Reduction of detailed chemical mechanism using reaction class

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Reduction of detailed chemical mechanism using reaction class-based global sensitivity and path sensitivity analyses Yachao Chang, Ming Jia, Bo Niu, Mao-Zhao Xie, and Chengyu Zhou Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.9b02249 • Publication Date (Web): 22 Aug 2019 Downloaded from pubs.acs.org on August 28, 2019

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Reduction of detailed chemical mechanism using reaction class-based global sensitivity and path sensitivity analyses Yachao Chang, Ming Jia *, Bo Niu, Maozhao Xie, Chengyu Zhou Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, Dalian University of Technology, Dalian 116024, PR China Keywords: mechanism reduction; global sensitivity analysis; path sensitivity analysis; reaction class

Abstract

The reduced chemical mechanisms with small size and good performance are very important for the simulation of advanced combustion engines. In the present study, a new reduction method of detailed chemical mechanism was proposed using the reaction class-based global sensitivity and path analyses. During the reduction process, the influence of the species and reactions was determined according to the contribution of their corresponding reaction classes on the prediction uncertainties by calculating the nominal sensitivity index and the path sensitivity coefficient of each reaction class from the detailed mechanism. Furthermore, the dependence of the prediction target on the operating temperature, pressure, and

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equivalence ratio was studied. After establishing the initial reduced mechanism, the refinement of the rate coefficients in the fuel-specific sub-mechanism was conducted to improve the nominal predicted value of the reduced mechanism covering broad temperature conditions. Based on the proposed method, a reduced n-heptane mechanism with 89 species and 276 reactions is obtained from a detailed one composing of 645 species and 2827 reactions. By comparing the calculated value of the reduction targets from the reduced mechanism and the detailed mechanism over broad operating conditions, the reliability of the reduced mechanism was examined. Good agreements for the predicted data between the reduced and detailed mechanisms indicate the advantages of the present reduction method. Compared to the other methods, the reduced mechanism built using the present method was capable of better reproducing the prediction performance of the detailed mechanism with a more compact size.

1. Introduction Global energy crisis and stricter emission regulations promote the development of internal combustion engines with better thermal efficient and reduced pollution emissions. The computational fluid dynamics (CFD) modeling is a powerful approach for the investigation and design of engines. In CFD simulations, reliable chemical kinetic mechanisms act an important role in understanding the ignition and emission characteristics of fuels. The in-cylinder combustion is an extremely complex process, including ignition, flame propagation, fuel oxidation, pollution formation, etc. To accurately reproduce these behaviors, the size of the chemical kinetic mechanism is usually huge. For example, the 2


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reaction mechanism of gasoline surrogate fuel composes of 1533 species and 8764 reactions 1.

It requires extremely long computational time for numerical simulations using the

large-scale mechanisms 2, even for 1-D flame simulations. Thereby, researchers reduced the scale of the detailed chemical mechanisms focusing on specific operating conditions. Many mechanism reduction approaches have been presented over the past decades 3, 4. Lu and Law 3 divided these methods into four classes, i.e., skeletal reduction, lumping, time-scale analysis, and stiffness reduction. The skeletal reduction method aims to delete the insignificant species/reactions in the detailed mechanism. The representative methods contain sensitivity analysis 5-7, directed relation graph (DRG) 8 and the DRG derived methods 9-12, genetic algorithm 13, and so on. In the lumping method, the species or reactions with similar structure are merged 14-16. In the time-scale analysis method, the species are divided into slow and fast subsystems. The species in the fast subsystem are calculated using the algebraic equations rather than the differential equations, which can effectively diminish the computation time. Many methods have been introduced to distinguish the fast species and reactions, e.g. partial equilibrium assumption (PEA) 17, quasi-steady-state approximation (QSSA) 18, computation singular perturbation 19-21, etc. The stiffness reduction method usually includes on-the-fly systems. The time-scale method is typically employed for the stiffness reduction of the ordinary differential equation (ODE) related to reaction mechanism 22.

The scale of the reduced mechanisms of large-molecule fuels is generally still huge using the above methods if the low-temperature chemistry is included 23. To further diminish the scale, 3


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a set of methodologies are coupled together for mechanism reduction. Pei et al. 24 constructed a reduced mechanism for diesel surrogate fuel with 163 species and 887 reactions from a detailed one including 2885 species and 11,754 reactions by coupling the DRG with expert knowledge, DRG-aided sensitivity analysis, and lumping methods. The previous mechanism reduction methods only consider the contribution of every single species or reaction on the reduction target. The detailed mechanisms of large hydrocarbons are generally developed based on the hierarchical method, and the reactions can be divided into different reaction classes in each hierarchy 25, 26. It might be more effective for mechanism reduction if the contribution of each reaction class or hierarchy on the reduction target is considered. However, few mechanism reduction methods were proposed focusing on the reaction class in the detailed mechanism. Only recently, Chang et al. 27 applied the reaction class-based method to reduce a detailed oxidation mechanism of n-heptane focusing on ignition delay times. In the present study, a novel mechanism reduction method by coupling the global sensitivity and path analyses is proposed, in which the contributions of each reaction class and hierarchy on the prediction uncertainty of the detailed mechanism are calculated. Different from our previous work 27, the importance of the reaction classes in detailed mechanisms is assessed by considering the global sensitivity coefficient and path sensitivity coefficient simultaneously. Moreover, both the computed ignition delay times and the species profiles using the detailed mechanism over broad conditions are employed as the reduction targets. The paper is arranged as follows. First, the methods of mechanism reduction and refinement 4


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are introduced in Section 2. Then, the mechanism reduction method is employed to reduce a detailed n-heptane mechanism built by Mehl et al. 28. And the performance of the method is evaluated by examining the calculated data from the detailed and reduced mechanisms obtained by the present method and the DRG with error propagation and sensitivity analysis (DRGEPSA) in Section 3. Finally, the main conclusions are presented. 2. Computational method 2.1 Global sensitivity analysis The sensitivity analysis (SA) is a powerful tool to explore the correlation between the input parameters and the model predictions. It is usually utilized to find the reactions dominating the mechanism prediction during the mechanism reduction process 29, 30. Compared to the local sensitivity analysis, the global sensitivity analysis (GSA) is capable of better capturing the interactions among different reactions and the non-linear relationship between the rate coefficient and the model prediction 30. Thus, GSA is used to determine the significance of reaction classes in this work. The GSA is performed by the Monte Carlo simulation in the research. In theory, the temperature dependence of the uncertainty in the reaction rate constant should be properly treated by considering the correlation between the Arrhenius parameters and their uncertainties 31. However, that is hard to achieve for a detailed mechanism of large-hydrocarbon fuels due to the deficiency of the uncertainty information on the rate coefficient of the reactions involving large-molecule species. Moreover, Yao and Wang 32 found that the co-optimization on the pre-exponential factor (A) and activation energy (Ea) of 5


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a reaction did not obviously reduce the prediction uncertainty compared to the results of only A factor being considered. They indicated that the mechanism optimization only on A-factor is acceptable for the uncertainty quantification of chemical mechanisms. Thus, in this study, only the uncertainty from the A-factor of each reaction is considered as the input variable in the Monte Carlo calculation. For all reactions, the uncertainty is described using a log-uniform probability following the work of Fridlyand et al. 33, as displayed in Equation (1). 𝑃log ― uniform(𝑋𝑖) =

{

1/log (𝑓𝑖)2 log (1/𝑓𝑖)) ≤ 𝑋𝑖 ≤ log (𝑓𝑖) 0 others

(1)

where Xi=log(Fi), and Fi represents the disturbed value of the ith reaction in an individual calculation; fi is its uncertainty factor, and it is defined as:

𝑓𝑖 =

𝐴max 𝐴0

𝐴0

= 𝐴min

(2)

where A0, Amax, and Amin represent the normal, maximum, and minimum value of the pre-exponential factor, respectively. In the research, fi of the reactions in the C0–C4 sub-mechanism is determined from the related database of chemical kinetics 34-36. For the reactions in the other sub-mechanisms, fi is set as 4 following the work of Cai and Pitsch 37. To investigate the source of the prediction uncertainty, a normalized sensitivity index (NSI) is employed. The NSI is calculated in Equation (3) as

NSIC𝑖 =

IDRC𝑖 IDRall

NSIRC𝑖 =

IDRRC𝑖 IDRall

6


(3)

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in which IDR is the interdecile range; Ci represents the sub-mechanism including the hydrocarbons with i carbon number; RCi represents the ith reaction class; and all represents the detailed mechanism of the fuel. The larger NSI, the more contribution of the sub-mechanism or reaction class on the prediction uncertainty. 2.2 Path sensitivity analysis In our previous study 27, it was found that some important reaction classes showed low NSI, while they are crucial paths in the detailed mechanism. If these reactions were removed, a fragmentary reduced mechanism would be obtained, in which the species and reactions cannot be well coupled 38. Consistently, Hantouche et al. 29 indicated that a reaction with low sensitivity coefficient might be important in the path analysis. To solve this problem, the path sensitivity analysis is introduced to identify the important reaction classes with low NSI in the present study. The path sensitivity coefficient (PSC) is calculated using Equation (4) |𝑄RC𝑖 ― 𝑄all| PSCC𝑖 = max (𝑄RC , 𝑄 𝑖

all

)

(4)

where 𝑄RC𝑖 represents the predicted value of the reduction target from the reduced mechanism without the ith reaction class; and 𝑄all denotes the predicted value of the reduction target from the full detailed mechanism. In Equation (4), max (𝑄RC𝑖, 𝑄all) is employed in order to ensure that the path sensitivity coefficient is within the range of 0 to 1. Because more than one reduction target is considered in this work, the final path sensitivity coefficient of the ith reaction class is determined as the maximum path sensitivity coefficient among the calculated PSCC𝑖 for different reduction targets.

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2.3 Spearman’s rank correlation coefficient For each reaction class in the detailed mechanism for large-molecule fuels, a mass of isomers exist. To determine the dominant isomers, the Spearman’s rank correlation coefficient (SCC) is introduced in this study. SCC represents the statistical correlation between the rankings of two variables 39. It represents the monotonic degree between two variables. The value of SCC is between -1 and 1. The SCCs of -1 and 1 represent the two variables are a monotonic correlation. The SCC coefficient of two variables X and Y with n raw scores is computed using Equation (5):

SCC =

COV(rg𝑋,rg𝑌) 𝜎rg𝑋𝜎rg𝑌

(5)

where rg𝑋 and rg𝑌 represent the ranks of variables X and Y, COV(rg𝑋,rg𝑌) represents the covariance of the rank variables, and 𝜎rg𝑋 and 𝜎rg𝑌 represent the standard deviations of the rank variables. In this research, variable X is the rate coefficient, and variable Y is the predicted value of the reduction targets, i.e., the ignition delay time and the concentrations of n-C7H16, C3H6, C2H2, and CO, as introduced in Section 2.5. 2.4 Optimization of the chemical mechanism using genetic algorithm Because rate constants have uncertainty, the optimization of the chemical mechanism is usually employed to improve its performance 40-44. Many automated methods were offered for chemical mechanism optimization 45. Among these methods, genetic algorithm (GA) shows promising performance 45, 46. In this research, the Non-dominated Sorting-based 8


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Genetic Algorithm II (NSGA-II) 47 was utilized for the refinement of the rate coefficients for the reduced mechanism. More details about the optimization can be found in Ref. 48. 2.5 Development of reduced mechanism Table 1. Reaction classes (RCs) in the C7 sub-mechanism 28

RC 1. RH pyrolysis

RC 2. RH+X=R+HX

RC 3. R pyrolysis

RC 4. R+O2= Q+HO2

RC 5. R=R’

RC 6. H-atom abstraction from olefin

RC 7. Alkenyl radical pyrolysis

RC 8. C7H13O pyrolysis

RC 9. Q pyrolysis

RC 10. R+O2=RO2

RC 11. R+RO2=RO+R′O

RC 12. R+HO2/CH3O2=RO+OH/CH3O

RC 13. RO2=Q+HO2

RC 14. RO2=QOOH

RC 15. RO2+HO2/H2O2=RO2H+O2/HO2

RC 16. RO2+CH3O2=>RO+CH3O+O2

RC 17. RO2+ R′O2=>RO+ R′O+O2

RC 18. RO2H=RO+OH

RC 19. RO decomposition

RC 20. Q+HO2=QOOH

RC 21. QOOH=>QO+OH

RC 22. QOOH decomposition

RC 23. O2QOOH=QOOH+O2;

RC 24. O2QOOH=C7ket+OH

RC 25. C7ket decomposition

RC 26. Reaction of QO with OH and HO2

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Figure 1. Flow diagram of the mechanism reduction procedure The potential of the method using the reaction class-based global sensitivity and path sensitivity analyses for mechanism reduction is evaluated by reducing a detailed n-heptane mechanism from Mehl et al. 28, which is a representative mechanism built using a systematic way. According to the research of Curran et al. 26, the reactions in the C7 sub-mechanism can be categorized into 26 reaction classes, as listed in Table 1. The mechanism reduction procedure is described in Figure 1. First, the importance of each sub-mechanism in a detailed mechanism on the target of mechanism reduction is analyzed using GSA. To cover broad running conditions during the mechanism reduction process, the relationship of the reduction target on the operating temperature, pressure, and equivalence ratio is investigated. Second, the important reaction classes in the C7 sub-mechanism are identified using global sensitivity and path sensitivity analyses, and a skeletal C7

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sub-mechanism is built. Third, a reduced C2−C6 sub-mechanism is constructed according to Spearman’s Rank Correlation Coefficient (SCC) and path sensitivity coefficient (PSC). Fourth, an initial reduced mechanism is built by coupling the skeletal C7, reduced C2−C6, and detailed C0−C1 sub-mechanisms. Finally, the final reduced mechanism is achieved by optimizing the rate coefficients of the initial reduced mechanism using genetic algorithm. In this study, the ignition delay times and the concentrations of n-C7H16, C3H6, C2H2, and CO in JSRs are employed as the reduction targets covering temperature (T) from 600 K to1500 K, pressure (p) from10 atm to 80 atm, and equivalence ratio (φ) from 0.3 to 1.5, which is related to the operating conditions in advanced low-temperature engines 49. As a representative olefin, C3H6 inhibits the reactivity of fuel and is an important intermediate species for formation of polycyclic aromatic hydrocarbons (PAHs) 50, thus it is included for the reduced mechanism construction. Moreover, C2H2 is included as a reduction target because of its significant role in the PAH and soot formations 51, 52. In the work, the ignition delay time is simulated using the zero-dimensional, constant-volume, and adiabatic reactor with the definition of the time that the temperature rises 400 K from the unreactive temperature. The species concentrations in JSRs are calculated based on a homogeneous model with constant pressure and temperature at a residence time of 0.04 s.

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Figure 2. Example of the temperature partitions for (a) ignition delay time and (b) CO concentration Although the large-molecule fuel shows distinctive oxidation behaviors under various temperatures, it is found that similar reactions dominate the fuel oxidation in a specific temperature zone 26, 53. Thus, the operating temperature is divided into four representative zones in this research according to the predictions from the detailed mechanism in Figure 2. The ignition delay time reduces as the temperature decreases at the NTC zone, while it increases with the decreased temperature at the other zones. At the high temperature (HIGH) zone, ignition delay time becomes shorter as the equivalence ratio increases, while an inverse trend is observed at the extremely high-temperature (EX-HIGH) zone. For the species concentrations in JSRs, the partition is performed based on the CO concentration profile, as shown in Figure 2(b). The CO concentration increases when the temperature increases at the low temperature (LOW) and HIGH zones, while the CO concentration decreases as temperature increases at other zones. In the next research, the investigation of the detailed mechanism is carried out using the global sensitivity and path analyses at a representative temperature in each zone. 12


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3. Results and discussion 3.1 Determination of operation conditions for mechanism reduction

Figure 3. Convergence of the Monte Carlo calculation for (a) ignition delay time and (b) CO concentration For the Monte Carlo calculation, it needs to judge the convergence of the calculations. In this study, for both the simulations of shock tubes and JSRs, 10,000 cases at a representative temperature in the NTC zone at p=10 atm and φ=1.0 were first calculated by perturbing the all A-factor from the detailed mechanism. As seen in Figure 3, a reasonable convergence is obtained with the case number of 5,000 for both ignition delay time and CO concentration. Thus, the case number is set as 5,000 in the subsequent Monte Carlo simulations.

Figure 4. NSIs of the Ci sub-mechanism for ignition delay time at different conditions. 13


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Figure 5. NSIs of the Ci sub-mechanism for CO concentration at different conditions. The NSIs of the C0−C7 sub-mechanism for ignition delay time and CO concentration over broad operating conditions are shown in Figures 4 and 5, respectively. It is found that the NSIs of the same sub-mechanism for ignition delay time and CO concentration are similar in the same temperature zone at different pressures and equivalence ratios. This indicates that the function of one sub-mechanism for the prediction uncertainty is similar in the same temperature zone and is nearly independent of pressure and equivalence ratio under the test conditions.

Figure 6. Spearman’s rank correlation coefficients (SCCs) of the dominant reactions for ignition delay time at different conditions. Furthermore, the SCCs of the dominant reactions in the C7 sub-mechanism for ignition delay

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time are calculated at different conditions. As mentioned above, SCC can be used to assess the dependency relationship between the rate coefficient and the target. In the present study, the smaller the absolute value of SCC, the less important the reaction is. The SCCs of the top 15 most crucial reactions for ignition delay time in the NTC temperature zone over broad pressures and equivalence ratio range are displayed in Figure 6. It can be found that the SCCs of a reaction are similar in the NTC zone and are insignificantly affected by pressure and equivalence ratio. Overall, from Figures 4−6, it can be concluded that consistent reactions and reaction classes govern the ignition delay time and species concentration in the same temperature zone over broad pressures and equivalence ratio range. In contrast, the NSI of the Ci sub-mechanism varies considerably with the temperature zone at the identical pressure and equivalence ratio, as shown in Figures 4(c) and 5(c). Thus, the mechanism reduction in the next research only focuses on the representative operating conditions of 600–1500 K at φ=1.0 and p=50 atm for ignition delay times in shock tubes, as well as φ=1.0 and p=10 atm for species concentrations in JSRs. 3.2 Development of reduced mechanism

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Figure 7. NSIs of the Ci sub-mechanism in different temperature zones for (a) n-C7H16, (b) C3H6, (c) C2H2, (d) CO, and (e) ignition delay time. For understanding the importance of the different sub-mechanisms in the n-heptane mechanism, Figure 7 illustrates the NSIs of the Ci sub-mechanisms for ignition delay time, n-C7H16, C3H6, C2H2, and CO concentrations in various temperature zones. At the LOW, NTC, and HIGH zones, the C7 sub-mechanism shows large NSI, while the NSIs of the C0 and C1 sub-mechanisms are relatively large in the EX-HIGH zone for ignition delay time and all species concentrations. It is noteworthy that the C3−C5 sub-mechanism also illustrates large NSI for C3H6 and C2H2 in the LOW zone in Figures 7(b) and 7(c). Because the concentrations of C3H6 and C2H2 are very low in the LOW zone, the C3H6 and C2H2 concentrations in the LOW and NTC zones are not considered as the reduction targets in the present study. Moreover, as displayed in Figure 7, if the species concentrations are considered as the target of mechanism reduction, the sub-mechanism involving the corresponding species also has large NSI. For example, the C3 sub-mechanism shows large 16


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NSI for C3H6. In general, the C0, C1, and C7 sub-mechanisms and the sub-mechanisms involving the species used as the reduction targets, i.e., the C2 sub-mechanism for C2H2 and the C3 sub-mechanism for C3H6, dominate the prediction uncertainty.

Figure 8. NSIs of 26 RCs in the C7 sub-mechanism for (a) n-C7H16, (b) C3H6, (c) C2H2, (d) CO, and (e) ignition delay time. As mentioned in Section 3.1, the reactions in the C7 sub-mechanism are separated into 26 reaction classes. Because of the huge scale of the C7 sub-mechanism, it needs to remove the unimportant reactions to acquire a compact reduced mechanism. The contribution of each reaction class in the C7 sub-mechanism on the prediction uncertainty is quantified by calculating the NSI of different reaction classes, as displayed in Figure 8. In this research, the different reaction classes are identified manually following the work of Curran et al. 26 As seen, the reaction classes with the most contributions on the prediction uncertainty of the reduction targets include RCs 1−3, 6, 13, 14, 20−22, 24−26, i.e., the high-temperature decomposition of fuel (RH) and fuel radical (R), the low-temperature reaction of alkylperoxy (RO2) and hydroperoxide (QOOH), and the carbonyl-hydroperoxide (C7ket) decomposition. 17


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Figure 9. PSCs of the 26 RCs in the C7 sub-mechanism for (a) C7H16, (b) C3H6, (c) C2H2, (d) CO, and (e) ignition delay time. In our previous study 27, it was found that, if a reduced mechanism was developed based only on NSI, the reaction paths could be incomplete. This is because the fact that some important reactions with low impact on the prediction uncertainty are excluded in the reduced mechanism, whereas these reactions are critical to couple the important species or reactions in the reduced mechanism 38. To identify these reactions, the path sensitivity coefficients (PSCs) of the 26 reaction classes in the C7 sub-mechanism are calculated in Figure 9. It can be found that the critical reaction classes include RCs 2, 6−8, 10, 13, 14, 17, and 23−25 in the C7 sub-mechanism. Comparing to the dominant reaction classes based on NSI, the reaction classes of R+O2=RO2 (RC 10), RO2+R’O2=>RO+R’O+O2 (RC 17), and QOOH+O2=O2QOOH (RC 23), and the alkene-related reaction (RCs 7 and 8) display high PSC, while their NSIs are low.

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Figure 10. Evolution of the accumulated error with the increased number of removed reaction class To guarantee the fidelity of the final reduced mechanism, the function of the reaction class is evaluated according to both PSC and NSI using Equation (6) (6)

𝜉𝑖 = max (PSCC𝑖, NSIC𝑖)

According to 𝜉𝑖, the reaction class is deleted from the detailed mechanism one after another. The evolution of the accumulated error caused by the deletion of the reaction class is shown in Figure 10. The accumulated error is calculated using Equation (7) as

Accumulated Error = max

(

|𝑄R𝑗 ― 𝑄D𝑗 |

R D 1 ≤ 𝑗 ≤ 𝑛 max (𝑄𝑗 , 𝑄𝑗 )

) × 100%

(7)

where 𝑄R𝑗 and 𝑄D𝑗 represent the calculated value of the jth reduction target with the reduced and detailed mechanisms, respectively. In the present study, the threshold of 0.2 is employed, which introduces an accumulated error of 20.9%. In the reduced mechanism, 17 reaction classes are retained, i.e., RCs 1−3, 6−8, 10, 13, 14, 17, and 20−26. Moreover, as indicated in Figures 7−9, obvious differences exist in the dominant sub-mechanisms and reaction classes 19


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with the reduction target of ignition delay time and species concentrations. As a consequence, the final reduced mechanism considerably relies on the reduction targets 5.

Figure 11. SCCs of the reactions in the C7 sub-mechanism for (a) n-C7H16, (b) C3H6, (c) C2H2, (d) CO, and (e) ignition delay time In each reaction class identified above, a large number of isomers still exist. Thereby, further reduction of the mechanism is performed by determining the important isomers in the C7 sub-mechanism. This process is realized by calculating the SCCs of all the reactions in the C7 sub-mechanism in Figure 11. As seen, the reactions containing the C7H15-2 and C7H15-3 radicals, as well as the related intermediates with the lowest bond energy (i.e., C7H15O2-2, C7H14OOH2-4, C7ket24, etc.), dominate the prediction uncertainty of ignition delay time and the concentrations of n-C7H16 and CO. For C3H6 and C2H2, the reactions involving the 20


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C7H15-1, C7H14OOH2-6, C7H14O2-6, and C7H133-6 radicals also have high SCC at the LOW and NTC zones, as displayed in Figures 11(b) and 11(c). Because the concentrations of C3H6 and C2H2 are extremely low at the LOW and NTC zones under the test operating conditions, the C3H6 and C2H2 concentrations at the LOW and NTC zones are not considered as the reduction targets in this study. Thus, only two isomers with the lowest bond are retained for the C7 species included in the dominant reaction classes. Furthermore, the reduced C2−C6 sub-mechanism is developed by calculating the SCC and PSC of each reaction in the C2−C6 sub-mechanism. From Figure 7, it can be seen that the C2−C6 sub-mechanism only slightly impacts the prediction uncertainty of the different reduction targets. Thus, a relatively large threshold value of 0.07 is set for the absolute value of SCC and PSC to select the important reactions in the C2−C6 sub-mechanism. In the work, the extremity of the mechanism reduction is determined when the predicted result of any reduction target using the reduced mechanism reaches the corresponding uncertainty boundary of the predicted results using the detailed mechanism. There are 34 species and 150 reactions in the C0−C1 sub-mechanism, and they all have high NSI for ignition delay time and species concentrations at the HIGH and EX-HIGH zones. Therefore, the whole C0−C1 sub-mechanism is maintained. By coupling the skeletal C7, reduced C2−C6, and detailed C0−C1 sub-mechanisms, the final reduced mechanism includes 89 species and 276 reactions. In the reduced chemical mechanism, most of the reactions are reversible reactions, and only the pyrolysis reactions from the C7 sub-mechanism are irreversible 21


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3.3 Evaluation of the initial reduced mechanism To compare the performance of the above methods on mechanism reduction, we built another reduced mechanism utilizing the DRGEPSA approach 11 with the reduction targets including the ignition delay times at p= 50 atm and φ=1.0 with error tolerance of 50% and the concentrations of n-C7H16, C3H6, C2H2, and CO at p=10 atm and φ=1.0 with error tolerance of 50% for n-C7H16 and CO, and 100% for C3H6 and C2H2. The reduction targets used in the DRGEPSA approach are consistent with those introduced in Section 3.2. The final reduced mechanism developed based on the DRGEPSA method includes 139 species and 543 reactions. Figure 12 compares the predicted results with the detailed mechanism (Detailed Mech.), the present initial reduced mechanism (Initial Reduced Mech.), and the reduced mechanism using DRGEPSA (DRGEPSA Reduced Mech.). Concerning ignition delay time, both the two reduced mechanisms display similar performance as the Detailed Mech. at the LOW, HIGH, and EX-HIGH zones. In the NTC zone, the predicted results using the Initial Reduced Mech. are slightly longer than that of the Detailed Mech., while the DRGEPSA Reduced Mech. shows shorter ignition delay time than the Detailed Mech. For the species profiles, compared to the DRGEPSA Reduced Mech., the predictions of the Initial Reduced Mech. agree better with that of the Detailed Mech. In general, the Initial Reduced Mech. can better reproduce the chemical kinetic behavior of the Detailed Mech. than the DRGEPSA Reduced Mech., although the scale of the Initial Reduced Mech. is smaller than that of the DRGEPSA Reduced Mech. 22


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Figure 12. Comparison of the predicted ignition delay times and major species concentrations using Detailed Mech. (symbols), Initial Reduced Mech. (dashed lines), and DRGEPSA Reduced Mech. (solid lines) To explain the different predictions of the two reduced mechanisms, the structures of the Initial Reduced Mech. and the DRGEPSA Reduced Mech. are compared in Figure 13. It can be found that obvious discrepancies exist in the structure of the two mechanisms. For the C7 sub-mechanism, only 18 species are retained in the Initial Reduced Mech., while 48 species are retained in the DRGEPSA Reduced Mech. In the DRGEPSA Reduced Mech., a large number of isomers are considered in the C7 sub-mechanism, such as 11 isomers being included for the QOOH radicals. Consistently, Luo et al. 10 indicated that the DRG method could not effectively remove the large groups of isomers. In the present study, it is found that the isomers with the lowest bond energy contribute the most to the prediction uncertainty. Thus, only two isomers of QOOH radicals in the dominant reaction classes are retained in the C7 sub-mechanism of the Initial Reduced Mech., which leads to the much more compact size of the Initial Reduced Mech.

23


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Figure 13. Comparison of the structures of Initial Reduced Mech. and DRGEPSA Reduced Mech. From the comparison of the C2−C6 sub-mechanism, it can be found that the scale of the C2−C6 sub-mechanism in the DRGEPSA Reduced Mech. is still larger than that of the Initial Reduced Mech. One of the reasons is that a mass of reactions involving the production of C2H2 and C3H6 from n-C7H16 are preserved in the DRGEPSA Reduced Mech. to reproduce the C2H2 and C3H6 evolutions, especially in the LOW and NTC temperature zones. Consistently, Wang and Gou 12 found that a mass of species should be included in the reduced mechanism to reproduce the low-temperature prediction behavior of the detailed mechanism. In contrast, the C2H2 and C3H6 concentrations in the LOW and NTC zones are not considered as the reduction targets in this research. Thus, the scale of the C2−C6 sub-mechanism in the Initial Reduced Mech. is smaller than that of the DRGEPSA Reduced Mech. For the DRGEPSA Reduced Mech., the C0−C1 sub-mechanism is also reduced, which results in worse performance than that of the Initial Reduced Mech., especially at the HIGH and EX-HIGH zones shown in Figure 12, because the C0−C1 sub-mechanism plays a significant role on the fuel oxidation 44 (see Figure 7). 24


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3.4 Mechanism optimization As seen in Figure 12, some discrepancies can be found from the calculated data between the Initial Reduced Mech. and the Detailed Mech. It is worth noting that there are some uncertainties in the rate coefficients, especially for the reactions involving the large-molecule species, which results in large prediction uncertainty of the mechanism 54. To improve the prediction performance, the Initial Reduced Mech. is optimized based on the genetic algorithm using the NSGA-II code introduced in Section 2.4. As indicated in Figure 7, the C7 sub-mechanism contributes the most to the prediction uncertainty. Thus, only the A-factors of the reactions in the reaction classes with large NSI in the C7 sub-mechanism are optimized within their uncertainties. The optimized reactions include the reactions in Classes 1−3, 6, 13, 14, 20−22, and 24−26, as listed in Table 1. More introduction about the optimization process can be found in the paper by Niu et al. 48 In the present study, two objective functions are considered, i.e., the ignition delay times (𝑓obj ― st) and species concentrations (𝑓obj ― jsr), as shown in Equations (8) and (9). 𝑁

|𝜏R𝑖 ― 𝜏D𝑖 |

𝑓obj ― st = ∑𝑖 = 1 𝑁

𝑀

𝜏D𝑖

|𝑋𝑖,𝑗 R ― 𝑋𝑖,𝑗 D|

𝑓obj ― jsr = ∑𝑖 = 1∑𝑗 = 1

𝑋𝑖,D 𝑗

(8) (9)

where R and D represent calculated results from the reduced and detailed mechanism, respectively. τi describes the predicted ignition delay time in the ith condition; 𝑋𝑖𝑗 denote the predicted concentration of species j under the ith condition; N is the sum of the test conditions, and M is the sum of species considered in the reduction targets. In each objective function, 25


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six operating conditions are employed as the optimized targets including four representative temperatures respectively at the LOW, NTC, HIGH, and EX-HIGH temperature zones, and two inflection temperatures between the LOW and NTC zones, and the NTC and HIGH zones at p=50 atm and φ=1.0 for ignition delay times and p=10 atm and φ=1.0 for species concentrations.

Figure 14. Developments of the objective functions for the mechanism optimization The value of the objective function of ignition delay time versus the value of the objective function of species concentration in JSRs at different generations during the evolution process is shown in Figure 14. As can be observed, both objective functions decrease with the increase of generation, which means that the performances of the reduced mechanism are improved continuously. Comparing the objective functions in the generations of 1000 and 1500, only slightly discrepancy exists. Thus, it can be concluded that convergent solutions are obtained in the generation of 1500. In the present study, 52 optimal reduced mechanisms are obtained through the genetic algorithm. The final optimized mechanism is determined according to the discrepancy between the results using the detailed and optimal reduced

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mechanisms for all the ignition delay times and species concentrations through Equation (10).

𝑁

(

𝛷 = ∑𝑖 = 1

𝜂R𝑖 ― 𝜂D𝑖 2 IDRD𝑖

)

(10)

where 𝜂R𝑖 and 𝜂D𝑖 are respectively the predicted value from the reduced and detailed mechanism at the ith condition; IDRD𝑖 represents the interdecile range of the detailed mechanism at the ith condition. The optimized mechanism with the minimum Φ is chosen as the final mechanism. 3.5 Mechanism validations Figure 15 compares the calculated results using the optimized reduced mechanism (Optimized Reduced Mech.) and the detailed mechanism (Detailed Mech.). It can be observed that the Optimized Reduced Mech. satisfactorily reproduces the predictions of the Detailed Mech. for both ignition delay times and species concentrations over low-to-high temperatures. Especially, the final products including CO2 and H2O are also rightly predicted by the Optimized Reduced Mech. due to the detailed C0−C1 sub-mechanism being used (see Figure 15(b)). It can be also found that the DRGEPSA Reduced Mech. shows relatively worse performance at NTC and HIGH zones. In the NTC and HIGH zones, the C0 and C1 sub-mechanisms show high NSI in Figure 7. The reduction on the C0−C1 sub-mechanism is responsible for the low performance of the DRGEPSA Reduced Mech. in the NTC and HIGH zones. Furthermore, the predictions on the concentrations of HO2, OH, H, and O radicals are assessed in Figure 15(c), though the concentrations of these free radicals are not employed as the reduction targets. The calculated results using the detailed mechanism agree well with 27


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that of two reduced mechanisms. Only there are some inconsistencies in the computed results on the HO2 radical between the Detailed Mech. and the DRGEPSA Reduced Mech.

Figure 15. Comparison of the predicted results using Detailed Mech. (symbols), Optimized Reduced Mech. (solid lines), and DRGEPSA Reduced Mech. (dashed lines) for (a) ignition delay times, (b) and (c) species concentrations Further validations are performed by examining the computed ignition delay times and species concentrations with the Optimized Reduced Mech. with that of the Detailed Mech. covering T=550−1500 K, p=10−80 atm, and φ=0.3−1.5, as shown in Figures 16 and 17. Overall, good agreements on the predicted results between the two mechanisms are obtained covering the whole operating conditions. The increase of ignition delay time with the reduction of equivalence ratio and pressure, especially in the NTC zone, is well reproduced by the Optimized Reduced Mech. in Figure 16. The Optimized Reduced Mech. also satisfactorily captures the variations of the species concentrations investigated in this study with equivalence ratio and pressure, as displayed in Figure 17. In the research, only one pressure and one equivalence ratio are included for the mechanism reduction and optimization. The good agreements on the predictions between the Optimized Reduced Mech. and the Detailed Mech. indicate that consistent reactions dominate the predicted result 28


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of the Detailed Mech. at a specific temperature over broad pressure and equivalence ratio ranges.

Figure 16. Comparison of the computed ignition delay times from Detailed Mech. (symbols) and Optimized Reduced Mech. (lines) at (a) p=50 atm and (b) φ=1.0

Figure 17. Comparison of predicted species concentrations using Detailed Mech. (symbols) and Optimized Reduced Mech. (lines) over different pressures and equivalence ratios 4. Conclusions A novel mechanism reduction method is introduced by using the reaction class-based global 29


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sensitivity and path analyses in this research. The importance of the sub-mechanisms for reduction targets under broad pressure, temperature, and equivalence ratio conditions is first investigated using the GSA. According to the characteristics of fuel oxidation, the operating temperature is divided into four zones. The results indicate that the function of a sub-mechanism on the prediction uncertainty is consistent at the same temperature zone over broad pressure and equivalence ratio. Thus, the mechanism reduction in the present study focuses only on the operating conditions of different temperature zones at a representative pressure and equivalence ratio. By calculating the contribution of the reaction classes on the prediction uncertainty of the reduction targets, the dominant reactions in different temperature zones are determined. Moreover, the path analysis is conducted to detect the significant reaction paths with low influence on the prediction uncertainty. Then, the important isomers are identified using Spearman’s rank correlation coefficient in each reaction class of the large-molecule sub-mechanism, which is capable of dramatically reducing the number of the species in the large-molecule sub-mechanism. After establishment of the initial reduced mechanism, the rate coefficients of the reactions owing to high contributions to the prediction uncertainty from the fuel-related sub-mechanism are refined within their uncertainties to improve the nominal predicted value of the reduced mechanism. Using the above methods with the reduction targets of the ignition delay times and the concentrations of n-C7H16, C3H6, C2H2, and CO, a reduced n-heptane mechanism containing 89 species and 276 reactions is constructed by reducing a detailed one with 645 species and 30


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2827 reactions. Comparisons the calculated results between the reduced and detailed mechanisms show that the final reduced mechanism well reserves the prediction behavior of the detailed one on the reduction target over broad operating conditions. Supporting Information The following files are available free of charge. The mechanism in CHEMKIN format (chem.txt) The thermodynamics data (therm.txt)

Corresponding author: Ming Jia Address: Key Laboratory of Ocean Energy Utilization and Energy Conversion of Ministry of Education Dalian University of Technology Dalian, 116024 P.R. China Tel: +86-411-84706722 Fax: +86-411-84706722 31


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Email: [email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding Sources This study is supported by the National Science Foundation of China (Grant Nos. 51706033, 51961135105, and 91641117).

Acknowledge This study is supported by the National Science Foundation of China (Grant Nos. 51706033, 51961135105, and 91641117). References 1. Puduppakkam, K. V.; Liang, L.; Naik, C. V.; Meeks, E.; Kokjohn, S. L.; Reitz, R. D., Use of detailed kinetics and advanced chemistry-solution techniques in CFD to investigate dual-fuel engine concepts. SAE Int. J. Eng. 2011, 4, 1127-1149, Doi: 10.4271/2011-01-0895. 2. Zhang, S.; Broadbelt, L. J.; Androulakis, I. P.; Ierapetritou, M. G., Comparison of biodiesel performance based on hcci engine simulation using detailed mechanism with on-the-fly reduction. Energy Fuels 2012, 26, 976-983, Doi: 10.1021/ef2019512. 3. Lu, T.; Law, C. K., Toward accommodating realistic fuel chemistry in large-scale 32


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Reduction of detailed chemical mechanism using reaction class

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