PengZhang- HKPolyU 7/7/17
2017Globex- PekingUniversity 1
CombustionScienceandEngineeringInstructor: ZHANG,Peng(張鵬)
AssociateProfessor
Address: FG608,DepartmentofMechanicalEngineering
TheHongKongPolytechnicUniversity
Email: [email protected]
2017Globex JulmesterCollegeofEngineering,PekingUniversity
ChemicalKinetics
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Introduction
• Chemical thermodynamics: relates the initial to thefinal equilibrium states of a reactive mixture; doesnot distinguish the path and time in the process (e.g.cycle analysis and NOx in i.c. engines)
• Chemical kinetics describes the path and rates ofindividual reactions and reactants; can be extremelycomplex ® O(103) intermediates and O(104)elementary reactions.
Outline
• Phenomenologicallawofreactionrates:– LawofMassAction(dependenceontemperatureandconcentration)
• Theoriesofreactionrateconstants– Arrheniuslaw
• Chainmechanisms– StraightandBranched
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LawofMassActionforReactionRates
LawofMassAction
• Forasingle-stepforwardreaction:
• Molarrateofchange:
• i andjrelatedby:
,1 1
fN N
ki i i i
i iv v M
= =
¢ ¢¢M ¾¾®å å
ˆˆ,ji
i i i jv v v vww w= =
¢¢ ¢ ¢¢ ¢- -
𝜔"# =𝑑𝑐#𝑑𝑡
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LawofMassAction
• Lawofmassaction:reactionrateproportionaltoproductofconcentrations;scaledreactionrategivenby
• Proportionalityconstantkf(T):reactionrateconstant;primarilyfunctionoftemperature;forelementaryreaction
1
= ( ) c ,iN
vf i
i
k Tw ¢
=Õ
LawofMassAction
Example:
Question:Whatistheexpressionforthereversereaction?
H+HO2→OH+OH
2[HO ][H] 1 [OH] = = = ,2
dd ddt dt dt
w - -
2= [H][HO ].fkw
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ReverseReactions
• Everyforwardreactionhasabackwardreaction:
• Netreactionrate:
1 1M Mb
N Nk
i i i ii iv v
= =
¢¢ ¢¾¾®å å
1 1
.i i
N Nv v
f i b ii i
k c k cw ¢ ¢¢
= =
= -Õ Õ
𝜔"# = 𝜔"#,) + 𝜔"#,+ = 𝜈#-- − 𝜈#- 𝜔) − 𝜔+ = 𝜈#-- − 𝜈#- 𝜔
ReverseReactions
• Atequilibrium:
• Implying:
• Irreversiblereactionapproximation:
( )
1
= .i
Nf v v
i cib
kc K
k¢¢ ¢-
=
=Õ
1
1 1
.i i
N Nv v
f i c ii i
k c K cw ¢ ¢¢-
= =
æ ö= -ç ÷
è øÕ Õ
.ii
Nv
fi
k cw ¢» Õ
0w º
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MultipleReactions
• Practicalreactionsinvolving
Reactants→Products
e.g.:2H2+O2→2H2O
rarely(never!)occurinonestep
betweenreactants(e.g.twoH2
andoneO2)
• Reality:ForH2-O2:(atleast)19
reversiblereactionsand8species
(H2,O2,H,O,OH,H2O,HO2,H2O2)
MultipleReactions
• Generalizedexpression:
,
,, ,
1 1, 1,2, , ,k f
k b
N Nk
i k i i k iki iv M v M k K
= =
¾¾®¢ ¢¢ =¬¾¾å å K
, ,, ,
1 1
, 1,2,..., ,i k i k
i i
N Nv v
k k f k bi i
k c k c k Kw ¢ ¢¢
= =
= - =Õ Õ
Ø Identifyingtheelementarystepsaswellastheirreactionrateconstantsisgenerallydifficult.Ø Solvingacombustionflowfieldbyincludingallreactionsisanextremelydifficulttask.
𝜔"# = /(𝜈#,1-- − 𝜈#,1- )3
145
𝜔1
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ApproximationMethods:QSS&PE
RationalApproximations
• Approximationsbasedoncomparisonofratesofcertainreactionentities– Quasi-steady-state(QSS)speciesapproximation– Partialequilibrium(PE)reactionapproximation
• Facilitatethesolutionprocedureofmultiplereactionsystem.
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QSSSpeciesApproximation• Somechaincarriersare
generatedandconsumedatrapidratessuchthattheirconcentrationsremainatlowvaluesandtheirnetchangeratesareverysmall.
• Consequence:(implicit)algebraicinsteadofdifferentialsolution
• Note:dci/dt maynotbenegligiblecomparedtootherrates
• dci/dt ≠0
• For
• If
• Then
Warning:thedeterminationofci mightinvolvealgebraiciterations,whichcouldstillbecomputationallytaxing.
PartialEquilibriumApproximation
• Ifboththeforwardandbackwardratesofareactionkismuchlargerthanitsnetreactionrate,thenwecanset:
• suchthat
• whichyieldsanalgebraicrelationbetweentheci’s.• Warning:wk notnecessarilysmallcomparedto
(2.1.17)
, ,, ,
1 1
0i k i kN N
v vk k f i k b i
i i
k c k cw ¢ ¢¢
= =
= - »Õ Õ, ,
, ,1 1
i k i kN N
v vk f i k b i
i i
k c k c¢ ¢¢
= =
»Õ Õ
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ApproximationbyGlobalandSemi-globalReactions
• SuccessiveapplicationofQSSspeciesandPEreactionswilleventuallyleadtoaone-stepglobalreaction(atleasttheoretically!).Theprocessistedious,withtheresultsdependontheindividualreactionrateparametersmostofwhicharenotknown.
• Mayaswelljuststartwithaone-stepreaction
describedby
whereni iscalledthereactionorder,andisempiricalinnature.
Fuel + Oxidizer Products k¾¾®
1
,ii
Nn
i
k cw=
= Õ
Therangeofapplicabilityofsuchempiricalglobalandsemiglobal reactionmechanismsdependsontheparticularcombustionphenomenon,aswellastherangeoftheparameterscharacterizingthephenomenon,throughwhichtheempiricalconstantsaredetermined.
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ReactionOrderandMolecularity
• Molecularity,ni:– numberofcollidingmoleculesinanelementaryreaction;
– afundamentalparameter;ni =1,2,3.• Reactionorder,ni:
– influenceofconcentrationofi onthereactionrate;– anempiricalparameter;– ni <2;– ni canalsobenegative;– ni =ni(p)
ArrheniusLawforReactionRateConstants
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TheArrheniusLaw• TheArrheniusLawstatesthedependenceof
thereactionrateconstantontemperature:
• ForconstantEa:
• Modifiedform:
2
ln ( ) ,ao
Ed k TdT R T
=
/( ) = o
aE R Tk T Ae-
= = A A(T) BTa
Svante AugustArrhenius(1859-1927)
NobelPrizeforChemistry in1903Today,Arrheniusisbestknownforhisstudypublishedin1896,onthegreenhouseeffect.
TheActivationEnergy
• For(exothermicreaction),Ea,f <Ea,b,andforwardreactionispreferred.• Example:
a, f c a,bE + Q = E
> 0 cQ
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TheArrheniusNumber,Ar
• ThefactthatAr >>1renderschemicalreactionstemperaturesensitive;
• 5<Ar <10
Ex:
max max
,a ao
E TArR T T
= =
max
max
exp ( / ) = exp 1exp ( / )
a
a
T T TArT T T- é ùæ ö-ç ÷ê ú- è øë û
/ .o
aE R Tdc Bcedt
-=-
24
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CollisionTheoryofReactionRate (1/3)
Assumptions:• EquilibriumMaxwellvelocitydistribution• Two-bodyhard-spherecollision• Reactionoccursifcollision(translational)energyexceedsactivationenergy
Maxwellian Distribution
26
This probability density function f gives the probability, per unit speed, of finding the particle with a speed near v
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• Reducedmass:Collisiondiameter:
• Collisionvelocity:
• Collisionfrequencypervolume:
• Boltzmannvelocitydistribution:
• Collisionfrequencywithenergyinexcessof(Ei+Ej=Ea)
CollisionTheoryofReactionRate(2/3)
1/ 2
,8 ,
o
i j
i
k TVmp
æ ö=ç ÷è ø
1/2
2, ,
,
8 = .o
i j i j i ji j
k TZ n nm
pspæ öç ÷ç ÷è ø*
*/ = .oE R Tn e
n-
, /( )i j i j i jm mm m m= +
= ( + )/ 2 i, j i js s s
,
1/2
/* 2 * *, ,
,
8 = =o
a
i j i j
ojE R T i
i j i ji j
dndnk TZ n n Z em dt dt
psp
-æ ö
= = - -ç ÷ç ÷è ø
• Relating
• Comparing:
CollisionTheoryofReactionRate (3/3)
0i ic n /A=
1/2
/ /2,
,
8 = = ( ) .o o
a a
oE R T E R Toi
i j i j i ji j
dc k TA c c e A T c c edt m
pw s - -æ ö
- =ç ÷ç ÷è ø 1/ 2
2,
,
8( ) , 1/ 2o
oi j
i j
k TA T Amps a
æ ö= =ç ÷ç ÷
è ø~6×1023 mol-1 ~10×10-16cm2
~5×104 cm/sat600K
~3×1013 cm3 mol-1s-1 ≈5×10-11 cm3 molecule-1s-1
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CriticismsonSimpleCollisionTheoryofReactionRateConstant
29
ExperimentallydeterminedAthatareconsiderablysmallerbyafactorof104 toeven108.
Ø Thecollisionbetweentwomoleculesinvolvemorethanjustthetransferoftranslationalenergies.® polyatomicmolecules
Ø Thestructureofpolyatomicmoleculesisnotsphericallysymmetric,thecollisionefficiencyshouldalsodependontheorientation
ØDeviationfromtheoryaccountedbystericfactor,whichcannotbedeterminedbysimplecollisiontheory
A Zy®
UnimolecularReaction(Pressure-dependenceofk)
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UnimolecularReaction
31
A gas-phase unimolecular reaction is apparentlysimple process:
An isolated molecule undergoes a chemical change
e.g. C2H6® CH3 + CH3
which has a rate expression
However, to understand this process in detail,particularly the behavior of k, we need almost thewhole package of modern theoretical chemistry.
This is where our story starts…
32ComprehensiveH2/O2 KineticModelforHigh-PressureCombustionM.P.BURKE,M.CHAOS,Y.JU,F.L.DRYER,S.J.KLIPPENSTEIN,Int JChem Kinet 44:444–474,2012
Unimolecularandrelatedrecombinationreactionsareimportantcomponentsofareactionmechanismoffuelcombustion.
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WhatareUnimolecularReactions?
33Inordertoundergosuchareaction,thereactantmoleculemustenergeticallyactivated(moreorless).Wheredoestheenergycomefrom?
RadiationHypothesis
34
J. Perrin (1919) made the first attempt in his written-in-Frenchpaper “Ann. Phys. (Paris) 11, (1919) 1”“He shows that Arrhenius equation for reaction rate (velocity) can bederived from the Planck or Wien radiation law upon the assumptionthat the chemical action depends on the absorption of a nearlymonochromatic radiation”
§ Experiments suggested
§Wien’s radiation law
where I is the radiation intensity, n radiation frequency, hPlanck constant, and kB Boltzmann constant.§ In an analogy with van’t Hoff’s equation
where Q is the required activation energy and R gas constant.§ Q = Nhv is in agreement with Haber’s quantum theory (1911)
JeanBaptiste PerrinFrenchPhysicist(1870-1942)
TheNobelPrizeinPhysics1926"forhisworkonthediscontinuousstructureofmatter,andespeciallyforhisdiscoveryofsedimentationequilibrium".
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IsCollisionIrrelevant?IsRadiationEnergySufficient?
35
I.Langmuir(1920)debatedPerrin’shypothesisinhispaper“J.Amer.Chem.Soc.42(1920)2190”“Perrin has given one of the strongest argument in favor of thehypothesis. In a unimolecular reaction the amount of substancereacting in a given time is proportional to the amount present. Thechance that any molecule shall react during any small time interval isthus independent of the pressure. But the number of collisions whichthe molecule makes with others is proportional to the pressure.Therefore, as Perrin points out, the reaction of any molecule must be aphenomenon which is independent of collisions between molecules. Ifthe reaction is not due to collision, it seems almost necessary toconclude that it is caused by radiation”
IrvingLangmuirAmericanChemist
(1881-1957)TheNobelPrizeinChemistry1932"forhisdiscoveriesandinvestigationsinsurfacechemistry
78YearsLater…
36
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DebatesContinuedin1922
37
F.A.Lindemann“Trans.Farady Soc.17(1922)598”
FrederickAlexanderLindemann(1886-1957)
Englishphysicistandaninfluentialscientificadviser
toWinstonChurchill
ExperimentalObservation
• Aunimolecular reaction
isreallythehighpressurelimitofasecondorderreaction
whereMisacollisionpartner
• Ex:
R P,k¾¾®
R+M P + M®
4CH +M P + M¾¾®¬¾¾0 0
constant as
~ as 0
RR
R
dc kcdtk k pFirst order reaction
k k p k c pSecond order reaction
¥
• = -
• ® = ®¥Þ -
• ® ®Þ -
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Lindemann-HinshelwoodTheory(1/3)
FrederickAlexanderLindemann1886-1957
Englishphysicistandaninfluentialscientificadviser
ToWinstonChurchill
FrederickLindemann discoveredtheconceptin1921andCyrilHinshelwood developedit
EnglishphysicalchemistNobelPrizeinChemistry1956
SirCyrilNormanHinshelwood1897-1967
Lindemann Theory(2/3)
•
•
1,
1,
*R M R Mf
b
k
k¾¾®+ +¬¾¾
2*R Pk¾¾®
*R
1, R M 1, MR+f b
dc k c c k c cdt
= -
*
* *R
1, R M 1, M 2R R.f b
dck c c k c c k c
dt= - -
Lindemann assumedthattheratecoefficientprocessfortheactivationcanbecalculatedfromsimplecollisiontheory
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• AssumesteadystateforcR* ,
• Then
Lindemann Theory(3/3)*R/ 0,dc dt =
*1, R M
R1, M 2
f
b
k c cc
k c k=
+
( )2 1, 1,R
2 1 M,
/ ,
1 /( ) f b
Rb
k k kdc cdt k k c
= -+
2 1, 1,
2 1, M
2 1, 1,
0 1, M
0
( / )
1 /( ) ( / ) ;
0;
1 1 1 +
f b
b
f b M
f M
k k kk
k k ck k k k as c First orderk k c as c Second order
k k k
¥
¥
• =+
® = ®¥ -
® = ® -
• =
ChainReaction
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ChainReactions
43
44
Semenov'soutstandingworkonthemechanismofchemicaltransformationincludesanexhaustiveanalysisoftheapplicationofthechain theorytovariedreactions(1934–1954)and,moresignificantly,tocombustion processes.Heproposedatheoryofdegeneratebranching,whichledtoabetterunderstandingofthephenomenaassociatedwiththeinductionperiodsofoxidationprocesses
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AnExample:H2 +Br2 ≡2HBr
45
MaxErnstAugustBodenstein(1871-1942)
MaxErnstAugustBodenstein:Germanphysicalchemistknownforhisworkinchemicalkinetics.Hewasfirsttopostulateachainreaction mechanismandthatexplosions arebranchedchainreactions,laterappliedtotheatomicbomb
However,healsofound
StraightChainReactions
• Theconsumptionofoneradicalleadstotheproductionofanotherradical
• Example:Hydrogen-halogensystemX2:I2,Br2,Cl2,F2
•1,
2,
3,
1,
2,
2
2
2
X + M X + X + M Chain initiation (X1 )
X + H HX + H Chain carrying (X2 )
H + X HX + X Chain carrying (X3 )
X + X + M X2 + M Chain termination (X1 )
H + HX X+ H2 Chain carrying (X2
f
f
f
b
b
k
k
k
k
k
f
f
f
f
f
¾¾®
¾¾®
¾¾®
¾¾®
¾¾® )(X2b)
Bondenergy:Br2:189KJ/mol,H2:427KJ/mol
(X1b)
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Halogen-HydrogenSystem(1/3)• Reactionrates:
• Steady-stateassumptionforHandX:
•[H] [X] 0 and 0d ddt dt
= =
2
222 2
2 2
[H ] = - [X][H2] + [H][HX]
[X ] - [X ][M] - [H][X ] + [X] [M]
[H] [X][H ] - [H][X ] - [H][HX]
2, f 2,b
1, f 3, f 1,b
2, f 3, f 2,b
d k kdtd k k kdtd k k kdt
=
=
2 2 2
2
2 2
[X] = 2 [X ][M]- [X][H ] + [H][X ]
+ [H][HX] - 2 [X] [M][HX] [X][H ] + [H][X ] - [H][HX]
1, f 2, f 3, f
2,b 1,b
2, f 3, f 2,b
d k k kdtk kd k k kdt
=
1/2 1/22, 1, 1, 2 2
2, 3 2,
2 ( / ) [H ][X ][HX] = .1+( / )[HX]/[X ]f f b
b f
k k kddt k k
• Fromdetailedanalysis
• Assumeone-stepreaction
• Detailedanalysisshows:– Complexinsteadoflineardependenceon[x2]– Inhibitingeffectof[HX]
Halogen-HydrogenSystem(2/3)
1/2 1/22, 1, 1, 2 2
2, 3 2,
2 ( / ) [H ][X ][HX] = .1+( / )[HX]/[X ]f f b
b f
k k kddt k k
02 2H + X 2HX (X0)k¾¾®
0 2 2[HX] = 2 [H ][X ], d kdt
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Halogen-HydrogenSystem(3/3):HeatsofReaction
• Overallreactions,(X0),areexothermic,henceself-sustaining;F2highlyexothermic,I2 least
• Initiationreactions,(X1f),areendothermic• (X1f )requireslessheatthan (104kcal/mol),
hencechaininitiating• Chaincarryingsteps:(X2f) and(X3f)
– HighlyexothermicforF2– ExtremelyweaklyexothermicforI2
2H +M 2H+M®
HydrogenandChlorineReaction
Flameinhibitors• HalogenatedcompoundsaregoodinhibitorsbecauseXscavengesH• HalogenradicalpresenteitherasahalogenacidorhalogenatedHC• EasiertodissociatethehalogenfromthecompoundthanH
• TheHCsegmentcanreleaseheatuponoxidation• Largecompoundcanabsorbmoreheat
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ChainBranchingReaction
BranchedChainReactions:H2-O2 System
• Theconsumptionofoneradicalgeneratesmorethanoneradical
H + O2 → OH + O Chainbranching (H1)O + H2 → OH + H Chainbranching (H2)OH + H2 → H2O + H Chaincarrying (H3)
• Thenetof(H1) to (H3) is:3H2 + O2 → 2H2O + 2Hshowing2Hproducedpercycle
• ChaincarryingstepscanbeweakeningH + O2 + M→ HO2 + MCH4 + H → CH3 + H2
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BranchedChainReactions:PressureEffect(1/2)
1
2
InitiationChain branching cycle
Gas termination
Wall termination
g
w
k
k
k
k
nR CR C aC P
C R R PC P
¾¾®
+ ¾¾® +
+ + ¾¾®
¾¾®
21 2
1 2 c
d[C] = [R] + ( 1) [R][C] [R] [C] [C]
[R] + [R]( )[C]
ng w
n
k a k k kdt
k k a a
- - -
= -
2
2
[R] + = 1 + .
[R]g w
c
k ka
k
BranchedChainReactions:PressureEffect(2/2)
•
– blowsupfora>ac– delaysfora<ac
•
1 2 cd[C] = [R] + [R]( )[C],nk k a adt
-
2
2
2
2
[R] + = 1 +
[R] 1 + 0[R][R]
1 +
g wc
w
g
k ka
kk as pkk
as pk
® ®¥ ®
® ®¥ ®¥
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55
Note:theactualfireontheHindenburgairshipwastheresultofasparkthatignitedtheouterskinandnotthespontaneousexplosionofhydrogen
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