PengZhang- HKPolyU 7/7/17

2017Globex- PekingUniversity 1

CombustionScienceandEngineeringInstructor： ZHANG,Peng(張鵬)

AssociateProfessor

Address: FG608,DepartmentofMechanicalEngineering

TheHongKongPolytechnicUniversity

Email: pzhang@polyu.edu.hk

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