An efficient approach for the optimization of skeletal chemical

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An efficient approach for the optimization of skeletal chemical mechanisms with multi-objective genetic algorithm Bo Niu, Ming Jia, Guangfu Xu, Yachao Chang, and Mao-Zhao Xie Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b00356 • Publication Date (Web): 30 Apr 2018 Downloaded from http://pubs.acs.org on May 2, 2018

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An efficient approach for the optimization of skeletal chemical mechanisms with multi-objective genetic algorithm Bo Niu,† Ming Jia,*,† Guangfu Xu,† Yachao Chang,† and Maozhao Xie† †Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, Dalian University of Technology, PR China

Abstract For low-temperature combustion (LTC) engines, the ignition and combustion processes are dominantly controlled by the chemical kinetics of fuels. In order to simulate the working process of LTC engines using the multi-dimensional model, the skeletal or reduced oxidation mechanisms of fuels are urgently required. In this study, a new approach was proposed for the construction and optimization of the skeletal chemical mechanism of primary reference fuel (PRF) by coupling the decoupling methodology with the multi-objective genetic algorithm (GA). An initial skeletal chemical mechanism for PRF was first proposed according to the decoupling methodology. Then, the non-dominated sorting genetic algorithm II (NSGA-II) code was employed to optimize the rate constants of the large-molecule reactions in the skeletal PRF mechanism. In NSGA-II, two objective functions based on the experimental data including ignition delay times in shock tubes (ST) and major species profiles in jet-stirred reactors (JSR) were introduced. Compared with the initial mechanism, the performance of the optimized mechanism is improved considerably. Furthermore, the optimized mechanism was used to predict the laminar flame speed, and the major species evolution in premixed laminar flames, as well as the in-cylinder pressure, heat release rate, and emissions in a homogeneous charge compression ignition (HCCI) engine. Good agreements between the predicted and measured results indicate that the integration of the decoupling methodology with genetic algorithm is an efficient approach for the construction of skeletal mechanisms. Furthermore, the influences of the selection and weight factor of the experimental data for mechanism optimization were discussed. The results indicate that the available experimental data on the ignition delay times under extensive operating conditions should be considered in the optimization process. Heavier weight factor should be assigned to the ignition delay times in the negative temperature coefficient (NTC) and low-temperature zones to improve the performance of the optimized mechanism.

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1 Introduction Primary reference fuel (PRF) composed of n-heptane and iso-octane has been widely treated as the surrogate fuel for practical diesel and gasoline due to the similar ignition and combustion characteristics1. For low-temperature combustion (LTC) engines, the ignition and combustion processes are dominated by the chemical kinetics of fuels. Therefore, a robust chemical reaction mechanism of PRF is urgent for the numerical simulation of LTC engines. Although a detailed mechanism, consisting of hundreds of species and thousands of reactions, can provide accurate kinetics information about the pyrolysis and oxidation of the fuel, its size is too huge to be applied in the multi-dimensional computational fluid dynamics (CFD) simulation of engines. Therefore, the simplification of the detailed mechanism is necessary. By removing the unimportant species and reactions in a detailed mechanism, the scale of the mechanism can be effectively reduced with little or even no accuracy loss. Many methods have been proposed for the reduction of detailed mechanisms, including sensitivity analysis2, 3, directed relation graph (DRG)4-6 and the methods based on time scales analysis, such as quasi-steady state approximation (QSSA)7, computational singular perturbation (CSP)8 and intrinsic low-dimensional manifolds (ILDM)9. Recently Liu et al.10 and Chang et al.11 proposed the decoupling methodology for constructing the skeletal mechanisms of heavy hydrocarbons, and a series of skeletal mechanisms for various fuels were developed. In the decoupling methodology, a skeletal mechanism is divided into three parts, i.e., a detailed H2/CO/C1 sub-mechanism, an extremely simplified skeletal sub-mechanism for large-molecule species and a reduced C2–C3 sub-mechanism. Compared with the other methods of mechanism reduction4-9, the decoupling method10, 11 has two advantages. First, for the previous methods of mechanism reduction4-9, a global reduction of the detailed mechanism is utilized by removing the unimportant species and reactions under specified operating conditions. As a result, the reduced mechanism strongly depends on the reduction targets. Mehl et al.12 indicated that the reduced mechanism obtained based on the auto-ignition characteristics alone cannot give accurate predictions of laminar flame speeds. In the decoupling method, a detailed H2/CO/C1 sub-mechanism is used as the core in the skeletal mechanism to describe the oxidation process after ignition and the flame propagation characteristics. Second, the size of the reduced mechanisms constructed by the previous methods4-9 could be relatively large for engine CFD simulations because a relatively large amount of species and reactions involving the large-molecule oxidation are still reserved in the reduced mechanisms5, 13. For the decoupling method, the large-molecule sub-mechanism can be constructed compactly, which is suitable for CFD modeling in practical engines. However, it must be noted that the 2


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decoupling method still involves some degree of empiricism and user input during the mechanism construction process. Because the reaction pathways are significantly simplified in the reduced or skeletal mechanisms, the optimization of the reaction rate constants is usually necessary in order to match the experimental data under wide operating conditions14. Moreover, the uncertainty exists in the rate coefficient for almost all kinetic models15, thereby further refinement of the reaction rate constants is required. Ra and Reitz16 pointed out that the optimization of a reduced mechanism was essential to reproduce the ignition delay characteristics satisfactorily. Wang et al.17 specially emphasized the necessity of the adjustment of the reaction rate constants to better match the measurements of ignition delay times. After developing a reduced chemical kinetic mechanism for the gasoline surrogate fuel based on the sensitivity analysis and rate of production methods, Lee et al.18 tuned the rate coefficients of the important reactions to match the experimental data. Recently, Pei et al.13 constructed a skeletal multicomponent mechanism for n-dodecane and m-xylene, with 163 species and 887 reactions, from the detailed mechanism, consisting of 2,885 species and 11,754 reactions. The rate coefficients of several important reactions in the skeletal mechanism were modified to further improve the prediction performance. In the decoupling methodology, by utilizing sensitivity analysis, path analysis, and rate of production method, Chang et al.19 manually adjusted the reaction rate constants of the large-molecule species until satisfactory agreements between simulation results and experimental data were achieved. The methods for the optimization of the reduced mechanism mentioned above are empirical and time-consuming. Therefore, an efficient and automatic approach for the mechanism optimization is in urgent need. Several methods have been proposed for the optimization of chemical mechanisms to give the best fit between the measured and calculated data. Recently, Wang and Sheen20 applied the method of uncertainty quantification to the mechanism optimization through minimization of the deviation of simulated results from experimental observations. Because the objective functions for mechanism optimization usually have complex and highly structured landscapes21, traditional ways such as gradient-based algorithms22 and the solution mapping method23 usually failed in the complicated non-linear optimization of chemical mechanisms, while genetic algorithm (GA) demonstrates prominent advantages. The GA was introduced first by Polifke et al.24 for the optimization of relatively simple two- or three-step mechanisms. Harris et al.25 employed GA to search the optimum rate constants for a H2 oxidation reaction mechanism by matching the net species production data in a perfectly stirred reactor (PSR) under various conditions. Afterwards, Elliott et al.21 constructed an improved 3


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objective function including the species concentrations in PSRs and laminar premixed flames for GA to enhance the performance of the H2 mechanism proposed by Harris et al.25 Furthermore, with the same method, Elliott et al.26 developed an optimized mechanism for aviation-fuel/air combustion, which was validated with the experimental data of PSRs and laminar premixed flames. In order to improve the performance of the fitness function in GA for mechanism optimization, Wade et al.27 coupled the GA with a simplex algorithm to optimize a pseudo-detailed thermal oxidation mechanism of jet fuel. Recently, Sikalo et al.28 introduced a penalty term into the GA to constraint the change range of the reaction coefficients in the optimization progress for the optimization of an extremely reduced methane mechanism, a reduced tert-butanol mechanism, and a hydrogen mechanism. Perini29 proposed an approach for the realization of the automatic reduction and optimization of the detailed chemical mechanism, in which the element-flux analysis was used to reduce the detailed mechanism and then the GA was employed to optimize the reduced mechanism. Furthermore, focusing on the homogeneous charge compression ignition (HCCI) combustion, Neshat et al.30 developed an optimized PRF mechanism by integrating the multi-zone model of HCCI with the GA. It should be noted that all the above mechanism optimization approaches using the GA belong to the single-objective optimization problem by minimizing or maximizing the value of one specified objective function. The single-objective optimization of the chemical mechanism is based on the assumption that the constructed objective function considering the different experiments under wide operating conditions is harmonious. However, there are usually uncertainties in the experimental data, and even errors exist21. As indicated by Elliott31, if the experimental data for the mechanism optimization are incompatible, the sub-items representing different experimental data in the objective function would conflict with each other, resulting in a defective-optimized reaction mechanism. Therefore, the muli-objective GA should be considered for the mechanism optimization. Recently, Lapene et al.32 applied the multi-objective GA to optimize the reaction rate parameters of the oxidation mechanism of heavy oil. Two objective functions including the O2 and CO2 concentrations were considered in the optimization process, and a set of optimized mechanisms were gained. However, only considering the product concentrations in the mechanism optimization is not sufficient29, 33. The purpose of this study is to provide an automatic and efficient method for the construction and optimization of skeletal mechanisms based on the decoupling methodology10,

19

. By intergrading the

multi-objective GA using the non-dominated sorting genetic algorithm II (NSGA-II) code34 and the decoupling methodology, a skeletal PRF mechanism is developed. There are four major parts in the following study. First, the 4


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approach for the mechanism development and optimization is introduced, and the objective functions in GA are constructed. Then, a set of optimized mechanisms are obtained and three representative optimized mechanisms are investigated in detail. Meanwhile, the influences of the selection and weight factor of the experimental data on the optimized results are discussed. Next, the final optimal mechanism is extensively validated by the experimental data including laminar flame speeds, major species profiles in premixed laminar fames, as well as the in-cylinder pressures, heat release rates and emissions in a HCCI engine. Finally, major conclusions are drawn.

2 Computational Approach 2.1 Mechanism construction The decoupling methodology has been successfully applied for the construction of the skeletal mechanisms for various fuels11, 19, 35. Due to the reliable performance and compact size of the mechanism constructed using the decoupling methodology, it is introduced in this study for the development of the skeletal PRF mechanism. In the decoupling methodology, a detailed H2/CO/C1 sub-mechanism36,

37

is introduced to provide the accurate

predictions of the flame propagation and extinction behaviors, the evolutions of small species, and the major heat release process11. An extremely simplified C4–C8 sub-mechanism including 30 reactions is adopted for the predictions of ignition delay times and the concentrations of the large-molecule species for PRF. As listed in Table 1, the C4–C8 sub-mechanism is mainly composed of the reactions of H-abstraction from alkanes by radicals of H, OH, HO2, and O2 producing alkyl radicals (R2–R5 and R13–R16), the addition of O2 with alkyl radicals (R6 and R18), the isomerization of alkylperoxy radical producing alkylhydroperoxy radicals (R7 and R19), the addition of O2 with proxy alkylhydroperoxy radicals (R8 and R20), the formation of ketohydroperoxide (R9 and R21), β-decomposition of alkyl radicals (R1, R10 and R22),and a set of decomposition reactions. As transitional reactions between the detailed H2/CO/C1 sub-mechanism and the skeletal C4–C8 sub-mechanism, a reduced C2–C3 sub-mechanism is utilized. The final PRF skeletal model contains 49 species and 163 reactions. In the skeletal PRF mechanism, the reaction rate constants of the detailed H2/CO/C1 sub-mechanism and the reduced C2–C3 sub-mechanism are directly taken from Refs. [36-38]. The reaction rate constants of the C4–C8 sub-mechanism (i.e., Reactions R1–R30 in Table 1) in the PRF mechanism are first estimated based on the detailed mechanism39. Because the large-molecule reaction paths are extremely reduced, it is necessary to optimize the reaction rate constants of these reactions. It has been widely accepted that the ignition behavior and the large-molecule species evolutions are very sensitive to the fuel-related reactions40-42. Thus, in the previous 5


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studies using the decoupling methodology43, the reaction rate constants of the fuel-related sub-mechanism is optimized by manual iteration until the predictions agree with the measurements of the ignition delays in shock tubes (ST) and the major species concentrations in jet-stirred reactors (JSR), which is time-consuming and strongly depends on the experience of the operator.

Table 1. Reactions in the C4–C8 sub-mechanism for optimization Reaction number

Reaction number

Reaction

Reaction

R1

C7H16=>C2H5+C2H4+C3H7

R16

C8H18+O2C8H17+HO2

R2

C7H16+HC7H15+H2

R17

C8H17iC4H8+iC4H9

R3

C7H16+OHC7H15+H2O

R18

C8H17+O2C8H17O2

R4

C7H16+HO2C7H15+H2O2

R19

C8H17O2C8H16OOH

R5

C7H16+O2C7H15+HO2

R20

C8H16OOH+O2O2C8H16OO

R6

C7H15+O2C7H15O2

R21

O2C8H16OOHC8KET+OH

R7

C7H15O2C7H14OOH

R22

C8KETCH2O+C6H13CO+OH

R8

C7H14OOH+O2O2C7H14OO

R23

C6H13CO+O2=>C3H7+C3H5+CO

R9

O2C7H14OOHC7KET+OH

R24

C8H17+O2C8H16+HO2

R10

C7KET=>C5H11CO+OH+CH2O

R25

C8H16+OH=>iC4H8+iC4H7+H2O

R11

C5H11CO+O2=>C3H7+C2H3+CO

R26

C8H16=>iC4H9+iC4H7

R12

C7H15=>C3H6+C2H5+C2H4

R27

iC4H8+HiC4H7+H2

R13

C8H18+HC8H17+H2

R28

iC4H8+OHiC4H7+H2O

R14

C8H18+OHC8H17+H2O

R29

iC4H9C3H6+CH3

R15

C8H18+HO2C8H17+H2O2

R30

iC4H7C3H4+CH3

2.2 Mechanism optimization In the present study, a multi-objective genetic algorithm, NSGA-II, is integrated with the decoupling methodology to realize the automatic and efficient optimization of the reaction rate constants of the PRF mechanism constructed above. As mentioned in Section 2.1, the rate constants of the fuel-related reactions (i.e., Reactions R1–R30 in Table 1) in the PRF mechanism are employed as the variables to be optimized.

2.2.1 Optimization process 6


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A three-parameter Arrhenius form is used to describe the rate of each reaction as = −

(1)

for i=1,2,…,N, where N is the number of reactions and R is the universal gas constant. The three parameters, Ai, bi and Eai, represent the pre-exponential factor, the temperature exponent and the activation energy, respectively. As indicated in the previous study33, if an optimized chemical mechanism is achieved based on only one category of experimental data, e.g., ignition delay times or major species concentrations, the predictions using the optimized mechanism usually cannot well agree with the other category of measurements. Therefore, wide sources of experimental data should be considered in the optimization of the chemical mechanism in order to achieve a satisfactory mechanism for the application in wide operating conditions. Although a single-objective function can be constructed by including all the experimental information into a mathematical expression, the relationships among the different objectives are hidden44, and it works worse when a large noise exists between the objectives33. To solve the problem, a multi-objective GA should be considered. For the decoupling methodology used in this study, only the measurements of the ignition delay times in STs and the major species in JSRs are introduced for the optimization of the developed mechanism19. This is primarily because that the ignition delay in STs and the major species concentrations in JSRs are very sensitive to the optimized reaction rate constants of the large-molecule reactions in the skeletal mechanism constructed by the decoupling methodology. On the other hand, Chang et al.45 indicated that if a mechanism can predict the ignition delay times accurately in shock tubes, the ignition delay of rapid compression machine (RCM) can also be well reproduced. Meanwhile, when the predicted concentrations of major species match well with the measured data in jet-stirred reactors, the corresponding species concentrations can also be satisfactorily captured in flow reactors, premixed flames and counter-flow flames. In this study, the minimizations of two-objective functions for the predictions of the chemical mechanism in STs (fST(A)) and JSRs (fJSR(A)) are introduced in the optimization process. To assess the accuracy of a given reaction mechanism in predicting ignition delay times in STs, Eq. (2) is employed. %& = ∑'(

$

!"

!"

#

(2)

where τjexp and τjsim are respectively the measured and predicted ignition delay times under the jth operating condition; NST is the total number of the operating conditions of ST tested in the optimization process; A represents the set of the pre-exponential factors of the reactions listed in Table 1, i.e., (A1,A2,…,AN); wj is the 7


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$%& weight factor of the jth operating condition, and ∑'( = 1.

Consistently, Eq. (3) is employed as a criterion to evaluate the precision of the chemical mechanism in predicting the major species concentrations in JSRs. * =

+∑$ 3

/

0 1/ , !"

,-,,/ -,,/ 1 !" 0 ∑/ / ,-,,/

∑ $ ∑2 " /

(3)

where 43,2,5 represents the measured mole concentration of the sth species at the tth temperature condition 678

29 from the rth set of JSR experiment, and 43,2,5 is the corresponding simulated result using the chemical

mechanism. For the JSR data used for the optimization of the mechanism, Nset sets of experimental data are utilized, including the concentrations of Nspe species under Ntem temperature conditions. In this study, the mole fractions of the reactants of O2 and fuels, as well as the products of CO, CO2, and H2 are considered. Moreover, the relatively uniform distribution of temperatures in JSR experiments makes Eq. (3) constructed without considering the temperature weighing. The CHEMKIN II code46 is used for the calculations of the ignition timings in STs and the species concentrations in JSRs using the PRF mechanism. By integrating the CHEMKIN II code with the multi-objective genetic algorithm, NSGA-II, an approach for the automatic optimization of the chemical kinetic mechanism is realized. The flow diagram of the optimization is illustrated in Figure 1, and the optimization steps are described as follows in detail. Step 1: According to the initial mechanism developed in Section 2.1, the initial PRF mechanisms are generated randomly with the pre-exponential factors of the fuel-related reactions (i.e., Reactions R1–R30) in the pre-defined boundaries from Ai,0/16 to Ai,0×16, where Ai,0 is the original rate constant of the ith reaction. Step 2: The ignition delay times in STs and the major species concentrations in JSRs are calculated for the cases used for mechanism optimization using the CHEMKIN II code. Step 3: The values of the objective functions are obtained through Eqs. (2) and (3) for STs and JSRs, respectively. Step 4: The next-generation chemical mechanisms are generated through the operations of selection, crossover, and mutation using NSGA-II based on the above calculated results. Step 5: The ignition delay times in STs and species concentrations in JSRs are predicted using the new chemical mechanisms generated in Step 4 by the CHEMKIN II code. 8


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Step 6: Steps 3–5 are repeated until the convergent criteria are achieved.

2.2.2 Multi-objective optimization For realization of the multi-objective optimization of the chemical mechanism, NSGA-II proposed by Deb et al.47 was employed in this study, which has been widely applied in various engineering applications because of its high computational efficiency48. For understanding the results discussed in Section 3, a brief introduction of NSGA-II is presented below, and more details can be found in the work of Deb et al.49 The target of NSGA-II is to find a series of Pareto optimal solutions, namely the Pareto Front, which has better objectives than the other solutions50. The Pareto Front is a set of un-dominated solutions. The non-dominated solution is defined as follows: for a non-dominated solution i and an arbitrary solution j, there is at least one objective k, where ki ≤ kj. As shown in Figure 2 for a two-objective optimization, the solutions A, B, C, and D are the Pareto Front solutions and the other ones are the dominated solutions. The Pareto Front solutions are un-dominated with each other and share a trade-off relationship. As a result, compared with one Pareto Front solution, any other Pareto Front solution with a better performance in one objective must have a simultaneous deterioration in the other objective(s). The widespread application of NSGA-II is benefited from its good convergence towards the Pareto Front and the acquisition of multifarious solutions by introducing the crowding tournament selection to the selection operator. The selection sequence of the solutions depends on their rank and crowding distance. In NSGA-II, every solution is ranked based on the number of its dominated solutions. The more cases a solution dominates, the lower rank it will be assigned, as shown in Figure 3. The crowding distance of solution i is the average length of the dashed cuboid illustrated in Figure 3. Solution i can defeat solution j in a selection tournament when it has one of the following priorities. First, solution i has a lower rank than solution j (ridj) when they share the same rank (ri=rj), which can avoid similar solutions and keep the diversity of the solutions. After the selection operation, pairs of solutions are chosen as the parent ones to cross over under a probability, pc, producing the possible better solutions as the offspring ones. For the real-coded NSGA-II, a simulated binary crossover (SBX) operator

49

was used in the study, in which the crossing over is based on the linear combination

of the optimized parameter in two different solutions. For example, for the ith optimized parameter in two parent solutions (e.g., xi and yi), the corresponding parameter through crossing over (′ and ;′ ) can be calculated by 9


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=

′ = 0.5A1 + C + 1 − C ; D ;′ = 0.5A1 − C + 1 + C ; D

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(4)

where αi is a function about the crossover distribution index ηc, and the function of αi=f(ηc) is described in Ref. [49] in detail. It should be mentioned that a smaller ηc makes ′ and ;′ produced farther away from xi and yi, while a larger ηc allows ′ and ;′ more near xi and yi.

As a follow-up operation, a mutation operator is used for increasing the diversity of the population by perturbing the optimized parameters in solutions under a probability, pm. In NSGA-II, a parameter-based mutation operator49 is used to perturb a parameter, xi, in one solution. After mutation, the regenerated value, zi, is determined by E = + F GH − I J

(5)

where xiu and xil represent the upper and lower constraints of the ith parameter (xi), respectively; δi is a function of a mutation distribution index, ηm, and the introduction of the function of δi=g(ηc) can be found in Ref. [49]. It should be noted that a larger value of ηm leads to a larger change of zi compared with xi. By introducing ηc and ηm, the spread of offspring solutions around parent solutions can be controlled effectively. After sequential operations of tournament crowding selection, crossing over, and mutation, the offspring solutions with good performance and diverse spread can be produced.

3 Results and discussion For understanding the factors affecting the multi-objective optimization of the skeletal PRF mechanism, the determination of the parameters in NSGA-II, the selection of experimental data for optimization, and the influence of the weight factor on the optimization results are discussed in this section. In the following study, if there is no special statement, the default parameters used in NSGA-II are as follows: the maximum generation is 2000, the size of individuals is 52, crossover probability (pc) is 0.9, crossover distribution index (ηc) is 15, mutation probability (pm) is 0.1, and mutation distribution index (ηm) is 20. And all the available experimental data are used and the weight factor of 2:1.5:1 for the low-temperature, NTC, and high-temperature zone is employed for optimization, if there is no special explanation.

3.1 Influence of the parameters of NSGA-II on optimization

10


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Table 2. Important parameters of NSGA-II Parameter value Parameters

Reference range case①

case②

case③

case④

case⑤ ⑤

case⑥

Crossover probability, pc

(0–1.0)

0.9

0.7

0.9

0.9

0.9

0.9

Mutation probability, pm

(0–1.0)

0.1

0.1

0.033

0.2

0.1

0.1

Crossover distribution index, ηc

(5–20)

10

10

10

10

15

10

Mutation distribution index, ηm

(5–50)

20

20

20

20

20

40

The parameters used in NSGA-II significantly affect the optimized results and convergent speed51. To study the influence of the parameters of NSGA-II on the chemical mechanism optimization, four vital parameters including crossover probability (pc), crossover distribution index (ηc), mutation probability (pm), and mutation distribution index (ηm) are investigated herein by designing six representative cases, as shown in Table 2. The variation ranges of the test parameters are determined according to the work of Deb34. The size of individuals is 52 for each population, and it is found that further increase of the individual size does not significantly affect the optimization results. The cases with different GA parameters are assessed through two criteria, i.e., the convergent speed and the performance of the predictions from the optimized mechanism. The comparisons of the two objective functions among the six cases in four typical generations, i.e., the 50th, 400th, 700th and 1000th generations, are shown in Figure 4. From the perspective of the convergent speed, it can be seen that Cases 2 and 5 illustrate the fastest convergent speed, while Case 6 converges the most slowly. In terms of the optimized results, the predictions of Cases of 2, 3, 4, and 5 are much better than those of Case 6. Moreover, it can be seen from Figure 4 that Case 5 converges slightly faster than that of Case 2 before the 400th generation. This is primarily because the fact that Case 5 has a larger crossover and mutation probability with pc=0.9 and pm=0.1, providing a higher possibility to create new offspring. Meanwhile, a larger ηc and a smaller ηm used in Case 5 than Case 2 make the offspring produced or altered within relatively tighter bounds near their parents. However, a rather large pm, such as pm=0.2 in Case 4 results in the very slow convergent speed because a large amount of outstanding genes in parents are changed. Thereby, the genetic operators and parameters from Case 5 listed in Table 2 are chosen for the mechanism optimization in the following study. The convergence of NSGA-II with the parameters from Case 5 is summarized in Figure 5, represented by the evolution of the two objective functions which are built based on the experimental data in ST and JSR. It can be 11


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seen that the history of the convergence can be divided into three stages. At the first stage (from the 1st generation to the 500th generation), the algorithm converges very quickly and the fitness in the 500th generation is greatly improved. The convergent speed becomes slow at the second stage (from the 500th generation to the 1000th generation), but the fitness of individuals still improves. At the end of the second stage, the individuals reach the desired Pareto Front. At the last stage of the 2000th generation, the population starts to spread along the Pareto Front, which corresponds to the developments of the extreme, but no more improvement of the fitness can be achieved. However, much computational time is required at the final stage to get a diversified population, which is not necessary in this study. In this study, only the individuals with the simultaneous minimization of the two objective functions of fST(A) and fJSR(A) are desired. As shown in Figure 5, in the 2000th generation, the values of fJSR(A) are closed to each other, which are limited to 0.923~1.133. But there are apparent discrepancies for the values of fST(A) of the individuals in the 2000th generation, ranging from 0.129 to 1.135. This is primarily because the fact that the ignition delay times of STs are more sensitive to the optimized large-molecule reactions compared with the major species in JSRs.

3.2 Selection of experimental data The oxidations of n-heptane, iso-octane, and PRF in STs and JSRs have been conducted extensively by experiment under wide operating conditions in the past decades. Because most of the operating conditions in the experimental data concentrated on the fuel/air mixture with stoichiometric ratio, the optimization of the PRF mechanism mainly focused on the experimental conditions with a stoichiometric ratio in previous studies16, 52. Because of a large quantity of available experimental data and the non-uniform distribution of the operating conditions, it is necessary to clarify the influence of the selection of experimental data on the mechanism optimization. As mentioned above, a great deal of experimental data for PRF oxidation in STs and JSRs are available. Fieweger et al.53 measured the ignition delay times of various PRFs in a shock tube under the temperatures (T) from 700 to 1190 K at a high pressure (p) of 40 atm, which have been widely used for the validation of the PRF mechanisms later. Shen et al.54, 55 experimentally investigated the ignition characteristics of the lean n-heptane/air mixture at p=12.5–45 atm and T=786–1396 K, as well as iso-octane with φ=0.25, 0.5, and 1.0 at p=12–50 atm and T=885–1200 K in a heated shock tube. More measurements about the ignition delay times of n-heptane/air were conducted by Ciezki and Adomeit 56, Heufer and Olivier57, and Gauthier et al.58 under a wide range of operating 12


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conditions with T=700–1450 K, p=1.85–60.6 atm, and φ=0.25–2.0. Meanwhile, Davidson et al.59 measured the ignition delay times of iso-octane/air with a stoichiometric ratio at p=10–60 atm and T=850–1200 K.

Table 3. Experimental data for the ignition delay times of PRF in shock tubes Temperature [K] Pressure [atm]

Equivalence ratio

Reference

Fuel

Fieweger et al.53

PRF0–PRF100

700–1190

40

1

Shen et al.54

n-heptane

786–1396

12.5–45

0.25 and 0.5

Ciezki and Adomeit56

n-heptane

1050–1350

3.2–13.5

0.5, 1 and 2

Heufer and Olivier57

n-heptane

700–1250

13 and 38

1

Gauthier et al.58

n-heptane

800–1450

1.85–60.6

1

Fieweger et al.53

iso-octane

700–1300

13 and 40

1

Shen et al.55

iso-octane

885–1200

12–50

0.25, 0.5 and 1

Davidson et al.59

iso-octane

850–1200

10–60

1

Table 4. Experimental data for the major species concentrations in JSRs for PRF Reference

Dilute gas

Fuel

Temperature [K]

Pressure [atm]

Residence time [s]

Equivalence ratio

Dagaut et al.60

N2

PRF50

550–1150

10

1

1

Herbinet et al.61

He

n-heptane

500–1100

1.05

2

1

Dagaut et al.62

N2

iso-octane

550–1150

10

1

1

For the experimental data of PRF in JSRs, Dagaut et al.60 investigated the oxidation of the blends of n-heptane and iso-octane in a high-pressure JSR under a wide temperature range of 550–1150 K at 10 atm with a stoichiometric ratio and a residence time (τ) of 1 s. Furthermore, the oxidation of iso-octane/air was measured in the JSR at the same operating conditions62. Recently, Herbinet et al.61 studied the low-temperature oxidation of n-heptane diluted in helium in a JSR at T=500–1100 K, p=1.05 atm, φ=1.0 and τ=2 s. The summary of experimental data in ST and JSR are listed in Tables 3 and 4, respectively, which are introduced for the optimization of the PRF mechanism in the present study.

13


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As can be found from Table 3, most of the available experiment of the ignition delay times of PRF was performed at the stoichiometric ratio in shock tubes. Therefore, as a preliminary attempt, only the experimental data at the stoichiometric ratio are employed for the PRF mechanism optimization, and a wide range of temperature of 700–1450 K and pressure of 1.85–60 atm is covered. Based on the measurements of the ignition delay times at φ=1.0, an optimized mechanism, Mechanismpart, is first achieved. Meanwhile, by including all the measurements from lean to rich fuel/air mixtures in shock tubes listed in Table 3, the other optimized mechanism, Mechanismall, is obtained for comparison. Comparison of the ignition delay times of n-heptane/air between the predictions with Mechanismpart and Mechanismall at the conditions of the lean and rich fuel/air mixtures is displayed in Figure 6. Compared with Mechanismpart, the predictions of Mechanismall agree better with the measured ignition delay times for the lean and rich n-heptane/air. Especially, the ignition behavior at low temperatures, φ=0.25, and p=45 atm, as well as at the negative temperature coefficient (NTC) range, φ=2 and p=13.5 atm, is reproduced by Mechanismall rather well. Therefore, for the application of the final optimized mechanism in sufficiently wide operating conditions, the available experimental data about ignition delay times under extensive conditions with various equivalence ratios, pressures, and temperatures should be considered for the optimization process. Furthermore, as listed in Table 5, only the experimental data of JSRs at stoichiometric ratio is considered for mechanism optimization in the present study. This is because the fact that the improvement of the performance of the optimized mechanism considering all the equivalence ratios in JSRs is insignificant, whereas the computational time increases considerably. Through the sensitivity analysis of major species concentrations at different equivalence ratios in JSRs, it is found that the important reactions affecting the predicted species concentration of the skeletal mechanism in JSRs under different equivalence ratios are quite similar. Thereby, if the optimized skeletal mechanism can satisfactorily reproduce the concentrations of the major species under stoichiometric ratio, those at lean and rich equivalence ratios can also be well predicted. From the perspective of physics, the fuel is usually highly diluted in JSR experiments with fuel mole fraction less than 0.5% in the mixture. Thereby, the main reaction pathways in JSRs might be consistent under various equivalence ratios63, 64. Therefore, the JSR data at lean and rich equivalence ratios are not considered in the optimization process in the present study.

3.3 Effect of weight factor The distribution of the available experimental data of the ignition delays of PRF covers a wide temperature 14


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range, and the temperature considerably affects the ignition behavior of various PRFs. In this study, the available experimental data of the ignition delay times for PRF are divided into three groups according to the measured temperature, including the low-temperature, NTC, and high-temperature zone. As shown in Figure 7, at a specific pressure and equivalent ratio for n-heptane, the ignition delay time first decreases when temperature increases in the low-temperature zone, then increases when the temperature reaches the first threshold value in the NTC zone. Finally, the ignition delay time decreases again when the temperature passes the second threshold value in the high-temperature zone. In Figure 7, the low-temperature, NTC, and high-temperature zones are recognized by two dashed lines for n-heptane. The consistent definitions of the three zones can be applied to the experimental data of iso-octane and PRF. It is indicated in previous studies, the NTC behavior is a distinctive feature for the oxidation kinetics of large hydrocarbons65. Thereby, a heavier weight factor in Eq. (2) is used in the NTC zone in order to satisfactorily reproduce the ignition characteristics of PRF. Furthermore, as can be seen from Figure 7 and Table 3, a limited number of experiments were conducted at low temperatures because of the low reactivity of PRF, especially for the ones with high octane42, 66. It should be noted that, for advanced LTC engines, such as HCCI, the auto-ignition is dominated by the low-temperature oxidation67, 68. Therefore, for the reproduction of the ignition behavior of PRF, the weight factor of 2:1.5:1 is respectively assigned for the ignition data in the low-temperature, NTC, and high-temperature zones. The effects of the weight factor of the temperature for the ignition delay measurements on the optimized mechanism are depicted in Figure 8. It can be seen that the optimized mechanism obtained with the weight factor of 2:1.5:1 can reproduce the ignition delay times rather well, while the mechanism using the weight factor of 1:1:1 underestimates the ignition delay times of the fuels with high octane number by around 20% at the NTC zone, and the situation even worsens at low temperatures. The optimized results using the weight factor of 1:1:1 depend more on the measurements at high temperatures because of the much larger fraction of the experimental data at high temperatures than those at low and NTC temperatures. As a result, when the weight factor of 1:1:1 is used, the prediction accuracy of the optimized mechanism in the low-temperature zone is sacrificed, whereas the improvement in the high-temperature zone is insignificant. Therefore, the weight factor of 2:1.5:1 with heavier weights on the low-temperature and NTC zones is used in this study in order to improve the performance of the optimized mechanism for engine simulations. Furthermore, it must be noted that the experimental data plays a key role in the optimization process, because 15


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the optimization aims at achieving an optimized skeletal mechanism that can match well with the measurements. However, the uncertainty of the measured ignition delay times could affect the final optimized mechanism. Chaos and Dryer69 emphasized the perturbative effects, including the non-idealities of fluid dynamics and the deflagrative processes in mild ignition events, on the measured ignition delays can reduce chemical induction times and shorten ignition delays, especially at low-temperature and high-pressure conditions. In order to reduce the perturbative effects, it is necessary to introduce the pressure-time history reported in shock-tube experiments into the CHEMKIN calculation. For the earlier shock-tube experiments, the history of pressure-time is not available, such as the work of Fieweger et al.53, thus it is assumed that the pressure does not change with time before autoignition for simplicity. Moreover, as many experimental data as possible were used in this study for the mechanism optimization to weaken the uncertainty of the measurements of ignition delays. Therefore, the perturbative effects and the uncertainty of the measurements on the predicted ignition delay time can be decreased to some extent. As for the experimental data of JSR, the experimental data are relatively uniformly distributed over the whole temperature range. The consideration of the temperature weighting or not does not significantly affect the final optimized mechanism. Therefore, by using Eq. (3) without considering the temperature weighing, the predictions from the optimized mechanism agree well with measurements in JSR, as shown in Section 3.4.

3.4 Comparison of the optimized mechanisms Different from the single-objective GA with only one optimized mechanism, 52 optimized mechanisms can be obtained by NSGA-II in the present study. Three representative optimized mechanisms are selected for comparisons in this section. As shown in Figure 5, the three ones include Mechanism #1 with the minimal fST(A), Mechanism #3 with the minimal fJSR(A), and Mechanism #2 with a balance between fST(A) and fJSR(A). Moreover, the numerical results of the detailed PRF mechanism from Mehl et al.39 and the initial PRF mechanism constructed by the decoupling methodology are compared with those of the optimized mechanisms. As shown in Figure 9, the numerical results of the three typical optimized mechanisms are compared with the measurements from Fieweger et al.53 The predictions with the detailed PRF mechanism and the initial skeletal mechanism without optimization are also shown in Figure 9(a) and Figure 9(b), respectively. As can be seen, the numerical results from Mechanism #1 match the measurements very well, even better than the detailed model. This is primarily because the fact that the optimized mechanism targets on the specific experimental data used for 16


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optimization, while the detailed mechanism pays much attention to the comprehensive performance. Mechanism #2 also has a remarkable improvement on the predicted ignition delay times compared with the initial mechanism. However, Mechanism #3 considerably over-predicts the ignition delay times of PRFs over the tested conditions, as shown in Figure 9(c). The three optimized mechanisms are also applied to predict the oxidation of the stoichiometric PRF50/air mixture at 10atm in a JSR60. Figure 10 compares the numerical results of the major species concentrations using the three optimized mechanisms with the measurements. Moreover, the predicted results with the initial model and the detailed model39 are also displayed in Figure 10(a) and 10(c), respectively. From Figure 10(a), it can be seen that the predictions in JSR are improved by Mechanism #1compared to those of the initial model, especially for the concentrations of O2 and fuel at the cool-flame and NTC regimes. For Mechanism #2, the deviation between the prediction and the measurement of CO around 750 K reduces than Mechanism #1. Furthermore, as shown in Figure 10(c), remarkable improvements on the predictions of n-heptane and iso-octane profiles around 750 K is obtained using Mechanism #3, which also offers more accurate information about the concentrations of CO and CO2 even compared with the detailed model39. Through the above analysis, Mechanism #3 has the excellent prediction capability on the major species concentrations in JSRs but with the poor performance on the prediction of the ignition delay times in STs. In contrast, Mechanism #1 gives good predictions on the ignition delay times, whereas it cannot reproduce the fuel concentrations at the NTC regime very well, though obvious improvements are found compared with the initial mechanism. The prediction capability of Mechanism #2 is similar to Mechanism #1. As mentioned above, the performance of the optimized mechanisms is determined by the objective functions in the multi-objective optimization. It can be seen in Figure 5 that a mechanism in the Pareto solutions with a smaller value of one objective function, the other objective function is usually larger. Although the experimental information in STs and JSRs could supplement each other, it is still a challenge to obtain a global optimized mechanism with satisfactory predictions on both the ignition delay times and the major species concentrations. For further evaluate the performance of Mechanism #2 and Mechanism #1, the measured major species profiles of the n-heptane61 and iso-octane62 oxidation in JSRs are compared with the predicted results using Mechanism #1, Mechanism #2, and the initial mechanism in Figure 11. It can be seen that both the optimized models improve the predictions in JSRs significantly compared with the initial model. Especially, the predictions of Mechanism #2 are better than Mechanism #1 at temperatures below 800 K, where Mechanism #2 captures the 17


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consumption of oxygen and fuels more accurately. Because the predicted ignition delay times in STs from Mechanism #2 is very similar to Mechanism #1 (see Figure 9). Mechanism #2 is regarded as the final optimized PRF mechanism in this study, named as Mechanismopt. It is always in a dilemma to choose one solution as the final optimized PRF mechanism in the multi-objective optimization, and future work is still needed to provide a way to determine the best solution quantitatively. In order to reveal the advantage of Mechanismopt in predicting the ignition delay times under an extensively wide range of operating conditions, Figure 12 compares the measured and predicted ignition delay times of the n-heptane/air and the iso-octane/air mixture using Mechanismopt at φ=1.0. It can be found that Mechanismopt can reproduce the measurements rather well for both fuels under the broad range of temperatures and pressures, despite there are slight discrepancies at temperatures above 1100 K and pressures below 15 atm. It should be noted that the conditions of ignition at very high temperatures and low pressures are far from those in engines, thus no further optimization of Mechanismopt is performed in the present study. Furthermore, Mechanismopt is employed to predict the ignition delay times for lean and rich fuel/air blends, as shown in Figure 13. As can be seen, Mechanismopt reproduces the ignition delay times of n-heptane very well, in spite of a slight underestimation in the NTC zone at φ=0.5. For iso-octane, Mechanismopt can also satisfactorily predict the ignition delay times, except for a slight overestimation at temperatures above 1150 K at φ=0.5. Overall, considering the uncertainties in the experiment of STs and JSRs, the predictions of Mechanismopt can be considered satisfactory, and extensive validations of Mechanismopt are presented in Section 4. As discussed above, a series of optimized mechanisms are obtained through the multi-objective optimization, which is different from the single-objective optimization with only one optimized mechanism. Most of this series of optimized mechanisms have good predictions on ignition delay times and major species concentrations. The prediction capability of the optimized mechanism obtained by the single-objective optimization might be similar to only one of those optimized mechanisms obtained by the multi-objective optimization. Therefore, the multi-objective optimization provides more flexible and alternative solutions for mechanism optimization.

3.5 Characteristics of the optimized mechanisms To investigate the difference of the above three typical mechanisms, the pre-exponential factors of the mechanism are normalized by the form of logarithm, log16(Ai/Ai,0), in which sixteen means the maximum perturbation factor assigned to the pre-exponential factor of Reactions R1–R30 in the initial mechanism. As can 18


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be seen from Figure 14, although the values of the pre-exponential factors of the three mechanism are similar to each other for some reactions, significant discrepancies exist in the reactions of R3, R4, R6, R13–15, R17, R21– 25, R27, and R29, which dominate the diversity of the optimized mechanisms. The differences of the rate constants in the optimized mechanisms result in the different prediction performance in STs and JSRs, shown in Figures 9 and 10. To illuminate factors affecting the performance of the optimized mechanisms, the sensitivity analysis is performed to Mechanisms #1, #2, and #3. The sensitivity coefficient of each reaction from the large-molecule sub-mechanism is calculated as K =

P ,0 S P ,Q.R 0 LMN Q.R

LMNO

(7)

where τi,2 and τi,0.5 are the ignition delay times computed with the rate constant of the ith reaction multiplied and divided by a factor of 2, respectively, and σi is the calculated sensitivity coefficient of the ith reaction. Figure 15 shows the sensitivity analysis of the ignition delay times for Mechanisms #1, #2, and #3 for stoichiometric PRF60/air at 40atm and T=700, 850, and 1000 K, which represents the low-temperature, NTC, and high-temperature regimes, respectively. As shown in Figure 15 at 700 K, only n-heptane specific reactions display a high sensitivity for Mechanism #1, whereas the iso-octane related reactions are highly sensitive for Mechanism #2, such as the H-atom abstraction from iso-octyl radical via O2 (R24) and the addition of O2 to the iso-octyl radical (R18). As for Mechanism #3, the pyrolysis reaction of C8H16 (R25) shows an obviously positive sensitivity. At 850 K, the sensitive reactions for three mechanisms are identical, such as the highly positive sensitive reaction including the β-decomposition of n-heptyl radical (R12) and the negative sensitive reactions including the H-atom abstraction from n-heptane via OH and HO2 radicals (R3 and R4). However, the ignition delay times predicted by Mechanisms #1 and #2 at 850 K are more sensitive to reaction R24 while the reaction of the H-atom abstraction from iso-octane via OH (R14) is more sensitive for Mechanism #3. Since reaction R12 plays a key role in predicting the ignition delay time at T=850 K for all the three mechanisms, Mechanism #3 with a larger rate constant of reaction R12 (see Figure 14) overestimates the experimental data around T=850 K, as shown in Figure 9(c). In terms of 1000 K, the ignition delay times are more sensitive to the iso-octane specific reactions for Mechanisms #1 and #2 compared with Mechanism #3. As shown in Figure 14, many reactions reach their prescribed upper or lower bounds. Moreover, consistent with the results shown in Figures 14 and 15, the reactions which reach the bounds are almost insensitive to the 19


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ignition delay times in shock tubes, such as R1, R5, R7, R9, R11, R17, R20, R22, R26–R28, and R30. Therefore, it can be concluded that the reactions with the optimized pre-exponential factors near their bounds are the unimportant ones for mechanism optimization, thus they are varied in a wide range in NSGA II. Therefore, the factor of 16 is used as the bounds of the pre-exponential factors based on the trial that the smallest factor can be introduced to obtain the overall good performance of the final optimized mechanism. Overall, the parameters of Reactions R1–R30 vary significantly in different optimized mechanisms, which demonstrates the diversity of the optimized mechanisms obtained by the multi-objective optimization. Furthermore, through the above investigation of the optimized parameters in different mechanisms, a deep understanding of the large-molecule reactions in the constructed skeletal mechanism can be obtained, which cannot be directly offered by the single-objective optimization.

4 Extensive validations To further access the performance of the optimized mechanism, Mechanismopt is validated by the experimental data of laminar flame speed, species concentrations in laminar flames, and the in-cylinder pressure, heat release rate, and emissions in a HCCI engine. It is worthy emphasizing that, even though the laminar flame speeds and species concentrations in premixed laminar flames are dominated by the small-molecule sub-mechanism, the effects of the fuel-related reactions on laminar flame speeds and species concentrations cannot be completely ignored70, because the products from the initial fuel decomposition affect the evolution of C1-C4 intermediates, and the subsequent C1-C4 reactions are responsible for flame reactivity.

4.1 Laminar flame speed Laminar flame speed is an important parameter for the combustion of fuel/oxidizer, which has been extensively used for the validation of a chemical reaction mechanism71. Dirrenberger et al.72 measured the laminar flame speeds of the atmospheric n-heptane/air in a flat-flame adiabatic burner over the equivalence ratios from 0.55 to 1.6 with the unburned gas temperatures from 298 to 398 K. Moreover, the laminar flame speeds for both n-heptane/air and iso-octane /air were measured by Kumar et al.73 in a counterflow flame setup at atmospheric pressure over the equivalence ratios from 0.7 to 1.4 with the unburned gas temperatures from 298 to 470 K. To calculate the laminar flame speed in the simulation, the PREMIX module in the CHEMKIN-PRO software46 was employed in this section. 20


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Figure 16 compares the measured laminar flame speeds and the numerical results with Mechanismopt and the initial mechanism without the optimization of the large-molecule reactions. It can be seen that the predicted results using both models agree well with the measurements at various temperatures and equivalence ratios, and the predictions of Mechanismopt are almost same as those of the initial model. This indicates that the modification of the rate constants on the large-molecule sub-mechanism has little effect on the laminar flame speeds of PRFs. It has been acknowledged that the laminar flame speeds of heavy hydrocarbons are mainly sensitive to the H2/CO/CO2 and small-hydrocarbon mechanism, while the fuel-related reactions have nearly no influence on the laminar flame speeds11, 70. Since a detailed H2/CO/CO2 sub-mechanism is used in the present PRF mechanism based on the decoupling methodology, satisfactory predictions on the laminar flame speeds can be achieved in Figure 16. Furthermore, because of the insensitivity of the predicted laminar flame speeds to the large-molecule reactions, it is convenient to optimize the rate constants of Reactions R1–R30 focusing only on the experimental data of ignition delay times and species concentrations without affecting the flame propagation characteristics in this study.

4.2 Species concentrations in premixed laminar flame Bakali et al.74 measured the concentrations of the reactants and major products concentrations of n-heptane/O2/N2 and iso-octane/O2/N2 in a flat-flame burner at atmospheric pressure with the equivalence ratio of 1.9. Simulations were performed with the PREMIX module in the CHEMKIN-PRO software46. Figure 17 shows the comparison between Mechanismopt and the initial model on the predicted major species concentrations. As seen in Figure 17, both mechanisms predict the mole fractions of the reactants and major products quite well, which are within the experimental uncertainty. Because the major species concentrations in the premixed laminar flame are dominated by the small-molecule sub-mechanism, the predictions of Mechanismopt and the initial mechanism are almost identical by adopting the same detailed H2/CO/CO2 sub-mechanism in the PRF mechanism.

4.3 HCCI engine The main purpose of this study is to construct a skeletal mechanism for the CFD simulation of the combustion and emission characteristics of practical engines. In this section, the final optimized mechanism, i.e., Mechanismopt, is coupled with the improved KIVA-3V code75, 76 and the CHEMKIN II code77 to simulate the 21


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combustion characteristics of a HCCI engine fueled with PRFs. A new generalized re-normalization group (gRNG) κ-ε turbulence model proposed by Wang et al.78 is used to describe the in-cylinder turbulent flow. An updated heat transfer model79, 80 is employed to describe the wall heat transfer process. The HCCI experiment was managed by Dempsey et al.81, 82 at the intake temperatures of 45 and 75 oC and the equivalence ratios of 0.26 and 0.30. The detailed specifications about the engine are listed in Table 5.

Table 5. Engine specifications and operating conditions Parameter Displacement (L)

0.477

Bore (mm)×Stroke (mm)

82×90.4

Compression ratio

16.7:1

Intake valve closing (IVC)

–132°ATDC*

Exhaust valve opening (EVO)

112°ATDC

Engine speed (rev/min)

1500

Intake temperature (oC)

45 and 75

Intake pressure (kPa)

110

Equivalence ratio

0.26 and 0.30

*ATDC: after top dead center

Two sets of 1/7th cylinder mesh used for HCCI engine simulations are displayed in Figure 18. At bottom dead center, the base coarse mesh (see in Figure 18(a)) contains 7,515 cells and the finer mesh (see in Figure 18(b)) includes 33,991 cells. Based on Mechanismopt, the predicted in-cylinder pressure and heat release rate using the two meshes for PRF30 at φ=0.26 and T=45 oC are shown in Figure 19. As can be seen, the predictions using the two meshes are almost identical and agree with the measurements rather well. This is primarily due to the fact that the fuel/air mixture in the cylinder is almost uniform during the whole combustion process, thus the computational results of HCCI combustion are not sensitive to the grid density83, 84. In order to reduce the computation time, the base mesh is employed in the following simulations. Figure 20 shows comparison between the measured and predicted in-cylinder pressure and heat release rate, as well as the exhaust emissions at two operating conditions for various PRFs. As can be seen, the computational model using Mechanismopt reproduces the combustion characteristics for all PRFs rather well, and the variations 22


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of the in-cylinder pressure and heat release rate with different PRFs are satisfactorily captured. In the simulation, the heat release rate is derived from the net heat release from the chemistry and the wall heat transfer, which is different from the experiment based on the in-cylinder pressure. Thus, the predicted heat release rates are higher than the measurements. Furthermore, the exhaust emissions of CO, CO2, and H2O are also captured well by the model, except for the discrepancy of the CO emissions for the high octane number fuels at φ=0.26 and the underestimation of the CO2 and H2O emissions at φ=0.3. Furthermore, for understanding the influence of mechanism optimization on the HCCI predictions, comparison of the in-cylinder pressure and heat release rate predicted using the initial mechanism and Mechanismopt, and the measurement in the HCCI engine fueled with PRF30 at φ=0.26 and T=45 oC, are shown in Figure 21. As can be seen, no obvious ignition behavior is captured by the initial mechanism for the test case because of its low reactivity. The low reactivity of the initial mechanism also can be observed in the predictions of the ignition delay times in shock tubes, as shown in Figure 9(c). In contrast, the predictions of HCCI combustion are improved considerably by Mechanismopt on the basis of the initial mechanism. Although the predicted in-cylinder pressure and heat release rate using the initial mechanism could match the measurements by significantly elevating the initial in-cylinder temperature and pressure at intake valve closing (IVC), the excessive oxidation of fuel results in the over-predicted CO2 and H2O concentrations.

5 Conclusions By integrating the decoupling methodology with the multi-objective genetic algorithm, an approach for the construction of skeletal mechanism is proposed in this study. Compared with the empirical adjustment of the rate constants of the large-molecule reactions in the previous decoupling methodology, the optimization of skeletal mechanism is much more automatic and efficient. Two objective functions are built for the mechanism optimization based on two strategies of experiments including the ignition delay times in shock tubes (ST) and the major species concentrations in jet-stirred reactors (JSR). A series of optimized mechanisms are achieved with significantly different performance in the predictions of ignition delay times in STs and species concentrations in JSRs. Compared with the single-objective optimization which offers only one optimized mechanism, the multi-objective optimization provides more flexible and alternative solutions for mechanism optimization. The computational results indicate that, compared with the initial mechanism without parameter optimization, the predictions on the ignition delay times and the species profiles are remarkably improved by the optimized 23


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mechanisms. The final optimized mechanism, Mechanismopt, is selected because of its comprehensive prediction performance in STs and JSRs. Further validations indicate that laminar flame speeds and major species profiles in the premixed laminar flames are reproduced rather well with Mechanismopt due to the introduction of the detailed small-molecule sub-mechanism. Finally, by validating against the experimental data of the in-cylinder pressures, heat release rates, and exhaust emissions in a HCCI engine, it is found that the combustion and emission characteristics can be satisfactorily reproduced by Mechanismopt. The reliable performance of the final mechanism proves the effectiveness of the present approach for the construction of the skeletal mechanism for engine simulations.

Acknowledgments The financial support from the National Natural Science Foundation of China (Grant Nos. 91641117 and 51706033) is greatly appreciated.

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Figure 1. Flow diagram of NSGA-II for mechanism optimization 279x215mm (300 x 300 DPI)


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Figure 2. Definitions of Pareto Front in a two-objective optimization 279x215mm (300 x 300 DPI)


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Figure 3. Definitions of rank and crowing distance 279x215mm (300 x 300 DPI)


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Figure 4. Comparisons of six representative cases at different generations 254x199mm (300 x 300 DPI)


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Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



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Figure 5. The evolution of values of the two objective functions 279x215mm (300 x 300 DPI)


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Figure 6. Comparison of the measured and predicted ignition delay times of n-heptane/air in shock tubes with Mechanismpart and Mechanismall (Symbols are measurements from Shen et al.51 and Ciezki and Adomeit53; Dashed lines are the numerical results with Mechanismpart; Solid lines are the numerical results with Mechanismall). 355x254mm (300 x 300 DPI)


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Figure 7. Distribution for the available experimental data of ignition delay times of n-heptane with φ=1.0 in shock tubes (Solid points represent the experimental data listed in Table 3). 284x199mm (300 x 300 DPI)


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Figure 8. Effects of the weight factor of the temperature of the experimental data in shock tubes on the optimized results (Dashed lines represent the predictions using the optimized mechanism with the weight factor of 1:1:1; Solid lines represent the predictions using the optimized mechanism with the weight factor of 2:1.5:1). 330x203mm (300 x 300 DPI)


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Figure 9. Comparison of ignition delay times of PRFs in STs (Symbols are measurements from Fieweger et al.50 Solid lines are the numerical results with three typical optimized mechanisms, namely Mechanisms #1, #2, and #3; Dash-dotted lines show the results with the detailed mechanism36; Dashed lines show the results with the initial mechanism). 284x199mm (300 x 300 DPI)


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Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 10. Comparison between the measured and predicted major species concentrations in JSRs for PRF50/air (Symbols are the measurements from Dagaut et al.57; Solid lines are the numerical results using Mechanisms #1, #2, and #3; Dashed lines are the numerical results using the initial mechanism; Dashdotted lines are the numerical results using the detailed mechanism36). 254x215mm (300 x 300 DPI)


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Energy & Fuels



Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



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Energy & Fuels

Figure 11. Comparison between the measured and predicted major species concentrations for (a) n-heptane and (b) iso-octane in JSRs (Symbols are the experimental data58-59; Dashed lines are the simulation results using Mechanism #1; Solid lines are the simulation results using Mechanism #2; Dash-dotted lines are the simulation results using the initial mechanism). 279x215mm (300 x 300 DPI)


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 11. Comparison between the measured and predicted major species concentrations for (a) n-heptane and (b) iso-octane in JSRs (Symbols are the experimental data58-59; Dashed lines are the simulation results using Mechanism #1; Solid lines are the simulation results using Mechanism #2; Dash-dotted lines are the simulation results using the initial mechanism). 279x215mm (300 x 300 DPI)


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Energy & Fuels

Figure 12. Comparison of the ignition delay times of stoichiometric (a) n-heptane and (b) iso-octane at various pressures in STs (Symbols are the measured data50, 53-55, 52, 56; Solid lines are the numerical results using Mechanismopt). 279x215mm (300 x 300 DPI)


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 12. Comparison of the ignition delay times of stoichiometric (a) n-heptane and (b) iso-octane at various pressures in STs (Symbols are the measured data50, 53-55, 52, 56; Solid lines are the numerical results using Mechanismopt). 279x215mm (300 x 300 DPI)


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Energy & Fuels

Figure 13. Comparison of the measured and predicted ignition delay times of (a) n-heptane and (b) isooctane at various equivalence ratios and pressures in STs. Symbols are the measurement51-53, 56. Solid lines are the numerical results using Mechanismopt. 304x215mm (300 x 300 DPI)


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 13. Comparison of the measured and predicted ignition delay times of (a) n-heptane and (b) isooctane at various equivalence ratios and pressures in STs. Symbols are the measurement51-53, 56. Solid lines are the numerical results using Mechanismopt. 304x215mm (300 x 300 DPI)


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Energy & Fuels

Figure 14. Comparison of the pre-exponential factors among the three typical mechanisms 406x152mm (300 x 300 DPI)


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Figure 15. Sensitivity coefficient of the ignition delay times for stoichiometric PRF60/air at p=40 atm and T=700, 850, and 1000 K. 457x215mm (300 x 300 DPI)


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Energy & Fuels

Figure 16. Comparison between the measured and predicted laminar flame speeds for n-heptane and isooctane. (Symbols are measurements65, 66; Solid lines and dashed lines are the numerical results using Mechanismopt and the initial mechanism, respectively). 279x215mm (300 x 300 DPI)


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 16. Comparison between the measured and predicted laminar flame speeds for n-heptane and isooctane. (Symbols are measurements65, 66; Solid lines and dashed lines are the numerical results using Mechanismopt and the initial mechanism, respectively). 279x215mm (300 x 300 DPI)


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Energy & Fuels

Figure 17. Comparison between the measured and predicted species profiles in n-heptane/O2/N2 and isooctane/O2/N2 premixed laminar flame (Symbols are measurements68; Solid lines and dashed lines are the numerical results for Mechanismopt and the initial mechanism, respectively). 279x215mm (300 x 300 DPI)


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 17. Comparison between the measured and predicted species profiles in n-heptane/O2/N2 and isooctane/O2/N2 premixed laminar flame (Symbols are measurements68; Solid lines and dashed lines are the numerical results for Mechanismopt and the initial mechanism, respectively). 279x215mm (300 x 300 DPI)


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Energy & Fuels

Figure 18. Computational mesh of 1/7th cylinder at top dead center 698x531mm (120 x 120 DPI)


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Figure 18. Computational mesh of 1/7th cylinder at top dead center 698x531mm (120 x 120 DPI)


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Energy & Fuels

Figure 19. Verification of grid independence in the HCCI simulation 698x465mm (120 x 120 DPI)


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Figure 20. Comparison between the measured and predicted in-cylinder pressure and heat release rate as well as the exhaust emissions in a HCCI engine fueled with PRFs (Solid lines and solid points are the measurement from Dempsey et al.79, 80; Dotted lines and hollow points are the simulation data). 698x465mm (120 x 120 DPI)


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Energy & Fuels

Figure 20. Comparison between the measured and predicted in-cylinder pressure and heat release rate as well as the exhaust emissions in a HCCI engine fueled with PRFs (Solid lines and solid points are the measurement from Dempsey et al.79, 80; Dotted lines and hollow points are the simulation data).

698x465mm (120 x 120 DPI)


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Energy & Fuels



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Figure 21. Comparison of in-cylinder pressure and heat release rate in the HCCI engine between the simulations using the initial mechanism and Mechanismopt, and the measurement. 698x465mm (120 x 120 DPI)


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An efficient approach for the optimization of skeletal chemical

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