Paper # 070RK-0090 Topic: Reaction Kinetics

8th U. S. National Combustion Meeting

Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah

May 19-22, 2013

Multi-target Global Sensitivity Analysis of n-Butanol Combustion

Dingyu D. Y. Zhou1,2 , Rex T. Skodje1, Wei Liu2, and Michael J. Davis2

1Department of Chemistry and Biochemistry, University of Colorado, Boulder CO 80309

2 Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, IL 60439

Global sensitivity analysis is applied to the combustion of n-butanol. The sensitivity of both ignition delay times and species concentrations are studied with respect to uncertainties in the reaction rate coefficients. It is demonstrated that the ignition delay times are sensitive to only a few rate coefficients, as are each of the species we studied. Comparisons are made to reaction pathway analysis. In the speciation studies, it is shown that two types of reactions have large global sensitivity coefficients. The first type promotes the formation of a given species and is identified in a reaction pathway analysis. The second type inhibits the formation of the species and is not directly identified in reaction pathway analysis. In order to accurately predict the formation of a given species, both of these types of reactions need to be accurately modeled when their global sensitivity coefficients are large. In this paper we also describe recently developed techniques to reduce the number of samples that are necessary for producing accurate global sensitivity coefficients.

1. Introduction

Chemical kinetic models of combustion processes consist of hundreds or thousands of reactions describing the chemistry of dozens to hundreds of chemical species [Westbrook et al (2005)]. Practitioners who develop these models make every attempt to include the most accurate information available for the rate coefficients of each reaction, as well as the thermochemistry and transport properties of all the species. The information used for such model building includes experimental measurements and theoretical calculation of chemical kinetics rate coefficients. However, even for the most studied chemical reactions the rate coefficients are not known with a great deal of precision, and in many cases most of the rate coefficients are estimated [Curran et al (1998)]. Even in cases where there is significant experimental evidence for a rate coefficient, it may have been measured under conditions that are quite different than what is needed for a particular chemical model; for example, it was measured in a different temperature range. Its rate coefficient at other types of conditions are estimated from an Arrhenius fit to the rate coefficient.

The well known uncertainty of rate coefficients has led to extensive use of sensitivity analysis to highlight specific reactions whose uncertainties may contribute significantly to uncertainties in target quantities such as ignition delay times, species concentrations, flames speeds, and other important aspects of the combustion process. Traditionally sensitivity analysis has been carried out in a local and linear fashion, generated from information near the estimated rate coefficient [Zador et al (2006)]. If the rate coefficient is sufficiently different from the nominal value the local sensitivity may be an inaccurate estimation of the role of the uncertainty of the rate coefficient in the uncertainty of the target. Sometimes sensitivity analysis is done in a more global way, for example for ignition delays the sensitivity is often estimated by changing rate coefficients by a more significant amount, for example, by increasing and decreasing the rate coefficients by a factor of two [Qin et al (2001)]. This is a more global approach and is much closer to what we describe here and what is referred to as global sensitivity analysis. Global sensitivity analysis as described in [Saltelli et al (2008); Tomlin and Ziehn (2010)] and implemented here samples the full range of uncertainty of all the reactions at once. So that nonlinearities and correlations may be described and would not be detected in the one at a time, linear approach often used for ignition delays.

We also present here initial work [Davis and Liu (2013)] on using the sparsity of the sensitivity indices to significantly reduce the sample size needed to get a ranking of the sensitivity of reactions in a chemical model. The implementation of sparse techniques allows us to apply global sensitivity analysis to the chemistry in complex reactive flows and devices, where each individual calculation is very time consuming. Using sparse techniques, the sample sizes can be much smaller than the number of reactions and thus can require less computational effort than linear one-at-a-time methods that were just described for ignition delays.

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This paper will describe our application of global sensitivity analysis to butanol combustion using the recently developed mechanism of [Black et al (2010)]. We extend the global sensitivity analysis we have done previously using ignition delay as a target to also include species targets, leading to multi-target global sensitivity analysis. We will study the combustion under the conditions used in the RCM experiments reported in [Karwat et al (2011)] as initially presented in [Zhou et al (2013)]. We will demonstrate that is possible to do global sensitivity for a large mechanism such as butanol that consists of 1446 reactions. The study of speciation leads to a discussion of the relationship between global sensitivity analysis and reaction pathway analysis, also. 2. Methods

Like our previous work [Skodje et al (2010); Klippenstein et al (2011), Davis et al (2011)], the present paper studies

how the uncertainties in the reaction rate coefficients affect the uncertainty in combustion processes. We present results here based on the work in [Zhou et al (2013)], which extended our previous work on ignition delay targets to species targets. In order to ascertain the role of the uncertainty of the reaction rate coefficients on the ignition and species targets, a series of computer runs are undertaken for which each run uses a different set of reaction rate coefficients generated by varying all the rate coefficients within their uncertainty ranges. The uncertainty ranges are taken from published sources [for example, Baulch et al (2005)] and expert elicitation. For the n-butanol mechanism studied here only about 25% of the reactions can be given uncertainties in this manner and the rest of the reactions are given an uncertainty factor, f, of 5. The rate coefficients used in the calculations are between k/f and kf, for a rate coefficient k.

In what follows, we will describe how global sensitivity analysis is implemented in [Skodje et al (2010); Davis et al (2011)], which in turn was based on the work of Sobol [Sobol (2001)], Li, Rabitz and co-workers [Li et al (2010)], and Tomlin and co-workers [Tomlin and Ziehn (2010)]. In the development presented here, we use the example of ignition delay time, which is one of the targets studied in this paper. It is straightforward to extend the analysis to species targets, as was done in [Zhou et al (2013)], and results for species sensitivities are also presented in this paper.

To generate global sensitivity coefficients, the uncertainty is scaled to be between 0 and 1, with “u” the designation for this quantity. The targets (ignition delay here) are written in terms of the following response surface: τ ({ui}) = τ 0 + τ i∑ (ui ) + τ ij(ui, u j)

i< j∑∑ + ... (1)

Equation 1 is referred to as an HDMR expansion (high-dimensional model representation) in the global sensitivity literature [Saltelli et al (2008)]. The quantity “τ0” is the mean ignition delay time. The functions τi and τij are often referred to as the “main effects” and “interaction terms”, respectively. It is possible to use the expansion in Eq. (1) [Sobol (2001)] to generate an expansion of the variance of the ignition delay times:

�

V = Vi∑ + Viji< j∑∑ + ..., (2)

where V refers to the usual variance that is estimated numerically using the random sampling described above. Vi and Vij in Eq. (2) are calculated from the expansion in Eq. (1). Global sensitivity indices are defined by dividing the right hand side by V:

�

Si = ViV

, Sij = VijV

, etc (3)

In this paper we will only be concerned with the main effect sensitivity coefficients, the Si’s. In all our previous

work, and in most of the results presented here these were generated from the following set of expansions:

τ i (ui ) = aikuik

k=0

n

∑ (4)

where n = 4. It is straightforward to infer the Vi’s from this expansion [Davis et al (2011)]. Recently [Davis and Liu (2013)] replaced the HDMR expansion with the following two expansions:

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τ (1) ({ui}) = aikuik

k=0

n

∑i=1

m

∑ (5a)

τ (2) ({ui}) = aikuik

k=1

n

∑i=1

m

∑ + b jkruv(j,1)k uv(j,2)

r

k+r≤sk,r > 0

∑s=1

n-1

∑j=1

p

∑, (5b)

where m refers to the number of reactions and n refers to the order of the expansion. In Eq. (5b), p is a number determined from the expansion in Eq. (5a). Only reactions in Eq. (5a) whose main effects are above a threshold are used in the second set of terms in Eq. (5b). The value of p is generally less than 200 in our calculations. We typically use thresholds of Si = 0.005 or Si = 0.01. If there are 10 reactions from Eq. (5a) above this value, p = 45 (1/2*10*9). The indices “v(j,1)” and “v(j,2)” in Eq. (5b) refer to a pair of reactions that are used in the interaction term based on their Si’s. For example, if the 300th and 400th reactions each have Si’s above the threshold, one set of the indices will be v(j,1) = 300 and v(j,2) = 400. It is straightforward to use Equation (5a) to extract the main effects (there are no interaction terms). Extraction of main effects from Eq. (5b) requires integration, which can be done analytically for the polynomial expansions. In this paper Eqs. (5a) and (5b) are used only for the results at the end of the next section. For future reference, Eq. (5b) is written in the following compact form:

τ ({ui}) = c jg ji=1

L

∑, (6)

where the g’s refer to the terms involving the u’s in Eq. (5b). There are L = 4m + 6p terms of this form for n = 4.

In Eqs. (5a) and (5b) the u’s are “centered”, so that their mean values are 0.0. This gives an expansions that starts at k = 1, rather than k = 0. We have found that the expansions in Eqs (5) and (6) require smaller samples to accurately ascertain sensitivity coefficients [Davis and Liu (2013)]. For n-butanol combustion, sample sizes of 10,000 are sufficient to duplicate the results from 50,000 samples using the expansion described in Eq. (4), as shown below in the results section.

The sample sizes used for our old computer codes were typically 20-40 runs per reaction for the screening of reactions [Davis et al (2011)], as described in the results section. Many of the results shown in [Zhou et al (2013)] were generated from 50,000 Monte Carlo runs for n-butanol, whose combustion was described by a reaction mechanism with 1446 reactions. However, more detailed studies of the uncertainty imposed on the targets by the uncertainties of the reaction coefficients required many more runs. In [Zhou et al (2013)], a probability density function for the uncertainty of ignition delay times for n-butanol combustion was generated from 1.1 million runs.

As noted, the sample size of 10,000 used for Eqs. (5a) and (5b) is a significant improvement over our previous sample size of 50,000. We have used the expansion in Eq. (1) in many studies and found that convergence was achieved much more rapidly for the large Si’s than for small Si’s. The slowness of the convergence is a manifestation of a phenomenon called “overfitting” in the statistics and machine learning communities [Bishop (2006); Hastie et al (2009)], which is alleviated to a significant extent by the expansion in Eq. (6). In addition, the expansion in Eq. (6) gives accurate values for the interactions terms (Sij’s) for the same sample of 10,000, compared to 1.1 million from the HDMR expansions in Eq. (1). The reduction of effort is achieved using ordinary least squares regression for the fits in Eqs. (1) and (6). Even further reductions can be found by starting from Eq. (6) and taking into account the sparseness of the results - most sensitivity coefficients are very small. The additional reduction is achieved by using sparse regression techniques [Hastie et al (2009)]. As will be shown below we can reproduce the results generated from 50,000 samples using Eq. (1) with less than 1,000 samples using Eq. (6) and sparse regression techniques. The savings from sparse regression results from the fact that typically the uncertainty of only a few reactions contribute significantly to the uncertainty in a given target. The sparse regression techniques take advantage of this without any prior knowledge about which reactions contribute. This is accomplished by adding a “penalty” term to the usual error function for least squares fits: ,

E(c) = 12

tk  -  c jg j(uk )i=1

L

∑⎛⎝⎜

⎞⎠⎟

2

k=1

M

∑  + λ | c jj=1

L

∑ | , (7)

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where tk refers to the ignition delay time for the kth sample run (each sample run has a different set of rate coefficients sampled from the uncertainties of all the reactions), and there are M runs. Equation (7) expands on the notation in Eq. (6) by explicitly indicating that the g’s are evaluated at a set of uncertainties for each run (uk designates a vector of uncertainties at the kth point). The first term on the right hand side of Eq. (7) is the ordinary least squares error term and the second is a penalty that depends on the c’s, the expansion coefficients. As with ordinary least squares, Equation (7) is minimized with respect to the c’s. However, unlike ordinary least squares, the minimization is not linear with respect to coefficients and special algorithms are used for the minimization, aided by the fact that Eq. (7) is convex [Tibshirani (1996)]. The algorithm we use is described in the upcoming paper [Davis and Liu (2013)], and is based on [Mosci et al (2010)] and [Chiquet et al (2012)]. The expansion described in Eq. (7) is referred to as the LASSO, first developed in [Tibshirani (1996)] [see Hastie et al (2009) for a review]. The penalty term serves two purposes, it “regularizes” the solution, reducing overfitting, but more importantly, for our purposes, it selects coefficients, resulting in many coefficients that are zero for specific values of λ. There are a number of ways of picking good values of λ [Hastie et al (2009)], and typically we use cross-validation or an empirical rule we discerned from a number of applications. Equation (7) is a simplification of the procedure used in [Davis and Liu (2013)], the sparse group lasso [Simon et al (2013)], which recognizes the grouping of expansion coefficients based on individual reactions. There is an additional adjustable parameter in the sparse group lasso, designated α, which is set to 0.5 in this paper (it can range from 0.0 to 1.0).

3. Results and Discussion In previous applications we have studied the

sensitivity of ignition delay to the uncertainties of individual reactions using global sensitivity analysis. Figure 1 extends this analysis to butanol ignition. We used the mechanism of [Black et al (2010)] that consists of 1446 reactions. Figure 1 results from the auto-ignition of a mixture of butanol and air as described in [Karwat et al (2011)]. The line representing the reaction n-C4H9OH + HO2 is the sum of four sensitivities, abstraction of H from butanol from any of the carbon atoms. This calculation, along with the results in [Zhou et al (2013)], demonstrates that it is possible to use global sensitivity analysis on a large mechanism. Figure 1 was generated from a set of 50,000 ignition-delay calculations run with different sets of the 1446 rate coefficients that were

varied within their uncertainty ranges. Reference [Zhou et al (2013)] shows how the global sensitivities change as the rate coefficients and their uncertainties were updated based on new theoretical calculations in the literature for the following sets of reactions: 1) n-C4H9OH + HO2 = C4H8OH + H2O2 (four reactions) [C.-W. Zhou et al (2012)], 2) HO2 + HO2 = H2O2 + O2 [Zhou et al (2012)], and 3) n-C4H9OH + OH = C4H8OH + H2O (four reactions) [C.-W. Zhou et al (2011)]. Figure 2 lists all reactions with Si > 0.01.

Besides extending the work in [Skodje et al (2010); Klippenstein et al (2011); Davis et al (2011)] to a larger mechanism, we have also extended the work to the study of species sensitivities. Our motivation for studying species sensitivities is [Karwat et al (2011)] where species time histories were generated

Fig. 1. First-order sensitivity coefficients (main effects) for n-Butanol ignition using the same initial conditions as in [Zhou et al (2013)] and [Karwat et al (2011)].

Fig. 2. Reaction Selection for n-Butanol ignition is shown.

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experimentally and with modeling using the same mechanism [Black et al (2010)] as we have used here [Zhou et al (2013)]. Figure 3 displays the results we obtained with global sensitivity analysis for three of the species studied in [Karwat et al (2011)], ethylene (top panel), propylene (middle panel) and acetaldehyde (bottom panel), under the same conditions reported there. Figure 3 shows the sensitivities of each of these species at a set of 10 times, which are fractions of the time to ignition. The sample size was 50,000. The plots show all reactions that accounted for at least 10% of the variance at any of the 10 scaled times.

We attempted to reconcile the differences observed in [Karwat et al (2011)] between the modeling and experiments for the time history of ethylene. Figure 4 has a list of the top 7 reactions at t/tign = 0.5. In [Karwat et al (2011)] the modeling of the time history of ethylene was done for two different sets of rate constants for the reaction C4H9OH + OH = C4H8OH + H2O. The discrepancy observed between modeling and experiments for the time history of ethylene got worse when the rate coefficients for these four reactions were updated with new rate coefficients from [C.-W. Zhou et al (2011)]. Global sensitivity analysis based on the top panel of Fig. 3 indicates that the time history of ethylene should be most sensitive to two of these reactions and indeed the modeling results indicated that new rate

coefficients for these reactions changed the time history of ethylene significantly, but made the disagreement between the experiments and modeling even worse. While the global sensitivity analysis anticipated the trend [Zhou et al (2013)], it cannot give direct information concerning why this occurs. However, it is possible to look for other reactions that may contribute to this discrepancy and Fig. 4 indicates that the ethylene concentrations should be sensitive to one of the beta scissions, which is the fifth most sensitive reactions at t/tign = 0.5. However, because the sensitivity coefficient is relatively small (0.022) the sensitivity analysis does not point to this reaction as one that would cause the discrepancy. One possible reason that the global sensitivity analysis is inconclusive is that the rate coefficients of some of the reactions, such as the H-atom abstractions and the beta scissions are so far from their accurate values they are beyond the uncertainty limits set in the analysis, which was 5 for all the reactions using the original mechanism of [Black et al (2010)]. We intend to pursue this further in the future.

We made a detailed comparison between global sensitivity analysis and the reaction pathway analysis of [Karwat et al (2011)]. There are two significant differences between the global sensitivity analysis and the reaction pathway

Fig. 3. Main effect sensitivity coefficients for three species at a series of times are shown. These were generated with the same initial conditions as Fig. 1. Only those coefficients that were larger then 0.1 at some time are shown.

Fig. 4. Reaction selection for ethylene formation is shown at t/tign = 0.5.

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analysis for ethylene formation related to the H-abstraction from n-butanol: 1) abstraction from the α-carbon is a significant pathway to ethylene formation but does not have a high global sensitivity coefficient and 2) the γ-abstraction is not a pathway to ethylene but has a high global sensitivity. Our explanation for these two differences is as follows. The α-abstraction is fast enough over the whole range of its uncertainty that the formation of ethylene is not sensitive to this uncertainty, demonstrating that reactions that are important do not have to be sensitive. The second discrepancy can be understood by examining more carefully the results of the global sensitivity analysis.

Figure 5 shows results for the two H-abstraction reactions that are most sensitive in the top panel of Fig. 3. The scatter of points shows some of the 50,000 calculations that were used to generate the global sensitivity analysis. The black curves in each panel show the fourth order polynomial fits to all 50,000 points, as described in Eq. (4). The x-axis describes the full uncertainty ranged scaled to be between 0.0 and 1.0, which is customary practice. The x-axis label on the bottom panel indicates that the rate

coefficient is increasing as u goes from 0.0 to 1.0. The plots in Fig. 5 indicate that the top reaction promotes formation of ethylene and the bottom reaction inhibits the formation of ethylene.

The results in Fig. 5, along with the analysis in Fig. 3 indicate that global sensitivity of species formation often leads to pairs of reactions, one that promotes formation of a species and the other that inhibits formation, primarily by formation of a different species. For example, the δ-abstraction in the bottom panel of Fig. 5 leads to formation of propylene. It has the highest sensitivity in the middle panel and is the main path to propylene in the reaction pathway analysis of [Karwat et al (2011)]. We leave out any detailed discussion of the acetaldehyde case in the bottom panel of Fig. 3, but only remark that the most sensitive reactions in that case occur further down the reaction pathway than the initial hydrogen abstraction. The calculation of the sensitivity coefficients outlined in the previous discussion is time consuming, because it requires the generation of information about the response of the chemical model over many randomly chosen realizations of the rate coefficients. The top two panels of Fig. 6 demonstrate that by modifying the previous algorithm [Skodje et al (2010); Davis et al (2011)] using Eqs. (5a) and (5b) one can save considerable computational time, reducing the number of computer runs from 50,000 to 7,000, as comparison of the top and middle panels of

Fig. 5. First-order functions [Eqs. (1) and (4)] for two reactions that contribute to the sensitivity of ethylene formation at t/tign = 0.5 (see Fig. 2).

Fig. 6. The top and middle panels show results for the old algorithm and new algorithm, respectively. The bottom panel shows a calculation using sparse regression.

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Fig. 6 demonstrate. For the n-butanol mechanism studied here the elapsed time to solution is reduced from a few processor days to one half day on a single processor for the ignition target. We believe that such a computational burden is not excessive and allows for relatively straightforward computation of the sensitivity coefficients for auto-ignition under homogeneous 0-D conditions, and can lead to the validation of chemical mechanisms in an efficient manner, as originally outlined in [Skodje et al (2010)].

The situation is much different for combustion modeling when a single computation is time consuming, as it is in device modeling, which is what is purposed in [Liu et al (2013)]. There a single computation of ignition in a compression-ignition engine model takes on the order of several processor weeks, which is generally accomplished in a single day using a few dozen processors. For such a case generating 7,000 computer runs is excessive. However, as noted in Sec. 2 and is obvious in many of the figures here (Figs. 1, 3, and 6), global sensitivity analysis often leads to sparse results, with most sensitivity coefficients small. There have been a number of techniques developed in the statistics literature to take advantage of such sparsity in an efficient manner, and we use one of them here, which is a variation on the lasso [Tibshirani (1996)] discussed in Sec. 2, called the sparse group lasso [Simon et al (2013)].

Global sensitivity coefficients for butanol ignition are presented in the bottom panel of Fig. 6 using the sparse group lasso. This panel demonstrates that good agreement with the larger scale calculations of 50,000 (top panel) and 7,000 runs (middle panel) can be obtained with only 800 runs, which is significantly less than 1446, the number of reactions. The sparse result was obtained using fourth polynomials for both the main effect and interaction terms. A limited number of interaction terms were included in the calculation, with only those reactions included whose main effects were greater than 0.01. The parameter λ was chosen based on an empirical rule we found that gave similar results to the cross-validation technique [Davis and Liu (2013)] for fourth order fits with interaction terms using the sparse group lasso. A full regularization path [see, for example, Hastie et al (2009)] was generated and the value of λ was chosen so that the number of expansion coefficients was ~700 (800-100).

A careful examination of Fig. 6 indicates that, while the basic structure of the sensitivity spectrum is retained using sparse regression with the lasso penalty, the sensitivity coefficients are shrunk from the ordinary least squares values in the top two panels. It is this shrinkage that leads to reduction of overfitting [Bishop (2006)], it “regularizes” the solution. Another way to regularize the least squares solution is called “ridge regression”, which replaces the absolute value in Eq. (7) with the square of the expansion coefficients. Ridge regression is much easier to implement than the lasso, requiring only a slight modification of the least squares solutions. Like the lasso, ridge regression also shrinks the expansion coefficients to reduce overfitting. However, unlike the lasso, ridge regression shrinks the coefficients in a uniform manner. The lasso shrinks coefficients in a preferential manner, setting very small expansion coefficients to zero. In the calculations we perform here and in [Davis and Liu (2013)] this results in the selection of a series of reactions whose sensitivity coefficients are non-zero. While these sensitivity coefficients are shrunk from their values using ordinary least squares they tend to keep the same ordering. Statisticians refer to this process as “variable selection”. In our case we are not selecting variables but reaction rate coefficients.

We end this section by presenting two sets of detailed results for reaction selection in Figs. 2 and 4. Figure 2 compares the order of reactions with sparse regression to the ordinary least squares (OLS) from the middle panel of Fig. 6. All reactions from that calculation with Si > 0.01 (1% of the variance) are shown in the first column of the table, followed by the number of the reaction in the mechanism of [Black et al (2010)] in the second column, Si from the OLS calculation in the third column along with the rank in the fourth column. The sparse regression rankings for three sample sizes are shown in columns 5-7. The sparse rankings which are different from the OLS values are shown in red in the table. Figure 2 shows that there is very good agreement between all the sample sizes and the much larger OLS result. The smallest sample size in the figure is 400 which is less than 6% of the OLS sample size of 7000. The only mis-rankings in the table involve the 6th and 7th most sensitive reactions, each of which account for only about 3% of the variance and have very close values.

A second ranking is shown in Fig. 4 for the species study in the top panel of Figure 3. These are the reactions for which the sensitivity to formation of ethylene is highest at t/tign = 0.5. The columns have the same ordering as they do in Fig. 2. Once again the sparse regression gives good agreement with the OLS. All reactions that account for more than 3% of the variance are accurately ordered. The agreement is somewhat worse than in Fig. 2, as the 10th most sensitive reaction is listed as the 6th most sensitive for a sample size of 400 and there is some mis-ranking among those reactions for 5-7 with sample sizes of 800 and 1200. The mis-ranking for a sample size of 1200 is minor as the 5th and 6th most sensitive reactions have fairly close values. 4. Conclusion

We have shown that it is possible to perform global sensitivity analysis for large mechanisms, by studying the butanol mechanism of [Black et al (2010)] that contains 1446 reactions. Our previous work on ignition delay targets was

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expanded to species targets, which allowed us to investigate species time histories that were studied in the experimental and modeling work of [Karwat et al (2011)]. The discussion in this paper concentrated on ethylene where we showed that two of the OH hydrogen abstraction reactions contributed most to the global sensitivity, with one promoting formation of ethylene and the other inhibiting it. This result showed the difference between global sensitivity analysis for species and reaction pathway analysis, where only the reactions that promote formation are listed as important.

The second half of the paper presented methods for doing global sensitivity analysis with small sample sizes. We showed that it was possible to get very reasonable orderings for the sensitive reactions with only 400 samples, which is a small fraction of the number of reactions in the mechanism. Such small sample size calculations will prove useful for device modeling, where the number of runs is limited. It was noted above that these small sample sizes are smaller than what would typically be used for linear sensitivity analysis for ignition delay problems [Qin et al (2001)] if all the reactions were studied.

Acknowledgements This work was supported by the Division of Chemical Sciences, Geosciences, and Biosciences, the Office of Basic Energy Sciences, the U.S. Department of Energy, under contract number DE-AC02-06CH11357. The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. References

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