut-ing this expressioninto the equationfor x gives:dx+dtCHAPTER1 TheBasicsof ReactionKineticsfor Chemical ReactionEngineering 45o ~v 0.50-0,