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Article Cite This: Energy Fuels XXXX, XXX, XXX−XXX

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Skeletal Kinetic Mechanism Generation and Uncertainty Analysis for Combustion of Iso-octane at High Temperatures Rui Li, Guoqiang He, Duo Zhang, and Fei Qin* Science and Technology on Combustion, Internal Flow and Thermal-Structure Laboratory, Northwestern Polytechnical University, Xi’an Shaanxi 710072, PR China S Supporting Information *

ABSTRACT: A detailed mechanism for combustion of iso-octane with 116 species and 754 reactions has been reduced using a directed relation graph with error propagation (DRGEP) and DRGEP with sensitivity analysis (DRGEPSA) methods under high-temperature conditions. Two skeletal mechanisms, i.e., a 63-species mechanism with a maximum error of 7.2% and a 51species mechanism with a maximum error of 28.5% on autoignition delay times have been generated. These two skeletal mechanisms are shown to reproduce ignition delays, laminar flame speeds, species and temperature profiles in good agreement with those of the detailed mechanism. Uncertainty in the ignition predictions by detailed and two skeletal mechanisms induced by the uncertainties in reaction rate coefficients has been studied. Probability distribution of autoignition predictions demonstrated that the 63-species mechanism can still keep the uncertainty characteristics, while the 51-species mechanism has significant discrepancy compared with the detailed one. Further analysis of autoignition shows that the structure and integrality of the reaction system in the 51-species mechanism has changed. Global sensitivities of 63-species and detailed mechanisms on ignition have been investigated using the high-dimensional model representation (HDMR) method. The highly important reactions for ignition in the detailed mechanism are the same as those in the 63-species mechanism, and sensitivity coefficients of the listed reactions agree well with each other. The most important reactions in the first-order sensitivity on autoignition in the detailed mechanism are the same as those in the 63-species mechanism, especially for the five most important reactions. The most important 10 reactions contribute almost 75% to the overall variance in ignition delay under the present conditions, while the second-order effects are quite small and almost negligible. The top ranked reactions show that small-molecule chemistry (C0−C4) contributes significantly to uncertainties in the ignition predictions at high temperatures.

1. INTRODUCTION Detailed chemical kinetic models are increasingly used to predict the formation of varieties of products and pollutants and to describe important chemical and physical characteristics such as autoignition times and flame speeds.1 Generally, the detailed mechanisms would thus be reduced to accommodate the demands of flame simulations and computational fluid dynamics (CFD) calculations.2 Over the last few decades, various strategies for mechanism reduction were proposed to generate reduced mechanisms that simplify the detailed chemical kinetics while still retaining the essential features of the reaction system. These mechanism reduction methods are generally categorized into skeletal reduction and time scale analysis.3 Among these mechanism reduction approaches, the automated and theory-based methods, e.g., directed relation graph (DRG),4−7 DRG with error propagation (DRGEP),8−11 path flux analysis (PFA),12−15 flux projection tree method (FPT),16 and sensitivity analysis combining DRG-based approaches17−22 have been widely used in CFD applications and are receiving more and more attention. With the development of CFD technologies and the computing capacity, more accurate and robust reduced mechanisms are required in order to obtain simulation results with higher fidelity. As a result, the uncertainty issues of chemical kinetic models become crucial and are receiving more and more attention.23−25 The uncertainty of a large mechanism is usually enlarged by the great number of reactions whose rate © XXXX American Chemical Society

coefficients are not well characterized due to the limitations of experimental data and theoretical calculation levels. Recently, the uncertainty quantification of the chemical kinetic mechanism was studied using optimization methods.3,26−28 Various sensitivity analysis and uncertainty quantification methods in combustion kinetics were discussed in detail by Turányi29 and Wang et al.3,23 Reduced mechanisms of different sizes for combustion of n-butane and iso-butane were generated and evaluated and then improved by optimization through polynomial chaos expansion (MUM-PCE).3 The high-dimensional model representation (HDMR) method proposed by Tomlin et al.30,31 was a powerful tool for global sensitivity analysis of complex chemical mechanisms and was widely used in the evaluation of combustion mechanisms. Iso-octane is an important surrogate for practical fuels, and its chemical kinetic models have attracted much attention.1,32−38 Ranzi et al. proposed a semi-detailed reaction model which includes about 150 species and 3000 reactions to simulate iso-octane oxidation in a wide range of conditions.35 An updated reaction model for the combustion of iso-octane for the simulation of gasoline surrogate fuels was developed by Mehl et al., which contains about 1550 species and 6000 reactions.32 Davis and Law used a fairly compact kinetic model Received: September 20, 2017 Revised: January 22, 2018 Published: February 6, 2018 A

DOI: 10.1021/acs.energyfuels.7b02838 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels which includes 69 species and 406 reactions to simulate laminar flame speeds of iso-octane.36 A detailed kinetic model for isooctane oxidation consisting of 857 species and 3606 reactions was developed by Curran et al., which can be applied to both high- and low-temperature ranges.37 A semi-detailed chemical reaction model consisting of 116 species and 754 reactions was developed by Chaos et al. to describe the high-temperature oxidation of iso-octane.34 Recently, Atef et al.38 developed a comprehensive iso-octane combustion reaction model which includes 2768 species and 9248 reactions with improved thermochemistry and chemical kinetics by updating the chemical kinetic models of Curran et al.37 and Mehl et al.32 using new experimental and theoretical results. A skeletal mechanism for combustion of iso-octane presented by Xin et al.39 was generated using the DRGASA method which contains a two-stage DRG strategy, and the mechanism consisting of 112 species and 481 reactions was finally obtained. Another skeletal mechanism consisting of 196 species and 1762 irreversible reactions was generated using the DRGEP method with a maximum error of 15% over a wide range of conditions by Niemeyer et al.19 However, uncertainty quantification of the detailed and reduced chemical kinetic models for combustion of iso-octane has not been investigated, nor has the HDMR method been used in global sensitivity analyses of the reduction process for autoignition of iso-octane. The present work aims to achieve compact skeletal mechanisms for combustion of iso-octane under high-temperature conditions and tries to improve the efficiency and performance of the skeletal reaction models that are used in the CFD codes. Then to evaluate the effects of uncertainties of skeletal mechanisms on ignition delay, and finally to find out the reactions which uncertainties contribute the most to the overall mechanism behavior through local and global sensitivity methods. In the present work the semi-detailed reaction model for combustion of iso-octane developed by Chaos et al.34 will be reduced and analyzed using the global sensitivity analysis method. This reaction model contains a moderate number of reactions and can be affordable in global sensitivity analysis. Moreover, this semi-detailed reaction model has been validated against experimental data including shock tube ignition delays, premixed laminar burning velocities, and speciations in a variable pressure flow reactor and jet-stirred reactor. This paper is organized as follows. In Section 2, skeletal mechanisms for combustion of iso-octane are constructed using DRGEP and DRGEPSA methods. In Section 3, probability distributions of ignition delay times of detailed and skeletal mechanisms are analyzed. In Section 4, global sensitivity analyses of detailed and skeletal mechanisms are carried out using brute-force and HDMR approaches.

∑i = 1, nR vi ,Aωiδ Bi

rAB(DRGEP) =

∑

PA =

max(PA , CA )

(1)

max(0, vi ,Aωi) (2)

i = 1, nR

∑

CA =

max(0, −vi ,Aωi) (3)

i = 1, nR

where rAB represents dependence of species A upon another species B, vA,i is the net stoichiometric coefficient of species A in the ith reaction. ωi is net reaction rates calculated by (ωf,i − ωb,i), in which ωf,i and ωb,i are the forward and reverse reaction rates, respectively. In DRGEP, the indirect coupling relationship between two species is also calculated, and species B can affect species A through an indirect pathway. A path-dependent interaction coefficient (PIC) is taken into consideration and determined by using n−1

rAB, p =

∏ rS S

i i+1

(4)

i=1

The overall interaction coefficient RAB is defined as the maximum of all possible rAB,p: RAB = max (rAB, p)

(5)

(P = 1, p)

A large-size and a small-size mechanisms are first generated by choosing two different thresholds using the DRGEP method. In sensitivity analysis, all the species in the small-size mechanism also appear in the large-size mechanism, and they are treated as important species.19,20,22 Sensitivity of the species in the uncertain set, i.e., not appearing in the small-size mechanism but contained in the large-size mechanism will be tested in a one-by-one manner. Sensitivity of ith species in the current skeletal mechanism is defined as the maximum absolute simulation error induced by removing this species compared with the detailed mechanism, as shown in eq 622 Ei = max j

|δi , j − δdetail, j| δdetail, j

,

j = 1,2, ...N (number of conditions)

(6)

where δi,j is the parameter, e.g., ignition delay or other combustion characteristics, obtained from the current mechanism through removal of the ith species in the uncertain set at the jth condition and δdetail,j is the corresponding parameter achieved from the detailed mechanism at the jth condition. Species with smallest Ei of the uncertain set will be removed in a one-by-one way as the process is repeated until the maximum absolute simulation error of the skeletal mechanism is larger than a user-specified threshold. A more detailed discussion about the DRGFEPSA method and their applications can be found in refs 19, 20, and 22. 2.2. Skeletal Reduction Using DRGEP. The kinetic model of iso-octane used here is from a detailed primary reference fuel (PRF) mechanism for high-temperature conditions developed by Chaos et al.,34 which contains 116 species and 754 elementary reactions and has been validated against extensive sets of experimental data. We still call this 116-species reaction model a detailed mechanism in order to conveniently compare it with skeletal ones. The selected conditions for the skeletal mechanism generation cover an equivalence ratio (ϕ) = {0.5,

2. SKELETAL MECHANISM GENERATION FOR COMBUSTION OF ISO-OCTANE USING DRGEPSA 2.1. DRGEPSA Method. DRGEPSA19,20 is a reduction method by combining DRGEP with sensitivity analysis. DRG with the error propagation method (DRGEP) proposed by Pepiot-Desjardins and Pitsch8 considers a coupling relationship between two species through all possible pathways, unlike in the DRG method, i.e., all species are assumed to be equally important and only the direct relation between two species is checked. The following interaction coefficient rAB for evaluating a coupling relationship between species is proposed in DRGEP8 B


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ratios together with the available experimental results.41−43 The two skeletal modeling results are in good agreement with those of the detailed mechanism and experimental data. In order to examine the reliability of the skeletal mechanisms further, species and temperature profiles are also investigated in a 1-D premixed flame using the detailed and 63-species and 51species skeletal mechanisms, which are plotted in Figure 4. One can see that concentrations of iC8H18, O2, CO, and H2O and some important intermediate radicals, e.g. OH and CO, are reproduced satisfactorily with the 63-species and 51-species reduced mechanisms as compared to the detailed mechanism. Validation of various combustion characteristics indicates that the two skeletal mechanisms for combustion of iso-octane are well reproducible.

1.0, 2.0}, P = {1 atm, 5 atm, 15 atm}, and T = {1000 K, 1100 K, 1200 K, 1300 K, 1400 K, 1500 K, 1600 K, 1700 K, and 1800 K} for autoignition. Oxygen and iso-octane are selected as target species in the DRGEP reduction. A skeletal mechanism containing 63-species and 384-reactions (large-size mechanism) is obtained with the DRGEP method using the automatic mechanism reduction program ReaxRed,40 and the maximum error is 7.2% for ignition delay times over the above simulation conditions. A small-size mechanism containing 30 species and 106 reactions is also generated using the same method for further reduction using sensitivity analysis. Figure 1 presents

3. UNCERTAINTY ANALYSIS As compared to the detailed mechanism, the two skeletal mechanisms i.e., 63-species (maximum error 7.2%, autoignition) and 51-species (maximum error 28.5%, autoignition) mechanisms, reproduced satisfactorily the ignition delay times, flame speeds, and temperature as well as species profiles, and these combustion characteristics are usually used to evaluate the skeletal mechanisms. Therefore, these two skeletal mechanisms would be regarded as reliable reduced mechanisms in mechanism reduction work. Recently, the uncertainty of combustion chemical models has received great attention, and global uncertainty analysis methods are used to evaluate the kinetic models.3 The uncertainties in the rate coefficient can be given typically by the uncertainty factor in eq 7. Then the rate coefficients are varied by changing the A-factors in the modified Arrhenius form of the rate expression over the range of their uncertainties:23

Figure 1. Maximum error in ignition delay times against species number of skeletal mechanisms using DRGEP and DRGEPSA methods.

the maximum error against species number of the skeletal mechanisms generated by DRGEP and DRGEPSA, and specific results from the two approaches are summarized in Table 1.

fi =

Table 1. Skeletal Mechanisms of Iso-octane Generated with DRGEP and DRGEPSA Methods method

threshold

species

reaction

max. error %

DRGEP (large-size) DRGEP (small-size) DRGEPSA

0.030 0.225 -

63 30 51

384 106 298

7.2 >100 28.5

ki0 kilower

=

kiupper ki0

(7)

where k0i is the nominal rate coefficient of the ith reaction with its lower and upper bounds given as kilowerand kiupper, respectively. f i is uncertainty factor of the ith reaction. All rate coefficients are assumed to have a uniform distribution by defining the upper and lower limits of the parameter. Uncertainty factors are assumed conservatively to be of a factor of 2 above and below the nominal value for each A-factor in the case of overestimation, while large uncertainty factors of 5 were used in the global sensitivity study of cyclohexane oxidation under low-temperature fuel-rich conditions using HDMR methods by Ziehn et al.44 and in the correlation studies of uncertainty quantification of transportation relevant fuel models by Fridly et al.45 Sobol sequence is adapted for random sampling, which is one of the best known low discrepancy sequences proposed by Sobol.46 It should be noted that the choice of the distribution is not crucial when making forward uncertainty quantification projections.23 Autoignition delay times in air are analyzed under the conditions of ϕ = 1.0, P = 1 atm, and T = 1000 K and 1500 K. Probability distributions of autoignition predictions for skeletal and detailed mechanisms for iso-octane under the simulation conditions are plotted in Figure 5. Each of the 16 384 runs involved the sampling of 754 parameters for detailed, 384 parameters for 63-species, and 298 parameters for 51-species mechanisms. It can be seen from this figure that the 63-species mechanism agrees well with the detailed one for two temperature conditions. The two curves not only have the

Obviously, the ignition delay times of obtained skeletal mechanisms show a strong nonlinear correlation to retained species number. Finally, a skeletal mechanism with 51-species and 298-reactions is obtained with the maximum error in autoignition of 28.5%. Moreover, results from Figure 1 and Table 1 also demonstrate that the DRGEPSA approach can yield a more compact and accurate reduced mechanism compared with DRGEP, which has been pointed out in ref 19. Ignition delay times calculated with the 63-species skeletal mechanism from DRGEP and the 51-species skeletal mechanism from DRGEPSA as well as the detailed mechanism are illustrated in Figure 2. One can see from this figure that these two skeletal mechanisms reproduce ignition delay times of the detailed mechanism rather well. Error of the worst-case for autoignition with the 63-species mechanism is 7.2%, while it is 28.5% with the 51-species mechanism. Both of these cases appear at an initial temperature of 1000 K. Figure 3 compares the calculated laminar flame speeds of isooctane + air mixtures obtained by using the detailed and skeletal mechanisms over a range of pressures and equivalence C


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Figure 2. Autoignition delays of detailed 63-species and 51-species mechanisms over a range of initial temperatures and pressures and at varying equivalence ratios.

Figure 4. Temperature and species profiles in a 1-D premixed flame with 63-species and 51-species skeletons as well as the detailed mechanisms at initial temperature of 298 K, equivalence ratio of 1, and pressure of 1 atm.

Figure 3. Laminar flame speeds with the detailed and 63-species and 51-species mechanisms as well as experimental data41−43 at initial temperature of 298 K and pressure of 1 and 5 atm.

small. To further explain this point and explore the contribution of the same reactions in different size mechanisms, the 298 reactions in 51-speceis are extracted from the 63species skeletal mechanism as well as from the detailed one and then studied. This means that only 298 reactions (the same as 298 reactions in the 51-species mechanism) of the 63-species mechanism and the detailed mechanism are taken into account, however, the other reactions which retain their rate constants are included when calculating autoignition delays. The probability distributions of ignition delays in this case are plotted in Figure 6. It can be seen from this figure that the uncertainty of ignition delays obtained using 298 reactions in

same shape but also peak on the same position. However, the curves of the 51-species mechanism with 298 reactions indicate significant discrepancy compared to the detailed and 63-species mechanisms at temperatures of 1000 K and 1500 K. There could be different reasons leading to this discrepancy: e.g., some reactions that play roles in uncertainty analysis may be missed in the 51-species mechanism, and those reactions present in the 63-species mechanism while absent from the 51species mechanism are important to support the integrality of the reaction system. These missing reactions may also help retain a coupling relationship between reactions, though the actual coupling relationship between two reactions may be D


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Figure 5. Distribution of ignition delay times obtained from random sampling at different initial temperatures, 1000 K and 1500 K. The sample number is 16 384 for each reaction model.

Figure 6. Distributions of ignition delay times obtained from random sampling. The sample number is 16 384 for each reaction model.

Figure 7. Sensitivity of autoignition for 51-species, 63-species, and detailed mechanisms at pressure of 1 atm, equivalence ratios of 1.0, and temperature of 1000 K. E


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Figure 8. Normalized sensitivity coefficients for the laminar burning velocity of iso-octane + air flame for detailed, 63-species, and 51-species mechanisms.

agree well with each other. An obvious discrepancy in sensitivity coefficients between results with the detailed and those with the 51-species mechanisms exists. The 51-species mechanism overestimates sensitivity coefficients significantly on some reactions, especially for the four reactions, i.e., iC8H18 + OH = aC8H17 + H2O, iC8H18 = yC7H15 + CH3, iC8H18 = yC7H15 + CH3, H + O2 = O + OH, and HO2 + CH3 = OH + CH3O. It is therefore possible that removal of the 12 species in the 63-species mechanism while absent from the 51-species mechanism has an effect on the reaction pathways for autoignition. One may expect that these 12 species may affect the reaction pathways in laminar flame speed modeling as well. To clarify this matter, a sensitivity analysis for laminar flame speeds is performed to reveal key reactions controlling flame propagation. The sensitivity analysis in the present work is performed at an initial temperature of 298 K, Φ = 1.0, and pressure for 1 and 5 atm. The sensitivity coefficients of the laminar flame speeds with respect to the rate constants are presented in Figure 8. Reactions with a sensitivity coefficient larger than 2% are illustrated in Figure 8 together with their normalized sensitivity coefficients for the three models. One can see from the figure that the sensitivity coefficients and reaction ranking for the three mechanisms are highly consistent with each other. The results indicate that the laminar burning velocity is sensitive to reactions from small-molecule reactions (C0−C1), which is different from the sensitivity for autoignition. It is concluded that maximum error of 30% for ignition delays may remove some species and then break the integrality of a reaction system for autoignition of iso-octane combustion. Otherwise, a small threshold, e.g., maximum error of 10% for skeletal mechanism, can retain uncertainty characteristics of ignition delays. Consequently, one should exercise caution when dealing with the uncertainty analysis using a skeletal mechanism obtained with a large threshold.

63-species reaction system and 298 reactions in the detailed mechanism are reproduced closely with 754 reactions of the detailed mechanism. Thus, the reactions that have not been sampled in 63-species and detailed mechanisms still play a role in reaction pathways and autoignition modeling, and that is why these reactions affect the probability distribution of autoignition predictions. These results also demonstrate that the important reactions which contribute mostly to uncertainties in autoignition are not missed in the 51-species mechanism. However, some reactions that may play roles in retaining integrality of a reaction system are missed in the 51-speceis mechanism without largely sacrificing the capacity of predicting ignition delays, flame speeds, and species profiles. It is also possible that sensitivity for autoignition has changed for some important reaction in the 51-species mechanism. To illustrate this point, a brute-force method of local sensitivity analysis is employed to examine the accuracy of the skeletal mechanisms and to identify these important reactions.22,47 The local sensitivity analysis for autoignition is achieved by adjusting the A-factors for each input reaction in the kinetic mechanisms by +1% from their nominal values, then ranking the reactions in terms of the % change in output response44 sensitivity (i) =

i i τign(k new + Δ) − τign(k new ) i Δ × τign(k new )

i k new = k 0i /2, k 0i , k 0i × 2 i Δ = (1%) × k new

(8)

where ki0 is the rate constant of the ith reaction and τign is the ignition delay time. In order to avoid misleading information from only considering calculations around the nominal parameter values, three parameter values are adopted here, i.e., ki0/2, ki0, and ki0*2. Brute-force sensitivity analysis for each reaction is thus carried out for the 51-species, 63-species, and the detailed mechanisms at temperature of 1000 K. Reactions with a sensitivity coefficient topped 10 for the detailed mechanisms are illustrated in Figure 7 together with their sensitivity coefficients. Corresponding reactions and their sensitivity coefficients in 51-species and 63-species mechanisms are also depicted in Figure 7. According to these sensitivity spectra, the most important reactions on autoignition in the detailed mechanism are the same as those in the 63-species mechanism, and sensitivity coefficients of the listed reactions

4. GLOBAL SENSITIVITY ANALYSIS The ignition delays thus vary substantially depending on which reactions and rate coefficients are adopted according to the results above. These 298 reactions are responsible almost exclusively for contributing to the uncertainty of the chemical kinetic mechanism for combustion of iso-octane. Obviously, not each reaction makes equal contribution to distributions of ignition delays even in those 298 reactions, which are, however, regarded as important ones. It is interesting to figure out which F


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Table 2. Sensitivity Indices of Important First and Second Order Using the Detailed Mechanism (N = 16 384) and 63-Species Mechanism (N = 16 384) first-order ranking 1 2 3 4 5 6 7 8 9 10 ∑Si ∑Sij ∑Si + ∑Sij

first-order

second-order

754-inputs

Si

R154 R694 R115 R712 R152 R572 R570 R114 R268 R1

0.1639 0.1270 0.1101 0.0672 0.0510 0.0498 0.0456 0.0353 0.0341 0.0325 0.9558

754-inputs R694 R115

R712 R694

second-order

Si,j

384-inputs

Si

0.0051 0.0012

R154 R694 R115 R712 R152 R572 R570 R114 R576 R1

0.1654 0.1249 0.1115 0.0601 0.0529 0.0518 0.0468 0.0392 0.0346 0.0330 0.9545

Si,j

384-inputs R694 R570 R115

R712 R572 R694

0.0040 0.0018 0.0014

0.0063

0.0072

0.9620

0.9618

Table 3. Top Five Reactions in First-Order Sensitivity Analysis Using 63-Species and Detailed Mechanisms (N= 16 384) and Local Ranking first-order ranking

first-order ranking

local ranking detailed

local ranking 63-species

number

reactions

detailed

63-species

1/2k0

k0

2*k0

1/2k0

k0

2*k0

R154 R694 R115 R712 R152

2CH3(+M) = C2H6(+M) iC4H7O = tC3H5 + CH2O HO2 + CH3 = OH + CH3O iC4H6OH + HO2 = iC4H7OH + O2 CH3 + O2 = OH + CH2O

1 2 3 4 5

1 2 3 4 5

2 1 3 4 7

1 2 3 4 6

2 10 3 4 1

2 1 3 4 7

1 3 2 4 5

1 12 4 6 2

over the uncertainty ranges of the 384 inputs (for 63-species mechanism) and 754 inputs (detailed mechanism) are used, and each uncertainty factor of these parameters is assumed to be of a factor of 2 as has been described in section 3. Finally, 37 important reactions for the 63-species skeletal mechanism and 38 important reactions for the detailed mechanism are identified for simulations of ignition delays. Table 2 shows the ten highest ranked first-order sensitivity indices using a sample size of N = 16 384 for the detailed and 63-species mechanism. One can see that ∑Si apparently converges to about 0.95 for N = 16 384. No significant reaction interactions are found according to the presence of second- and higherorder effects, as shown in Table 2. The second-order effects are quite small and almost negligible. According to Table 2, the most important reactions in the first-order sensitivity on autoignition in the detailed mechanism are the same as those in the 63-species mechanism (9 of 10), and sensitivity coefficients of the listed reactions agree well with each other, especially for the five most important reactions. The most important 10 reactions contribute almost 75% to the overall variance in ignition delays under the present conditions. The top 10 reactions are listed in the Supporting Information. The comparative rankings based on the HDMR first-order sensitivities and the local sensitivity coefficients from section 3 are shown in Table 3. The top ranked reactions in the HDMR first-order are almost all present in the local highly ranked group (almost top 10). These results from Table 3 demonstrate that the ranking evaluated by the local sensitivity method varies substantially depending on the values chosen. A similar conclusion has been noted by Xin et al.39 Local ranking using the base case of the nominal value can give closer results compared with HDMR first-order ranking.

parameters should be prioritized. The global uncertainty analysis method has been used widely in analyzing combustion kinetics. A global sensitivity approach based on high dimensional model representations (HDMR) developed by Tomlin et al. is applied in order to identify those reactions which make the largest contributions to the overall uncertainty of the kinetic mechanisms.30 The HDMR method can deal with a complex model consisting of a large amount of input parameters and was used to investigate the cyclohexane oxidation under lowtemperature fuel-rich conditions44 and methane oxidation process.31 Recently, an accelerated global sensitivity analysis, the ANN-HDMR, developed by Li et al.48 by combining an artificial neural network with HDMR, has been used in the sensitivity analysis of the master equation kinetic model and the premixed H2/O2 ignition model. Further exposition on the HDMR method and their applications in uncertainty quantification have been discussed in great detail in previous work by Tomlin et al.30,31,44 The 63-species mechanism with 384 reactions is used to investigate the global sensitivity due to its good performance in probability distributions of ignition delays and local sensitivity analysis compared with those of the detailed mechanism. Calculation of a large number of inputs is difficult to lead to high quality of the HDMR results.44,46 Therefore, to reduce the number of terms in the HDMR expansion, a threshold of 0.1% is applied so that only component functions that have a significant contribution toward the overall output value are considered.31 Alternatively, the Morris screening method49,50 and a global screening approach based on Spearman Rank Correlation Coefficients (RCCs) can be adopted as prescreening step.46 Simulations for autoignition predictions were carried out under the conditions of ϕ = 1.0, P = 1 atm, and T = 1000 K. A uniform Sobol sequence distribution of 16 384 samples G


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From the results presented in Table 2 and Table 3, all the top 5 reactions are small-molecule reactions for the detailed and 63species skeletal mechanism of iso-octane, which means that small-molecule chemistry (C0−C4) contributes significantly to uncertainties in the ignition predictions at the high-temperature conditions. A similar conclusion has been made in ref 45. In contrast to the present high-temperature region, 12 important reactions were identified by Monte Carlo simulations according to their contribution to uncertainty at 725 K (NTC region) for ignition of iso-octane, and most of these 12 reactions are H atom abstractions and isomerization steps.45 In summary, the 63-speceis skeletal mechanism can retain the uncertainty features, i.e., probability distribution and global sensitivity analysis of ignition prediction in the high-temperature conditions. However, a 51-species mechanism has significant discrepancy in probability distribution of ignition predictions compared with those of the detailed mechanism.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.energyfuels.7b02838. Additional tables. Abbreviations and full name of reduction methods used in this paper and top 10 reactions in global sensitivity analysis (PDF) 51-species skeletal reaction model for iso-octane oxidation in CHEMKIN format (TXT) 63-species skeletal reaction model for iso-octane oxidation in CHEMKIN format (TXT)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

5. CONCLUSIONS DRGEP and DRGEPSA methods have been applied to reduce the detailed mechanism for combustion of iso-octane with 116 species and 754 reactions at high temperatures. Two skeletal mechanisms, i.e., a 63-species mechanism with a maximum error of 7.2% and a 51-species mechanism with a maximum error of 28.5% on autoignition, have been generated. These two skeletal mechanisms were validated for both the ignition and laminar flame speeds as well as species profiles, and the results are in good agreement with those of the detailed mechanisms. Uncertainty of the detailed and two skeletal kinetic mechanisms induced by the uncertainties in reaction rate coefficients was studied, and probability distribution of autoignition predictions was investigated at the conditions of ϕ = 1.0, P = 1 atm, and T = 1000 K and 1500 K. The results demonstrate that the 63species with 384 reactions mechanism agrees well with the detailed mechanism, and it can still retain the uncertainty characteristics, while the 51-species mechanism with 298 reactions has significant discrepancy compared with the detailed one. Further analyses show that sensitivity coefficients of some important reactions for autoignition have changed. However, laminar flame speed modeling looks less sensitive compared with autoignition because the key reactions controlling flame propagation are mostly small-molecule reactions (C0−C1), a possible reason that some species are missed in the 51-species reaction model that results in an incomplete reaction system. Global sensitivity of the 63-species and detailed mechanisms for ignition was investigated further and compared with the local sensitivity analysis. The highly important reactions on ignition in the detailed mechanism are the same as those in the 63-species mechanism, and sensitivity coefficients of the listed reactions agree well with each other. The global uncertainties of the 63-species and detailed mechanisms are checked using the HDMR method. The most important reactions in the firstorder sensitivity on autoignition in the detailed mechanism are the same as those in the 63-species mechanism, and sensitivity coefficients of these reactions agree well with each other, especially for the five most important reactions. The most important 10 reactions contribute almost 75% to the overall variance in ignition delay under the present conditions, while the second-order effects are quite small and almost negligible. The top ranked reactions show that small-molecule chemistry (C0−C4) contributes significantly to the uncertainties in the ignition predictions at high temperatures.

Fei Qin: 0000-0002-1440-1521 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China (Contract No. 91541110). The authors thank Prof. Alexander Konnov for advice on sensitivity analyses of autoignition and grammar modifications.

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