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A Comprehensive Modeling Study of n-Heptane Oxidation

H. J. CURRAN,* P. GAFFURI, W. J. PITZ, AND C. K. WESTBROOKLawrence Livermore National Laboratory, Livermore, CA

A detailed chemical kinetic mechanism has been developed and used to study the oxidation of n-heptane in flowreactors, shock tubes, and rapid compression machines. Over the series of experiments numerically investi-gated, the initial pressure ranged from 1–42 atm, the temperature from 550–1700 K, the equivalence ratio from0.3–1.5, and nitrogen-argon dilution from 70–99%. The combination of ignition delay time and speciescomposition data provide for a stringent test of the chemical kinetic mechanism. The reactions are classed intovarious types, and the reaction rate constants are given together with an explanation of how the rate constantswere obtained. Experimental results from the literature of ignition behind reflected shock waves and in a rapidcompression machine were used to develop and validate the reaction mechanism at both low and hightemperatures. Additionally, species composition data from a variable pressure flow reactor and a jet-stirredreactor were used to help complement and refine the low-temperature portions of the reaction mechanism. Asensitivity analysis was performed for each of the combustion environments. This analysis showed that thelow-temperature chemistry is very sensitive to the formation of stable olefin species from hydroperoxy-alkylradicals and to the chain-branching steps involving ketohydroperoxide molecules. © 1998 by The CombustionInstitute

INTRODUCTION

There is continued interest in developing abetter understanding of the oxidation of largehydrocarbon fuels over a wide range of operat-ing conditions. This interest is motivated by theneed to improve the efficiency and performanceof currently operating combustors and reducethe production of pollutant species emissionsgenerated in the combustion process. This studyparticularly focuses on the effect of elevatedpressures on the oxidation of n-heptane. Manyimportant practical combustion systems such asspark-ignition, diesel and gas-turbine enginesoperate at pressures well above 1 atm. n-Hep-tane is a primary reference fuel (PRF) foroctane rating in internal combustion enginesand has a cetane number of approximately 56,which is very similar to the cetane number ofconventional diesel fuel. Therefore, a betterunderstanding of n-heptane oxidation kinetics isuseful in studies of engine knock and autoigni-tion. Recent experimental studies of n-heptaneoxidation have focused on shock tubes [1–5],jet-stirred reactors [6–8] performed under sta-tionary conditions, rapid compression machines[9–12], engines [13–19], plug flow reactors [20–

22], and jet-stirred flow reactors [23, 24] inwhich a dynamic behavior is observed. All ofthese systems exhibit phenomena including self-ignition, cool flame, and negative temperaturecoefficient (NTC) behavior. Furthermore, vari-ation in pressure from 5–40 bar changes thetemperature range over which the NTC regionoccurs.

Recent modeling studies of the premixedsystems such as stirred reactors and shock tubescited above [2–6] have helped in the develop-ment of detailed chemical kinetic mechanismsthat describe n-heptane oxidation. These publi-cations have been complemented by the work ofChevalier et al. [25, 26], Muller et al. [27],Foelsche et al. [28], and Lindstedt and Maurice[29]. In addition, Bui-Pham and Seshadri [30]carried out a numerical study of an n-heptanediffusion flame. More recently, Ranzi et al. [32]have used a semi-detailed chemical kineticmodel to simulate n-heptane pyrolysis and oxi-dation. In addition, this semi-detailed modelwas used to simulate the oxidation of primaryreference fuel (n-heptane and 2,2,4-trimethyl-pentane) mixtures [22]. Come et al. [33] haveused a computer package to generate chemicalkinetic mechanisms for n-heptane and iso-oc-tane.

In this study we include all of the reactionsknown to be pertinent to both high- and low-temperature kinetics. We show how the detailed

Corresponding author.Current address of H. J. Curran, L-407, Lawrence Liver-more National Laboratory, Livermore, CA 94550.

COMBUSTION AND FLAME 114:149–177 (1998)© 1998 by The Combustion Institute 0010-2180/98/$19.00Published by Elsevier Science Inc. SSDI 0010-2180(97)00282-4

kinetic model reproduces the measured resultsin each type of experiment, including the fea-tures of the NTC region. We discuss the specificclasses of elementary reactions and reactionpathways relevant to the oxidation process, howwe arrived at the rate constants for each class ofreaction, and indicate which reactions are themost important in consuming the fuel at bothlow and high temperature. In addition, a sensi-tivity analysis was carried out on each set ofexperimental results by changing the rate con-stants for different classes of reaction in thekinetic mechanism. The results of this analysisindicate the relative importance of each class ofreaction and also the variation in contributionof these classes of reactions to the changingconditions of the experiments.

MODEL FORMULATION

Computer modeling of n-heptane oxidation wasperformed using the HCT (hydrodynamics,chemistry, and transport) program [31], whichsolves the coupled chemical kinetic and energyequations and permits the use of a variety ofboundary and initial conditions for reactivesystems, depending on the needs of the partic-ular system being examined. The present de-tailed reaction mechanism was constructedbased on the hierarchical nature of hydrocar-bon–oxygen systems. The mechanism was builtin a stepwise fashion starting with small hydro-carbons and progressing to larger ones. Much ofthis work has been documented previously [34–38] but has required extensive refinements. Asemi-detailed kinetic scheme developed byRanzi et al. [32, 39] in which both the low- andthe high-temperature reaction submechanismsare reduced to a lumped kinetic model involvinga limited number of intermediate steps wasemployed. This lumped reaction model wasfound to be extremely valuable in identifyingportions of the detailed mechanism that wereespecially sensitive and that required modifica-tion and improvement.

To cover the complete range of temperatureand pressure typical of n-heptane oxidation, itwas important to include both the low- and thehigh-temperature mechanisms. At higher tem-peratures, unimolecular fuel and alkyl radical

species decomposition and isomerization reac-tions are especially important, while, at lowtemperatures, H-atom abstraction from the fuelmolecule and addition of alkyl radicals to mo-lecular oxygen followed by reactions of thealkylperoxy radicals dominate the oxidationmechanism.

The low-temperature submechanism that wasdeveloped previously [35–38] has undergoneseveral major changes that were necessary toexplain the combustion of n-heptane in thetemperature range 550–900 K. A b-decomposi-tion reaction path for hydroperoxy-alkyl radi-cals, leading to the formation of smaller olefinsand aldehydes, was included. This step hashelped explain the selectivity for lower alkenes[7, 8, 12] and also increases the number of chainpropagation pathways that compete with thechain-branching channel in the NTC region.

Furthermore, ketohydroperoxide specieshave been identified during the oxidation ofn-heptane in a motored CFR engine [15]. Con-sequently, we have added a pathway leading tothe formation of ketohydroperoxide compoundsfrom the isomerization reactions of O2QOOHradicals. These ketohydroperoxide species sub-sequently decompose, producing one other hy-droxyl radical (chain-branching step) in addi-tion to other oxygenated compounds. Theinclusion of this step has had a large influenceon the reproduction of the observed NTC be-havior and two-stage ignition of the fuel. Thelumped reaction model was particularly valu-able in identifying this reaction pathway asbeing necessary in the modeling of alkane fuels.

Finally, the THERM program [40] of Ritterand Bozzelli, which uses group additivity rulesdeveloped by Benson [41], was used to evaluatethermochemical quantities for all chemical spe-cies for which there were no available data. Inaddition to improving the specific heats andenthalpies of formation for many C7 com-pounds, it was found that reverse rate constantsof many reactions in the low-temperature re-gime were quite important, and improved ther-modynamic parameters for these species pro-vided better reverse reaction rate constants.H/C/O groups and bond dissociation groupswere updated based on recent work by Bozzelliand coworkers [42].

150 H. J. CURRAN ET AL.

CLASSES OF REACTIONS

We have developed our chemical kinetic modelin a systematic way. The oxidation of any fueltakes place through a series of steps. For exam-ple, at both low and high temperature, n-hep-tane undergoes H-atom abstraction, leading tothe formation of four possible, structurally dis-tinct alkyl radicals. At high temperatures theseradicals decompose via b-scission to yield asmaller olefin and another radical species. How-ever, at low temperatures, these four alkyl rad-icals undergo addition to O2 leading to theformation of four heptylperoxy radicals. There-fore, we can categorize each step in the oxida-tion process as a class of reaction, including all thepossible reactions taking place. These classes ofreactions are listed below and indicate the com-plexity of the model. The complete reaction mech-anism for n-heptane oxidation included 2450 ele-mentary reactions among 550 chemical species.The entire mechanism is not represented here dueto its length, but we discuss below its contents, anda complete copy can be obtained from the authorsin either printed or electronic form.

We have found many reaction types to beimportant, and particular attention and carehave been taken in developing rate constantexpressions for these reaction classes. However,not all reactions have been found to be impor-tant, and we have at times been expedient in ourtreatment of certain reaction types. For exam-ple, in our treatment of H-atom abstractionfrom C7 alkene species, we assume only onealkenyl radical is produced, which is taken as an“average” over the species possible for n-hep-tane. Furthermore, we have also simplified alk-enyl consumption to consist only of unimolecu-lar decomposition to products we have selectedas being reasonable for the fuel. Thus in n-heptane, alkenyl radical decomposition is as-sumed to lead to allyl radical and olefinic prod-ucts. This treatment is very approximate, andfurther attention may be warranted, but this hasproven adequate for our current applications.The major classes of elementary reactions con-sidered in the present mechanism include thefollowing:

1. Unimolecular fuel decomposition2. H-atom abstraction from the fuel

3. Alkyl radical decomposition4. Alkyl radical 1 O2 to produce olefin 1 HO2

directly5. Alkyl radical isomerization6. Abstraction reactions from olefin by OH, H,

O, and CH37. Addition of radical species to olefin8. Alkenyl radical decomposition9. Olefin decomposition

10. Addition of alkyl radicals to O211. R 1 R9O2 5 RO 1 R9O12. Alkyl peroxy radical isomerization (RO2 º

QOOH)13. RO2 1 HO2 5 RO2H 1 O214. RO2 1 H2O2 5 RO2H 1 HO215. RO2 1 CH3O2 5 RO 1 CH3O 1 O216. RO2 1 R9O2 5 RO 1 R9O 1 O217. RO2H 5 RO 1 OH18. RO decomposition19. QOOH 5 QO 1 OH (cyclic ether forma-

tion via cyclization of diradical)20. QOOH 5 olefin 1 HO2 (radical site b to

OOH group)21. QOOH 5 olefin 1 carbonyl 1 OH (radical

site g to OOH group)22. Addition of QOOH to O223. Isomerization of O2QOOH and formation

of ketohydroperoxide and OH24. Decomposition of ketohydroperoxide to

form oxygenated radical species and OH25. Cyclic ether reactions with OH and HO2

The naming conventions used above are Rand R9, denoting alkyl radicals or structures,and Q, denoting CnH2n species or structures.For each of these classes of reactions we use thesame reaction rate constant for analogous oc-currences in different molecules. Thus, we as-sume that the abstraction of a tertiary H atomby reaction with OH radicals has exactly thesame rate in 2-methyl butane, 2-methyl pentane,3-methyl pentane, and in iso-octane. Corre-spondingly, the total rate of tertiary H-atomabstraction by OH in 2,3-dimethyl butane and in2,4-dimethyl pentane is twice that in 2-methylpentane, since the two former fuels have twosuch H atoms at tertiary sites. The n-heptanemechanism contains C4, C5, and C6 submecha-nisms. We treat all of the different reactionclasses provided above in exactly the same wayregardless of whether the fuel is n-butane, n-

151MODELING STUDY OF n-HEPTANE OXIDATION

pentane, n-hexane, or n-heptane. Our treat-ment of these reaction classes in described inthe following sections.

HIGH-TEMPERATURE MECHANISM

Reactions in classes 1–9 are sufficient to simu-late many high-temperature applications of n-heptane oxidation. We have made a number ofad hoc assumptions and approximations thatmay not be suitable for some problems involvingalkene and alkyne fuels, and further analysis isneeded to refine details for these fuels. How-ever, under the conditions of this study, n-heptane oxidation is relatively insensitive tothese assumptions.

Reaction Type 1: Unimolecular FuelDecomposition

These reactions produce two alkyl radicals orone alkyl radical and one H atom. Because thepaths producing H atoms have very high activa-tion energies, they are only important in thereverse direction, where they serve as sinks of Hatoms. Type 1 reactions serve as initiation stepsand as fuel consumption reactions but only atrelatively high temperatures, such as thosefound in shock tubes.

We calculate the rate constant expressions forunimolecular decomposition of n-heptane fuelfrom the reverse reaction, the recombination oftwo radical species to form the stable parentfuel, and from microscopic reversibility. Forproducts of alkyl 1 H atom, we assume a rateconstant for recombination of 1 3 1014 cm3

mol21 s21, based on the recommendation ofAllara and Shaw [43]. There is very little infor-mation available on the rate of this reaction forC2 alkyl radicals and larger. For decompositionswhere the smallest product is CH3, we assumethe reverse recombination rate to be 1.0 3 1013

cm3 mol21 s21, similar to that recommended byBaulch et al. [44] for CH3 1 CH3 5 C2H6.When the smallest product is an alkyl radicalsuch as ethyl or larger, we assume the recombi-nation rate constant to be 8.0 3 1012 cm3 mol21

s21.

Reaction Type 2: H-Atom Abstraction

At both low and high temperatures, H-atomabstraction takes place at both primary andsecondary sites of n-heptane, which leads to theformation of four distinct heptyl radicals. Thereare no tertiary sites on the n-heptane molecule,but we have included these rate constant expres-sions so that we can provide a complete set, andthese will be used in modeling iso-octane oxida-tion.

We assume that the rate constant for abstrac-tion at any particular site (1°, 2°, or 3°) to beequal to that at the same type of site in othermolecules. For example, we chose the rateconstant for H-atom abstraction by OH radicalsat both primary and secondary sites to be iden-tical to those recommended by Droege andTully [45] for propane fuel. The rate constantfor tertiary H-atom abstraction by OH radicalswas taken from a similar study by the sameauthors [46] with isobutane fuel. We summarizethese rate constant expressions in Table 1, andcalculate the reverse rate constants from ther-mochemistry.

Reaction Type 3: Alkyl Radical Decomposition

Alkyl radical decomposition is important only atrelatively high temperatures (T $ 850 K) underthe conditions of this study, as the addition ofalkyl radicals to molecular oxygen, even thougha bimolecular reaction, is faster than b-scissiondue to the relatively high activation energybarriers for alkyl radical decomposition (there isno energy barrier for the addition to O2). Gen-erally, we have chosen products based on theprinciple that b-scission will be the dominantdecomposition path for alkyl radicals. In manycases there are two or more pathways possiblefor an alkyl radical, with different products, andall such paths have been included in the presentmechanism.

Previously, we treated this type of reaction inthe forward direction, estimating the rate con-stant for each b-scission by analogies with sim-ilar reactions. However, because alkyl radicalb-scission is endothermic, we now calculate therate constant in the reverse, exothermic direc-tion, i.e., the addition of an alkyl radical (or Hatom) across the double bond of an alkene. In


this way we avoid the additional complexity ofthe enthalpy of reaction, allowing the forward,b-scission rate constant to be calculated fromthermochemistry.

Rate constants for the addition of radicalsacross a double bond are reasonably well knownand are very similar depending on (1) the site ofaddition (terminal or internal C atom) and (2)the type of radical adding on. We use rateconstants for these addition reactions based onthe recommendations of Allara and Shaw [43].Typically, the rate of addition of a H atomacross a double bond has a pre-exponential

!-factor of 1 3 1013 cm3 mol21 s21 with anactivation energy of 1200 cal/mol if the H atomadds to the terminal C atom of the alkene and2900 cal/mol if the H atom adds to an internal Catom. The rate constant for the addition of analkyl radical has a lower !-factor and higheractivation energy than for the addition of a Hatom. For the addition of an alkyl radical, the!-factor is approximately 8.5 3 1010 cm3 mol21

s21 with an activation energy of approximately7800 cal/mol if addition occurs at the terminal Catom and 10,600 cal/mol if addition occurs at aninternal C atom. We assume these reactions are

TABLE 1

Rate Constant Expressions for H-Atom Abstraction from the Fuel (cm3-mol-s-cal Units)

Radical Site

Rate expression per H atom

Reference! n %a

H Primary 9.33 3 106 2.0 7,700 47H Secondary 4.55 3 106 2.0 5,000 47H Tertiary 1.26 3 1014 0.0 7,300 48OH Primary 1.75 3 109 0.97 1,590 45OH Secondary 2.34 3 107 1.61 235 45OH Tertiary 5.73 3 1010 0.51 63 46O Primary 7.33 3 105 2.4 5,500 49O Secondary 2.35 3 105 2.5 2,230 49O Tertiary 1.10 3 1013 0.0 3,280 49CH3 Primary 2.17 3 1011 0.0 11,600 50CH3 Secondary 2.00 3 1011 0.0 9,500 50CH3 Tertiary 1.00 3 1011 0.0 7,900 51HO2 Primary 1.34 3 1012 0.0 19,400 52a

HO2 Secondary 1.22 3 1012 0.0 17,000 52a

HO2 Tertiary 2.16 3 1012 0.0 14,400 52a

CH3O Primary 5.27 3 1010 0.0 7,000 53CH3O Secondary 5.48 3 1011 0.0 5,000 53CH3O Tertiary 1.90 3 1010 0.0 2,800 53O2 Primary 4.17 3 1012 0.0 49,000 53b

O2 Secondary 1.00 3 1013 0.0 47,600 53b

O2 Tertiary 2.00 3 1013 0.0 41,300 53b

C2H5 Primary 1.67 3 1010 0.0 13,400 43C2H5 Secondary 2.50 3 1010 0.0 10,400 43C2H5 Tertiary 1.00 3 1011 0.0 7,900 43C2H3 Primary 1.67 3 1011 0.0 18,000 54C2H3 Secondary 2.00 3 1011 0.0 16,800 54C2H3 Tertiary 2.00 3 1011 0.0 14,300 54CH3O2 Primary 2.02 3 1012 0.0 20,430 55c

CH3O2 Secondary 2.02 3 1012 0.0 17,700 55c

CH3O2 Tertiary 2.00 3 1012 0.0 14,000 55c

RO2 Primary 2.02 3 1012 0.0 20,430 55c

RO2 Secondary 2.00 3 1012 0.0 17,700 55c

RO2 Tertiary 2.00 3 1012 0.0 16,000 55c

a !-factors adjusted from original values of 1.0 3 1012.b As recommended by [53], %a ' DH. Overall !-factor of 4.0 3 1013 [53] was partitioned between 1°, 2°, and 3°.c Analogy with RH 1 HO2. !-factor has been adjusted down from 1.8 3 1012.


in their high-pressure limit for the conditionsconsidered in this study.

Reaction type 4: Alkyl radical 1 O2 5olefin 1 HO2

The reaction of alkyl radicals with O2 proceedsthrough many reaction channels. Most of thesechannels can be represented by the addition ofan alkyl radical, R, to O2, reaction type 10,followed by alkylperoxy, RO2, radical isomeriza-tion to a hydroperoxy-alkyl radical, QOOH,reaction type 12, and subsequent decompositionor further addition of the hydroperoxy-alkylradical to O2, reaction types 19–22. Reactiontype 4 represents the only chemically activatedchannel of the R 1 O2 reaction system that isconsidered in the reaction mechanism, proceed-ing through a vibrationally excited alkylperoxycomplex, RO*

2, and leads to the formation of an

olefin and HO2 radical.

R 1 O23 RO*23 olefin 1 HO2

The reaction mechanism leading to the forma-tion of conjugate olefin from alkyl plus molec-ular oxygen is a topic of considerable currentresearch. Quelch and coworkers [56] have pro-posed that C2H5 1 O2 reacts through a cyclictransition state and then proceeds to ethyleneplus hydroperoxy radical, HO2, through a con-certed elimination. This proposed reaction se-quence does not proceed through the hydroper-oxy-alkyl radical, QOOH. This is in agreementwith the earlier proposals of Walker and co-workers [57, 58], in which they suggested that apathway involving a cyclic quasi-stable structuremust exist, without the prior formation ofC2H5O2, which can either decompose back intoC2H5 1 O2 or unimolecularly decompose intoC2H4 1 HO2. However, Wagner et al. [59]argue that, were this cyclic intermediate to exist,HO2 would also react with C2H4 forming C2H5

1 O2 to a significant extent instead of exclu-sively forming CH2CH2O2H, followed by thesubsequent formation of ethylene oxide,C2H4O 1 OH. No evidence of such behaviorwas observed by Walker and co-workers. Thus,questions concerning the kinetic behavior of theC2H5 1 O2 3 C2H4 1 HO2 reaction sequenceremain. However, all proposed alkyl radical plus

molecular oxygen reactions do lead to the for-mation of the conjugate olefin and HO2.

Koert et al. [60] showed that only about 10%of the propene produced from propane oxida-tion occurs through reaction type 4, i.e., addi-tion of an alkyl radical to O2 to form RO**

2,

isomerization of RO*2

to QOOH*, and dissoci-ation of QOOH* to olefin and HO2 radical,over the temperature range 650–800 K and at10–15 atm. The remaining 90% of the propeneis formed through the addition of the alkylradical to O2, isomerization of propyl-peroxyradical to hydroperoxy-propyl radicals, and sub-sequent decomposition to form propene plusHO2.

C3H7 1 O23 C3H7O*2

C3H7O*23 C3H7O2

C3H7O23 C3H6OOH

C3H6OOH3 C3H6 1 HO2

It is expected that the contribution of reactiontype 4 will decrease significantly with increasingnumber of carbon atoms in the alkyl radical.These alkyl radical plus O2 chemical activationreactions put approximately 34.0 kcal mol21 ofenergy into the molecule, which is quickly dis-tributed to all the possible vibrational modes.There are 3N-6 modes of vibration possible fora non-linear molecule, where N is the numberof atoms in the molecule. Thus, as the numberof atoms in a molecule increases, the probabilitythat a critical number of quanta will reside inthe one vibrational mode needed for reactionwill decrease significantly. Alternatively, theRO*

2radicals can undergo collisional stabiliza-

tion, which occurs nearly 100% of the time atthe high pressure (13.5–40.0 bar) and low tem-perature (550–800 K) conditions investigated inthis study. When the temperature greatly ex-ceeds 800 K, the equilibrium of the R 1 O2 5RO2 reaction favors R 1 O2, and large alkylradicals (R) are mostly consumed by decompo-sition (reaction type 3). For these reasons, wehave not included reaction type 4 in our mech-anism for alkyl radical containing more thanfour carbon atoms. We have obtained goodagreement between experimental and computa-


tional results for both C5 and C6 species usingthis assumption [61–63].

Reaction Type 5: Alkyl Radical Isomerization

Alkyl radicals can transfer H atoms from onesite to the radical site at rates that depend onthe type of C-H bond (primary, secondary, ortertiary) broken and the ring strain energy bar-rier involved. This process has been well knownfor many years and was summarized by Benson[41]. The rate constants for isomerization aredescribed in terms of the number of atoms inthe transition state ring structure (including theH atom) and the type of site at which thetransferred H atom was initially located. Thus,we estimate the activation energy, %a, using theexpression,

%a 5 DHrxn 1 ring strain 1 Eabst (1)

where DHrxn is taken to be the enthalpy ofreaction and is only included if the reaction isendothermic. The activation energy for abstrac-tion is determined, following the analysis ofBozzelli and Pitz [64], from an Evans-Polanyiplot, Eabst vs. DHrxn (taken in the exothermicdirection) of similar H atom abstraction reac-tions, RH 1 R9 5 R 1 R9H, leading to thefollowing expression:

Eabst 5 12.7 1 ~DHrxn 3 0.37!

The !-factors were obtained using RADI-CALC [65], a computer code that implementstransition state theory. RADICALC calculatesthe change in entropy of the radical to thetransition state due to loss or gain of internalrotors, of specific vibrations, and of opticalisomers. A more in-depth description of the useof RADICALC has been published by Bozzelliand Pitz [64]. We consider only H-atom trans-fers, excluding CH3 or larger radical transfers.The rate constants employed for heptyl radicalisomerizations are summarized in Table 2.

Reaction Type 6: H-Atom Abstraction fromOlefin

Although smaller olefin species are always im-portant in virtually all combustion environ-ments, larger olefin species are generally much

less important. At high temperatures, alkyl rad-icals decompose rapidly to smaller olefins, whileat lower temperatures the RO2 reaction paths,which produce only small quantities of largeolefins, tend to dominate. As a result, we havechosen not to include a great deal of detail inthe reactions of the larger olefins.

We have assumed that, for olefins larger thanC4, each alkene can have H atoms abstracted byH, O, OH, and CH3. However, because (1)site-specific abstraction rate constants from ole-fins are more difficult to estimate than fromparaffins, (2) the number and complexity of theproduct species become difficult to follow, (3)the fate of those products are not very wellunderstood, and (4) the sensitivity of the com-puted results to variations in these steps is verysmall, we decided to assume a single rate ex-pression for reactions of all of these largeolefins with each of the radical species. Anapproximate rate value is assumed for eachabstracting radical, intended to provide a rateconstant averaged over primary, secondary, al-lylic, and vinylic C™H sites. As olefins get largerin carbon number, the double bond affects onlya small portion of the molecule, the rest ofwhich remains paraffinic in character. Thus, forlarge olefins, we expect the rate constants forH-atom abstraction to look more like those for

TABLE 2

Rate Constant Expressions for C7 Alkyl RadicalIsomerization Reactions. (cm3-mol-s-cal Units)

IsomerizationRingSize

Rate Expression

! n %a

1C7H15 º 2C7H15 3 5.48 3 108 1.62 38,760reverse 1.74 3 107 2.01 41,280

1C7H15 º 3C7H15 4 1.39 3 109 0.98 33,760reverse 4.41 3 107 1.38 36,280

1C7H15 º 3C7H15 6 4.28 3 1011 21.05 11,760reverse 1.36 3 1010 20.66 14,280

1C7H15 º 4C7H15 5 2.54 3 109 0.35 19,760reverse 1.61 3 108 0.74 22,280

2C7H15 º 3C7H15 3 9.59 3 108 1.39 39,700reverse 9.59 3 108 1.39 39,700

2C7H15 º 3C7H15 5 3.22 3 109 0.13 20,700reverse 3.22 3 109 0.13 20,700

2C7H15 º 4C7H15 4 1.76 3 109 0.76 34,700reverse 3.50 3 109 0.76 34,700

3C7H15 º 4C7H15 3 6.04 3 108 1.39 39,700reverse 1.20 3 109 1.39 39,700


alkanes than smaller olefins. Furthermore, onlyone radical is assumed to be produced fromeach large olefin, which we designate as an“alkenyl” radical, an “average” of the possiblevinylic, allylic, primary, and secondary radicalspecies formed from n-heptane fuel.

This type of treatment is expedient in thecurrent study, and modifications would beneeded in cases of specific interest. For exam-ple, if the study considered a large olefin as theprimary fuel, this approach might be inappro-priate and refinements would be necessary. Therate constant expressions estimated in thismechanism are reported in Table 3.

Reaction Type 7: Addition of Radical Speciesto Olefin

Similar to the discussion of type 6 reactions, wehave tried to account for the possible additionreactions of small radicals with large olefinspecies. We have already considered the addi-tion reactions of H and CH3 to olefins, as part oftype 3, b-scission reactions above. HO2 additionto olefins is considered later as the reverse ofreaction type 20. RO2 addition to olefins wasnot considered.

OH and O addition occurs at the doublebond, which occupies only a small portion of alarge olefin. Thus, the !-factors below have

been reduced to reflect the steric factor associ-ated with the probability of an OH or O radicalreacting with the olefinic rather than the paraf-finic part of the molecule.

The addition of OH or O radical to a largeolefin forms an adduct not specifically includedin the reaction mechanism. Instead, we haveexamined the likely b-scission products of theadduct and used those as products for reactiontype 7. We will add more detail to the treatmentof OH and O radical addition reactions aschemical understanding increases and computa-tional capabilities grow. We have found thisprocedure to be sufficient over the range ofconditions applicable to this study. The specificreaction rate constants we have assumed arereported in Table 4.

Reaction Type 8: Alkenyl RadicalDecomposition

Since H-atom abstraction from an olefin hasbeen greatly simplified, we have also chosen tosimplify the subsequent consumption of theaverage alkenyl radical formed to consist only ofunimolecular decomposition to products we se-lected as being “reasonable” for the fuel. Thusin n-heptane, alkenyl radical decomposition isassumed to lead to products of allyl radicals andolefins. The rate constants of these decomposi-

TABLE 3

Rate Constant Expressions for H-Atom Abstraction from C7 Olefin(cm3-mol-s-cal Units)

Reaction

Rate Expression

Reference! n %a

Olefin 1 H 5 Alkenyl 1 H2 1.00 3 1012 0.00 3,900 66Olefin 1 OH 5 Alkenyl 1 H2O 1.00 3 1012 0.00 1,230 66Olefin 1 O 5 Alkenyl 1 OH 1.00 3 1012 0.00 4,000 66Olefin 1 CH3 5 Alkenyl 1 CH4 2.00 3 1011 0.00 7,300 66

TABLE 4

Rate Constant Expressions for OH and O Radical Addition to C7 Olefin(cm3-mol-s-cal Units)

Reaction

Rate Expression

Reference! n %a

Olefin 1 O 5 Products 2.00 3 1010 0.00 1,050 66Olefin 1 OH 5 Products 2.00 3 1010 0.00 4,000 66


tions are all assumed to be 2.5 3 1013 exp(245,000/RT) s21. This treatment is very ap-proximate, and further attention is warranted.

Reaction Type 9: Olefin Decomposition

We have found that, in model computations, thelarge olefin species decompose at appreciablerates. We have used, for all of these reactions, arate constant expression of 2.5 3 1016 exp(271,000/RT) s21 as recommended by Edelsonand Allara [50] for 1-hexene decomposition,and we have tried to select product distributionsthat would be expected on structural and ther-mochemical grounds. This is another area inwhich further work would improve the mecha-nism.

LOW-TEMPERATURE MECHANISM

At temperatures lower than approximately 900K, the high activation energies (27–40 kcalmol21) associated with the b-scission of alkylradicals and internal H-atom abstraction (H-shift) reactions make these processes ratherslow. Under such conditions, the most impor-tant reactions for alkyl radicals, R, consist ofaddition to molecular oxygen:

R 1 O2 5 RO2

followed by internal H-atom abstraction, a sec-ond addition to O2, H-atom abstraction, andsubsequent decomposition to yield two reactivehydroxyl radicals and a carbonyl radical. Inmany ways, the first addition of an alkyl radicalto O2 is the most important reaction for low-temperature oxidation, even though it does notimmediately determine the overall rate of chainbranching. Pollard [67] carried out an extensivekinetic analysis of hydrocarbon oxidation underlow-temperature conditions and identified themost important features of the low-temperaturesubmechanism. Many recent modeling studieshave incorporated these features in their mech-anisms. Most of these models assume that thechemical details of fuel autoignition are socomplex that many simplifications are essentialin order to be able to simulate the oxidationprocess. Some of the most prominent of thesesimplified model treatments of hydrocarbon ig-

nition are the “Shell Model” [68] and relateddevelopments by Hu and Keck [69] and Cox andCole [70]. A recent survey and critical analysisof these simplified approaches by Griffiths [71]have summarized the strengths and limitationsof these models. However, recent studies indetailed kinetic modeling of hydrocarbon oxida-tion [36, 61, 62] have made it possible to addressa wide variety of issues related to ignition, mostof them leading to improved descriptions ofignition in internal combustion engines andengine knock. In the following section we de-scribe our low-temperature submechanism,which includes detailed chemistry associatedwith the most important reactions associatedwith low-temperature oxidation of hydrocarbonfuels as identified by earlier workers above.

Reaction Type 10: R 1 O2 Addition

Following Benson [72], additions involving alkylradicals larger than C4 to O2 were assumed tohave the same bimolecular rate constant of 2 31012 cm3 mol21 s21. The reverse decompositionrate constants are calculated from microscopicreversibility. We assume that the RO2 radical israpidly stabilized to its ground state as there are66 (i.e., 3N-6) modes of vibration in theC7H15O2 radical, so the energy released duringC™O bond formation is easily dissipatedthroughout the molecule. In addition, colli-sional stabilization is fast under the high-pres-sure (13.5–40 bar) conditions of this study. Theactivation energy for the addition reaction istaken to be zero but is quite large ('30 kcal/mol) in the reverse dissociation direction.Therefore, the equilibrium constant for thisreaction is very strongly temperature depen-dent. At very low temperatures, this reactionproceeds rapidly to produce the alkylperoxyspecies very efficiently; at high temperatures,RO2 dissociates rapidly and the concentrationof RO2 is very small. This point is clearly relatedto the concept of “ceiling” temperature [73],defined as that temperature at which [RO2]/[R]5 1. As the total pressure is increased, theceiling temperature increases, so when the alkylradical is 2C7H15 the ceiling temperature wascalculated to be 897 K at a partial pressure ofO2 equal to 0.1 atm and becomes 1066 K whenthe partial pressure of O2 is 2 atm. Under diesel


engine conditions assuming a stoichiometricmixture of n-heptane in air and a total pressureof 50 atm, the partial pressure of O2 would beapproximately 10 atm, corresponding to a ceil-ing temperature of 1188 K.

Reaction Type 11: R 1 R*O2 5 RO 1 R*O

Reactions of alkyl radicals with alkylperoxyradicals are assumed to occur at

1.9 3 1012 exp (11200/RT) cm3 mol21 s21

forward

1.0 3 1010 cm3 mol21 s21 reverse

The forward rate is based on the value of Kaiseret al. [74] for the reaction CH3 1 CH3O2 5CH3O 1 CH3O. They recommend an !-factorof 3.8 3 1012 with an activation energy of 21200cal/mol equal to that given above. We havereduced their !-factor by two as the C7 R andR9O2 radicals are larger than the CH3 andCH3O2 radicals, and therefore we expect areduced rate of reaction. The reverse rate con-stant is a strong function of temperature. Thisreaction is approximately 30 kcal mol21 exo-thermic, and therefore the reverse is 30 kcalmol21 endothermic. Thus, the reverse rate con-stant presented is only correct over a narrowtemperature range, 600 # T # 900 K.

Reaction Type 12: RO2 Isomerization

Our treatment of these reactions has beenadopted from Benson [72], Pollard [67], andBozzelli and Pitz [64]. In addition to the fourRO2 species formed by the addition of heptylradicals to O2, there are 18 possible QOOHisomers produced by internal H-atom abstrac-tion of these RO2 radicals. Figure 1 depicts2-heptylperoxy radical undergoing one of fivepossible isomerization reactions. This six-mem-bered transition state (TS) structure undergoesrapid stabilization, transfers a secondary Hatom, and forms 2-hydroperoxy-4-heptyl radi-cal. Isomerization reactions including interme-diate TS structures as large as eight membersare included in the mechanism.

The way in which we calculate the activationenergy for RO2 isomerization is identical to that

described for alkyl radical isomerization. Rateconstants depend on (1) ring strain energybarriers, (2) the type and location of the ab-stracted H atom (1°, 2°, or 3°), and (3) thedegeneracy or number of H atoms at that site.However, it was found that, in order to correctlypredict relative concentrations of heterocyclicproducts reported in the literature [7, 12], it wasnecessary to lower the ring-strain energy for thechange in going from a six-membered (1,5p) toa seven-membered, (1,6p) TS ring by 2.8 kcalmol21 and not 4.8 kcal mol21 as recommendedby Baldwin et al. [75]. We arrive at a loweractivation energy for the seven-membered ringas do Baldwin et al., which is in some contrast toalkyl radical isomerizations, where six-mem-bered rings are preferred. Nonetheless, in keep-ing with the alkyl radical isomerization analogy,we do increase the ring strain energy in goingfrom a (1,6) to a (1,7) TS ring choosing a valueof 2.8 kcal mol21. The approximate ring strainenergies used in this study are reported in Table5. Thus for example, we calculated the activa-tion energy for the five-membered, primary(1,4p) TS to be 29.2 kcal mol21 using equation

Fig. 1. (1,5) H-atom isomerization via transition state ringstructure.


1, which is similar to 30.0 kcal mol21, the valuereported by Wagner et al. [76] for the (1,4p) TSinternal H-atom abstraction in the ethylperoxyradical leading to the formation of ethylhy-droperoxide. We chose a value of 29.7 kcalmol21, a number that lies between both of theabove values. The preexponential !-factor isassumed to decrease with increasing ring sizedue to the loss of internal rotors. This consid-eration is the main contributor to the decreasein entropy of the transition state [64]. The!-factor depends linearly on the number ofequivalent H atoms being abstracted. The!-factor for the (1,4p) TS ring was chosen to be8.9 3 1012 s21, a value approximately the sameas that recommended by Baldwin et al. [75].This value is reduced for the (1,5p) TS ring by amultiple of 12, assuming a change in entropy,DS ' 5.0 cal mol21 K21, and in multiples of 12for all singular changes in TS ring size thereaf-ter.

These reactions are reversible, and again wecalculate the reverse isomerization using ther-mochemistry. Although Pollard and Benson dis-cuss rate constant expressions in a general way,detailed study of this class of reaction has onlyrecently received considerable attention, andthere is still some question about the correctenergy barriers to these processes. It is possiblethat some of the disagreements or lack ofunanimity is due to analyses at different pres-sures and temperatures that might affect stabi-lization of the excited RO*

2radical. However, as

already noted, for the applications in this study,we expect stabilization of RO*

2to be very rapid

and so our treatment should be appropriate. Wesummarize our recommended rates of RO2

isomerization in Table 6.

Reaction Type 13: RO2 1 HO2 5 RO2H 1 O2

It should be noted that this class of reaction isbimolecular, the rate of reaction depending onk[RO2][HO2]. The concentration of HO2 radi-cal species, although high relative to other rad-ical species in the oxidation process, will typi-cally be of the order of 1026 mol cm23 or less.This infers that the unimolecular isomerizationRO2 º QOOH (type 12) will inevitably bemuch faster than this bimolecular reaction. Theexception is when R is CH3 because CH3O2does not isomerize at a significant rate. None-theless, the balance between HO2 and otherradicals is often of great importance in theseproblems and so this reaction type is included inour mechanism. This reaction, when followed bythe decomposition of RO2H, converts HO2 toOH, which can accelerate the overall rate ofreaction. However, the RO2H species is quitestable, and, at sufficiently low temperature, thisreaction terminates chain branching and reducesthe overall rate of reaction. There is not muchinformation available on this reaction type exceptfor the case when R is CH3. For this class ofreaction we have assumed a rate expression:

1.0 3 1011 cm3 mol21 s21 forward

3.0 3 1011 exp (239,000 cal/RT)

cm3 mol21 s21 reverse

TABLE 5

Number of Atoms in TS Ring Structure versus RingStrain Energy

Number ofRing Members

Ring Strain(kcal mol21)

5 8.66 2.87 0.08 2.8

TABLE 6

Rate Constant Expressions for RO2 IsomerizationReactions (cm3-mol-s-cal Units)

Ring Size Site

Rate Expression (per Hatom)

! n %a

5 Primary 2.98 3 1012 0.0 29,700Secondary 2.98 3 1012 0.0 27,900Tertiary 2.59 3 1012 0.0 25,400





Our rate choice is somewhat lower than therecommendation of Wallington et al. [77] whogive a rate constant expression of 3.1 3 1011 exp(11272 cal/RT) cm3 mol21 s21 in the tempera-ture range 228–573 K, where R is CH3. Thereaction showed no pressure dependence overthe pressure range of 25–760 torr. This rateconstant yields 7.74 3 1011 cm3 mol21 s21 at 700K. However, we expect the rate constant todecrease with increasing size of the R radical,and therefore our recommended rate constantis lower. Because of lack of data, we haveneglected the small negative activation energyin our choice.

Reaction Type 14: RO2 1 H2O2 5 RO2H 1HO2

This is an interesting reaction sequence thatconverts one stable species and a peroxy speciesinto another stable species and another peroxyspecies. The subtlety in the reaction and itsinfluence on the overall reaction sequence isdue to the differences in the temperatures atwhich the RO2H and H2O2 species decompose.RO2H decomposes at a lower characteristictemperature than H2O2. Conversion of H2O2 toRO2H leads to an enhanced overall reactivity atlower temperatures. We have used rate constantexpressions that are the same in both forwardand reverse directions,

2.4 3 1012 exp (210,000 cal/RT) cm3 mol21 s21

There is little information available for thisreaction rate constant except for the case whereR is CH3. Our choice of rate constant is basedon Tsang’s recommendation [78] for CH3O2 1H2O2 5 CH3O2H 1 HO2 of k 5 2.41 3 1012

exp (29940 cal/RT) cm3 mol21 s21 from 300–2500 K with an uncertainty of a factor of 5. Forthis isoergic reaction, the reverse rate is thesame as the forward rate by analogy.

Reaction Type 15: RO2 1 CH3O2 5 RO 1CH3O 1 O2

This sequence converts peroxy radicals to otherradicals that decompose more readily. The re-action rate constants are not particularly wellknown, but we have estimated the forward rate

constant of the reaction to be 1.0 3 1011 cm3

mol21 s21 and the trimolecular reverse reactionrate is set to zero, since the three-body reactionis not expected to proceed. There is little infor-mation available for R other than CH3. Our rateconstant choice agrees well with the recom-mended rate constant of CH3O2 1 CH3O2 5CH3O 1 CH3O 1 O2 from the recent review ofWallington et al. [77]. They give a total rate ofCH3O2 1 CH3O2 leading to products, in addi-tion to a branching ratio in the temperaturerange 250–600 K. No pressure effect in the totalrate was observed from 10–760 torr. Using thisinformation, Wallington et al. recommend arate constant expression of 6.9 3 1010 exp (122cal/RT) cm3 mol21 s21 for the CH3O 1CH3O 1 O2 product channel, which yields 8 31010 cm3 mol21 s21 at 700 K. We have neglectedthis small, negative activation energy in ourextrapolation to other analogous reactions.

Reaction Type 16: RO2 1 R*O2 5 RO 1R*O 1 O2

This is another reaction that interconverts ROand RO2 radical species. The rate is estimatedto be 1.0 3 1011 cm3 mol21 s21 with a zeroreverse reaction rate. We were unable to findany information for cases where R and R9 aredifferent alkyl radicals. When R 5 R9, Walling-ton et al. give 6.81 3 1010 exp (2664 cal/RT)cm3 mol21 s21 for R 5 C2H5 and 260–380 K.This rate constant gives 4.2 3 1010 at 700 K.Wallington’s review also recommends 1.39 31012 exp (25090 cal/RT) cm3 mol21 s21 for R 5iC3H7 and 300–373 K, which yields 3.6 3 1010

cm3 mol21 s21 at 700 K.

Reaction Type 17: RO2H 5 RO 1 OH

It had been thought in the past that this reactionstep was important as it produces two veryreactive radical species. However, as alreadyexplained for type 13 reactions above, RO2radical preferentially undergoes the unimolecu-lar isomerization to QOOH. Therefore, verysmall concentrations of stable RO2H species areformed in the low-temperature oxidation pro-cess. RO2H species can be used as sensitizers orcetane improvers, and so its decomposition inthose circumstances will be very important.


Thus, for completeness, these reactions havebeen maintained in our mechanism. UsingTHERM, we have generated more reliablethermodynamic data for both reactant andproduct species of this reaction class and haveassigned a rate constant of k 5 1.0 3 1013 cm3

mol21 s21 for the reverse radical–radical addi-tion reaction, calculating the forward decomposi-tion rate constant from microscopic reversibility.

Reaction Type 18: RO Decomposition

The alkoxy radicals that are produced in theoxidation process are quite unstable and tend todecompose quite readily. Large alkoxy radicalscan decompose into smaller stable oxygenatedspecies, primarily aldehydes or ketones, in ad-dition to an alkyl radical species. This smalleralkyl radical will subsequently undergo b-scis-sion to form an even smaller alkyl radical and astable olefin species. Thus, the reaction actuallyincorporates two steps, each with its own acti-vation energy. We have used an analogy withb-scission to identify the product distributionand estimate a rate constant expression of 2.0 31013 exp (215,000 cal/RT) s21 for these reac-tions. With this activation energy, this reactionwill be rapid only in the higher ranges oftemperature in our applications, where theproduct alkyl radical should also be expected todecompose. As a result we have combined thesetwo steps into one overall global reaction, and asthe reverse reaction contains three species its rateconstant is set to zero. Since aldehydes are gener-ally quite reactive, and the alkyl products also leadto radical species production, the process of ROdecomposition can contribute to reaction acceler-ation. However, we have found that RO radicalsare relatively unimportant in the low-temperatureoxidation process and so this type of treatment hasbeen sufficient for this study.

Reaction Type 19: QOOH 5 QO 1 OH

This reaction sequence involves the breaking ofthe O™O bond, coupled with the formation of acyclic compound including the remaining Oatom. We have followed the recommendationsof Pollard [67], in which the activation energybarrier depends on the size of the cyclic speciesring that is formed. However, we have altered

the rate constants of these reactions from thosepublished previously [35–37]. Taking into ac-count loss in entropy as described earlier, weagain reduce the pre-exponential factor by amultiple of 12 as one extra rotor is tied up ingoing from a three to 4 and progressively largerring heterocycles. The activation energies havealso been altered in order to model correctly cyclicether distributions measured experimentally. Thesix-membered ring tetrahydropyran species areassumed to have the lowest ring-strain energy,while the oxirane species are assumed to have thegreatest strain energy. The rate parameters we usefor these reactions are reported in Table 7.

Reaction Type 20: QOOH 5 Olefin 1 HO2

QOOH species that have a radical site b to thehydroperoxy group can decompose to yield aconjugate olefin and HO2 radical. The rateconstant for this reaction was considered in thereverse direction, i.e., the addition of an HO2

radical at an olefinic site, in the same way asalkyl radical decomposition (type 3 above). Theenergy barrier for the addition reaction is betterknown than is the decomposition of the alkyl-hydroperoxide to products. Thus, followinganalogy rules for addition of a radical to adouble bond, a rate constant of 8.5 3 1010 exp(27800 cal/RT) cm3 mol21 s21 was chosen forolefin 1 HO2, similar to the value recom-mended by Tsang [79] for the reaction CH3 1C3H6 5 isoC4H9. The forward rate constant iscalculated from thermochemistry. This reactionhas proven to be quite sensitive and is respon-sible for a large part of the NTC behavior inn-heptane oxidation kinetics.

TABLE 7

Rate Constant Expressions for Cyclic Ether Formationfrom QOOH Radicals (cm3-mol-s-cal Units)

Cyclic EtherRing Size

Rate Expression

! n %a

3 3.00 3 1011 0.0 22,0004 2.50 3 1010 0.0 15,2505 2.08 3 109 0.0 6,5006 1.50 3 108 0.0 1,800


Reaction Type 21: QOOH 5 Olefin 1Carbonyl 1 OH

QOOH species, produced by an RO2 isomeriza-tion with an intermediate ring structure of sixatoms, can undergo b-scission. Scission prod-ucts were chosen considering the weakest inter-atomic bonds in the molecule. We have chosena rate constant of 5.0 3 1013 exp (225,500/RT)s21, which is similar to that recommended byPollard [67], for each of these b-scission reac-tions. Since the reverse reaction is trimolecular,its rate is assumed to be negligible.

Reaction Type 22: Addition of QOOH to O2

Finally, QOOH can react with molecular oxygento form peroxyalkylhydroperoxide, O2QOOH,species. A rate constant of 2 3 1012 cm3 mol21 s21

was chosen, consistent with the value chosen foralkyl plus O2, type 10. The reverse dissociationrate was then calculated from thermochemistryusing THERM.

Reaction Type 23: O2QOOH Isomerization toKetohydroperoxide 1 OH

One of the main additions to the current mech-anism is the treatment of the decomposition ofthe O2QOOH species formed. Based on theexperimental observations reported by Sahetchianet al. [15] in which heptylketohydroperoxides areobserved, we assume that the O2QOOH isomer-izes, releasing OH and forming different ketohy-droperoxide species, depending on the O2QOOHinvolved. For example, Fig. 2 depicts 2-hydroper-oxy-5-heptylperoxy radical undergoing isomeriza-tion through a seven-membered transition statering structure, forming 5-hydroperoxy-2-hep-tanone. The rate constant for this and otherisomerizations via an internal H atom transfer areanalogous to those for RO2º QOOH isomeriza-tion and are reported in Table 8. However, theactivation energy has been reduced by 3 kcalmol21 as the hydrogen atom being abstracted isbound to a carbon atom, which is bound to ahydroperoxy group and should be more easilyremoved. In addition, the !-factor has also beenreduced by a factor of 0.5, considering sterichindrance due to the OOH group. These reac-

tions are very important in the modeling of two-step ignition phenomena observed in shock tubes[5], CFR engines [13–15], and rapid compressionmachines [9, 12].

Reaction Type 24: KetohydroperoxideDecomposition

One reactive hydroxyl radical is formed duringthe production of ketohydroperoxide species byreaction type 23. The subsequent decomposi-tion of ketohydroperoxide molecules leads tothe formation of two radicals, a carbonyl radicaland another OH radical, providing chainbranching as it produces two radical speciesfrom one stable reactant. It is especially impor-tant that formation and decomposition of keto-hydroperoxide molecules produce two OH rad-icals in the present mechanism. Griffiths [71]reported that multiplication of reaction chains iscurtailed by displacement from branching tonon-branching reaction modes, that the OHradical is considerably more reactive than the

Fig. 2. (1,6) H-atom isomerization forming ketohydroper-oxide 1 OH.


HO2 radical, and that considerable exothermic-ity is associated with OH radical propagation,while HO2 propagation is virtually thermoneu-tral.

A single rate constant of 1.0 3 1016 exp(243,000 cal/RT) s21 was chosen for the de-composition of each ketohydroperoxide mole-cule even though each ketohydroperoxide spe-cies has slightly different thermodynamicproperties. Our rate constant expression isbased on the recommendations of Sahetchian etal. [80] for 1-heptyl and 2-heptyl hydroperoxidedecomposition. This set of reactions is espe-cially important at low temperatures as the highactivation energy ensures an induction periodduring which ketohydroperoxide concentrationbuilds up. The subsequent decomposition prod-ucts help accelerate the overall rate of fueloxidation, raising the temperature and allowingthe remaining ketohydroperoxide species to de-compose more easily.

Reaction Type 25: Cyclic Ether Reactions withOH and HO2

The detailed reaction mechanisms of most cy-clic ether species are not well established. Lif-shitz et al. [81, 82] investigated the pyrolysis ofethylene oxide and oxidation of ethylene oxide,propylene oxide, and two epoxybutanes behindreflected shock waves. However, very littlechemical kinetic data are reported other thanArrhenius-type correlation equations pertainingto the global oxidation of fuel/oxygen mixturesdiluted with both argon and nitrogen. Baldwinet al. [83] reported rate constant expressions for

H-atom abstraction from ethylene oxide by H,OH, and CH3, including the subsequent decom-position of the oxiranyl radical to CH3 1 CO.More recently, Dagaut et al. [84] have publisheda paper including speciation data and kineticparameters of reactions associated with ethyl-ene oxide oxidation. However, this study wascarried out mainly at high temperatures whereethylene oxide isomerizes to acetaldehyde.

Cyclic ethers are produced under low-tem-perature conditions during hydrocarbon oxida-tion and in the present study are relatively largeC7 species with an O atom embedded in themolecule. Therefore, it is reasonable to assumethat the reaction proceeds by means of H-atomabstraction. We have made some assumptionsrelating to site of H-atom abstraction and thenature of the product species formed in thesereactions. We assume that the ease of H-atomabstraction will be in the following order 3° . 2°. 1°. In addition, a H atom bound to a C atomthat is bound to the O atom in the ring structurewill be more easily abstracted, and we haveassumed that only H-atom abstraction by OH andHO2 radicals will be of any great importancebecause these radicals are in highest concentra-tions at low temperature. After abstraction, weassume the ring opens immediately, leading to theformation of an alkyl-aldehyde or alkyl-ketoneand either water or hydrogen peroxide. The alkyl-aldehyde or ketone is then assumed to undergob-scission to a stable aldehyde or ketone and asmaller alkyl radical species.

We have adopted two sets of product distri-butions for abstraction reactions by OH radicalsand two more for abstractions by HO2. We have

TABLE 8

Rate Constant Expressions for O2QOOH Isomerization Reactions(cm3/mol per s per cal Units)

Ring Size Site

Rate Expression (per H atom)

! n %a

5 Primary 1.49 3 1012 0.0 26,700Secondary 1.49 3 1012 0.0 24,900





tried to include rate constants for these variousH-atom abstraction reactions by drawing anal-ogies with similar reactions published in theliterature. Abstraction of primary and tertiary Hatoms from a cyclic ether by OH radical wastaken to be equal to the values for 1° and 3°H-atom abstraction from isobutane recom-mended by Tully et al. [46]. Secondary H-atomabstraction was taken to be equal to that rec-ommended by Droege and Tully [45] for 2°H-atom abstraction from propane. Abstractionof a H atom bonded to a 1° C atom which inturn is bonded to an O atom was taken to befour times faster than abstraction from a typical2° C atom. Abstraction of a H atom bonded toa 2° C atom which in turn is bonded to an Oatom was taken to be approximately three timesfaster than the rate recommended by Walker[85] for 3° H-atom abstraction from isobutane.These rate constant expressions are summa-rized in Table 9. Primary and tertiary H-atomabstraction from a cyclic ether by HO2 wastaken to be equal to the values for 1° and 3°H-atom abstraction from isobutane recom-mended by Tsang [86]. Secondary H-atom ab-straction was taken to be equal to that recom-mended by Tsang [87] for 2° H-atom abstractionfrom propane. Abstraction of a 2° H atom bondedto a C atom that is bonded to an O atom wastaken to be four times faster than a typical 2° Hatom, and abstraction of a 3° H atom by HO2 at asimilar site was taken to be approximately threetimes faster than a typical 3° H atom.

REACTION MECHANISM

The overall flux diagram for n-heptane oxida-tion can be shown schematically in a particularlysimple way as seen in Fig. 3. At high tempera-tures, the overall reaction pathway proceeds viab-scission of the alkyl radicals R proceedingrapidly to a smaller olefin and other species,with chain branching due primarily to the reac-tion H 1 O2 5 O 1 OH. At low temperatures,chain branching is mainly due to the reaction

TABLE 9

Rate Constant Expressions for H-Atom Abstraction from Cyclic Ether by OH and HO2

(cm3/mol per s per cal Units)

Type ofHAtom Radical Site

Rate Expression (per H atom)

! n %a

Primary 3.83 3 107 1.53 775OH Secondary 2.34 3 107 1.61 235

H™C™C Tertiary 5.73 3 1010 0.51 64Primary 3.33 3 103 2.55 15,500

HO2 Secondary 7.40 3 103 2.60 13,910Tertiary 3.61 3 103 2.55 10,532Primary 9.50 3 107 1.61 235

OH Secondary 8.84 3 109 1.00 2149H™C™O Primary 3.00 3 104 2.60 13,910

HO2 Secondary 1.08 3 104 2.55 10,532

Fig. 3. Lumped kinetic scheme of the primary oxidationreactions.


pathway leading through the ketohydroperoxidespecies. As the temperature increases, the chainpropagation reactions of QOOH species in-crease because the energy barrier to their for-mation is more easily overcome, leading to theformation of cyclic ether species, conjugate ole-fins, and b-decomposition products at the ex-pense of the reaction pathways through theketohydroperoxide species. The increasing im-portance of these propagation channels leads toa lower reactivity of the system, which is ob-served as the NTC region. The difference in thechain-branching mechanisms at low and hightemperatures leads to varying reactivity, de-pending on the fuel to air equivalence ratio.Because the chain-branching mechanism at hightemperatures is due to the H 1 O2 5 O 1 OHreaction, fuel lean mixtures are more reactive inthis regime. However, at low temperatures, be-cause chain branching is dependent on radicalspecies formed directly from the parent fuel,fuel rich mixtures are oxidized more quickly.

MECHANISM VALIDATION

An explanation of the chemical kinetic mecha-nism formulation has been given in the preced-ing section. In order to validate this mechanism,it is necessary to carry out simulations of exper-imental measurements available in the litera-ture. In the following section we describe howthis mechanism was used to simulate experi-mental results obtained in a plug flow reactor[20, 22], a jet-stirred reactor [6–8], shock tubes[1–5], and rapid compression machines [9–12].Except for the shock tube studies of Vermeer etal. [1] and Coats and Williams [2], these studiesinvolve n-heptane oxidation over temperaturesat which NTC behavior can be clearly observed.

Variable Pressure Flow Reactor

Experiments, carried out in an adiabatic flowreactor, provide a well-characterized environ-ment that is designed to minimize mixing anddiffusion effects. Details of the experimentalapparatus are provided by Dryer et al. [20, 22].n-Heptane was studied under stoichiometric airto fuel ratios; the initial concentration of fuelwas approximately 0.14%, with a high amountof nitrogen diluent, approximately 99%. Exper-iments were performed under adiabatic condi-tions over an initial temperature range of 550–850 K and at a constant pressure of 12.5 atm.Data were obtained for the mole fractions ofCO, CO2, H2O, O2, and the temperature at thefixed sampling location. Simulations were per-formed under the assumption of plug flow: thevelocity and temperature profile in the reactor isradially uniform and axial diffusion of speciesand energy is negligible. Constant pressure of12.5 atm and adiabatic walls were also assumed.

The experimental results indicate quiteclearly the characteristic NTC behavior of n-heptane oxidation. Figure 4 shows the O2 andCO concentrations measured at a fixed resi-dence time of 1.8 s as a function of initialtemperature. Plotted O2 concentrations below1.6% indicate consumption of O2 in the reactor.Consumption of O2 and production of COindicate that the n-heptane (not reported) wasconsumed. The consumption of n-heptane be-gins at about 560 K and increases until fuelconsumption peaks at 600 K. The simulationsshow that the reactivity from 560–600 K is dueto the consumption of ketohydroperoxide spe-cies leading to the formation of reactive OHradicals. At initial temperatures less than 600 K,the QOOH radical preferentially adds to mo-

Fig. 4. 0.14% n-heptane oxida-tion at 12.5 atm, f 5 1.0, t 51.8 s in a pressure flow reactor.Experimental results (points)[20, 22] versus model predic-tions for O2, CO and heat re-lease. Dashed line correspondsto open circles.


lecular oxygen leading to chain branching,rather than undergoing decomposition. At ini-tial temperatures above 600 K, fuel consump-tion gradually decreases up to a temperature ofabout 730 K. This decrease in reactivity, char-acteristic of NTC behavior, is caused by thecompetitive decomposition of the QOOH radi-cal to form olefin and HO2, cyclic ether, andOH or b-scission products, all of which arechain propagation paths and provide loweroverall reactivity than the chain branching ke-tohydroperoxide pathway. At initial tempera-tures greater than approximately 760 K, theradical pool becomes well established, and dis-sociation of hydrogen peroxide into two hy-droxyl radicals leads to rapid consumption ofthe remaining fuel.

Jet-Stirred Reactor

Recently Dagaut et al. [7, 8] reported experi-mental results on the oxidation of n-heptane ina jet-stirred reactor at 10 and 40 atm coveringthe low- and the high-temperature regimes(550–1150 K) with equivalence ratios from 0.3to 1.5 and 99% dilution by nitrogen. Theseexperimental results are especially valuablesince both the reactant concentrations and in-termediate and final product concentrationswere measured. A series of comparisons be-tween computed and experimental results is

shown in Fig. 5 for stoichiometric mixtures of0.1% n-heptane in O2/N2 at 10 atm and aresidence time of 1 s. Simulations were per-formed under isothermal, constant pressureconditions, and assumed perfect mixing of thereactants. Time-dependent calculations wererun for long times until a steady-state solutionwas obtained. None of these calculations exhib-ited dynamic behavior (periodic cool flames,multistage ignitions, etc.) [88]. The results ofthese calculations indicate that, below 750 K,oxidation takes place through a low-tempera-ture mechanism leading to the formation of CO,CO2, CH2O, CH3CHO, C2H5CHO, and majorintermediates including heptenes and cyclicethers. All of these species concentrations arereproduced quite well by the model. The accu-rate prediction of these species levels reliesheavily on the relative rates of alkylperoxy rad-ical isomerization reactions and the fate of theQOOH radical, ultimately leading to the forma-tion of the different heptenes and cyclic ethers.It is interesting to note that most of the CO2 isbeing produced by a reaction sequence that hasnot been previously reported in the literature:

HCO 1 O2 5 HCO3

RH 1 HCO3 5 R 1 HCO3H

HCO3H 5 HCO2 1 OH

HCO2 1 M 5 H 1 CO2 1 M

Fig. 5. 0.1% n-heptane oxida-tion at 10 atm, f 5 1.0, t 5 1 sin a jet-stirred reactor. Experi-mental (points) [7, 8] and mod-el-predicted mole fraction forsome oxygenated compounds,n-heptenes, and furans. Dashedline corresponds to opencircles.


where RH is the fuel or a stable hydrocarbonintermediate.

As seen in Figs. 5 and 6, the distinct NTCbehavior measured between 640 and 750 K isalso reproduced well by the model. However,the model shows a lower overall reactivity thanthe experiments. In the reaction mechanism,product species distributions for the thermaldecomposition reactions of the ketohydroperox-ide species were postulated, since experimentalevidence for these product distributions wasunavailable. Additional refinements in this areaof the mechanism could improve the modelresults at the low temperatures.

Above 750 K oxidation takes place via anintermediate-temperature mechanism in whichthe most important branching reaction is thedecomposition of hydrogen peroxide H2O2 ºOH 1 OH. The model description of thissystem could not be accurately reproduced with-out including fall-off corrections for this reac-tion rate constant. We have used the recent rateconstant expression of Marinov and Malte [89],which gives a nine-parameter, pressure-depen-dent fit. Variations in equivalence ratio at con-stant pressure were found to change the overallreactivity of this system but did not change thetemperature range of the NTC region, as isevident from Fig. 6(a). Model calculations at ahigher pressure of 40 atm and a residence timeof 2 s showed an apparent reduction of the NTCregion (Fig. 6(b)), in agreement with experi-mental observation.

Shock Tube

The autoignition of n-heptane can be studiedconveniently at high temperatures in shocktubes and at lower temperatures in rapid com-

pression machines. We have used the model,assuming constant-volume, homogeneous, adia-batic conditions behind the reflected shockwave, to examine high-temperature shock tubeexperiments of Vermeer et al. [1] and Coats andWilliams [2]. Vermeer et al. studied the autoig-nition of n-heptane–oxygen mixtures behindreflected shock waves over the pressure range of1–4 atm and the temperature interval of 1200–1700 K. Stoichiometric fuel-oxygen mixtureshad to be diluted with 70% Ar to reduce theinfluence of the boundary layer. Coats andWilliams studied the ignition of n-heptane/O2/Ar mixtures behind both incident and re-flected shock waves with equivalence ratios of0.5 to 4.0 in the temperature range 1300–2000K. Poor agreement between simulated and mea-sured ignition delay times was observed for theexperimental results of Vermeer et al., Fig. 7(a),but we attained good agreement with the mea-surements of Coats and Williams, Fig. 7(b). Theexperimental trend that fuel-lean mixtures an-toignite more easily than fuel-rich mixtures athigh temperature (1300–2000 K) is also repro-duced. We are unclear as to why we have pooragreement with the results of Vermeer et al.and at the same time obtain good agreementwith the results of Coats and Williams in whichthe fuel and oxygen is highly diluted in 98% Ar.The reason for this discrepency is currentlyunder investigation. It may be that includingfall-off for these decomposition reactions wouldlead to longer predicted ignition delay timesand better agreement with the long ignitiondelay times measured in the Vermeer experi-ments.

Ciezki and Adomeit [5] also carried out re-flected shock tube experiments but at somewhatlower temperatures and rather high pressures,

Fig. 6. 0.1% n-heptane oxidationat t 5 1 s in a jet-stirred reactor.Experimental (points) [7, 8] andmodel-predicted conversion withinfluence of (a) equivalence ratioand (b) pressure. Dashed line cor-responds to open circles.


observing NTC behavior at temperatures be-tween 750 and 1000 K. Computed results arecompared with data in Fig. 8, showing overallexcellent agreement between computed andmeasured results. The experimental trend thatfuel-rich mixtures autoignite more easily thanfuel-lean mixtures from 750–950 K is well re-produced by the model and is in contrast withthe trend observed both experimentally andcomputationally for ignition delay times at hightemperatures in which fuel-lean mixturesburned faster than fuel-rich mixtures, Fig. 7(b).The temperature range of 750–950 K and apressure of 13.5–40 bar are close to the condi-tions observed in the unburned gas of a sparkignition engine, where engine knock occurs.These experimental and modeling results agreewith automotive engine experience that fuel-rich mixtures have a greater tendency to autoig-nite and lead to knock than do stoichiometric orfuel-lean mixtures. The magnitude of the NTCregion is very closely reproduced by the reactionmechanism. Perhaps most importantly, the shiftin the NTC region to higher temperatures aspressure is increased is also accurately repro-duced. This shift is due to the influence ofpressure on the equilibria of the addition reac-tions of molecular oxygen to the alkyl and

hydroperoxy-alkyl radicals. Chevalier et al. [25,26] obtained similar results when they used theirmodel to simulate these same experiments.

Rapid Compression Machine

The experiments of Ciezki and Adomeit couldcover only a small part of the low-temperatureregion because of limitations in measuring longignition times. Similar conditions have beeninvestigated experimentally by Minetti et al. [12]in a rapid compression machine (RCM). Theseexperiments are characterized by ignition delaytimes of the order of 20–40 ms. Experimentswere carried out at a compression ratio of 9.8.Stoichiometric mixtures of fuel and “air” wereused, the air composing 21% O2 and 79%diluent. The diluent consisted of mixtures of N2,Ar, and carbon dioxide, which have much dif-ferent heat capacities. In this way, the com-pressed gas temperature could be varied byselecting different ratios of diluents; the highesttemperatures were obtained when the diluentconsisted entirely of Argon and the lowest tem-peratures with the greatest concentration ofcarbon dioxide. The initial gas pressure was 162torr at an initial temperature of 355 K. Minettiet al. [12] have reported that n-heptane did not

Fig. 7. Experimental (points) (a)[1] and (b) [2] and model-predictedignition delay times. Dashed linecorresponds to open circles.

Fig. 8. 1.79% n-heptane oxidationbehind reflected shock waves in80% N2. Comparison between ex-perimental (points) [5] and modelprediction. t is the ignition delaytime. Dashed line corresponds toopen circles.


react to any perceptible extent during compres-sion, and therefore we have simulated theirexperimental results assuming a homogeneousadiabatic reactor at top dead center (TDC). Inthe simulations, we used the experimentallymeasured pressure at the end of compressionand the temperature as the initial conditions fora constant volume calculation. The compressedgas temperatures are calculated from the initialconditions of pressure, temperature, and reac-tant composition, the pressure at TDC, and thetemperature dependence of the ratio of specificheats g, according to the equation:

ET0

Tc g

g 2 1dTT

5 lnP1

P0

In these experiments at compressed gas tem-peratures below about 850 K, a noticeable two-stage ignition process as shown in Fig. 9(a) isobserved. A typical example of the results isshown as the bottom curve in Fig. 9(a), wherethe temperature at the end of compression is661 K. This mixture ignites in two distinctstages, the first occurring at about 25.2 ms afterthe end of compression and the second occur-ring at 32.4 ms. As the ratio of Ar/N2 carbondioxide is increased, the post compression tem-perature increases and the overall ignition delaytime becomes shorter, as shown by the secondcurve (Tc 5 738 K) in Fig. 9(a), and both thefirst and second ignition stages occur earlier. Asthe Ar/N2 ratio is further increased, the systemfalls within the NTC region, as indicated by thethird curve at 829 K, and the total ignition delaytime actually increases with increasing com-pressed gas temperature. At this point, the firststage ignition is hardly discernible, and theincrease in total ignition delay is clearly equiv-

alent to an increase in the second stage ofignition.

The results in Fig. 9(a) actually representcomputed temperature versus time profiles cal-culated by the kinetic model, but the model-predicted cool flame and total ignition delaytimes have been shown to agree well with thosemeasured experimentally, Fig. 9(b). The modelreproduces the absolute delay times for both thefirst and second stages of ignition. Observe thatthe ignition becomes essentially a single-stageignition as the temperature exceeds 810 K,although the NTC region does not disappearuntil the temperature exceeds 850 K. The intri-cate interactions in the reaction mechanismresponsible for this complex behavior are se-verely tested by these experiments.

In rapid compression machine experiments,the question often arises as to whether or notthere is heat loss to the combustion chamberwalls and whether significant reaction takesplace during the compression stroke, especiallywhen the compression time is comparable to theinduction period as is the case of the experi-ments of Minetti et al., in which the compres-sion time is 60 ms.

Recent RCM experiments have been carriedout by Griffiths et al. [9–11] in order to addressthe two-fold question of reaction during thecompression stroke and heat loss to the cham-ber walls. Experiments were carried out at acompression ratio of 11:1 with a compressionstroke duration of 22 ms. Stoichiometric mix-tures of fuel and diluent were used, in the sameway as in the Minetti experiments. The initialtemperature of the gas mixture before compres-sion was 327 K at a pressure of 250 torr. Thetemperature of the chamber walls was assumedto remain constant at 340 K.

Fig. 9. (a) Model-predicted tempera-tures profiles in rapid compression ma-chine experiments [12] depicting NTCbehavior. (b) Experimental (points)[12] and model prediction (lines) igni-tion delay time (F) and first ignition(E). Time zero is the time at the end ofcompression and Tc is the temperatureat the end of compression. Dashed linecorresponds to open circles.


The model was used to simulate the overallprogress of reaction, beginning at the start ofthe compression stroke. The temperature andspecies concentrations were assumed to be uni-form across the combustion chamber. As in theexperiments, the initial mixture temperaturewas 327 K, and the diluent concentrations wereselected to vary the compression temperature.In this study, two sets of simulated compressionhistories were computed similar to the Minettiexperiments, both neglecting heat losses to thecombustion chamber walls. The first consistedof a full kinetic calculation including both reac-tions and heat release during the compressionstroke, followed by the constant volume calcu-lation throughout the ignition period. In thesecond set of calculations, it was assumed thatno reaction occurred during the compressionstroke, and a fresh charge of fuel/O2/diluent wasused for the constant volume ignition delaycalculations. The results of these two sets ofcomputations are shown in Fig. 10(a) and arelabeled reactive and unreactive. Below 780 K,the two sets of calculations show the sameignition delay time as a function of the com-puted mixture temperature at the end of com-pression, showing that essentially no fuel con-sumption or other reaction takes place duringthe compression phase. As the mixture compo-sition is adjusted further to increase the temper-ature following compression, the model showsthat the extent of reaction during compressionbegins to increase. In particular, the concentra-tions of ketohydroperoxide species increase. Ata temperature of about 830 K, these speciesstart to decompose, leading to rapid chainbranching and rapid heat release.

At this point, a second factor becomes impor-tant, the equilibrium of the addition reactionbetween alkyl radicals and molecular oxygen.

This reaction represents the transition betweenthe high- and low-temperature reaction re-gimes; at temperatures above some limitingvalue, the equilibrium of this reaction lies on theside of the alkyl radical and molecular oxygen,while at lower temperatures the alkylperoxyradical is formed and the low-temperature re-gime is important. It is interesting to note that,regardless of the ratio of Ar to N2 for all of themixtures in Fig. 9(a) that show a two-stageignition, the first stage ends when the gas tem-perature reaches 850–900 K. For each mixture,the first ignition is largely associated with keto-hydroperoxide species decomposition at a tem-perature between 800 and 850 K, and the end ofthe first stage occurs when the temperaturereaches a level where the equilibrium of theR 1 O2 5 RO2 reactions begin to shift towarddissociation at about 900 K, thereby shutting offthe low-temperature branching reaction paths.

An important conclusion of the present mod-eling study is that reaction during the compres-sion phase leads to a dependence of ignitiondelay time on compression temperature asshown in the reactive curve in Fig. 10(a). Thelong flat region from 750–850 K is due to fuelconsumption and heat release during the finalportions of the compression stroke. However, ifa modeling study assumes that no reactionoccurs during compression and that the initialtemperature for computation is that reached atthe end of compression, the computed ignitiondelay times would appear to be like the unreac-tive curve in Fig. 10(a). This familiar s-shapedcurve would be expected on the basis of manytheoretical studies, but it bears little resem-blance to actual experimental results over thetemperature range from 800–1000 K.

The question as to whether or not there wasany heat loss to the chamber walls was also

Fig. 10. (a) Experimental (points) [9,10] and model-predicted ignition delaytimes. The terms “reactive” and “un-reactive” refer only to modeling of thecompression stroke portion of eachsimulation. (b) Pressure histories forunreactive case with n-heptane:N2:Ar 5 1:20:32.5. Open circles refer toexperiments and the solid line to thecalculation using the heat transfer co-efficient selected, see text.


addressed. In order to determine an overall heattransfer coefficient, an unreactive mixture ofn-heptane and diluent in the absence of O2 wasanalyzed experimentally, undergoing both thecompression and constant volume portions ofthe experiment. The initial conditions wereidentical to those described for reactive experi-ments above, with the mixture containingn-heptaneN2/Ar in the ratio 1:20:32.5. The pres-sure profile for this experiment was recordedand is shown as the points in Fig. 10(b). Clearly,there is indeed some heat loss, indicated by apressure drop, the large part of which occurs atthe end of compression just as the piston comesto rest. This same mixture was simulated nu-merically, treating the combustion chamber as ahomogeneous reactor, and varying the volumewith time to simulate the compression. Spatialtemperature variations in the reactor were ne-glected, treating heat loss as a distributed heattransfer rate, proportional to the temperaturedifference between the average gas temperatureand the time-averaged wall temperature. Thecoefficient of proportionality is an effective heattransfer coefficient that was determined in thefollowing manner. We varied the heat transfercoefficient to reproduce as closely as possiblethe pressure history measured in the unreactiveexperiment of Griffiths et al., the result of whichcan be seen as the line in Fig. 10(b).

A third set of RCM simulations was thencarried out using the same heat loss submodeland also including the effects of chemical reac-tion during the compression stroke. The com-puted results are shown in Fig. 10(a) and showbetter agreement with the experimental resultsthan when either heat losses or reaction duringcompression are neglected. A final refinementin which spatial variations in compressed gastemperature are included in a coupled kinetics/fluid mechanics model was not carried out inthis study, although Griffiths [71] has previouslyexamined this issue, using a somewhat simpli-fied kinetic mechanism.

SENSITIVITY ANALYSIS

A detailed analysis was carried out to investi-gate the sensitivity of each class of reaction,denoted earlier, to the oxidation of n-heptane.

We have tried to focus our attention on low-temperature kinetics as these have the greaterimpact on engine knock chemistry, and so wedid not include sensitivity analyses of the shocktube experiments of Vermeer et al. [1] or Coatsand Williams [2], where only high-temperaturekinetics are important. In analyzing the chemis-try edits produced as output from the model, wewere able to develop a flux diagram of the majoroxidation pathways responsible for n-heptaneoxidation, as seen in Fig. 3. For this reason, ouranalysis has focused on the classes of the reac-tion mechanism that are represented in Fig. 3and we assume all others are of minor impor-tance. Furthermore, we did not include sensitiv-ity to other reactions such as OH 1 OH 1 M 5H2O2 1 M even though Koert et al. [60] haveobserved a high sensitivity to this reaction. Theyfound that this reaction is important in control-ling the overall reaction rate at the end of theNTC region and has comparatively little effectat the onset of the NTC region.

Sensitivity analyses were performed by multi-plying the rate constants of a particular class ofreaction by a factor of two (both forward andreverse rates) and then calculating the percentchange in reactivity. In the case of the shocktube experiments of Ciezki and Adomeit [5], forexample, we calculated the percent change inignition delay time compared with the baselinesimulation. A positive percent change indicatesa longer ignition delay and a decreased overallreaction rate, and a negative change indicatesan increased overall reactivity of the system.Three different temperatures were chosen tohelp indicate sensitivity of each class to theonset, middle, and end of the NTC region at anaverage pressure of 13.5 atm. The reaction rateconstants that exhibited the highest sensitivityare shown in Fig. 11. Reactions in which wemultiplied both forward and reverse rate con-stants by a factor of two are denoted with anequal to “5” sign between reactants and prod-ucts, and reactions in which we multiplied onlythe forward rate constant (i.e., effected achange in the equilibrium constant) are denotedwith an arrow “f” between reactants and prod-ucts.

The reaction class with the highest negativesensitivity and is therefore the most effective inpromoting the overall rate of oxidation is reac-


tion type 23, the isomerization of the peroxy-alkylhydroperoxide radical to form a ketohy-droperoxide molecule and OH radical:

xC7H14OOH™yO2º23

nC7 ket xy 1 OH

In xC7H14OOH™yO2, x refers to the number ofthe carbon with the OOH group attached and yrefers to the site where the O2 group is attached.In nC7ketxy, x refers to the number of the Cwith the keto group attached and y refers to thesite where the hydroperoxy group is attached.We have also observed identical sensitivity tochanging the equilibrium constant of this reac-tion by a factor of two, i.e., multiplying theforward rate by two but maintaining the reverserate at its usual value. This result is expected, asthe reverse rate of addition of OH radical to theketohydroperoxide is very slow and is not ob-served computationally.

Reaction type 23 leads to the formation ofreactive OH radicals. Subsequent decomposi-tion of the ketohydroperoxide molecule leads tothe formation of another OH radical and anoxygenated-alkoxy radical, reaction type 24,which is chain branching:

nC7 ket xy 324

oxygenated radical species 1 OH

We observe a high negative sensitivity coeffi-cient to this reaction type. At low temperaturesthe high activation energy barrier (43,000 cal/mol) associated with the decomposition of ke-tohydroperoxide molecules is difficult to over-come and ensures that this reaction occurs veryslowly. As fuel oxidation proceeds and the as-sociated heat release raises the reactor temper-

ature, these stable molecules decompose morereadily, relieving this “bottleneck” and ensuringgreater reactivity of the system. This behavior is,to a large degree, responsible for the first stageor cool-flame ignition at low temperatures.

We observed a large sensitivity to changingthe equilibrium constant of reaction type 22 bya factor of two (i.e., multiplying the reverseaddition rate by two but maintaining the for-ward rate at its usual value):

xC7H14OOH™yO2722

xC7H14OOH™y 1 O2

This sensitivity is easily understood, as theaddition of hydroperoxy-heptyl radical to O2

leads preferentially to chain branching and al-lows fewer hydroperoxy-alkyl radicals to decom-pose.

The sensitivity coefficients of reaction types22–24 decrease in importance as the tempera-ture increases because, at higher temperatures,heptyl and hydroperoxy-heptyl radicals decom-pose more readily via b-scission rather than gothrough the successive O2 addition channelsthat lead to the formation of two reactive OHradicals. These b-scission channels becomemore accessible because the activation energybarriers associated with these unimolecular re-actions are overcome more easily with risingtemperature. Also at higher temperatures, theinfluence of the equilibrium of R 1 O2º RO2

becomes more important as shown by the sen-sitivity in Fig. 11. The equilibrium shifts fromfavoring formation of RO2 to reverse reaction,R 1 O2. This equilibrium shift further increasesthe concentration of R that can react via b-scis-sion.

The reaction that is next greatest in promot-ing the rate of fuel oxidation at low tempera-tures is H-atom abstraction from the fuel by OHradicals. This high negative sensitivity was alsoobserved by Koert et al. in their study on theoxidation of propane. However, as the temper-ature increases to 950 K and above, this reactiontype decreases in importance because a de-crease in the rates of reaction classes 22–24effects a decrease in the production of OHradicals. Thus, a reduced sensitivity to H-atomabstraction by OH radicals is observed at highertemperatures.

The production of heptenes and HO2 radicals

Fig. 11. Sensitivity coefficients for shock tube simulations[5]. Stoichiometric fuel in air, P5 5 13.5 bar.


from hydroperoxy-heptyl radicals, reaction type20, shows the highest positive sensitivity at 750K and 950 K:

xC7H14OOH™y º20

C7H14 1 HO2

This reaction has a large inhibiting effect on theoverall oxidation rate [60] and plays an impor-tant role in producing NTC behavior as previ-ously described in the literature [72]. This reac-tion competes with the addition of molecularoxygen to QOOH described above, consumesone hydroperoxy-heptyl radical, and producesthe relatively unreactive hydroperoxyl radicaland a stable heptene species.

For this same reason, two related reactiontypes, which are also important in producingNTC behavior, show a relatively high positivesensitivity coefficient; (1) the production of acyclic ether and OH radical from QOOH, reac-tion type 19, and (2) the b-scission of QOOH toform an olefin, aldehyde and OH species, reac-tion type 21.

xC7H14OOH™y 319

x™yC7H14O 1 OH

xC7H14OOH™y 321

olefin 1 aldehyde 1 OH

In x™yC7H14O, x and y refer to the position ofthe C™O bonds in the cyclic ether. The positivesensitivity coefficient for both of these reactionsis quite similar but is not as great as that abovefor the formation of heptene and HO2 species.This is because one relatively reactive OH rad-ical is formed in these reactions which leads togreater overall reactivity than does the forma-tion of the HO2 radical.

Alkyl radical b-scission, reaction type 3, alsoshows a positive sensitivity coefficient at 750,950, and 1100 K. This reaction produces a stableolefin and a relatively unreactive methyl orlarger alkyl radical and competes with molecu-lar oxygen addition and the low-temperaturechain branching reactions which increase theoverall reactivity of the system. This reactionhas a high activation energy so that it exhibitshigher sensitivities at 950 K and 1150 K than at750 K. Whether the forward and reverse rateconstants are changed by a factor of two or justthe forward rate constant, the same sensitivity isobtained. This indicates the calculations are notsensitive to the reverse rate, R9 1 olefin.

One very interesting and quite unexpectedresult of this analysis was the sensitivity coeffi-cient observed for heptylperoxy radical isomer-ization, reaction type 12:

xC7H15O2º12

xC7H14OOH™y

When we multiplied both the forward andreverse rate constants by a factor of two, thisreaction type showed a positive sensitivity coef-ficient at 750 and 950 K and a small negativecoefficient at 1100 K. In order to understandwhy this happens one must consider the prod-ucts favored when we increase both of thesereaction rate constants. At 750 and 950 K weobserve more sensitivity to the reverse isomer-ization, the formation of heptyl-peroxy radicalfrom hydroperoxy-heptyl radical. Thus, we in-hibit the chain-branching pathway, which resultsin lower reactivity and a positive sensitivitycoefficient. At 1100 K the reverse is true, we seemore sensitivity to the forward isomerization. Inthis case hydroperoxy-heptyl radicals that arefavored can form (1) an olefin and HO2 radical,(2) b-scission to produce an OH radical and analkoxy radical, or (3) a cyclic ether and OHradical. Molecular oxygen addition does notoccur as much at 1100 K because this reaction isbimolecular, and the activation energy barriersto the QOOH decomposition reactions are eas-ily overcome. These decomposition reactionsform OH and HO2 radicals, which are morereactive than CH3 or other larger alkyl radicalsthat may be formed from alkyl radical b-scis-sion, reaction type 3, at 1100 K. Thus reactiontype 12 shows a negative sensitivity at 1100 K.This argument is supported by the fact that weobserve negative sensitivity coefficients at allthree temperatures when only the forward rateconstant is multiplied by two, i.e., when wechange the equilibrium constant as this favorschain branching at 750 and 950 K and leads tomore reactive OH and HO2 radicals at 1100 K.

A sensitivity analysis was also performed onthe rapid compression machine experiments ofGriffiths et al. [9, 10]. In carrying out thisanalysis we included both the compression andconstant volume portions of the experiment butdid not include any heat losses to the walls. Theresults are shown in Fig. 12 and report sensitiv-ity at three different ignition delay times. We do


not include a constant compressed gas temper-ature as the temperature and pressure variedslightly for each sensitivity calculation. The tem-peratures reached in the baseline simulationswere 754.16 K, 790.98 K, and 952.05 K, corre-sponding to ignition delay times of 3.39 ms, 3.64ms, and 1.99 ms, respectively. These sensitivityresults are almost identical to those observed inthe shock tube of Ciezki and Adomeit [5]. Wesee high negative sensitivity coefficients at allthree “temperatures” to the reactions that leadto chain branching.

nC7H16 1 OH32

xC7H15 1 H2O

xC7H15 1 O2310

xC7H15O2

xC7H14OOH 2 yO2722

xC7H14OOH 2 y 1 O2

xC7H14OOH 2 yO2323

nC7 ket xy 1 OH

nC7 ket xy324

oxygenated radical species 1 OH

Reactions that inhibit the low-temperaturechain-branching reactions show positive sensi-tivity coefficients.

xC7H1533

olefin 1 R9

xC7H14OOH 2 y 319

x 2 yC7H14O 1 OH

xC7H14OOH 2 y 320

xC7H14 1 HO2

xC7H14OOH 2 y 321

olefin 1 aldehyde 1 OH

It should be noted that, again, by multiplyingboth the forward and reverse rate constants of

peroxyalkyl radical isomerization, reaction type12,

xC7H15O2º12

xC7H14OOH 2 y

by a factor of two, we see a positive sensitivitycoefficient but, when we increase the equilib-rium constant to favor the forward isomeriza-tion, we see a negative sensitivity coefficient.This is also consistent with the sensitivity resultsfor Ciezki and Adomeit.

The sensitivity coefficients associated with theRCM experiments of Minetti et al. [12], variablepressure flow reactor experiments of Dryer etal. [20, 22], and the experimental results ob-tained by Dagaut et al. [7, 8] in a jet-stirredreactor are depicted in Figs. 13–15, respectively.Foremost, it is clear that the same families ofreactions have the same overall influence on the

Fig. 12. Sensitivity coefficients for rapid compression ma-chine simulations [9, 10]. Stoichiometric fuel in diluent air.Ti 5 340 K, Pi 5 250 torr, CR 5 11:1.

Fig. 13. Sensitivity coefficients for rapid compression ma-chine simulations [12]. Stoichiometric fuel in diluent air. Ti

5 355 K, Pi 5 162 torr, CR 5 9.8:1.

Fig. 14. Sensitivity coefficients for flow reactor simulations[21, 22], 0.14% n-heptane oxidation at 12.5 atm, f 5 1.0, t5 1.8 s.


simulations as discussed above for the shocktube results. There is a general trend, especiallyfor Figs. 13–15, where the sensitivity seems to begreatest for conditions in the middle of the NTCrange. This points out the fact that this is theregime in which the competition between theketohydroperoxide/chain-branching reactionpath and the hydroperoxyalkyl decomposition/chain-propagation paths is most closely bal-anced, so a perturbation in this balance has thegreatest effect.

CONCLUSIONS

The present study has developed a detailedreaction mechanism for n-heptane oxidationusing the best available kinetic data and soundthermochemical analyses. The mechanism hasbeen tested quite thoroughly by comparingcomputed results with a wide variety of experi-mental data reported by different authors usinga number of experimental techniques. Agree-ment between computed and measured resultswas generally very good and suggests stronglythat the great majority of the important reactionpaths and rate expressions are reasonably cor-rect.

Many of the reaction paths and rate expres-sions in the mechanism can still be improvedconsiderably. Many of the experimental studiesused here for model testing and validationinclude significantly more data than were usedhere, and we intend to carry out more thoroughanalyses in the near future using these data.

Further model refinements may be producedthrough these future efforts.

The model and the accompanying sensitivityanalyses have shown the reaction pathways thatare particularly important in each regime ofpressure, temperature, and equivalence ratio.As in previous kinetic modeling by many au-thors, the key to understanding the reactionmechanism is a careful and accurate descriptionof the major chain-branching reaction paths andthose kinetic processes that compete with thechain-branching paths. In the present mecha-nism, the production and decomposition reac-tions of ketohydroperoxide molecules werefound to provide the important low-tempera-ture chain branching, and the hydroperoxyalkyldecomposition reactions provide the majorcompetition.

The authors wish to thank Professor E. Ranzi ofPolitecnico di Milano for assistence with thelumped model and for support of P. Gaffuriduring the period of this work. We would like tothank Prof. F. Dryer, Dr. T. Held, and Mr. C.Callahan for providing experimental data prior topublication. Furthermore, we are grateful to Dr. R.Minetti, Dr. P. Dagaut, and Dr. J. Griffiths forproviding us with additional data and insightsfrom their experimental results. Finally, we wouldlike to thank Prof. Joseph Bozzelli for his insightsin the model development and updates on hisH/C/O and bond dissociation groups for THERM.This study was performed under the auspices ofthe U.S. Department of Energy by the LawrenceLivermore National Laboratory under contractNo. W-7405-ENG-48.

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3. Burcat, A., Farmer, R. F., and Matula, R. A. (1981).Thirteenth Symposium (International) on Shock Tubesand Waves, (Ch. E. Treanor, and J. G. Hall, Eds.), pp.826–833.

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Received 13 June 1997; accepted 10 September 1997


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