Higher-Order Correlation of Kinetic Parameters from
Global Sensitivity Analysis with Consideration of
Extinction Phenomena of Non-Premixed Flames
G Esposito ∗ and HK Chelliah †
Department of Mechanical and Aerospace Engineering, University of Virginia
Charlottesville VA 22904, USA
The experimental uncertainty factors associated with the chemical kinetic parameters ofelementary reactions have a considerable impact on the development of accurate detailedchemical models. Previous Global Sensitivity Analysis (GSA) investigations have shownthat, under certain conditions, the variation of chemical parameters within their uncer-tainty bounds affects combustion properties via both single and joint (second-order) effects.Most of the existing literature focuses on ignition and laminar flame propagation phenom-ena, whereas, to best of our knowledge, GSA of the extinction limits of non-premixedcounterflow flames has never been considered previously. In this work, we perform GSAof ethylene-air flames in near extinction conditions using the High Dimensional ModelRepresentation (HDMR) methodology. By analyzing the results of the HDMR with theemphasis on the extinction strain rate and maximum concentration of acetylene, we firstidentify the major sources of uncertainty of the chemical model and then quantify thepresence of higher-order interactions between parameters and target flame properties.
I. Introduction
The accurate modeling of finite-rate chemistry is widely recognized as a fundamental step in the process ofachieving a complete predictive capability of combustion phenomena occurring in aerospace applications. Thecompilation and optimization of detailed chemical kinetic models is a complex process that requires severalrefinements and iterations. The objective of a typical optimization procedure is to find the combinationof chemical kinetic parameters that reproduces a series of target combustion properties. The presence ofthe experimental uncertainties associated with the reaction rate constants defines the parameter space ofthe compiled chemical kinetic model. Because of the non-linear nature of most chemical phenomena, undercertain conditions, the influence of chemical parameters on combustion properties of interest must take intoaccount both single and joint (higher order) effects. Understanding how the uncertainty of parameters andtheir higher-order interactions affect target combustion properties is of paramount importance in chemicalkinetic modeling. Even though recent investigations have studied uncertainty propagation and high-orderinteractions of chemical parameters using several canonical combustion cases,1–3 the important phenomenonof non-premixed extinction has been overlooked. In this work we present a Global Sensitivity Analysis(GSA) of non-premixed flames in near-extinction conditions and our primary objective is to quantify themajor sources of uncertainty in the evaluation of the extinction strain rate, aext, of non-premixed ethylene-air flames. In addition, given the current interest in the analysis of soot precursor formation in ethylene-airflames, we also include the maximum concentration of acetylene, known to play a key role in the synthesisof the first PAH through the so-called HACA mechanism,4 as a further objective of the GSA.
I.A. Global Sensitivity Analysis
The identification of relevant reactions controlling combustion properties of interest is a crucial step in modelanalysis and reduction. The importance of a target combustion property, C, with respect to the j-th reaction
∗Member AIAA†Professor, Associate Fellow AIAA.
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parameter, kj , can be assessed by a simple perturbation of kj from its baseline values, i.e. kj = kj +∆kj .The ratio ∆C/∆kj is a measure of the influence of kj on C. Thus, for a model having NR reactions, all theparameters can be perturbed One-At-a-Time (OAT) from their baseline values to obtain ∆C/∆kj . For a casewith many independent variables, Zk, a more efficient and widely used strategy consists in calculating partialderivatives of the k-th variable Zk’s with respect to the j-th reaction rate kj . This method takes advantageof the formulation of simple combustion flow fields to decrease the computational cost of evaluating thesensitivity derivatives5 ∂Zi/∂kj . Since the sensitivity values are computed at a single point in the parameterspace, this strategy belongs to the category of Local Sensitivity Analysis (LSA). Even though LSA canefficiently identify important reaction pathways of a chemical model and be included in model reduction,6–9
it does not take into account how a chemical parameter influences combustion properties over its wholeuncertainty range. In addition, LSA cannot evaluate non-linear or higher-order effects.
In order to overcome the shortcomings of LSA, Global Sensitivity Analysis (GSA) strategies based on thesampling of the entire parameter space are routinely employed.10 Compared to LSA, the computational costof GSA can be extremely large because the numerical solutions must be obtained over a large set of points inthe parameter space. The common idea behind most of GSA methods is that the total variance, V (Y ), of afunction of interest, Y , can be decomposed in a sum of single and joint effects of the parameters included inthe model. Typically two fundamental pieces of information can be extracted by GSA, namely (i) the fractionof the variance of a specific target function due to single and joint effects, Sj = V (Y |kj )/V (Y ), or Sij =V (Y |ki,kj )/V (Y ) and (ii) the functional form of the interaction between parameters and target functions.In the first case, the ratios Sj and Sij , known as Global Sensitivity Indices, summarize the importance ofparameters over all the uncertainty range. Although there are several implementations available in literatureon GSA,11–13 in this study we employ the Random-Sampling High Dimensional Model Representation (RS-HDMR) technique.14,15 The basic assumption of RS-HDMR is that the model output can be expressed bya sum of hierarchical functions accounting for independent and joint parameter effects,
Y (k) = f0 +
NR∑i=1
fi(ki) +
NR∑i<j
fij(ki, kj) + h.o.t, (1)
where fi’s represent the individual effects of ki’s on Y and fij ’s represents the second-order effects of ki andkj on Y. The assumption that third- and higher-order effects can be negligible is generally valid for mostphysical problems16 and has been shown to be true in chemical kinetics studies as well.17–19 The HDMRcomponent functions, f0, fi and fij are defined by simple integral functions. They can be convenientlyapproximated by polynomials20 and calculated through Monte Carlo sampling.
I.B. The Screening Method of Morris
Sobol LPτ quasi-random low discrepancy sequences provide an unbiased approach of sampling the parameterspace over two levels (high-low).21 Thus, for a model ofNR parameters, a thorough sampling of the parameterspace requires 2NR model simulations. For this reason and associated computational cost, the HDMR analysisis typically performed on a subset of chemical parameters. The loss of information in performing HDMRon a subset of reactions is usually negligible. In fact, it has been shown that only a rather small numberof chemical parameters (10-15) controls global combustion properties of C1-C4 hydrocarbon flames.22,23
Instead of relying on LSA to identify such a subset of important reactions, some investigators have suggestedthe systematic implementation of the screening method of Morris (MM).10,15,24
The Morris Method is an extension of the One-At-a-Time (OAT) approach to include the screening ofparameters over the whole uncertainty bounded space.25 Instead of perturbing the initial set of parametersfrom the nominal values, one parameter at a time, Morris proposed to evaluate the increments of theparameters at distinct points spread over the parameter space along well designed trajectories. In this sense,the Morris Method is referred to as a Multiple-One-At-a-Time approach (MOAT), and it fills the gap betweenLSA and GSA.
The elementary effects of the j-th reaction on the output Y , EElj = ∆Y/∆kj , are averaged over l =
1, . . . , r trajectories to yield three important statistics, specifically (i) the mean, µj(EE), (ii) the mean ofthe absolute value of elementary effects, µ∗
j (|EE|), and (iii) standard deviation, σj(EE). Even though µj
and σj may provide some insight into the model response, µ∗j is known to accurately distinguish influential
parameters26 and it will be used as the statistics of reference here. It should be pointed out that Morrisindices, despite univocally determining unimportant parameters, provide only qualitative information on
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the model behavior. As a result, in this study the Morris method will be used only in the first phase ofidentifying reactions of importance to be successively analyzed by a more quantitative approach.
II. Computational Details of the Implementation of MM and RS-HDMR
A second-order polynomial approximation of the extinction curve, namely the peak flame temperaturevariation with local flow strain rate, has been chosen to evaluate the extinction conditions in non-premixedflames.27 This approximation requires the solution of the counterflow flame structure at three points near theflame extinction strain rate for each parameter set. This implies that the Morris method requires 3×r(NR+1)simulations, i.e. for r = 20 trajectories and NR = 784 parameters, with a total of 47,100 flame solutions.Considering that thorough sampling of the parameters space in HDMR requires 2NR points, the analysishere is limited to 14 parameters for a total of 3× 214 = 49, 152 flame simulations.
The numerical simulation of counterflow flames requires the solution of a non-linear system of equationswhich is typically performed using the iterative Newton-Raphson (NR) method.28 The computational timenecessary to reach convergence can be highly variable depending upon the distance between the initialguess for the NR iteration and the actual solution. In the present parametric study, given an initial guesscalculated at a point Pa in the reaction parameter space, the calculation of the solution at a point Pb can bea time consuming task. In fact, for certain combinations of chemical parameters, a sequence of intermediatesolutions along the line connecting Pa and Pb must be calculated before reaching final converged solution atPb. In the particular case of the HDMR decomposition performed using realistic uncertainty factors,22 a fewparameter combinations do not lead to converged solutions. However, since the accuracy of the Monte Carlointegrals, which define the HDMR component functions, is weakly dependent upon the number of samples,29
the global sensitivity indices are not affected.Based on the computational issues discussed above and on the exact specifics of the flame code host
machine, on the average about 100 minutes are needed for one counterflow simulation. The total estimatedcumulative computation time for the 47,100 flame simulations needed by the Morris Method, plus the 49,152for HDMR, results in over 6,000 days on a single core. Clearly, the availability of a large number of computerprocessors will determine the realization of parametric study described above. Fortunately, the Cross CampusGrid (XCG) feature available at the University of Virginia offers several hundreds of cores to the researchcommunity.30 The job queueing system on XCG provides a perfect solution for the present computationalneed of thousands of simulations with a rather high variability in total running time. Depending on the usageof the clusters, typically 200 to 250 cores are available, reducing the effective computational time down toabout 30 days.
III. Results and Discussion
III.A. Local Sensitivity Analysis vs. Multiple One-At-a-Time Sensitivity Analysis
A comparison between derivative based local sensitivities and Morris µ∗j indices is presented to further
underline the differences between the two methods and to identify the reactions of importance. In particular,the local sensitivity derivative of the flame eigenvalue (J ), i.e. ∂J /∂kj , is compared with Morris’ µ∗
j (J ).Here, J is a dependent variable that is solved for during the computation of counterflow flames and whichcan be directly correlated to the local strain rate, a.31 In contrast, the influence of reactions on the maximumconcentration of C2H2 to assess the soot formation propensity cannot be expressed by any local derivativeestimate. This underlines the additional importance of the Morris screening method adopted here.
Table 1 reports the top 20 reactions identified according to the rankings of local derivative of J withrespect to kj and µ∗
j (J ). The two rankings methods are seen to agree well, especially among the top10 reactions, except for the chain-propagation reaction (M6) and the carbon monoxide formation reaction(M18).
Considering the subtle differences of the two ranking approaches, the Morris screening method is usedhere to select a subset of reactions for the HDMR analysis. Because of the interest in identifying reactionsof importance for both nonpremixed flame extinction strain rate prediction and soot precursor formation inethylene-air flames, Tables 2 and 3 lists reactions ranked according to µ∗
j (aext) and µ∗j (maxC2H2). Based
on the rankings shown in Tables 2 and 3, 14 reactions to be considered in HDMR analysis are identified asindicated in the second column. It should be noted that the identified 14 important reactions are consistent
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Reaction LSA MM
(M1) H + O2 ↔ O + OH 1 1
(M2) CO + OH ↔ CO2+H 2 2
(M3) O + H2 ↔ H + OH 3 4
(M4) C2H3 + H ↔ C2H2 + H2 4 3
(M5) C2H3 + O2 ↔ HCO + CH2O 5 5
(M6) 2OH ↔ O + H2O 6 34
(M7) HCO + M ↔ CO + H + M 7 6
(M8) CO + OH ↔ CO2 + H 8 8
(M9) OH + H2 ↔ H + H2O 9 7
(M10) C2H3 + H ↔ H2CC + H2 10 9
(M11) CH2 + O2 ↔ HCO + OH 11 18
(M12) C2H4 + H ↔ C2H3 + H2 12 16
(M13) HCCO + O2 ↔ OH + 2CO 13 17
(M14) HCO + H ↔ CO + H2 14 11
(M15) HO2 + H ↔ 2OH 15 13
(M16) 2CH3 ↔ H + C2H5 16 19
(M17) HCCO + H ↔ CH2∗ + CO 17 12
(M18) HCO + O2 ↔ CO + H2O 18 10
(M19) C2H2 + O ↔ HCCO + H 19 14
(M20) CH2 + O2 ↔ CO2 + 2H 20 20
Table 1. Top 20 reactions of importance based on flame extinction strain rate eigenvalue (J ), according to localsensitivity analysis ∂J /∂kj and Morris method’s µ∗
j (J ) at near extinction conditions.
with the classical description of flame structures of hydrocarbon combustion in non-premixed counterflowflames, i.e. regions or layers of fuel decomposition, radical production/oxygen consumption, and intermediatespecies oxidation.32,33
III.B. First and Second-order Global Sensitivity Values for Non-premixed Extinction
Under the assumption that only first- and second-order interactions affect the combustion properties ofinterest, the global sensitivity indices for the extinction strain rate have been normalized and summarizedin Table 4. The total contribution to the variance of the extinction strain rate of first-order effects is90%, indicating that the second-order effects are not negligible. Scatter plots showing the first-order HDMRcomponent function effects on (a) the predicted extinction strain rate, and (b) maximum C2H2 concentrationfor reactions R3 and R14 are shown in Figs. 1(a) and 1(b), respectively. Considering all such first-ordercomponent function effects for the 14 reactions, the weak non-linear behavior may justify the adoption ofGSA techniques.
The present results confirm that, although associated with a small uncertainty factor,22 the branchingreaction H+O2 ↔ O + OH is responsible for a significant part of the total uncertainty of the model. On theother hand, the two highly uncertain reaction channels of a vinyl radical leading to the formation of C2H2
or H2CC are among the top three reactions in the extinction strain rate ranking. Based on the normalizedsensitivities Si(aext) listed in Table 4, the top two vinyl radical reactions (R9) and (R3), associated with anuncertainty factor of 5, accounts for 50 % of the variability of aext.
The vinylidene radical, H2CC, is characterized by a very short lifetime that explains the lack of directmeasurements involving the reaction C2H3+H ↔ H2CC+H2 (R9). In fact, only recently has an estimate forthe reaction rate of (R9) been proposed.22,34 However, under certain conditions, vinylidene plays a criticalrole in the acetylene chemistry35 which represents an important subset of the ethylene combustion pathway.For these reasons, reaction (R9) ranks very high in the global sensitivity indices listed.
A few experimental evaluations are available for the competing reaction C2H3+H↔C2H2+H2 (R3).However, the alternate pressure-dependent reaction channel H+C2H3(+M) → C2H4(+M) (RC1) createsnoise in the experimental data.36 In fact, reaction RC1 can be quite important because proper experimentaltechniques involve the presence of other species or atoms (iodine, fluorine, or other small hydrocarbons) thatenhance its reaction rate. Therefore, there is a large uncertainty because kinetic rates of both reactions (R3)and (RC1) must be simultaneously analyzed.
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Table 5. Normalized first-order sensitivity indices (Si) forthe maximum concentration of C2H2.
Reaction i Reaction j Sij(aext)
C2H3+H↔C2H2+H2 C2H3+H↔H2CC+H2 2.73E-2
C2H3+H↔C2H2+H2 HCO+M↔CO+H+M 8.03E-3
HCO+M↔CO+H+M C2H3+H↔H2CC+H2 7.31E-3
Table 6. Second-order sensitivity indices (Sij) for the extinction strain rate, aext.
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0 1 2 3 4 5
x 1014
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
A C2H3+H<=>C2H2+H2
f 0 + f
1
(a)
1 2 3 4 5 6 7 8
x 107
0.015
0.02
0.025
0.03
0.035
0.04
A C2H4+OH<=>C2H3+H2O
f 0 + f
1
(b)
Figure 1. Scatter plots showing the first-order component function of HDMR analysis, fi for: (a) aext for reactionC2H3 +H ↔ C2H2 + H2 (R3) and (b) maxC2H2, for reaction C2H4+OH ↔ C2H3+H2O (R14).
The second-order sensitivity between (R3) and (R9) confirms the importance of these reactions in theethylene combustion pathways in non-premixed flames in near-extinction conditions. As seen from Table 6,the value for Sij of this reaction pair ranks at the top, with a value comparable to some of top first-ordereffects (0.027) listed in Table 4. The above observations of the first- and second-order global sensitivity indicesinvolving the vynil radical reactions (R3) and (R9) suggest that chemical kinetic experiments should focuson clarifying the rate constants of the reaction channels C2H3+H ↔ C2H2+H2 and C2H3+H ↔ H2CC+H2.
The corresponding second-order HDMR component function, fij , for (R3) and (R9) presents a distinctshape (or surface) as shown in Fig. 2, indicating that better understanding of the interaction is necessaryfor the improvement of chemical kinetic models. The presence of maximum values of the interactions at theextreme limits of the parameters is a common feature found in most other HDMR studies.2,37
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 1014
0
0.5
1
1.5
2
2.5
3
x 1014
−50
0
50
100
150
A C2H3+H<=>H2CC+H2A C2H3+H<=>C2H2+H2
f ij
Figure 2. Second-order component function fij for aext, for the reactions C2H3+H ↔ C2H2 + H2 (R3) and C2H3+H↔ H2CC + H2 (R9).
In contrast to the extinction strain rate indices, first-order effects for the maximum concentration ofC2H2 represent 97% of the total variability (see Table 5), with top four reactions accounting for 75% of thetotal variability of the maxC2H2. Hence, it is evident that for this particular species, second-order jointcontributions can be neglected. Once again the large uncertainty factor affecting the three reactions involvingthe highly reactive vinyl radical C2H3 suggests that kinetic modeling and experimental effort should focus onincreasing the accuracy of the chemical pathways of vinyl radical creation and destruction. The rather smalluncertainty factor of the acetylene oxidation, C2H2+O ↔ HCCO+H, emphasizes the role of this reaction
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path in increasing the total concentration of C2H2.
IV. Conclusion
In this investigation, the analysis of uncertainty and second-order effects of selected chemical kineticparameters on target combustion properties was performed by considering the fundamental non-premixedextinction phenomenon. The normalized global sensitivity indices revealed that 10% of the total variance ofthe extinction strain rate is due to non-negligible second-order effects, whereas the variance of the maximumconcentration of acetylene is dependent mostly on first-order effects. In addition to the highly sensitiveoxygen consumption / radical production reaction H+O2 ↔ O + OH, two other highly uncertain reactionchannels involving the vinyl radical, C2H3, were found to be important contributors to the uncertainty ofthe extinction strain rate of ethylene-air nonpremixed flames. Similarly, the radical C2H3 participates in themost uncertain reactions controlling the maximum concentration of acetylene. For this reason, it is clearthat a greater accuracy of chemical modeling for ethylene combustion could be achieved by improving thereaction rates evaluations involving the vinyl radical.
Acknowledgments
This work was supported by the NASA under contract FA-9101-04-C-0016 and by US Air Force Oce ofScientific Research under contract FA9550-09-1-0611, under the technical monitoring of Dr. Richard Gaffneyand Dr. Julian Tishkoff, respectively.
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