Handbook of Combustion, Volume 1 (Fundamentals and Safety)

499 Pages·2010·30.32 MB·English

by  Maximilian Lackner

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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
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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,theproducts’temperature–alsocalledtheflame
temperature(T )–maybedifferentfromthetemperatureofreactants(T ).Theheat
f R
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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.
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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
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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
transcendentalequationsforc’s.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
a“full set”ofrate-equations forthesevariables based ona“full set”ofelementary
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
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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
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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
calledthe“temperatureexponent.”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
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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-called“inductionperiod”thereactions
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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
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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