temperature-dependent feature sensitivity analysis for combustion modeling

Temperature-DependentFeature Sensitivity Analysisfor Combustion ModelingZHENWEI ZHAO, JUAN LI, ANDREI KAZAKOV, FREDERICK L. DRYER

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544-5263

Received 7 July 2004; accepted 3 December 2004

DOI 10.1002/kin.20080Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: Sensitivity analysis is one of the most widely used tools in kinetic modeling. Typ-ically, it is performed by perturbing the A-factors of the individual reaction rate coefficientsand monitoring the effect of these perturbations on the observables of interest. However, thesensitivity coefficients obtained in this manner do not contain direct information on possibletemperature-dependent effects. Yet, in many combustion processes, especially in premixedflames, the system undergoes substantial temperature changes, and the relative importance ofindividual reaction rates may vary significantly within the flame. An extension of conventionalsensitivity analysis developed in the present work provides the means of identifying the temper-atures at which individual reaction rate coefficients are most important as a function of inputparameters and specific experimental conditions. The obtained information is demonstratedto be of critical relevance in optimizing complex reaction schemes against multiple experimen-tal targets. Applications of the presented approach are not limited to sensitivities with respectto reaction rate coefficients; the method can be used for any temperature-dependent propertyof interest (such as binary diffusion coefficients). This application is also demonstrated in thispaper. C© 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 282–295, 2005

INTRODUCTION

Modeling of combustion phenomena requires knowl-edge of reaction rates, diffusion coefficients, and otherphysical/chemical properties of the reactants as in-put parameters, and produces the observables (such asspecies concentration, temperature profiles, and lami-nar flame speed) as the output, with the input and theoutput connected by dynamic equations. Applicationof sensitivity analysis, a useful and efficient approach

Correspondence to: Frederick L. Dryer; e-mail: [email protected].

Contract grant sponsor: Chemical Sciences, Geosciences, andBiosciences Division, Office of Basic Energy Sciences, Office ofScience, U.S. Department of Energy.

Contract grant no. DE-FG02-86ER13503.c© 2005 Wiley Periodicals, Inc.

to examine the relationship of the underlying input andoutput, has been adopted in kinetic modeling sometimeago (e.g., [1,2]). Since sensitivity analysis methodswere introduced to combustion research (e.g., [3–5]),they have been widely used either to aid in qualitativeunderstanding the observed kinetic behavior or to assistin improving or optimizing kinetic models. Sensitivityanalyses are usually performed on the reaction rates,specifically the A-factors of the corresponding rate co-efficients, by perturbing them with a small constant,thus essentially perturbing the specific rate constantover the entire temperature range of the calculation(Fig. 1). However, in many combustion processes, es-pecially in flames, the system transformation inherentlyoccurs over a very wide range of temperature. Giventhe fact that reaction rates generally exhibit a widelydiffering and nonlinear dependence on temperature, the

TEMPERATURE-DEPENDENT FEATURE SENSITIVITY ANALYSIS FOR COMBUSTION MODELING 283

Figure 1 Schematic representation of the use of global andlocal perturbations to obtain sensitivity coefficients definedby Eqs. (4) and (6).

relative sensitivities of the reactions are also expectedto vary with temperature. Sensitivity methods that varyonly A-factors cannot provide direct information ontemperature dependence of the sensitivity spectrum.

Free-radical diffusion rates can be as critical as el-ementary reaction kinetics in accurately predicting acombustion process, and the main model input parame-ters (binary diffusion coefficients) are also temperaturedependent. Although the importance of species diffu-sion in flame kinetics has long been recognized, sen-sitivity analysis with respect to diffusion coefficientswas not commonly performed until very recently [6].Mishra et al. [6] applied sensitivity analysis to a burner-stabilized flame to qualitatively describe the influenceof diffusion coefficients on the predicted flame struc-ture. More recent efforts by Yang et al. [7] and Middhaet al. [8] have focused on the effects of binary diffusioncoefficients on laminar flame speed. In these works,sensitivity analysis was performed in a similar manneras for the reaction rates, i.e. no temperature-dependentinformation was obtained.

In this work, temperature-dependent sensitivity co-efficients with respect to elementary reaction rate coef-ficients and binary diffusion coefficients for premixedflames were obtained by applying a small temperature-dependent perturbation to the property of interest. Theperturbation was taken to be a Gaussian function oftemperature, and the center of the Gaussian profilewas varied with the assigned temperature mean. Bygradually moving the center of this Gaussian window,new, temperature-dependent sensitivity coefficients areobtained. A similar idea has recently been suggestedby Rumminger et al. [9]. Rumminger et al. used arectangular-shaped perturbation (which they called a“band”) in premixed flame simulations to identify the

temperature ranges where the flame suppression by in-hibitors is expected to be most effective. A series oftests were performed for atmospheric CH4/air laminarpremixed free flames, and, amongst the particular tests,Rumminger et al. [9] reported temperature-dependentresponses of predicted laminar flame speed to the per-turbations applied to two important reactions,

H + O2 = OH + O (R1)

and

CO + OH = CO2 + H (R2)

Here, we propose a more rigorous methodology andapply it in a systematic manner to a number of casesrelevant to detailed reaction model development, withbroader implications with respect to combustion mod-eling in general.

The sensitivity parameters for several elementaryreactions, such as H + O2 = OH + O, (R1), CO +OH = CO2 + H, (R2), HCO + M = CO + H + M,(R3), HCO + O2 = CO + HO2 (R4), and H + O2

(+M) = HO2 (+M) (R5) were obtained for H2, H2/CO, CH3OH, and C3H8 kinetic flame models. Thetemperature-dependent sensitivity of binary diffusioncoefficients of H-atom with He and H2O was alsodetermined for selected conditions. The obtained re-sults were analyzed to identify the physical/chemicalmechanisms responsible for the observed behavior. Thepractical examples through which we demonstrate theutility of the present methodology for development ofcomprehensive kinetic models are discussed in detailbelow.

METHODOLOGY AND IMPLEMENTATION

To develop a more general formulation, we consider adynamic system which yields an observable F(x, �)given a vector of variables x and vector of parameters� . Elementary sensitivity gradient (coefficient) Si withrespect to the parameter αi for the observable F isdefined as

Si = ∂ F

∂αi(1)

In combustion modeling, it is more common to use log-arithmic sensitivity gradients (or normalized sensitivitycoefficients),

si = ∂ log F

∂ log αi= αi

F

∂ F

∂αi(2)

to follow relative (rather than absolute) changes inproblem input parameters and observables. In practice,

284 ZHAO ET AL.

the system parameter αi is usually written in the form

αi = α0i (1 + pi ) (3)

where α0i is the baseline value of the parameter and

pi � 1 is the perturbation. Although αi itself may bea function of system variables x, the perturbation pi

is usually treated as an independent variable, i.e., itis applied in a uniform manner over the entire rangeof possible variation in αi . In such a case, Eq. (2) ispresented as

si = 1

F

∂ F

∂pi

∣∣∣∣pi →0

(4)

Next, we consider the situation when the system pa-rameter αi is, in fact, a function of one of the systemvariables, x j . In this case, the sensitivity coefficientgiven by Eq. (4) is still pertinent in providing, for ex-ample, relative importance of αi in comparison withother parameters in the system. However, it may alsobe of interest to know specific conditions (range of x j )where F is most sensitive to variation in αi . This in-formation is unavailable due to the global nature of theperturbation pi used to determine si in Eq. (4). Use oflocal (with respect to x j ) perturbation of αi appearsto be the logical step to address this issue. Here, wepropose a perturbation that is Gaussian shaped withrespect to x j ,

pi = xpj√

2πσexp

[−

(x j − xc

j

)2

2σ2

](5)

where xcj is the center of the perturbation, σ the width

of the perturbation, and xpj the magnitude of the pertur-

bation. Using the perturbation defined by Eq. (5), weintroduce a new sensitivity coefficient,

ξi(xc

j

) = 1

F

∂ F

∂xpj

∣∣∣∣∣xp

j →0,σ→0

(6)

which is a function of the center of perturbation, xcj . We

note that Eq. (6) corresponds to the limit when the func-tional form of the Gaussian perturbation is reduced to aDirac delta function. However, we retain the definitionof perturbation via Eq. (5) since it will be used in actualnumerical implementation of the proposed approach.

The physical meaning of the new sensitivity analysisis further illustrated in Fig. 1. The global perturbationused to define the sensitivity coefficient si , Eq. (4), isindependent of x j and shown with a straight horizon-tal line. The local perturbations shown with a series ofnarrow Gaussian peaks, on the other hand, are confined

to the regions around the corresponding centers of theperturbations. By moving the Gaussian probe along thex j axis and monitoring the response of F to this per-turbation, ξi is defined as a function of the center of theprobe. Having this information, one can quantitativelyidentify the range(s) of x j where the impact of variationin αi would be the greatest. In practical applications,system parameters αi always have uncertainties thatmay also vary with x j . The sensitivity coefficient ξi

can therefore be used as a guidance tool to determineconditions at which αi should be defined with highprecision in order to yield accurate predictions of F .

PRACTICAL IMPLEMENTATION: LAMINARFLAME SPEED

For the remainder of the paper, we demonstrate this newsensitivity analysis approach using a laminar premixedfree flame system as an example. The main observablefor this system is laminar flame speed sL, a fundamentalproperty often used to validate detailed reaction mech-anisms. Thus, sL is used as marker of the system re-sponse to the parameter change, i.e., as an observable Fin Eqs. (4) and (6). The primary parameters describingthe system are the rate coefficients of chemical reac-tions composing the detailed reaction model. The ratecoefficients are strongly nonlinear functions of one ofthe system variables, the temperature T . Furthermore,the rate coefficients are rarely defined with sufficientaccuracy over the entire range of temperatures encoun-tered in flames. More commonly, rate coefficients areobtained by correlation of experimental measurementsover a limited temperature range and then extrapolatedoutside this range using various analytical expressions.Accuracies of the original correlation as well as theextrapolation are difficult to assess and usually remainas open questions. On the other hand, flames repre-sent systems where the temperature variations duringthe combustion process are very large as compared toother systems normally used to study chemical kinet-ics (such as shock tubes and flow reactors). Therefore,the use of sensitivity coefficients based on local (withrespect to the temperature) perturbation, Eq. (5), offersan opportunity to identify the “temperature windowsof importance” for specific reactions controlling theflame propagation. Because the sensitivity coefficients(6) will be considered exclusively in terms of theirtemperature dependence in all of the following exam-ples, we redefine parameter notations in the Gaussianperturbation (5),

pi = Tp√2πTσ

exp

[− (T − Tc)2

2T 2σ

](7)


to make their meaning more clear, i.e., Tc is the center,Tσ the width, and Tp the magnitude of the perturbation.For the same reason, the sensitivity coefficient will bereferred to as simply ξ(T ), without a subscript.

A modified version of PREMIX package [10,11]was used in all flame calculations. As in the originalPREMIX formulation, the sensitivities reported hereare computed for the net reaction rates, i.e., equilibriumconstants for reversible reactions were unaltered whenthe perturbation was applied. An algorithm based onthe numerical Jacobian implemented in PREMIX wasadopted to compute the sensitivity coefficients. Daviset al. [12] have recently argued that direct brute-forcemethod yields more accurate results as compared tothe PREMIX’s numerical Jacobian method. We alsoperformed several tests with brute-force calculationsand reached similar conclusions; however, the accu-racy of the numerical Jacobian method was found tobe acceptable for the purposes of the present work andoffers substantial savings in computational time.

RESULTS AND DISCUSSION

Simulations were carried out for hydrogen, carbonmonoxide, methanol, and propane laminar premixed(free) flames. Different reaction models were chosenfor different fuels based primarily on the their abilityto accurately predict the flame speeds over the entireequivalence ratio range. For hydrogen flames, we usedthe updated hydrogen mechanism of Li et al. [13]. Themechanism of Mueller et al. [14] updated with the hy-drogen submechanism of Li et al. [13] was adopted forCO/H2 flame calculations. The sensitivity analysis formethanol flames was performed with the kinetic modelof Held and Dryer [15]. Finally, the optimized C1 C3

mechanism of Qin et al. [16] was utilized for propaneflames.

Sensitivity Analysis for HydrogenCombustion Mechanism

The “conventional” sensitivity spectrum based onEq. (4) for laminar flame speed of H2/air flame freelypropagating at 1 atm and room temperature is shown inFig. 2. The information presented in this figure is welldocumented in numerous prior studies, and the mainchain branching reaction (R1) is clearly most influen-tial. What is not clear from these results, however, ishow the perturbation of the specific rate constant valueat a particular temperature in the flame structure influ-ences the laminar flame speed. The significance of thisknowledge is that it indicates where uncertainties in therate constant as a function of temperature most affect

Figure 2 Conventional sensitivity analysis for H2/air lam-inar premixed flame at 298 K, 1 atm, and equivalence ratiosof 1.0 and 2.0.

the calculated flame speed. Reaction (R1) is in generalone of the most important elementary reactions in com-bustion chemistry and has been extensively studied ex-perimentally and theoretically. Recent assessments ofthe rate claim relatively small uncertainties over the en-tire temperature range of evaluation. The simplicity ofthe present system affords a powerful first example ofthe technique developed here in providing insights asto where uncertainties in this rate constant most signif-icantly affect the primary feature of interest. We there-fore start our demonstration of the methodology witha discussion of this reaction.

Because the sensitivity coefficient, Eq. (6), is eval-uated numerically, the choice of parameters in theGaussian perturbation, Eq. (7), which results in nu-merically converged values of the sensitivity coeffi-cient, needs to be verified in each application. Fig-ure 3 shows the results obtained for reaction (R1) usingdifferent values of Gaussian parameters. In this case,an acceptable convergence is observed for Tσ = 10 Kand Tp/

√2πTσ = 0.01. For all values of the input pa-

rameters, the sensitivity coefficient for reaction (R1)is observed to have a pronounced maximum at about1210 K, and approaches zero at low and high temper-atures. As the input parameters are varied to achieveconvergence, nearly the same temperature of peak sen-sitivity is found for even high values of Tσ . The asymp-totic convergence of the sensitivity coefficient ξ(T ) toits maximum value is illustrated in Fig. 4. The two dis-played curves correspond to results obtained by fixingeither Tσ at 10 K or Tp/

√2πTσ at 0.01 and observ-

ing the effect of variations in the remaining parameter

286 ZHAO ET AL.

Figure 3 Sensitivity coefficient for reaction (R1), H + O2= OH + O, in H2 kinetic model at stoichiometric, 1 atm, 298K with different Tσ and Tp/

√2π Tσ : (a) Tσ = 2500 K and

Tp/√

2πTσ = 0.01; (b) Tσ = 500 K and Tp/√

2πTσ = 0.01;(c) Tσ = 100 K and Tp/

√2πTσ = 0.01; (d) Tσ = 10 K

and Tp/√

2πTσ = 1; (e) Tσ = 10 K and Tp/√

2πTσ = 0.01;(f ) Tσ = 10 K and Tp/

√2πTσ = 10−4.

on the value of the ξ(T ). A robust numerical conver-gence is observed in both cases. In fact, the choice ofTσ = 10 K and Tp/

√2πTσ = 0.01 provided converged

solutions for all of the cases considered and thus, theseparameters were used in all of the remaining calcula-tions presented here.

Figure 5 presents the results of parametric calcula-tions for a number of hydrogen/air flames with differingambient pressures and fuel/air equivalence ratios. The

Figure 4 The effect of the perturbation magnitude andwidth on the obtained sensitivity peak value for (R1) in H2kinetic model at stoichiometric conditions, 1 atm and 298 K.

Figure 5 The sensitivity temperature window for reactionrate of (R1) in H2 kinetic model. The numbers in the figurerefer to equivalence ratio.

solid vertical lines indicate the entire temperature rangefor each specific flame (i.e., from the initial room tem-perature to the final adiabatic flame temperature), whilethe shaded bars on each solid line indicate the tempera-ture range where the value of the sensitivity coefficientξ(T ) for the reaction (R1) is larger than 10% of itsmaximum value (as illustrated in Fig. 3). We refer hereto this range of temperatures as the sensitivity windowfor the input parameter in question upon the observableof interest. The horizontal lines within each bar indi-cate the location of ξ(T )max. Here, we observe that thelaminar flame speed is sensitive to (R1) over only a nar-row subdomain of the overall temperature range withineach flame structure, and that this subdomain changesin both extent and location with equivalence ratio andambient pressure. The lowest temperature in each sub-domain is principally a function of pressure, with asecondary dependence on equivalence ratio at lean con-ditions, while the sensitivity window and temperatureat which the sensitivity is maximum, ξ(T )max, shiftstoward higher temperatures with increasing pressure.

The above results have immediate practical impli-cations. At atmospheric conditions, the sensitivity win-dow is confined to the range of about 850–1550 K andξ(T )max occurs at about 1210 K. As shown in Fig. 6,the rate coefficient expressions for reaction (R1) usedin several recent H2/O2 mechanisms [13,14,17] yieldnearly the same specific rate values within the tem-perature sensitivity window, with all correlations inter-secting close to the temperature of ξ(T )max. The lam-inar flame speed predictions for atmospheric H2/airflames are very similar for all three mechanisms andare in a good agreement with the experimental dataeven though different rate correlations were used.Furthermore, most of experimental measurements for


Figure 6 Rate coefficient of (R1), H + O2 = OH + O, usedin different H2 kinetic models.

this rate coefficient were conducted in shock tubes attemperatures much higher [18–22] than those withinthe sensitivity window for atmospheric conditions.However, differences associated with the correlationsat lower temperatures can become important in the pre-dictions. This condition can occur in flame speed ex-periments in low pressure flames, and the differencesin these correlations are also noticeable in other exper-imental configurations that are sensitive to the relativevalues of the rate constants for (R1) and (R5), H + O2+M = HO2 + M, at temperatures below 1000 K. Recentmeasurements of (R5) at temperatures below about 900K require the lower rate values for (R1) predicted bythe correlation used by Li et al.

At high pressures where the sensitivity windowmoves to higher temperatures, the disparities betweenthe rate coefficients used in different mechanisms be-come more apparent, and the sensitivity window over-laps with high-temperature shock-tube measurementsof this rate. As can be seen, the rate coefficient ofPirraglia et al. [20] used in the mechanism by Muelleret al. [14] becomes noticeably higher than the high-temperature consensus values of Yu et al. [21] usedby GRI-Mech 3.0 [17] and of Hessler [22] used by Liet al. [13] when extrapolated toward higher tempera-tures (Fig. 6). Mainly due to this reason, the computa-tional results obtained with the mechanism of Muelleret al. significantly overpredicts the experimental high-pressure laminar flame speed measurements of Tseet al. [23]. Further discussions of comparisons of thepredictions of these high-pressure flame speeds byMueller et al. appear in Li et al. [13].

Diluent addition/substitution is commonly utilizedin laminar flame research. For example, helium is sub-stituted for nitrogen in laminar flame speed experi-ments to reduce flame front instabilities [23]. As shownin Fig. 7, diluent changes have significant influences on

Figure 7 Sensitivity for (R1), H + O2 = OH + O, in H2 ki-netic model for different diluents at equivalence ratio 2.0 (N2:O2/N2 = 1/3.76, He1: O2/He = 1/3.76, He2: O2/He = 1/7.52,He3: O2/He = 1/11.28).

the sensitivity window for reaction (R1). For the samevolume percentage of diluent, the sensitivity windowsfor H2/He/O2 flames are wider than those for H2/airflames. When more diluent is used, the sensitivity win-dow shifts toward lower temperatures following thedecrease in the adiabatic flame temperature. These ob-servations indicate that kinetic models verified againstflame speed with one diluent may not correctly pre-dict the flame speed with another diluents dependingon differences in the correlations used to predict (R1)as a function of temperature. The temperature cor-responding to the peak value of ξ(T ) also increaseswith pressure. This occurs primarily because of the de-pendence of radical diffusion on ambient pressure. Athigher pressures, the diffusion of radicals generated inthe high-temperature portion of the flame zone is lesspronounced, and consequently, the maximum of thesensitivity coefficient moves to higher temperature.

As can be seen in Fig. 2, the reaction H2 + OH =H2O + H also strongly affects the predicted laminarflame speed. While similar sensitivity analyses havealso been conducted for this reaction, its rate coeffi-cient has comparatively smaller uncertainties withinthe corresponding sensitivity windows. As a result,variation of this rate coefficient did not produce appre-ciable changes in the predicted laminar flame speed.

Sensitivity Analysis for the CO/H2Combustion Mechanism

The CO/H2 flame system represents the next more com-plex dependence of flame speed on combustion chem-istry, and laminar flame speed studies for this systemhave been conducted for selected CO/H2 ratios [24,25].

288 ZHAO ET AL.

As shown in Fig. 8, it is well recognized that the burningvelocity of CO/H2 flames is very sensitive to the rateconstant of CO oxidation by OH, (R2), for CO/H2 vol-ume ratios greater than 1. Although the conventionalsensitivity of the flame speed to (R1) is considerablyreduced for these conditions, the functional characterand temperature windows defined by ξ(T ) for reac-tion (R1) were found to be similar to those observed inH2/air flames. The temperature-dependent sensitivitycoefficient ξ(T ) for reaction (R2) is very interesting inthat there are two maxima in the function at rich con-ditions that collapse into a single peak at lean condi-tions (Fig. 9). The sensitivity window for (R2) is muchwider and extends to much lower temperatures than forreaction (R1) (Fig. 5).

Sensitivity windows for a range of equivalence ra-tios and selected CO/H2 ratios for reaction (R2) arepresented in Fig. 10. The sensitivity patterns presentedin this figure provide immediate and direct guidanceregarding the choice of the rate coefficient correlationsappropriate for the specific rate of (R2) as a functionof temperature that are more appropriate for the pre-diction of the flame speed for these flames. Reaction(R2) is a chemical activation process well known to ex-hibit a strong non-Arrhenius behavior [26]. As one ofthe most important elementary reactions in combustionchemistry, it has been studied extensively, and there is alarge body of both experimental and theoretical resultsavailable in the literature. Figure 11 shows a compi-lation of most recent literature data. Of importance isthat most of reported experimental data are from high-temperature shock-tube studies, and the vast majorityof these data are outside of the sensitivity window for

Figure 8 Conventional sensitivity analysis for H2/CO flameat 298 K, 1 atm, and equivalence ratio 1.0.

Figure 9 Sensitivity coefficient for (R2), CO + OH =HO2 + H, in CO/H2 kinetic model at 1 atm and 298 K(CO/H2 = 95/5).

this reaction shown in Fig. 10. On the low-temperatureside, there are much fewer data points; however, thesensitivity of the flame speed to the specific rate con-stant at these temperatures still remains high. Two re-cent RRKM fits (Yu [27] and Senosian et al. [28])are intended to yield the rate coefficient for (R2) overthe entire range of relevant temperatures. However,both resulting correlations were primarily calibratedagainst high-temperature shock-tube data and signifi-cantly overpredict all available experimental determi-nations of this rate constant at intermediate tempera-tures around 900 K. These temperatures are not onlywithin the sensitivity window for this reaction but also

Figure 10 Sensitivity coefficient for (R2), CO + OH =HO2 + H, in CO/H2 flames at 298 K, atmospheric pressure.Black and white horizontal lines within each bar indicate lo-cations of first (highest) and second peak values, respectively(see Fig. 9). The numbers in the figure refer to equivalenceratio.


Figure 11 Rate coefficient of (R2), CO + OH = HO2 + H. The left-hand figure shows the compilation of all recent literaturedata, and the right-hand figure presents only the data within the sensitivity window for this reaction.

encompass the temperatures at which ξ(T )max occursas flame parameters of pressure, equivalence ratio, andCO/H2 ratio are varied. If one assumes that the theoret-ical fits misrepresent the specific rate of this reactionby comparison with the (presumably) more accurateexperimental data at these temperatures, one wouldexpect CO/H2 flame speeds to be overpredicted withthe reaction models based on these rate parameters. In-deed, as shown in Fig. 12, GRI-Mech 3.0 which usedthe values of Yu [27] overpredicts the experimentallymeasured CO/H2/air laminar flame speeds of McLeanet al. [24]. Based upon the observations reported here,Li et al. [29] recently performed a modified Arrhenius

Figure 12 Laminar flame speed of CO/H2/air mixture(CO/H2 = 95/5) at 1 atm and 298 K.

fit of all experimental data shown in Fig. 11, weightedby reported or estimated experimental errors. The re-sulting correlation is presented in Fig. 11. It should beemphasized here, as it is also noted in Li et al. [29],that error weighting of the experimental data is ab-solutely essential because at lower temperatures, theresulting fit gravitates toward the data of Ravishankraand Thompson [30]. However, if equal weighting is as-sumed, the correlated fit would be closer to the flamedata of Vandooren et al. [31], measurements that areknown to have much larger uncertainties. We also notethat the resulting fit is very close to the theoretical ex-pression of Troe [32] at low and intermediate tempera-tures. At high temperatures, however, the empirical fitproduces lower values, consistent with the experimen-tal data of Lissianski et al. [33]. Finally, the substitutionof this correlation for that used in the GRI-Mech 3.0mechanism for the specific rate constant of (R2) bringsthe model predictions into excellent agreement withthe experimental data (Fig. 12). It should be pointedout, however, that the result presented in Fig. 12 can-not be viewed as validation of this new fit within thescope of GRI-Mech 3.0; it serves only as an illustra-tion of the new method of sensitivity analysis. Detailedcomparisons of predictions of combustion phenomenainvolving CO oxidation kinetics inclusive of this cor-relation are discussed elsewhere [29].

Similar to the reaction (R2), the sensitivity windowsfor reaction HCO + M = CO + H + M, (R3) (Fig. 13),and HCO + O2 = CO + HO2, (R4), are very wideand start at a very low temperature. In CO/H2 flames,HCO is formed at very early stages by reaction of COand H atom, where H can be transported by diffusion

290 ZHAO ET AL.

Figure 13 Sensitivity coefficient for (R3), HCO + M =CO + H + M, in CO/H2 kinetic model at 1 atm and 298 K.Black and white line notations within shaded bars are thesame as in Fig. 10. The numbers in the figure refer to equiv-alence ratio.

from the high-temperature flame regions to the initialflame regions where CO, the initial reactant, is present.These reactions are not only of significance here, butare of essential importance in predicting flame speedsfor fuels where the principal source of HCO is CH2O,as discussed below.

Sensitivity Analysis for Methanol Flames

For methanol flames, the flame speed is most sensi-tive to HCO + M = CO + H + M (Fig. 14). Incontrast to CO flames, the temperature-dependent sen-sitivity for this reaction and the competing reaction,HCO + O2 = CO + HO2, are located at very hightemperatures (Fig. 15) in comparison to those for re-

Figure 14 Conventional sensitivity analysis for stoichio-metric methanol/air flame at 1 atm and 298 K.

Figure 15 Sensitivity coefficient for (R3), HCO + M = CO+ H + M, and (R4), HCO + O2 = CO + HO2, in methanolkinetic model at 1 atm and 298 K.

actions (R1) and (R2). In methanol flames, the majorsource for HCO is CH2O, which is generated in thehigh-temperature reaction zone. Note that the sensi-tivity window for reaction (R2) in hydrocarbon flamesystems displayed in Fig. 16 is considerably narrowerthan for CO flames, primarily because CO is an inter-mediate species formed at higher temperatures in theflame zone (not an initial reactant). Due to the differ-ent origins of CO and HCO, the temperature-dependentsensitivity spectra of the CO flame will generally differfrom those for hydrocarbon fuels.

As in the case of the comparisons of existing ratecorrelations for reactions (R1) and (R2) and their sig-nificance in predicting flame speeds, one finds similarimplications as to the temperature ranges over whichspecific rate constant correlations for (R3) and (R4)most affect flame speed predictions. These differencespoint to a need to develop a weighted correlation forthe specific rate constant measurements of (R3) to more

Figure 16 Sensitivity for reaction rate of (R2), CO +OH = CO2 + H, in methanol kinetic model at 1 and 10 atm,298 K. The numbers in the figure refer to equivalence ratio.


accurately predict flame speed results as well as otherkinetic phenomena. The specific reaction data availablefor (R4) are not sufficient to perform such a correla-tion over a wide temperature range, and thus a specificcorrelation of more limited data must be accepted. De-tails resulting from these procedures as well as com-parisons of the revised mechanistic behavior against awide range of kinetic experiments are presented in Liet al. [29].

Sensitivity Analysis for the C3H8Combustion Mechanism

Propane is generally considered to be the smallest hy-drocarbon molecule with combustion characteristicssimilar to larger hydrocarbon fuels. Because the flamespeeds for molecules larger than C3 are primarily con-trolled by the C3 submechanism (e.g., [38,39]), it isreasonable to expect that for large molecule hydro-carbon fuels, conventional sensitivity, temperature-dependent sensitivity, and temperature sensitivitywindow patterns for small species interactions will besimilar to those found for propane flames. Therefore,we use propane/air flame speed cases here as a finalapplication example.

For propane/air flames, temperature-dependent sen-sitivity analyses were performed on the five reactionsmost important to determining flame speed (Fig. 17):H + O2 = OH + O (R1), CO + OH = CO2 + H (R2),HCO + M = CO + H + M (R3), HCO + O2 =CO + HO2 (R4), and H + O2 (+ M) = HO2 (+ M) (R5)

Figure 17 Conventional sensitivity analysis for stoichio-metric propane/air flame at 1 atm and 298 K.

as identified by conventional sensitivity analyses. Theresults are compiled in Figs. 18–22, where the colorchanges in the bar indicate the sign change of the sen-sitivity coefficient. As compared to the same reactionin CO/H2 and hydrogen flames, it is again noted thatthe sensitivity windows are narrower and do not extendto as low temperatures as in either of these cases. It isalso interesting to note that for reactions (R2), (R4), and(R5), the sensitivity may change sign at high pressureand fuel-rich conditions (Figs. 20–22). Although HO2

is not important at low-pressure conditions, at higherpressures the enhanced formation rate of HO2 resultsin production of significant quantities of H2O2, whichin turn can decompose to produce OH. As a result, thesensitivities for reactions such as (R2) and (R5) maychange sign at those conditions.

Sensitivity Coefficientsand Reaction Rates

One might suspect that the reported temperature-dependent sensitivity coefficients with respect to a spe-cific reaction rate coefficient can be approximately cor-related with the corresponding reaction rates (at leastfor the laminar flame speed cases considered here). Fig-ure 23 shows comparisons of sensitivity coefficientsand the net reaction rates for (R1) and (R2). Indeed,the rate curves closely follow the corresponding sen-sitivity coefficients for the C3H8/air case. Clearly, insuch cases, one can simply evaluate the reaction ratesto approximately identify the sensitivity windows in-stead of following a complex series of steps involvedin the evaluation of the temperature sensitivity coeffi-cients themselves. A similar observation was reportedby Rumminger et al. [9]. However, this reasoning does

Figure 18 Sensitivity coefficient for (R2), CO + OH =CO2 + H, in propane/air flames at 1 and 10 atm. The numbersin the figure refer to equivalence ratio.

292 ZHAO ET AL.

Figure 19 Sensitivity coefficient for (R1), H + O2 =OH + O, in propane/air flames at 1 and 10 atm. The numbersin the figure refer to equivalence ratio.

not hold for all circumstances. As evidenced by the ex-ample of (R2) for CO/H2 flame, the reaction rate, whileclosely following the corresponding sensitivity coef-ficient at lower temperatures, extends to much highertemperatures where the actual sensitivity is nearly zero.Therefore, the sensitivity window estimated based onthe reaction rate curve would be substantially widerthan the exact result based on the temperature sensi-tivity analysis. Furthermore, the analyses for reaction(R1) in cases of H2 and CO/H2 flames show distinctdifferences between the net reaction rate and the tem-perature sensitivity coefficient results. While similarlyshaped, the curves exhibit peaks at very different tem-peratures, almost 500 K apart with the peak in the sen-sitivity coefficient occurring at a lower temperature.Because the sensitivity window width for these cases

Figure 20 Sensitivity coefficient for (R5), H + O2 + M =HO2 + M, in propane/air flames at initial T = 298 and 650K, 1 and 10 atm pressure. The numbers in the figure refer toequivalence ratio.

Figure 21 Sensitivity coefficient for (R3), HCO + M =CO + H + M, in propane/air flames at initial T = 298 and650 K, 1 and 10 atm pressure. Black and white line notationswithin shaded bars are the same as in Fig. 10. The numbersin the figure refer to equivalence ratio.

is about 700 K, the sensitivity window for (R1) esti-mated based on the reaction rate curve for these caseswould not only be misleading but also would have al-most no overlap with the actual sensitivity window.

Sensitivity Analysis for BinaryDiffusion Coefficient

The temperature-dependent sensitivity coefficients forbinary diffusion coefficients can be determined in amanner similar to that used in investigating kinetic pa-rameters. The temperature-dependent sensitivity fea-tures of binary diffusion coefficients are demonstratedhere using H2 flames.

Figure 22 Sensitivity coefficient for (R4), HCO + O2 =CO + HO2, in propane/air flames at initial T = 298 and 650K, 1 and 10 atm pressure. Black and white line notationswithin shaded bars are the same as in Fig. 10. The numbersin the figure refer to equivalence ratio.


Figure 23 Comparison of temperature-dependent sensitivity coefficients with the corresponding net reaction rates; solid lines—sensitivity coefficient, dotted lines—net reaction rate. All flame calculations were performed at stoichiometric conditions at 1atm and 298 K. For CO/H2/air flame, CO/H2 volumetric ratio is 95:5.

Figures 24 and 25 show temperature-dependent sen-sitivity of H/He and H/H2O binary diffusion coeffi-cients for laminar flame speeds of H2/O2/He mixtures.The sensitivity of binary diffusion coefficients ofH-atom to H2 and O2 and of He to H2 and O2 was alsostudied. The observed features are essentially the same

Figure 24 Sensitivity of H/He binary diffusion coefficienton H2/He/O2 laminar flame speeds at equivalence ratio 1.5,and the diffusion coefficient of H He for different expres-sions.

as those shown in Figs. 24 and 25, and detailed resultsare therefore not reported here. Similar to the case ofspecific rate analyses, the sensitivity with respect toa particular binary diffusion coefficient correlation isalso concentrated in a narrow temperature range andmay change sign. The shape of the computed ξ(T )function is a result of complex reaction/transport in-teractions within the flame structure. Figure 26 showsthe comparison of the normalized sensitivity case forH/He diffusion coefficient, (R1) reaction rate and theH-atom concentration gradient at the indicated con-dition. At lower temperatures, an increase in the dif-fusion coefficient causes an enhanced transport of Hatoms toward the cold region where they do not con-tribute to chain branching kinetics (primarily, via: (R1),

Figure 25 Sensitivity coefficient for H/H2O binary diffu-sion coefficient in H2/He/O2 flame at equivalence ratio 1.5.

294 ZHAO ET AL.

Figure 26 Sensitivity coefficient for H/He binary diffusioncoefficient (dotted line), reaction rate of (R2), H + O2 +OH + O, (solid line) and H concentration gradient (dashedline) for H2/He/O2 (He/O2 + 23/2) flame at equivalence ratio1.5 and 20 atm pressure. All presented quantities are normal-ized by the respective maximum values.

H + O2 = OH + O). Consequently, the sensitivitywith respect to the H/He binary diffusion coefficient isnegative. As the temperature increases, the H/He dif-fusion sensitivity becomes positive and peaks close tothe point where the rate of (R1) reaches a maximum.Here, the diffusive flux of H atoms to the region wherechain branching is active results in an increase in theflame speed. At even higher temperatures, the sensitiv-ity with respect to the H/He binary diffusion coefficientgoes through zero again (around the point where H con-centration has its peak, indicative of directional changein H diffusion flux). From this point, the diffusion fluxcarries H atoms away from the chain-branching regiontoward the postflame zone, and the sensitivity becomesnegative again. The temperature range and sensitivityvalue of this nonmonotonic shape may cause the overallbinary diffusion coefficient sensitivity to even changesign with initial conditions.

The nonmonotonic behavior of binary diffusion co-efficient sensitivity with respect to temperature hassignificant implications for detailed flame modeling.For example, Middha et al. [8] recently reported im-proved values of H/He binary diffusion coefficientwhich are generally higher than those computed withthe CHEMKIN II database over the entire temperaturerange of interest (Fig. 24). However, it was also re-ported that the resulting flame speeds did not changesignificantly. One of the reasons is clearly the rela-tively small difference (as compared to uncertainties inreaction rate coefficients) between the TRANFIT ex-pression and the result of Middha et al. Another reason

discovered in the present study is that the difference inbinary diffusion coefficient (which is uniformly pos-itive with slight increase with the temperature) maybe compensated by the sensitivity sign change withthe temperature, resulting in a minor difference in pre-dicted flame speed.

SUMMARY AND CONCLUSIONS

We have demonstrated here a novel applied sensitivityanalysis method using perturbation with a temperature-dependent Gaussian-shaped profile to determine therange of temperatures over which a predicted observ-able is sensitive to a particular input parameter. Todemonstrate the methodology and utility of the tech-nique, we presented results obtained in analyzing thetemperature sensitivity of flame speed predictions tothe most significant reactions in H2, H2/CO, CH3OH,and C3H8 oxidation systems. We also demonstratedthe methodology by analyzing the effects of temper-ature dependence of binary diffusion coefficients onpredicted H2/He/O2 flame speeds. The sensitivity pa-rameters for almost all reactions are very temperaturedependent. Our results also explain the cause of the signchange in the sensitivity coefficient of binary diffu-sion coefficient. The temperature-dependent sensitiv-ity approach not only provides new insights for flamemodeling, it also may serve as a tool to identify themost significant temperature range in which parame-ters should be experimentally and theoretically studiedto reduce uncertainties important to a particular set ofpredictions. Examples were also described to apply thetemperature-dependent sensitivity analysis in helpingto determine reaction rate for elementary reactions.

BIBLIOGRAPHY

1. Tomovic, R. Sensitivity Analysis of Dynamic Systems;McGraw-Hill: New York, 1963.

2. Bukhman, F. A.; Melamed, V. G.; Polak, L. S. Appli-cation of Computational Mathematics in Chemical andPhysical Kinetics; Nauka: Moscow, 1969; pp. 12–81 (inRussian).

3. Bowman, C. T. Proc Combust Inst 1974, 15, 869–877.4. Boni, A. A.; Penner, R. C. Combust Sci Technol 1977,

15, 99–106.5. Yetter, R.; Eslava, L. A; Dryer, F. L.; Rabitz, H. J Phys

Chem 1984, 88, 1497–1507.6. Mishra, M. R.; Yetter, R.; Reuven, Y.; Rabitz, H.;

Smooke, M. D. Intl J Chem Kinet 1994, 26, 437–453.7. Yang, Z.; Yang, B.; Wang, H. 2nd Joint Meeting of the

U.S. Sections of the Combustion Institute, Oakland, CA,2001; Paper No. 237.


8. Middha, P.; Yang, B. H.; Wang, H. Proc Combust Inst2002, 29, 1361–1369.

9. Rumminger, M. D.; Babushok, V. I.; Linteris, G. T. ProcCombust Inst 2002, 29, 329–336.

10. Kee, R. J.; Rupley, F. M.; Miller, J. A. CHEMKIN II:A Fortran Chemical Kinetics Package for the Analy-sis of Gas Phase Chemical Kinetics, Sandia Report no.SAND89-8009; Sandia National Laboratories, 1989.

11. Kee, R. J.; Dixon-Lewis, J.; Warnatz, J.; Coltrin, J. A.;Miller, J. A. A Fortran Program for Modeling SteadyLaminar One-Dimensional Premixed Flames, SandiaReport no. SAND85-8240; Sandia National Laborato-ries, 1985.

12. Davis, S. G.; Mhadeshwar, A. B.; Vlachos, D. G.; Wang,H. Int J Chem Kinet 2004, 36, 94–106.

13. Li, J.; Zhao, Z.; Kazakov, A.; Dryer, F. L. Int J ChemKinet 2004, 36, 566–575.

14. Mueller, M. A.; Yetter, R. A.; Dryer, F. L. Int J ChemKinet 1999, 31, 705–724.

15. Held, T. J.; Dryer, F. L. Int J Chem Kinet 1998, 30, 805–830.

16. Qin, Z.; Lissianski, V.; Yang, H.; Gardiner, W. C., Jr.;Davis, S. G.; Wang, H. Proc Combust Inst 2000, 28,1663–1669.

17. Smith, G. P.; Golden, D. M.; Frenklach, M.; Eiteneer,B.; Goldenberg, M.; Bowman, C. T.; Hanson, R. K.;Song, S.; Gardiner W. C., Jr., Lissianski, V. V.; Qin, Z.http://www.me.berkeley.edu/gri mech/, 1999.

18. Masten, D. A.; Hanson, R. K.; Bowman, C. T. J PhysChem 1990, 94, 7119–7128.

19. Du, H.; Hessler, J. P. J Chem Phys 1992, 96, 1077–1092.20. Pirraglia, A. N.; Michael, J. V.; Sutherland, J. W.;

Klemm, R. B. J Phys Chem 1989, 93, 282–291.21. Yu, C-L.; Frenklach, M.; Masten, D. A.; Hanson, R. K.;

Bowman, C. T. J Phys Chem 1994, 98, 4770–4771.22. Hessler, J. P. J Phys Chem A 1998, 102, 4517–4526.23. Tse, S. D.; Zhu, D. L.; Law, C. K. Proc Combust Inst

2000, 28, 1793–1800.24. McLean, I. C.; Smith, D. B.; Taylor, S. C. Proc Combust

Inst 1994, 25, 749–757.25. Huang, Y.; Sung, C. J.; Eng, J. A. Third Joint Meeting of

the U.S. Sections of the Combustion Institute, Chicago,IL, 2003.

26. Dryer, F. L., Glassman, I., Naegeli, D., Combust Flame1971, 17, 270–272.

27. Yu, C.-L. Chemical Kinteics of Reaction Systems,CO + OH and CH3 + O2, Ph.D. thesis, Department ofMaterials Science and Engineering, Pennsylvania StateUniversity, 1996.

28. Senosiain, J. P.; Musgrave, C. B.; Golden, D. M. Int JChem Kinet 2003, 35, 464–474.

29. Li, J.; Zhao, Z.; Kazakov, A.; Dryer, F. L.; Scire, J. J., Jr.Int J Chem Kinet submitted; Li, J.; Zhao, Z.; Kazakov,A.; Dryer, F. L.; Posters 1F1-04, 30th InternationalSymposium on Combustion, Chicago, IL, 2004.

30. Ravishankara, A. R.; Thompson, R. L. Chem Phys Lett1983, 99, 377–381.

31. Vandooren, J.; Peeters, J.; Van Tiggelen, P. J. ProcCombust Inst 1975, 15, 745–753.

32. Troe, J. Proc Combust Inst 1998, 27, 167–176.33. Lissianski, V.; Yang, H.; Qin, Z.; Mueller, M. R.; Shin,

K. S.; Gardiner W. C., Jr. Chem Phys Lett 1995, 240,57–62.

34. Wooldridge, M. S.; Hanson, R. K.; Bowman, C. T. ProcCombust Inst 1994, 25, 741–748.

35. Westenberg, A. A.; deHaas, N. J Chem Phys 1973, 58,4061–4065.

36. Wooldridge, M. S.; Hanson, R. K.; Bowman, C. T. Int JChem Kinet 1996, 28, 361–372.

37. Golden, D. M.; Smith, G. P.; McEwen, A. B.; Yu,C.-L.; Eiteneer, B.; Frenklach, M.; Vaghjiani, G. L.;Ravishankara, A. R.; Tully, F. P. J Phys Chem A 1998,102, 8598–8606.

38. Zhao, Z.; Kazakov, A.; Li, J.; Dryer, F. L. Combust SciTechnol 2004, 176, 1705–1723.

39. Zhao, Z.; Li, J.; Kazakov, A.; Zeppieri, S. P.; Dryer,F. L. Combust Sci Technol 2005, 177, 89–106.

40. Paul, P. H. DRFM: A New Package for the Evalua-tion of Gas-Phase Transport Properties, Sandia ReportNo. SAND98-8203; Sandia National Laboratories,1998.

41. Kee, R. J.; Dixon-Lewis, J.; Warnatz, J.; Coltrin, J. A.;Miller, J. A. A Fortran Computer Code for the Evalu-ation of Gas Phase, Multicomponent Transport Proper-ties, Sandia Report No. SAND86-8284; Sandia NationalLaboratories, 1989.

temperature-dependent feature sensitivity analysis for combustion modeling

Documents