Handbook of Combustion, Volume 1 (Fundamentals and Safety) PDF
Preview Handbook of Combustion, Volume 1 (Fundamentals and Safety)
j 1 1 Combustion Fundamentals MohammadJanbozorgi,KianEisazadehFar,andHameedMetghalchi 1.1 Introduction Combustion is a complex subject in chemical physics. A deep understanding of combustionsciencerequiresasolidgraspofawidespectrumofscientificdisciplines, suchasquantummechanics,thermodynamics,chemicalkinetics,andfluiddynam- ics. On the application level, combustion phenomena can be classified based on interactions between exothermic chemical reactions and fluid mechanics. Such aninteractiondependsheavilyontherelativeorderofmagnitudeofthetimeand spatial scales of each individual phenomenon, leading to different forms of com- bustion.Premixedcombustionoccurswhenthefluidmixingissufficientlyfastasto createanear-uniformdistributionoffuel/airmixtureinthereactor.Dependingon the thermodynamic conditions, premixed combustion can also be either strictly kineticallycontrolled(e.g.,autoignition),orconvection/reaction/diffusioncontrolled (e.g., premixed flames). The former condition underlies the operation of homoge- neous charged-compression-ignition (HCCI) engines, diesel engines and rapid compression machines (RCMs), and is essential in understanding the engine knock. The latter introduces a fundamental physico-chemical property for any premixed mixture, that is, laminar burning speed. Knowledge of this property is cruciallyimportantinspark-ignitionengines,partlytopreventautoignition.Inthe case of slow mixing and fast reaction, nonpremixed or diffusion flames will be observed.Thischapterisdevotedtoananalysisoftheabove-mentionedmodesof combustion, with special emphasis placed on laminar burning speeds and flame structures of different hydrocarbons at high pressures, and the experimental methodstomeasurethem.Suchdataareextremelyimportantforthevalidationof anyreliablechemicalkineticmechanism,andwillbeespeciallyusefulforinternal combustionenginedesigners. HandbookofCombustionVol.1:FundamentalsandSafety EditedbyMaximilianLackner,FranzWinter,andAvinashK.Agarwal Copyright(cid:1)2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:978-3-527-32449-1 j 2 1 CombustionFundamentals 1.2 CombustionThermodynamics Combustion is defined as an energy-evolving (exothermic) chemical transforma- tion[1].Whilestrictlyinvolvingtime-dependentchemicalreactions,thefinalyieldof combustion,andhowmuchenergycanbeextractedfromafuel/airmixtureundera specified process, are restricted by the laws of thermodynamics. A stoichiometric mixture of fuel and air is defined as a mixture containing just enough oxygen to theoreticallyburnthehydrocarbonfueltowaterandcarbondioxide(onlyhydrocar- bon fuels are considered in this chapter).The equivalence ratio is commonly used toindicatequantitativelywhetherafuel/oxidizermixtureisrich,lean,orstoichio- metric[2]: ðA=FÞ ðF=AÞ w¼ stoic ¼ act ð1:1Þ ðA=FÞ ðF=AÞ act stoic Fuel-rich,fuel-lean,andstoichiometricmixturesaredefinedbyw>1,w<1,and w¼1,respectively.InEquation1.1,Aisthemassofair,Fthemassoffuel,andthe stoic and act subscripts represent the stoichiometric and actual mixtures, respectively. 1.2.1 EnthalpyofReaction The enthalpy of reaction or enthalpy of combustion is defined as the net change of enthalpy due to a chemical reaction. This quantity takes a positive value for an endothermicreaction,andanegativevalueforanexothermicreaction.Thismeans thatintheformerreactiontheenergyisabsorbedbythereactingsystem,whereasin the latter reaction it evolves as a result of the reaction. Considering the global stoichiometriccombustionchemicalreactionofagenericnonoxygenatedhydrocar- bonwithair: (cid:1) (cid:2) y y C H þjðO þ3:76N Þ!xCO þ H Oþ3:76jN ; j¼xþ x y 2 2 2 2 2 2 4 ð1:2Þ thisstatementtranslatesto Dh ¼h ðT Þ(cid:1)h ðT Þ R prod f reac R (cid:1) (cid:2) y ¼xh ðT Þþ h ðT Þþ3:76jh ðT Þ CO2 f 2 H2O f N2 f (cid:1)h ðT Þ(cid:1)jh ðT Þ(cid:1)3:76jh ðT Þ ð1:3Þ CxHy R O2 R N2 R wherehistheenthalpyandTisthetemperature.Dependingonthethermodynamic processthatthesystemundergoes,theproductstemperature–alsocalledtheflame temperature(T )–maybedifferentfromthetemperatureofreactants(T ).Theheat f R j 1.2 CombustionThermodynamics 3 ofcombustionrepresentstheamountofenergyreleasedorabsorbedbythemixture duringanisothermalchemicalconversion.Forexample, COþH OðgÞ!CO þH ðgÞ; Dh ¼(cid:1)41:16kJðT ¼T ¼298:15KÞ 2 2 2 R f R meansthat41.16kJofenergywillbereleasedtothesurroundingsif1moleofcarbon monoxide reacts completely with 1 mole of water vapor at constant pressure to produce1moleofcarbondioxideand1moleofhydrogenmolecule[1].Thetotal enthalpyofaspeciesAisdefinedas: hA¼hf(cid:2);AðTrefÞþDhs;AðTref;TÞ ð1:4Þ ThefirstterminEquation1.4representstheenthalpyofformation,andisdefinedas the net change in enthalpy associated with breaking the chemical bonds of the standardstateelementsandformingnewbondstocreatethecompoundofinterest[2]. Here,thestandardstateelementsaretakentobethemoststablestateofthatelement atthetemperatureofinterestandthepressureof1atmosphere.Forthecommon elementsofcombustioninterestatT ¼298:15K,thesestatesarecarbon(C)as ref graphite, molecular hydrogen (H ), oxygen (O ),nitrogen (N )as idealgases, and 2 2 2 atomicsulfur(S)assolid[1].Thenaturalconsequenceofthisdefinitionisthatthe enthalpy of formation of, for example, an oxygen atom (O) is half of the bond dissociation energy of the oxygen molecule. The second term in Equation 1.4 representsthesensibleenthalpychangeandisdefinedas: ðT Dhs;AðTref;TÞ¼ cp;AðTÞdT ð1:5Þ Tref Clearly,anydeparturesfromthestandardstateenthalpyarereflectedinthisterm. Valuesofthespecificheatatconstantpressure,c ðTÞ,aretabulatedformanyspecies p intheCHEMKIN[3]database. 1.2.2 FlameTemperature Flametemperatureisthetemperaturereachedatthestateofchemicalequilibriumin areactingsystem.Theenergybalance,dE ¼dQ(cid:1)dW,impliesthat,forafixedtypeof workinteractionwiththesurroundingenvironment,anadiabaticprocess,dQ ¼0, hasthehighestflametemperature.Furthermore,dependingontherelativeorderof magnitudeofthechemicalenergyreleasetimescale,t ,andthatofboundarywork ch interactionthroughacousticwavepropagation,t ,thisadiabatictemperaturefalls a betweentwoextremes: (a) t (cid:3)t , resulting in dV ¼0. This further results in dW ¼pdV ¼0 and ch a therefore dE ¼0, known as constant energy–constant volume flame temper- ature,Tf;ðE;VÞ.Here,pisthepressureandVrepresentsvolume. j 4 1 CombustionFundamentals (b) t (cid:4)t ,resultingindp¼0.ThisfurthertranslatestodH¼dðEþpVÞ¼0, ch a knownasconstantenthalpy–constantpressureflametemperature,Tf;ðH;pÞ. Sinceinthelattercasepartoftheenergyisusedtodoworkagainstthesurrounding environment,itresultsinalowerflametemperature,thatis,Tf;ðH;pÞ <Tf;ðE;VÞ.The firstcaseisusuallyassumedtobetrueforautoignitioninclosedsystems(e.g.,shock tubes),whilethelatterisassumedtobethecaseforcombustioninopensystemsat lowMachnumbers(e.g.,gasturbinecombustors,Bunsenburners). 1.2.3 ChemicalEquilibrium There are two different approaches to determine the chemical equilibrium compo- sition of a reacting mixture. One is based on the application of the Law of Mass Action [1], and the other on the method of Lagrange multipliers [4]. If N is the sp numberofspeciesandN isthenumberofatomicelementsinthesystem,thenthe e firstmethodrequiresthecompilationofN (cid:1)N independentchemicalreactions, sp e followed by the simultaneous solution of the same number of equations, one for eachreaction.Obviously,asthenumberofchemicalspeciesincreases,theproblem becomes more tedious. An excellent coverage of this approach can be found in Ref. [1]. Thesecondmethod,whichdoesnothavetheshortcomingsofthefirstapproach, consistsofconsideringallconceivablereactionmechanismsbetweenthechemical species,andusingthemaximumentropyprinciplewithoutspecifyingexplicitlyany ofthereactionmechanisms[4].AspointedoutbyKeck[5],anequilibriumstateis meaningfulonlywhentheconstraintssubjecttowhichsuchastateisattainedare carefullydetermined,andallequilibriumstatesareinfactconstrainedequilibrium states.Attemperaturesofinteresttocombustion,nuclearandionizationreactions canbeassumedfrozen,andthefundamentalconstraintsimposedonthesystemare theconservationofneutralatoms.IfN,a andSðU;V;NÞrepresent,respectively, i ij i thenumberofmolesofspeciesi,thenumberofjthatomicelementinspeciesi,and theentropyofaclosedadiabaticsystem,then XNsp C ¼ a N; j¼1;...;N ð1:6Þ j ij i e i¼1 isthetotalnumberofmolesofatomicelementjinthesystem,whichisconserved during chemical conversion. Therefore, the problem reduces to determining a chemicalcompositionwhichmaximizesSðU;V;NÞsubjecttotherelationshipin i Equation 1.6. Using the method of undetermined Lagrange multipliers, it can be easilyshownthatsuchacompositionwillbegivenby[5]: ! XNe N ¼Q exp (cid:1) a c ; i¼1;...;N ð1:7Þ i i ij j sp j¼1 j 1.3 ChemicalKinetics 5 where c is the constraint potential (Lagrange multiplier) conjugate to elemental j constraintC,andQ isthepartitionfunctionofspeciesi,whichisdefinedasfollows: j i (cid:3) (cid:4) p V m(cid:2)ðTÞ Q ¼ (cid:2) exp (cid:1) i ; i¼1;...;N ð1:8Þ i R T R T sp u u In Equation 1.8, m(cid:2)ðTÞ represents the standard Gibbs free energy of species i at i temperatureT,andR istheuniversalgasconstant.Asthisfunctiontakesonfinite u valuesforeveryspecies,Equation1.7showsthat,inprinciple,allspeciesmadeofthe declaredatomicelementsarepresentatchemicalequilibriumstate,nomatterhow smalltheirconcentration.Thisfurtherexplainswhyfullconversiontocarbondioxide and water in a stoichiometric mixture, as given by Equation 1.2, is a hypothetical situation. Substituting Equation 1.7 back into Equation 1.6 forms a set of N e transcendentalequationsforcs.Thelevelofreductioninthenumberofequations j tobesolvedisdazzling;fromN whichcouldeasilygouptoseveralthousandsfor sp heavyhydrocarbonstoamaximumoffiveforatomicelementsofcarbon,oxygen, hydrogen,nitrogenandsulfurforalmostanyhydrocarbonfuels.Thismethodforms thebasisofthewidelyusedequilibriumcodesofSTANJAN[6]andNASA[7]. 1.3 ChemicalKinetics 1.3.1 CombustionChemicalReactions The development of models for describing the dynamic evolution of chemically reactingsystemsisafundamentalobjectiveofchemicalkinetics.Thistaskinvolves identifyingthechemicalreactionsinthemostelementarylevel,andalsotherateat whichsuchreactionsproceed.Theconventionalapproachtothisprobleminvolves firstspecifyingthestateandspeciesvariablestobeincludedinthemodel,compiling afull setofrate-equations forthesevariables based onafull setofelementary chemical reactions, and then integrating this set of equations to obtain the time- dependent behavior of the system [8]. Such models are frequently referred to as detailedkineticmodels(DKMs).Themostwidelyknownandusedsuchmodelis GRI-MECH 3.0 for the combustion of methane at high temperatures and low pressures [9]. A DKM can easily include several hundred chemical species and severalthousandchemicalreactionsforheavyhydrocarbons[10].(Anextensivestudy ofDKMsisprovidedinChapter2.)Combustionchemicalreactionsareingeneral chainreactions,whichmeansthattheproductsofonereactionserveasthereactantsof otherreactions.However,independentofthefuelmolecule,thesereactionscanbe classifiedintofourgroups: . Initiationreactions, whichstartthechain,involvecollisionwithfuelbyoxygen moleculeand,astheradicalpoolispopulated,byradicals.FU þ O ¼FR þ HO 2 2 j 6 1 CombustionFundamentals andFU þ X¼FR þ XH,inwhichFU,FR,andXrepresentthefuelmolecule, fuelradicalandradicalpool(O,OH,H,HO ,etc.),respectively,fallintothisclass. 2 Ifthetemperatureissufficientlyhigh,thecrackingofeitherC(cid:1)HbondsorC(cid:1)C bondsinthefuelmoleculebecomesalsoimportant. . Chain-branching reactions change the number of radicals and populate the radicalpool.H þ O ¼OH þ Oisoneofthemostimportantchain-branching 2 reactionsincombustionapplications. . Chain-propagating reactions change the type of radical, while conserving the number of radicals in the reactions. H þ H O¼H þ OH represents these 2 2 reactions. . Three-bodyreactions,sometimesknownaschain-terminatingreactionsorequiv- alently,dissociation–recombinationreactions,changethenumberofmolesofthe mixture; for example, H þ OH þ M¼H O þ M. As the recombination of 2 radicalsishighlyexothermicandusuallyinvolvesonlyasmallrotationalenergy barrier,thesereactionscannotbebimolecularandrequireinteractionwithathird body(M),towhichtheenergyofmoleculeformationisdisposed.Otherwise,this energywoulddissociatetheproductsintotheoriginalreactants.Dependingon theirmolecularsize,differentmoleculeshavedifferentthird-bodyefficiencies. AmoredetailedpresentationofthissubjectisprovidedinChapter2. 1.3.2 KineticRateEquations Assumingthatchangesinthechemicalcompositionofthesystemaretheresultsof elementaryreactionsofthetype: XNsp XNsp n(cid:1)x $ nþx ; k¼1;...;N ð1:9Þ ik i ik k r i¼1 i¼1 thenthemolarrateofchangeofeachchemicalspeciescanbeexpressedas 1dN XNr i ¼v_ ¼ ðnþ(cid:1)n(cid:1)Þðrþ(cid:1)r(cid:1)Þ; i¼1;...;N ð1:10Þ V dt i ik ik k k sp k¼1 where YNsp YNsp rkþ ¼kfþkðTÞ ½xl(cid:5)nlþk; rk(cid:1)¼k(cid:1)rkðTÞ ½xl(cid:5)n(cid:1)lk ð1:11Þ l¼1 l¼1 The temperature dependence of k and k are represented by the Arrhenius f r form as (cid:3) (cid:4) E kðTÞ¼A0exp (cid:1) a : ð1:12Þ R T u j 1.3 ChemicalKinetics 7 The pre-exponential factor or collision frequency, A0, is weakly temperature- dependent over the temperatures reached in combustion applications. Such tem- peraturedependenceisrepresentedbyamodifiedformofA0 ¼ATn,inwhichnis calledthetemperatureexponent.Theexponentialpart,onthecontrary,isstrongly temperature-dependent.Thispartrepresentsthefractionofmoleculespossessing enoughenergytosurmounttheactivationenergybarrier,E andundergochemical a reactions. The typical value of this parameter for combustion of hydrocarbons is 40–45 kcal mol–1. When considering Equation 1.10, at the state of dynamic equi- librium the forward and reverse reaction rates must balance, which leads to the PrincipleofDetailedBalancing; KcðTÞ¼kkfkððTTÞÞ¼YNsp ½xl(cid:5)ðnlþk(cid:1)n(cid:1)lkÞ ð1:13Þ rk l¼1 Although this relationship is obtained under chemical equilibrium, the first equalityinEquation1.13alsoholdsatnonequilibriumconditions.Thereasonfor thisisthatchemicalreactionsareassumedtobetooslowtodisturbtheMaxwell– Boltzmann distribution of energy among internal molecular degrees of freedom, hencelocalthermodynamicequilibriumamonginternaldegreesoffreedom.This furtherallowsthedefinitionofasingletemperatureduringchemicalrelaxation.The principleofdetailedbalanceprovidesatooltodeterminethereverserateofreaction, k basedontheforwardrateconstantandequilibriumcoefficient.Alesscommon r practice is also to assign the reverse rate independently of the forward rate [11]. However,thisapproachislessaccuratethanusingtheequilibriumconstant,forthe obviousreasonthatthethermodynamicdataareknownmuchmoreaccuratelythan thekineticdata. 1.3.3 ChemicalTimeScalesandNonequilibriumEffects Each species and reaction in a kinetic mechanism evolves based on definite time scales.Speciestimescalescanbedefinedasfollows: 1 dN 1 XNr t(cid:1)1¼ i¼ ðnþ(cid:1)n(cid:1)Þðrþ(cid:1)r(cid:1)Þ; i¼1;...;N ð1:14Þ i N dt ½N(cid:5) ik ik k k sp i i k¼1 inwhichusehasbeenmadeofEquation1.10.Inthisform,thechemicaltimescaleis determined based on the collective effect of all chemical reactions which either consumeorproducespeciesi.Ifonlyonereactionisconsidered,saythekthreaction, thenthereactiontimescalecanbedefinedast ¼Maxft g,inwhicht ¼N/nr is rk rki rki i k thereactiontimebasedonspeciesparticipatinginreactionk.Combustionchemical reactionsareusuallycharacterizedbyawidespectrumofchemicaltimescales.When achemicalsystemundergoeseitherheatorworkinteractionwiththesurrounding environment on a time scale t , depending on how the chemical time scales ext comparewitht ,thesystemcouldbeeitherinthestateoflocalthermodynamic ext j 8 1 CombustionFundamentals equilibrium(LTE),t (cid:4)Maxftg,nonequilibrium,Minftg<t <Maxftg,or ext i i ext i frozenequilibrium,t (cid:3)Minftg.Thesuddenexpansionofcombustionproducts ext i in an internal combustion engine, or through a hypersonic nozzle and sudden cooling of combustion products through a heat exchanger with constant area, are examplesofsuchinteractions. According to the principle of Le Chatelier, the internal dynamics shifts towards minimizing the effect of external change and re-establishing a new chemical equilibrium,consistentwiththenewvaluesofthestatevariables.Iftheinteraction lowers the gas temperature and density of a highly dissociated mixture, then the internaldynamicswillshiftintheexothermicdirectionsoastominimizethecooling effectofinteraction.Asaresult,three-bodyrecombinationreactions–forexample, H þ H þ M¼H þ MandH þ O þ M¼HO þ M–becomeanimportantpart 2 2 2 of the energy restoration process. Bimolecular reactions also shift towards the exothermicdirection.Fromakineticsstandpoint, three-bodyreactionshavesmall orzeroactivationenergies,whichmakesthemalmosttemperature-insensitiveand ratherhighlypressure-(density)sensitive,whereastherateofbimolecularreactions whichinvolveactivationenergiesaretemperature-sensitive[12].Therefore,sudden cooling to low temperatures and lowering of the density will depress the rate of recombinationandexothermicbimolecularreactionsmarkedly,andtheexothermic processeswilllagintheirattempttorestoretheequilibrium.Afailuretoreleasethe latentenergyofmoleculeformationenhancesthecoolingandputsthesystemfarther outofequilibrium.Iftheexpansionisfastenough,thentheexothermiclaggrows indefinitelyandthecompositionbecomesfrozen[13].Animportantsituationwhere predictionsbasedonequilibriumfailisthepredictionsofCOattheexhaustofan internalcombustionengine.Here,themainreactionstepinoxidationofCOtoCO 2 is CO þ OH¼CO þ H, which involves an activation energy of about 18kcal 2 mol(cid:1)1. This energy barrier makes the reaction temperature-sensitive such that, whenthetemperaturefalls,thereactionbecomesslower,andsodoestheenergy- restorationprocess.SuchaneffectshowsitselfindeparturesfromLTEpredictions. Janbozorgi et al. have examined is redundant the expansion stroke of an internal combustion engine with an intermediate piston speed, and compared the kinetic predictions with frozen and LTE predictions, as shown in Figure 1.1[12]. Clearly, duringtheearlystagesofexpansion,wherethepistonspeedisslow,thestateofthe gasfollowstheLTEpredictionsanddeparturesemergeasthepistonspeedincreases. 1.3.4 KineticsSimplificationandReduction ConsideringthefactthataDKMinvolvesawidespectrumofchemicaltimescales associatedwithchemicalspecies,thesystemofEquation1.10comprisesasetofstiff ordinarydifferentialequations(ODEs),whichcouldbecomputationallyexpensive forreactingflows.Asaresult,agreatdealofefforthasbeendevotedtodeveloping methodsforreducingthesizeofDMKs. Aquasi-steady-stateapproximation(QSSA)[14,15],whichisusuallyemployedfor short-livedradicals,assumesthatafteraso-calledinductionperiodthereactions j 1.3 ChemicalKinetics 9 Figure1.1 COpredictionsduringexpansionstrokeusingdifferentmodelsatanenginespeedof 3000rpm. consumingradicalsbecomemuchfasterthanthoseproducingthem;hencealow, stationary leveloftheseintermediates emerges.Mathematically,this isequivalent withzeronetrateofchangeinEquation1.10,whichisadeliberatetransitionfrom differentialtoalgebraicequationsfortheseintermediates.However,decidingwhich radicalsthisassumptioncanbeappliedtorequiresagooddealofknowledgeand physicalintuitiononthepartofthekineticist. Apartialequilibriumapproximation(PEA)[16]isinvokedforreactionswhichreach astateofdynamicequilibrium.Theentropygenerationduetoachemicalreactionk canbeexpressedas: ! 1XNsp dN 1 XNsp dSkjE;V ¼(cid:1)T mi dti ¼T minik dlk ð1:15Þ i¼1 i¼1 wherel istheprogressvariableofreactionk.Thenecessaryandsufficientcondition k for reaction k to be in equilibrium is that the entropy must be a maximum with respecttoallpossiblechangesdl ,andso k dS 1XNsp dlkjE;V ¼T minik¼0 ð1:16Þ k i¼1 is the constraint to be satisfied by a reaction to be in partial equilibrium. As mentionedinRef.[17],however,thecheckforwhetherareactionksatisfiespartial equilibriumassumptionornot,shouldbebasedontheorderofmagnitudeofthenet rateofchangeduetoreactionkcomparedtonetrateofproductionandnetrateof consumption due to that reaction individually. The more advanced methods are based on ideas from dynamical systems, computational singular perturbation (CSP) [18] and inertial low-dimensional manifold (ILDM) [19], to automatically j 10 1 CombustionFundamentals identify the species and reactions for which QSSA and PEA hold. Other elegant methods,suchasAdaptiveChemistry[20],DirectedRelationGraph(DRG)[21],the ICE-PIC method [22] and rate-controlled constrained-equilibrium (RCCE) [5, 23] have been proposed and developed. Whilst a detailed presentation of combustion kineticsmodelinghasbeenundertakeninChapter8,detailsoftheRCCEmethodare providedinthenextsection. 1.3.4.1 Rate-ControlledConstrained-Equilibrium(RCCE)Method TheideaofRCCEisalogicalextensionofchemicalequilibriumconstrainedtothe conservationofneutralatoms(asdiscussedinSection1.2.3).Owingtotheirveryhigh activation energies, slow ionization and nuclear reactions can be assumed frozen overtheenergiesandtimescalesencounteredincombustionapplications,leadingto conservationofatomicelements.Bythesametoken,thecascadeofconstraintsina chemicallyreactingsystemcanbeeasilyextendedbasedontheexistenceofclassesof slowchemicalorenergy-exchangereactionswhich,ifcompletelyinhibited,would prevent the relaxation of the system to the complete chemical equilibrium. For instance,aheavyhydrocarbondoesnotbreakdownintosmallerfragmentsunlessthe C–C bonds are broken; the total number of moles in a reacting system does not change unless a three-body reaction occurs; and radicals are not generated in the absence of chain-branching reactions, the definition of a single temperature in a chemically reacting system is based on the observation that thermal equilibration among translation, rotation and vibration is in general faster than the chemical reactions [12]. Consistent with the perfect gas assumption and definition (Equa- tion1.6),theconstraintsimposedonthesystembythereactionsareassumedtobea linearcombinationofthemolenumbersofthespeciespresentinthesystem: XNsp C ¼ a N; j¼1;...;N ð1:17Þ j ij i c i¼1 whereC includeskineticconstraintsinadditiontotheelementalconstraintsdefined j earlier,anda hasthesamemeaningasbefore;thevalueofconstraintsjinspeciesi. ij The mathematical work is exactly the same as that presented under chemical equilibrium,andtheconstrained-equilibriumcompositionofthesystemis,there- fore, expressed by Equations 1.7 and 1.8. By taking the time derivative of Equa- tion1.17andusingEquation1.10,itispossibleeasilytoobtain: C_ ¼XNr b ðrþ(cid:1)r(cid:1)Þ; b ¼XNsp a n ð1:18Þ j jk k k jk ij ik k¼1 i¼1 whereN isthenumberofreactionsinthemechanism.Clearly,anyreactionkthat r doesnotchangeallconstraintsjisinconstrained-equilibrium,andnotrequired.The workingequationsofRCCEintermsofconstraintpotentialshavebeenderivedfora constantvolume,constantenergysysteminRef.[8]. Sincetheoxidationofanyheavyhydrocarbonfuelischaracterizedbyessentially thecompletefragmentationoflargemoleculestoamixtureofsmallhydrocarbons,it
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