Nineteenth International Multidimensional Engine Modeling User's Group Meeting April 19, 2009, Detroit, Michigan

Towards Using Realistic Chemical Kinetics in Multidimensional CFD

Long Liang, Karthik Puduppakkam, Ellen Meeks

Reaction Design 6440 Lusk Blvd Suite D205, San Diego, CA 92121

This paper presents several advanced solution strategies for overcoming the CPU time barrier associated with

solving realistic fuel chemistry in multidimensional CFD. The strategies include a dynamic multi-zone (DMZ) partitioning scheme, a dynamic adaptive chemistry (DAC) scheme, and an advanced ODE solver developed at Reaction Design that takes advantage of the sparsity of detailed reaction mechanisms. All of the solution strategies were implemented in CHEMKIN-CFD/API and tested by using KIVA-3V as the CFD framework. Compared to conventional solution approaches, the use of these advanced strategies can improve the overall computational efficiency of solving the detailed fuel chemistry by two to three orders of magnitude in typical engine simulations, making the once-prohibitive computational tasks feasible with minimal or zero loss of accuracy.

1. Introduction

The next decade will witness a dramatic increase in the complexity of automotive engine design as companies strive to maintain a high level of engine performance, while continuing to reduce emissions with increasing use of wider variety of fuels. Detailed understanding of the fuel-combustion chemistry is crucial to unraveling the complexity associated with fuel effects. For example, adopting fuels with high ethanol content may reduce particulate emissions, but may also prompt new regulation of pollutants, such as aldehydes; soot formation in diesel engines is correlated with not only hydrocarbon molecular structure, but also engine operating conditions; knock behavior, which limits the peak thermal efficiency in spark-ignition engines, is also affected by fuel composition, such as aromatics or alcohol levels.

Although computational fluid dynamics (CFD) tools have advanced in recent years to more accurately address fluid mixing under turbulent-combustion conditions, they continue to lack sufficient chemistry detail to reliably predict emissions, ignition and fuel effects. Thus, coupling realistic combustion chemical kinetics with multidimensional CFD for engine combustion has been a long-standing goal. A significant barrier to this coupling is the time-intensive nature of solving the chemical kinetic equations. The use of detailed chemical kinetics gives rise to a large system of stiff nonlinear ordinary differential equations (ODEs), the solution of which within the CFD framework requires extremely long CPU times. Consequently, large-scale 3-D reactive flow simulations are computationally prohibitive with detailed kinetic mechanisms developed for most hydrocarbon fuels. However, there are opportunities to minimize the expenditure of computing resources in the calculation of chemical source terms and to do so with a minimal loss of accuracy. This paper presents several advanced solution strategies aimed at overcoming the CPU time barrier, including

(1) a dynamic multi-zone (DMZ) partitioning scheme, (2) a dynamic adaptive chemistry (DAC) scheme, and

(3) an advanced ODE solver developed at Reaction Design which takes advantage of the species-interaction sparsity that is typical of detailed kinetics mechanisms.

The use of these strategies can improve the computational efficiency of typical multidimensional engine combustion calculations by two to three orders of magnitude. In the present work, these solution strategies were implemented in CHEMKIN-CFD/API [1] and linked with KIVA-3V [2]. The rest of this paper first reviews the coupled reactive flow equations to be solved, and then briefly describe the advanced solution strategies. Test cases follow to demonstrate the performance of these strategies. Finally, some concluding remarks are drawn.

2. Governing equations of transient reactive flows

CFD simulation of engine combustion involves solving transient transport equations for each gas-phase species and for internal energy [2]:

( ) chemk

kk

k Dt

ρρρρρρ

&+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅∇=⋅∇+

∂∂ u ),...,1( Kk = (1)

( ) chemQpete &++⋅∇−⋅∇−=⋅∇+

∂∂ ρερρ Juu (2)

where ρk is the mass density of species k, K is the total number of species, ρ is the mass density of the mixture, u is the flow velocity vector, D is mass diffusivity, e is specific internal energy, p is pressure, J is the heat flux vector accounting for contributions due to heat conduction and enthalpy diffusion, ε is the dissipation rate of turbulent kinetic energy, is the rate of change of species density due to chemical reactions, and is the chemical heat release rate. and

are the chemistry source terms modeled by the kinetic equations.

chemkρ&

chemQ& chemkρ&

chemQ&

The transient reactive flow equations are typically solved using the operator-splitting method. This method splits the

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transport equation into two sub-equations and solves them with overlapping time-steps; the first step handles the chemistry source terms and the second step handles the transport (e.g. diffusion or convection). In the first step, each computational cell is modeled as a closed constant-volume perfectly stirred reactor (PSR), and the kinetic equations are solved on a cell-by-cell basis. The kinetic equations are formulated as an ODE set, typically including K equations for species mass fractions Yk and one equation for gas temperature T, that is,

( )

⎪⎪⎩

⎪⎪⎨

⎧

−=

==

∑=

K

kkkk

v

kkk

eWcdt

dT

KkWdt

dY

1

1

,...,1

ωρ

ρω

&

&

(3)

where kω& is the production/consumption rate of species k, Wk and ek are the molecular weight and specific internal energy of species k, respectively, and vc is the constant volume specific heat of the gas mixture. Using Eq. (3), the chemistry source terms of Eqs. (1) and (2) can be written as,

dtdYkchem

k ρρ =& , (4) ( Kk ,...,1=

2

)

( )∑=

Δ−=

K

k k

kfkchem

Wh

dtdYQ

1

0& , (5)

where ( is the molar heat of formation of species k. )kfh0Δ

3. Advanced solution strategies 3.1. Dynamic multi-zone (DMZ) partitioning scheme

Traditionally, the kinetics equation set (Eq. (3)) is integrated over a hydrodynamic time-step in each computational cell. However, we note that Eq. (3) is independent of the mass and volume of a specific cell. Thus, cells that have the same temperature, pressure, and species densities, will yield the same result upon solution of Eq. (3). This observation forms the basis for grouping a set of cells of high similarity into zones and solving the kinetic equations only once for each zone. A dynamic multi-zone (DMZ) partitioning scheme was proposed by Liang et al. [3] which takes advantage of this similarity. The DMZ scheme was built upon several multi-zone modeling ideas in the literature [4], but it distinguishes itself from the literature methods by dynamically determining the optimal number of zones using a rigorous data-clustering algorithm for each specific time-step. The zoning algorithm is solely based on the cells' thermochemical states and is independent of their locations in the CFD mesh. In addition, the zoning algorithm is highly automated and requires minimal user-provided inputs.

The DMZ scheme includes three major steps: (1) grouping cells into zones using an evolutionary zoning/clustering algorithm; (2) solving chemical kinetic equations based on zonal averaged state variables; and (3) mapping the zonal averaged solution back to the individual cells while preserving the initial temperature and species stratification. A detailed description of the DMZ scheme can be found in Ref. [3]. We present only a brief review here.

3.1.1. The evolutionary zoning algorithm The multi-zone partitioning can be classified as an

unsupervised data clustering problem. “Unsupervised” means that there are no pre-defined boundaries among the clusters and typically there is no unique solution. An important aspect of such a task is to determine the optimal number of clusters/zones that retain accuracy while reducing the computational time. In the present DMZ scheme, a novel evolutionary zoning algorithm is used to group computational cells into K zones of high similarity, as defined by dispersion metrics. This algorithm does not require the a priori specification of the number of zones. The evolutionary algorithm starts by initializing all the cells as one big zone. Then during the evolution, the algorithm iteratively checks the dispersion metrics in all zones and splits zones that have large parameter dispersion into smaller ones via a bisection-splitting method. If any new zones are generated from the splitting step, a K-means algorithm [5] is used to reallocate the cells into the zones. Following this procedure, the number of zones increases. The zoning algorithm is terminated when the dispersion metrics become smaller than user-specified thresholds in all zones.

Customized for engine combustion problems, the DMZ scheme uses temperature and equivalence ratio as features to measure the inter-cell similarity. The zoning algorithm uses the features sequentially so that the zoning is divided into two stages. In the first stage, the zoning is based on only temperature, and the zones evolve until the temperature dispersion in all the current zones is less than a user-specified threshold. In the second stage, the zones generated based on temperature are further split into smaller zones if the dispersion in equivalence ratio is larger than the corresponding threshold. 3.1.2. Solution of the kinetics equations at the zonal level

For each zone containing n computational cells, the zonal averaged temperature, pressure, and species densities are calculated. Using the averaged quantities as the initial conditions, the kinetic equations, Eq. (3), are integrated over a time-step, ∆t.

3.1.3. Backward-mapping formula

A general backward-mapping formula was proposed in the DMZ scheme. The formula preserves the initial species and temperature stratification among the cells. Using the zonal average species densities before and after time integration, we first calculate the total change of species mass in this zone:

∑ =Δ+ −=Δ n

i itk

ttkzonek Vm 1, )( ρρ , (6)

where n is the number of cells in the zone, and V is the cell volume. Then we distribute this total change of species mass to the individual cells based on the following criteria: if a species is consumed in the zone (

),...,1( Kk =

0, <Δ zonekm ), then the distribution of zonekm ,Δ among the cells is proportional to the mass distribution of species k at the initial time t; if a species is produced in the zone ( 0, ≥Δ zonekm ), then the distribution of

zonekm ,Δ is proportional to the mass distribution of the gas mixture at time t. Specifically, the density of species k in cell i is updated by

⎪⎪

⎩

⎪⎪

⎨

⎧

<ΔΔ+=

≥ΔΔ+=

∑

∑

=

Δ+

=

Δ+

0if)(

0if)(

,1 ,

,,,,

,1

,,,

zoneknj j

tjk

tik

zonekt

iktt

ik

zoneknj j

tj

ti

zonekt

iktt

ik

mV

m

mV

m

ρρ

ρρ

ρρρρ

(7)

The backward mapping formula not only conserves the total mass in a zone, it also guarantees that the updated species densities are non-negative, as shown in Ref. [3]. 3.2. Dynamic adaptive chemistry (DAC) scheme

To ensure validity over a wide range of thermochemical conditions, comprehensive chemical kinetic mechanisms for the combustion of realistic fuels typically include hundreds of species and thousands of elementary reactions. However, much smaller subsets of species and reactions often are adequate to capture the dominant reaction pathways for specific, local conditions over a short time span (typically taken to be the hydrodynamic time-step in CFD calculations). For example, including all the low-temperature hydrocarbon break-down reactions to simulate the kinetics in a burnt gas mixture, where mainly CO oxidation and NOx reactions are dominant, is a huge waste.

The dynamic adaptive chemistry (DAC) scheme proposed in Ref. [6,7] reduces the comprehensive detailed mechanism to locally valid smaller mechanisms on the fly. The DAC scheme was based on the directed-relation-graph with error propagation (DRGEP) method [8], which offers efficient linear-time reduction. We now briefly review the major steps of the DAC scheme. 3.2.1. Mechanism reduction using DRGEP

Given a detailed kinetic mechanism and a specific thermochemical state X(T, p, yk), where k is the species index (k=1, …, K) and yk is the mass fraction of species k, the directed relation graphs in the DRGEP method are constructed such that one vertex (species) is connected to all others by directed edges. These edges are weighted by the immediate dependence of one species on another. This dependence is quantified by the normalized contribution of species B to A, defined by

∑∑

=

=≡Ii iAi

Ii BiiAiABr

,1

,1

ωυδωυ

, (8)

where i is the reaction index (i=1, …, I), υAi is the stoichiometric coefficient of species A in the ith reaction, and ωi is the progress variable of reaction i. rAB is a measure of the error introduced to the production rate of A due to elimination of all the reactions that contain B. Certain species deemed of primary importance are selected as initial species in the reduced mechanism. Then starting from each of these pre-selected initial species, a breadth-first search (BFS) is performed to identify the species on which the initial species depends to form a subsidiary set. Consequently, the union of the subsidiary sets of all the initial species forms the active species set of the reduced mechanism. Thus the mechanism reduction is equivalent to identifying vertices to which there exist “strong” paths that connect them to a vertex in the initial set.

⎩⎨⎧

=otherwise0

involvesreaction th theif1 BiBiδ

The strength of the connection between the species being visited and the initial species diminishes as we proceed along a path. This diminution can be used to control the search depth. To quantify the decreasing dependence, an “R-value” is defined at each vertex V with reference to the initial vertex V0:

max)( 0 ijV rVR Π=

Ω, (9)

where Ω is the set of all possible paths leading from V0 to V, and Πrij is the chain product of the weights of the edges along the given path. Based on this definition, vertex V will be marked as “reachable” if RV0(V) is larger than a user-defined threshold value εR. Thus, all the reachable vertices starting from V0 constitute the subsidiary set of V0, and the union of such sets gives the species in the reduced mechanism. A pseudo-code for implementing this search algorithm can be found in Ref. [6]. This reduction method was tested on both diesel and gasoline surrogate fuels in Refs. [6,7]. The studies proved that fuel, CO and HO2 is an effective choice for the search-initiating species set, and suggested that εR=10-4 be used for all the surrogate fuels tested. 3.2.2. On-the-fly formulation of kinetics equations

The DRGEP method extracts a group of active species for the existing local thermochemical conditions. Consequently, a reaction is included in the reduced mechanism only if all the reactants and products are active species (third bodies are not counted as participants). Species not in the active set are treated as inactive, with their mass fractions kept fixed. The basis for the mechanism reduction gives assurance that, were they to be included, the small changes in the mass fraction of these species would have a negligible effect on the heat release rate and the evolution of key species.

Given a system that involves m active and n inactive species X(T, p, y1

a, …, yma, y1

i, …, yni), where the superscripts

“a” and “i” denote active and inactive species, respectively, the formulation can be expressed as

⎪⎪⎩

⎪⎪⎨

⎧

==

=

+ )).,,,,,,,(()),,,,,,,((

)),,,,,,,((

111

11

1111

in

iam

am

in

iam

am

am

in

iam

aa

yyyypTfTyyyypTfy

yyyypTfy

LL&LL&

M

LL&

XX

X

(10)

In Eq. (10), ordinary differential equations are formulated with respect to only active species, eliminating redundant equations due to inactive species and thus leading to a compact Jacobian matrix. When the rate functions are evaluated, however, all the species are considered, thus eliminating the need to include the third-body species into the reduced mechanisms. 3.3. Advanced ODE solver using sparse-matrix technology

Numerical integration of the stiff ODE system (Eq. (3)) is

commonly based on the backward differentiation formula (BDF), which typically results in highly super-linear scaling between the CPU time and number of equations (species). However, chemical kinetics systems are often intrinsically sparse. An advanced ODE solver that takes advantage of the sparsity of such reaction systems has been developed at Reaction Design. The sparse solver offers linear scaling between CPU time and number of equations with no loss of accuracy.

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4. Test cases

In this section, we first demonstrate the performance of

each of the solution techniques individually, and then discuss the feasibility of using large mechanisms in 3-D engine calculations with the help of all of the strategies. All the calculations were done on a Linux machine running a 2.4-GHz CPU. 4.1. PCCI and diesel combustion calculation using the DMZ scheme

The DMZ scheme was first tested in simulating a

gasoline-fueled premixed charge compression ignition (PCCI) engine with a compression ratio of 16:1. A hollow-cone swirl-type injector injects gasoline into the cylinder at -50 ºATDC, creating a highly stratified mixture. A 45-degree sector mesh is employed, which contains 15,000 cells at bottom dead center (BDC). A small 25-species 51-reaction isooctane oxidation mechanism extracted from a primary reference fuel (PRF) mechanism [9] was used to simulate gasoline combustion chemistry. The hollow-cone spray was modeled by the linearized instability sheet atomization (LISA) model [10]. The calculation was carried out from IVC (-143º ATDC) to EVO (130º ATDC). The maximum hydrodynamic time-step of 5 μs was used. Kinetic calculations and the DMZ scheme are invoked in a cell only when its temperature exceeds 600 K. In the DMZ calculations, the maximum dispersion of temperature and equivalence ratio in each zone are set to Tε =10 K and ϕε =0.05 respectively.

Time profiles of pressure and mass fraction of several key species computed using the DMZ scheme are compared with the results obtained from the fully resolved (cell-by-cell) calculation (Figs. 1 and 2). Computed temperature contours on a cut-plane are also compared between the DMZ solution and the fully resolved solution at -3 ºATDC when the major ignition occurs (Fig. 3). The legend bars in Fig. 3 are marked by the temperature range in the cylinder. In general, the DMZ calculation reproduced the fully resolved results with negligible loss of accuracy.

Fig. 1 Pressure traces computed using the DMZ scheme and the fully resolved solution (gasoline PCCI case)

Fig. 2 Species profiles computed using the DMZ scheme and the fully resolved solution (gasoline PCCI case)

(a) (b)

Fig. 3 Computed temperature contours on cut planes at CA=-3.0 ºATDC. (a) DMZ solution (b) Fully resolved solution. (gasoline PCCI case)

Figure 4 shows the number of computational cells that have temperatures exceeding the 600-K temperature threshold (i.e., those for which chemistry calculations are occur), the number of zones specified from the first stage of clustering (generated purely due to temperature stratification), and the final number of zones. On average, the final number of zones required is one order of magnitude smaller than the number of computational cells.

Fig. 4 Time histories of number of cells, final number of clusters, and first-stage number of clusters in the DMZ calculation (gasoline PCCI case)

The CPU times consumed by the dynamic zoning algorithm, τzoning, and by the chemistry calculation, τchem, are summarized in Table 1. τzoning measures the overhead introduced by the DMZ scheme. The zoning overhead of 51 minutes only amounts to ~4% of the time required for the fully resolved calculation, yet the greatly reduced number of chemistry calculations permits an 8-fold reduction in overall CPU time.

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Table 1 CPU time (minutes) consumed by zoning and by solving chemistry in the gasoline PCCI test case

τzoning τchem τtotal DMZ 51 92 142

Fully resolved 1180 1180

5

The DMZ scheme was further tested in simulating the

diffusion flames in conventional diesel engines. The same engine mesh used in the above PCCI case was recycled for this diesel case. In the diesel engine, diesel fuel is injected by a high pressure solid cone injector at -35 ºATDC, and a reduced 36-species n-heptane mechanism [11] is used to simulate the ignition behavior of diesel fuel. Other setup conditions are the same as those in the PCCI test case.

The computed pressure and heat release rate traces agree very well between the DMZ calculation and the fully resolved calculation, as shown in Fig. 5. This proves that the DMZ scheme can effectively resolve the stratification of chemistry parameters in a highly stratified diffusion flame.

Fig. 5 Pressure traces computed using the DMZ scheme and the fully resolved solution (diesel case)

Figure 6 shows the time profiles of number of cells and

zones in the diesel calculations. The corresponding CPU times consumed by zoning and solving chemistry are listed in Table 2. The use of the DMZ scheme reduced the time spent on solving chemistry from 39 hours to 2 hours. An overall speedup factor of 14 was achieved.

Fig. 6 Time histories of number of cells, final number of clusters and first-stage number of clusters in the DMZ calculation (diesel case)

Table 2 CPU time (minutes) in the conventional diesel engine case

τzoning τchem τtotal DMZ 38 130 168

Fully resolved 2350 2350 4.2. HCCI calculation using the DAC scheme and the sparse solver

The performance of the DAC scheme and the sparse ODE solver was assessed by calculating homogeneous charge compression ignition (HCCI) combustion of n-hexadecane using a single-cell mesh. A 381-species n-hexadecane mechanism was used as the baseline mechanism. The mechanism was obtained by reducing a 2116-species LLNL mechanism [12] using a guided mechanism reduction facility called Reaction Workbench [13], which is an add-on capability for CHEMKIN-PRO that is under development. The maximum time-step is set to 5 ms, a typical value constrained by the transport equations in engine calculations.

The DAC scheme has been comprehensively tested on PRFs and TRFs (toluene reference fuels) in Refs. [6,7]. Using the input parameters suggested by the previous studies, the DAC scheme performed very well on the n-hexadecane HCCI case. As an example, the pressure calculated using the DAC scheme perfectly matches the pressure obtained using the full mechanism, as shown in Fig. 7. Though not shown here, the sparse ODE solver also exactly reproduced the dense solver results as expected.

Fig. 7 Pressure traces computed using the DAC scheme and the full mechanism (n-hexadecane single cell HCCI)

The accumulated CPU time histories obtained by using different solution techniques are compared in Fig. 8 (note the logarithmic scale on the y-axis). In general, the use of individual or a combination of the advanced solvers achieved 14- to 28-fold speed-up compared to the full calculation using the conventional ODE solvers. When the sparse solver is used throughout the calculation, the local chemistry solution time is independent of the combustion phasing. When only the DAC scheme is applied, the calculation was slower than that using the sparse solver before ignition, where most of the hydrocarbon species were kept in the locally reduced mechanisms. However, the DAC calculation significantly speeds up after ignition because only a small fraction of species were kept in the reduced mechanisms for the high temperature kinetics. The speed-up is indicated by the reduced slope of the red curve at around -20ºATDC. The comparison

between the DAC scheme and the sparse solver suggests that computational efficiency is achievable if we combine the merits of the two strategies. This was proved by the blue time profile in Fig. 8, where a combination of solver approaches was used.

Fig. 8 Accumulated CPU time histories using different solution techniques 4.3. 3-D diesel combustion calculation using a combination of the solution strategies

Based on the tests of the individual solution techniques, we applied the 381-species n-hexadecane mechanism to simulating the diesel engine combustion presented in section 4.1. The purpose is to test the robustness of the solution techniques and obtain the CPU time needed for such a detailed calculation. This 3-D calculation combines the sparse solver and the DMZ scheme. The total CPU time spent solving the n-hexadecane chemistry is around 28 hours, which is 12.7 times the CPU time spent by the diesel calculation using the 36-species n-heptane mechanism (cf. Table 2). This indicates that with the help of the sparse solver the chemistry solution time roughly scales linearly with the number of chemical species in the kinetic mechanisms. The computed time profiles of number of cells and zones are shown in Fig. 9 (similar to the profiles shown in Fig. 6.) The cell/zone ratio varies from 10 to 100, giving an average speedup factor of 50. In addition, the sparse solver offers another 20-fold speed-up compared to the dense solver. Therefore, using a single CPU, the 28-hour computation would have taken more than 1000 days without the advanced solution strategies.

Fig. 9 Time histories of numbers of cells and clusters in the DMZ calculation (diesel case using the 381-species n-hexadecane mechanism)

5. Concluding Remarks

Several advanced strategies for accurately and efficiently solving detailed chemistry in multidimensional CFD are presented. The strategies include a dynamic multi-zone (DMZ) partitioning scheme, a dynamic adaptive chemistry (DAC) scheme, and a new ODE solver using sparse-matrix technology. Compared to conventional solution approaches, the use of these advanced strategies can improve the overall computational efficiency of solving the detailed fuel chemistry by two to three orders of magnitude.

Efforts to combine the best features of the solution techniques are in progress, which will further reduce the CPU time expenditure. All the solution techniques can be readily combined with parallel computation, enabling even faster turnaround speed in engine design. The CHEMKIN-CFD/API module is furthermore designed to be linked with any CFD software, though the demonstration was performed here using KIVA-3V. Acknowledgement The authors thank our colleagues, Drs. Abhijit Modak, Cheng Wang, and Chen-Pang Chou, for their help on the calculations. References 1. CHEMKIN-CFD, Reaction Design, San Diego, CA. 2. A. A. Amsden, KIVA-3V: A block-structured KIVA

program for engines with vertical or canted valves, Report LA-13313-MS, Los Alamos National Lab., 1997.

3. L. Liang, J.G. Stevens, J.T. Farrell, A dynamic multi-zone partitioning scheme for solving detailed chemical kinetics in reactive flow computations. Submitted to Combust. Sci. Technol.

4. A. Babajimopoulos, D.N. Assanis, D.L. Flowers, S.M. Aceves, R.P. Hessel, Int. J. Engine Res., 6, 497-512, 2003.

5. A.K. Jain, R.C. Dubes, Algorithms for Clustering Data, Prentice Hall, 1988.

6. L. Liang, J.G. Stevens, J.T. Farrell, Proc. Combust. Inst., 32, 527-534, 2009.

7. L. Liang, J.G. Stevens, S. Raman, J.T. Farrell, The use of dynamic adaptive chemistry in combustion simulation of gasoline surrogate fuels. Combust. Flame, in press, 2009.

8. P. Pepiot-Desjardins, H. Pitsch, Combust. Flame, 154, 67-81, 2008.

9. S. Tanaka, F. Ayala, J.C. Keck, J.C., Combust Flame, 133, 467-481, 2003.

10. D.P. Schmidt, I. Nouar, P.K. Senecal, C.J. Rutland, J.K. Martin, R.D. Reitz, and J.A. Hoffman, SAE Paper 1999-01-0496.

11. A. Patel, S.-C. Kong, R.D. Reitz, SAE Paper 2004-01-0558.

12. C. K. Westbrook, W. Pitz, O. Herbinet, H. J. Curran, E. J. Silke, A comprehensive detailed chemical kinetic oxidation mechanism for combustion of n-alkane hydrocarbons from n-octane to n-hexadecane, Combustion and Flame, in press, 2008.

13. Reaction Workbench, Reaction Design, San Diego, CA 2008.

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