Transported PDF modelling with detailed chemistry of pre- and auto-ignition in CH 4 /air mixtures

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1 Proceedings of the Combustion Institute 31 (27) Proceedings of the Combustion Institute Transported PDF modelling with detailed chemistry of pre- and auto-ignition in CH 4 /air mixtures K. Gkagkas, R.P. Lindstedt * Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2AZ, UK Abstract The pre- and auto-ignition behavior of methane under varying levels of preheat in a turbulent flow field has been studied through the combination of detailed chemistry with a transported PDF approach closed at the joint-scalar level. The study considers the Cabra Burner configuration, which consists of a central methane/air jet issuing into a vitiated co-flow. The aim of the work is to explore the detailed thermochemical flow structure and to substantially reduce uncertainties associated with the chemical kinetics. The applied chemistry features 44 solved species and 256 reactions and includes low temperature oxygen adducts. The mechanism has, in related work, been shown to reproduce the spontaneous temperature limit for methane and ethane along with ignition delays times at higher temperatures. Radiation is accounted for through the RADCAL method and the inclusion of enthalpy into the joint-scalar PDF. Molecular mixing is closed using the modified Curl s model and a set of time scale ratios (C / = 2.3, 2.5, 3. and 4.) have been used to explore the model sensitivity. The impact on predictions of variations in the pilot stream composition have been explored by varying concentrations of OH and H 2 over a wide range. A detailed analysis of the flame structure, focusing on the chemical processes occurring before and during the ignition, suggests that the burner conditions lead to a classical auto-ignition pattern with the early formation of HO 2 and CH 2 O prior to ignition. The work suggests that, under the current conditions, flame stabilization is dominated by turbulence chemistry interactions rather than by specific modes of flame propagation. The work shows a significant sensitivity to the pilot stream composition and that residual H 2 acts as an ignition promoter. However, the sensitivity to the time scale ratio C / is shown to be less than can be expected from studies of flame extinction using the same methodology. Ó 26 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Flame structures; PDF calculations; Detailed chemistry; Auto-ignition 1. Introduction The ability of calculation methods to reproduce pre- and auto-ignition phenomena in turbulent flow fields is of fundamental importance in the context of flame stabilization and emerging * Corresponding author. Fax: address: p.lindstedt@imperial.ac.uk (R.P. Lindstedt). technologies such as HCCI engines. Mastorakos et al. [1] performed DNS of auto-ignition in turbulent flows and compared the properties of the corresponding turbulent and laminar flames. Differences in the trends of auto-ignition were explained using the turbulent time- and lengthscales along with partial premixing. Cabra et al. [2,3] introduced a simple burner geometry that permits the experimental study of auto-ignition and flame lift-off in a well-defined flow configuration. The burner consists of a co-axial fuel jet /$ - see front matter Ó 26 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:1.116/j.proci

2 156 K. Gkagkas, R.P. Lindstedt / Proceedings of the Combustion Institute 31 (27) issuing into hot combustion products from a lean premixed hydrogen flame. The subsequent ignition occurs in the shear layer and the geometry permits the detailed study of turbulence chemistry interactions. Extensive measurements of H 2 /N 2 jet flames in the Cabra burner configuration [3 5] and related calculations using transported PDF-based closures have been performed by Masri et al. [6], Cabra et al. [3] and Cao et al. [7]. A range of mixing models and detailed or reduced chemical kinetic mechanisms have been applied. Good agreement between experimental and computational results has been achieved and the studies suggest that the flames are largely controlled by chemical kinetics. The transported PDF approach has been shown to be able to capture extinction and re-ignition processes in hydrocarbon flames [6,9 11]. However, Cao and Pope [13] have shown a significant sensitivity to the applied chemistry as part of a study of a range of piloted turbulent diffusion flames. Past studies of methane flames in the Cabra configuration include that of Cabra et al. [8] who explored the influence of mixing models and boundary conditions. A perhaps surprising result was that the modified Curl s model [12] was found to perform comparatively well. The focus of the current study is on a detailed examination of the thermochemical flow structure of methane/air flames in a vitiated co-flow [8] using a comprehensive detailed reaction mechanism that includes low-temperature chemistry. The sensitivity of predictions to boundary conditions, chemical rate constants and the time scale ratio are also explored. 2. The Cabra burner The Cabra burner [3] consists of a fuel jet nozzle and a surrounding perforated disk. The fuel jet has an inner diameter of D = 4.57 mm and a wall thickness of.89 mm. The disk has a diameter of 21 mm and 22 holes with a diameter of 1.58 mm. Each hole stabilizes a premixed flame and thus provides a hot co-flowing stream. The central fuel jet nozzle extends 7 mm downstream of the plane of the perforated disk to ensure uniform co-flow properties. The boundary conditions are presented in Table 1 along with other properties. Detailed single point measurements of temperature and composition were performed using the Raman Rayleigh- LIF technique and mass fractions of CH 4,N 2,O 2, H 2 O, OH, CO and CO 2 were measured [2,8]. 3. Computational details The transported PDF approach of Lindstedt and co-workers [1,11,2] is used in the present Table 1 Boundary conditions [2,4] Fuel jet Co-flow Re 28, 23,3 d (mm) Velocity (m/s) T (K) X O X N X H2O X OH (ppm) <1 2 X H2 ðppmþ 1 1 X NO (ppm) <1 <1 X CH /.4 work. The joint-scalar PDF may be written as the following random vector ~f / ð/ a ; f ; H; xþ ð1þ where / i with i =1,...,a are the species mass fractions of the gas phase and H is the enthalpy of the mixture. In the current hybrid approach, the flow field is closed at the second moment level, using the pressure strain correlation from Speziale et al. [14]. A transport equation for the composition PDF is coupled and solved using a Langrangian particle-based Monte Carlo method [21]. A full detailed mechanism that includes lowtemperature chemistry capable of reproducing the spontaneous ignition of methane is used. The chemistry is based on the work of Lindstedt and Meyer [15] and extended by Rizos [16]. The mechanism consists of 256 reactions and the following 44 species: CH 4,H,H 2 O, O 2,H 2 O 2,O, OH, HO 2, CO, CO 2,C 2 H 4,CH 3,C 2 H 2,CH 3 OH, C 2 H 6,H 2,CH 2 O, C 2 O, H 2 C 2,CH 3 CHO, CH 3 CO, CH 3 OO, C 2 H 4 O, CH 2 CHO, C 2 H 5 OO, CH 3 O, CHO, 1 CH 2, CH 2 OH, C 2 H 5 OOH, CH 3 OOH, C 2 H 5, C 2 H 4 OOH, C 2 H 5 O, CH, 3 CH 2, C 2 H, HCCO, CH 2 CO, C 2 H 3, C 1, C 2, CH 2 CHO and N 2 [17]. The radiative loss term is expressed on the basis of the optically thin assumption [18] as outlined by Lindstedt and Louloudi [19]. The turbulent transport of the PDF is modelled through a gradient diffusion approximation with the turbulent Prandtl number (r t ) set to unity. The second moment closure provides the scalar mixing time scale (s / ) used in the transported PDF solution procedure via the modified Curl s model [12] applied here. s 1 / ¼ e / ¼ C / ~ ¼ C / g / 2 2 ~ k 2 s 1 T ð2þ In the above equation, e / is the scalar dissipation rate and g / 2 the corresponding scalar variance. In

3 K. Gkagkas, R.P. Lindstedt / Proceedings of the Combustion Institute 31 (27) the present work, C / = 2.3 [11] is retained as a base case with C / 6 4. also explored Boundary conditions and solution details The flow is treated as axi-symmetric and the boundary conditions were based on experimental data [2,4] as shown in Table 1. Boundary values for minor species concentrations were computed based on chemical equilibrium. For H 2, the concentration was set to correspond to the measurements taken one diameter downstream from the jet exit plane. The axial domain extends from x/d = to x/d = 1 and the (adaptive) radial domain from = 5 to approximately = 15. A uniform velocity was used for the co-flow (r > D/2). The initial velocity profile (r < D/2) for the fuel jet was derived by assuming a fully developed turbulent pipe flow. The velocity co-variance was specified via the correlation coefficient (q uv ). The values from Cao et al. [7], defined in terms of R D/2, were used. Between 6 r/r 6 1 a linear variation of q uv from to.4 was assumed and followed by q uv =.4 for 1 6 r/r and q uv = for r/r > The temperature and composition profiles were set as step functions across the inner diameter (D = 4.57 mm) of the nozzle. 4. Computational results Computations were performed with 8 and 14 computational cells (N c ) in the radial direction, 19 axial steps and with 12, 2 and 4 expected stochastic particles (N p ) per cell. No significant differences were observed as a result of increased refinement. The results shown below correspond to N c = 8 and N p = 12 unless otherwise indicated Centerline profiles Comparisons between measurements and calculations of temperature, mixture fraction and various scalars along the central axis of the jet are shown in Figs. 1 and 2. The centerline profiles of the Favre-averaged temperature, mixture fraction and their fluctuations are shown in Fig. 1. Predictions of mean values generally agree well with measurements and scalar fluctuations also show a satisfactory trend. The mean values for the mass fractions of CH 4, O 2, CO, CO 2,H 2 and OH along the centerline are shown in Fig. 2. It can be seen that there arguably is an under-prediction of CO and an overprediction of OH. However, the discrepancies remain within experimental error. The flow studied exhibits a two-stage behavior. Initially, the mixing process is dominant and there is no apparent effect of chemical reaction. The temperature at the centre line is rising slowly, with x/d [-] low levels of fluctuations, whereas the concentrations of CO, CO 2 and OH remain negligible. As the flow progresses downstream, the first stage is followed by an ignition region, which is accompanied by strong temperature fluctuations and rapidly increasing radicals concentrations. The effects of ignition become obvious at the centerline at the axial position of x/d Radial profiles x/d [-] Fig. 1. Temperature and mixture fraction statistics along the centerline. Lines correspond to computations and symbols to measurements. Y CH4 [-] Y CO [-] Y OH [-] x/d [-] x/d [-] A better understanding of the ignition process requires examination of the radial profiles of the various scalars. The radial profiles of the Favreaveraged temperature are shown in Fig. 3. The agreement obtained for mean and rms values is generally good. It is obvious from the rise in temperature that ignition occurs between the axial distances of x/d = 3 and x/d = 4. Until x/d = 3 the effects of mixing dominate, whereas at x/d = 4 there is a distinct rise in the mean temperature above the co-flow level, as well as an increase in fluctuations T" [K] f" [-] Y O 2 [-] Y CO2 [-] Y H2 [-] Fig. 2. Species mass fractions along the centerline. Lines correspond to computations and symbols to measurements.

4 1562 K. Gkagkas, R.P. Lindstedt / Proceedings of the Combustion Institute 31 (27) x/d=1 x/d=15.6 x/d=3 x/d=4.4 x/d=3 x/d=4 PDF [-].2.6 x/d=5 x/d=7 2 x/d=5 x/d= Fig. 3. Radial profiles of Favre-averaged temperature and temperature fluctuations with increased axial distance. Solid lines correspond to calculated mean values, dashed lines to calculated RMS values, circles to measured mean values and squares to measured RMS values Fig. 5. The downstream evolution of the conditional PDF of temperature. Lines correspond to calculations and symbols to measurements (N p = 4)..3.2 x/d=1 x/d=15 The radial profiles of the mixture fraction are shown in Fig. 4. The level of agreement between the simulation and the experiment indicates a good representation of the mixing process. The radial profiles generally show acceptable agreement at most axial positions. However, significant differences are observed for molecular hydrogen around the point of ignition. The current calculations do not account for upstream diffusion of H 2 and an elliptic study of the flame could prove helpful. Nevertheless, the downstream evolution of the conditional PDF of temperature is satisfactorily predicted as shown in Fig. 5. The corresponding radial profiles for the hydroxyl radical are shown in Fig. 6, where the scale is changed to more clearly show the level of agreement obtained. The OH radical is a good indictor of x/d=1 x/d=3 x/d=5 x/d=15 x/d=4 x/d= Fig. 4. Radial profiles of Favre-averaged mixture fraction and mixture fraction fluctuations with increased axial distance. Lines and symbols as in Fig. 3. Y OH [-] x/d=3 x/d=5 x/d=4 x/d= Fig. 6. Radial profiles of Favre-averaged OH mass fractions with increased axial distance. Lines and symbols as in Fig. 3 (N p = 4). post-ignition chemical activity. In the calculations, the maximum OH concentration appears slightly shifted towards the co-flow region. The implication is that ignition appears to occur under slightly leaner conditions than recorded experimentally. It is also obvious that the OH levels are somewhat higher than those measured but, as shown for the centerline, the discrepancies are probably within experimental uncertainties Scatter and conditional plots in mixture fraction space Scatter plots of temperature versus mixture fraction (Fig. 7) show the state of particles during the evolution of the flow and provide further insight into the ignition process. At x/d =3 most particles reside on the mixing line between the fuel and the oxidizer stream, with a few particles on the lean side indicating the onset of

5 K. Gkagkas, R.P. Lindstedt / Proceedings of the Combustion Institute 31 (27) x/d=3 x/d=4 x/d=5 x/d= Fig. 7. Scatter plots of temperature versus mixture fraction with increased axial distance. The lines correspond to chemical equilibrium. chemical reaction. At x/d = 4, ignition is obvious, with particles on the lean side reaching adiabatic temperatures. Further downstream, the majority of the particles have been fully burnt. The conditional statistics can be very useful to depict the evolution of the ignition process by providing insight into the scalar structure. Generally the agreement obtained mirrors that shown in physical space. However, the results for the ensemble averages of YOH, shown in Fig. 8, do illustrate an interesting point. It can be seen that immediately after ignition, the computational results suggest that ignition initially occurs for lean to stoichiometric mixtures Sensitivity analysis The base case simulation outlined above suggests that overall good agreement can be 1563 obtained. However, it is also useful to investigate key parametric sensitivities. In the present work, the effects of changes in the chemical composition of the co-flow, variations of the co-flow temperature and a variation in the time scale ratio constant C/ are explored. Values of C/ = 2.3, 2.5, 3. and 4. yield the corresponding H/D = 35., 34.7, 34. and 31.2 at ignition. By increasing the value of C/, the mixing of the two streams is increased, therefore leading to a shorter lift-off height. However, the magnitude of change is such that it would appear that ignition is mainly chemically controlled with a comparatively modest sensitivity to changes in the time scale ratio. This finding is notably different from studies of flames undergoing extinction where comparatively small changes can have a significant impact [9 11]. A sensitivity analysis was also performed on the effect of the co-flow temperature on the liftoff height. Several criteria were used in order to define the ignition point, including reference concentrations of species like OH, CH2O and HO2. The results can be seen in Fig. 9, where comparisons are made to the measurements of Cabra et al. [8]. The trend is arguably captured satisfactorily and the choice of indicator does not affect the recorded ignition point strongly, especially for higher initial temperatures. The different co-flow temperatures can also be expected to impact the boundary conditions on species concentrations. The effects of increasing concentrations of H2 and OH in the co-flow were investigated systematically. Boundary condition values of X H2 = , , and yield the corresponding H/D = 49., 44., 35. and 1.5 at ignition. Similarly, boundary condition values of XOH = , and yield the corresponding H/D = 35., 32. and 2.. It is obvious that the ignition point is very sensitive to OH and H2 concentrations in the co-flow..2 x/d=3 x/d= x/d=5 x/d=7 H/D YOH [-] Fig. 8. Evolution of the conditional ensemble average of OH mass fraction. Solid lines correspond to calculations with Np = 12, dashed lines to Np = 4 and circles to measurements. Fig. 9. Effect of co-flow temperature on lift-off height as obtained with different indicator species: : CH2O, : OH, + ÆÆ +: HO2, : measurements.

6 1564 K. Gkagkas, R.P. Lindstedt / Proceedings of the Combustion Institute 31 (27) Analysis of pre- and auto-ignition chemistry Species of particular relevance to ignition include intermediates such as CH 2 O, HO 2 and H 2 O 2. The temperature is shown in Fig. 1 and the HO 2 radical field is shown in Fig. 11. The build up of the latter species in the mixing region is obvious, with the peak occurring in the area of the initial ignition position. The mass fraction is subsequently quickly reduced and it is obvious that HO 2, and H 2 O 2, which has a similar concentration pattern, are crucial during the pre-ignition phase. Their subsequent decomposition leads to the formation of the hydroxyl radical, which triggers the high temperature chemistry. It can be seen from Fig. 1 that the concentration of the OH radical has a similar pattern to the temperature field, shifted upstream and towards the centreline. The formation of formaldehyde is characteristic of the conversion of methane at lower temperatures as shown in Fig. 11. It can be seen that the maximum concentration of CH 2 O is reached just before the flame front, and then quickly drops as the high temperature burning process develops. It is evident that HO 2 and CH 2 O play a crucial role during the pre-ignition stage and create the proper conditions for the initiation of the high temperature chemistry. x/d Y CH2O 7.E-4 5.6E-4 4.2E-4 2.8E-4 1.4E-4.E+ YHO2 2.E-5 1.6E-5 1.2E-5 8.E-6 4.E-6.E+ Fig. 11. Computed CH 2 O (left plot) and HO 2 (right plot) mass fractions (N p = 4). x/d YOH 3.E-3 2.4E-3 1.8E-3 1.2E-3 6.E-4.E Reaction path analysis The pattern discussed above is consistent with a classical auto-ignition behavior and a study of the reaction paths and key sensitivities was performed to investigate the critical reactions controlling the pre- and auto-ignition process. The relevant reaction rate constants are shown in Table 2. In the pre-ignition region (2 6 x/d 6 5) the HO 2 radical is formed mainly by reactions (5) and (34) with relative contributions of 47% and 37%, respectively. O 2 + H (+ M) HO 2 ðþmþ ð5þ CHO + O 2 CO þ HO 2 ð34þ Consumption of the HO 2 radical is mainly through reactions (6), (8) and (74) which contribute 14%, 51% and 2%. HO 2 +H OH þ OH ð6þ HO 2 +OH H 2 O þ O 2 ð8þ Fig. 1. Computed temperature (left plot) and OH (right plot) mass fraction (N p = 4). CH 3 +HO 2 CH 3 O þ OH ð74þ Formaldehyde is formed via reactions (67), (81) and (85) which contribute 28%, 26% and 17%. Reaction (81) is the dominant channel for CH 3 O decomposition.

7 K. Gkagkas, R.P. Lindstedt / Proceedings of the Combustion Institute 31 (27) Table 2 Key reactions with rate coefficients in the form k = AT n exp( E/RT) Number Reaction A a n a E a Ref. 1 H+O 2 O + OH 3.55E [22] 4 OH+OH H 2 O + O 3.57E [23] 5 b O 2 +H+M HO 2 +Mk 1.41E+12.8 [24] k E+9.44 [25] 6 HO 2 +H OH + OH 1.68E [23] 8 HO 2 +OH H 2 O+O E [23] 17 CO + OH CO 2 + H 6.32E [24] 34 CHO + O 2 CO + HO 2 1.2E [26] 35 c CHO + M CO + H + M 1.86E [27] 6 CH 2 O+H CHO + H E [24] 62 CH 2 O+OH CHO + H 2 O 3.43E [28] 67 CH 3 +O CH 2 O + H 8.43E+1 [23] 74 CH 3 +HO 2 CH 3 O + OH 1.8E+1 [23] 81 CH 3 O+M CH 2 O + H + M 5.45E [23] 85 CH 2 OH + O 2 CH 2 O+HO E [29] 9 CH 4 +OH CH 3 +H 2 O 1.56E [28] a Units are kmol, m 3, s, K and kj/mol. b Chaperon efficiencies are 2. for H 2, 11. for H 2 O, 1.9 for CO, 3.8 for CO 2,.4 for N 2 and 1. for all other species. Troe parameter is F c =.5. c Chaperon efficiencies are 1.89 for H 2, 6.5 for H 2 O, 2.5 for CO, 2.5 for CO 2 and 1. for all other species. CH 3 +O CH 2 O þ H CH 3 O+M CH 2 O þ H þ M ð67þ ð81þ CH 2 OH + O 2 CH 2 O þ HO 2 ð85þ In the second region (3 6 x/d 6 6), where the maximum concentration of CH 2 O can be found, HO 2 is consumed mainly by reactions (6), (8) and (74) which contribute 27%, 35% and 21%. HO 2 +H OH þ OH ð6þ HO 2 +OH H 2 O þ O 2 ð8þ CH 3 +HO 2 CH 3 O þ OH ð74þ Consumption of CH 2 O is mainly via reactions (6) and (62) which contribute 12% and 75%. Lindstedt and Meyer [15] found in their study of CH 3 OH oxidation that reactions (6) and (62) exhibited similar sensitivities, but with opposite signs. Finally, in the high temperature region (45 6 x/d 6 8) CO is formed through reactions (34) and (35) which contribute 12% and 53%. Reaction (17) is virtually the only consumption path. CO + OH CO 2 þ H ð17þ CHO + O 2 CO þ HO 2 ð34þ R85 R67 R62 R8 CH 2 O+H CHO þ H 2 CH 2 O+OH CHO þ H 2 O ð6þ ð62þ R6 The OH radical is formed predominantly by reactions (1) and (4) which contribute 37% and 22%. H+O 2 O þ OH ð1þ OH + OH H 2 O þ O ð4þ R ln (H/Ho) / ln 5 Fig. 12. Lift-off height logarithmic response sensitivities.

8 1566 K. Gkagkas, R.P. Lindstedt / Proceedings of the Combustion Institute 31 (27) CHO + M CO þ H þ M ð35þ The main reaction paths indicate the crucial role of CH 2 O and HO 2 at the different pre-ignition stages. A sensitivity analysis was performed by multiplying and dividing the key reaction rate constants identified above by a factor of 5. The results are shown in Fig. 12. It is evident that the ignition point is particularly sensitive to reactions (6), (8) and (62). 6. Conclusions In the present work, a transported PDF approach, closed at the joint-scalar level, is coupled with a second moment closure for the velocity field and a detailed chemical mechanism featuring 44 independent scalars to compute a CH 4 /air lifted flame. Extensive comparisons of the computed results with experimental data illustrate the ability of the current modeling approach to accurately predict the detailed thermochemical structure of the flame studied and the potential to predict auto-ignition phenomena. A comprehensive examination of the different scalar fields was conducted, providing useful insight on the processes occurring during the different stages of the flow. The most important minor species during the pre-ignition phase were identified and the crucial role of the H 2, HO 2 and CH 2 O chemistry was illustrated. The current approach is using a parabolic formulation and hence there is no mechanism for upstream propagation of a turbulent edge flame. The results thus indicate that under the current conditions the flow may be described as a classical auto-ignition event in a turbulent flow field. The sensitivity of results to parametric variations in modeling constants and boundary conditions reveal a sensitivity to the latter. In particular, it appears that the presence of H 2 in the shear layer has a significant role in promoting the onset of ignition. Acknowledgment The authors acknowledge the financial support of BP Global Fuels Technology and are grateful for the interest shown by Dr C. Goodfellow, Dr J.T. Joseph, Dr R. Kay, Dr D.B. McLeary, Dr I.A.B. Reid and Dr J.S. Rogerson. References [1] E. Mastorakos, T.A. Baritaud, T.J. Poinsot, Combust. Flame 19 (1 2) (1997) [2] R. Cabra, Turbulent Jet Flames in a Vitiated Coflow, Tech. Rep. CR , NASA (24). [3] R. Cabra, T. Myhrvold, J.Y. Chen, R.W. Dibble, A.N. Karpetis, R.S. Barlow, Proc. Combust. Inst. 29 (22) [4] R. Cabra, R.W. Dibble, available from: < (22). [5] A.R. Masri, R.W. Bilger, available from: < (23). [6] A.R. Masri, R. Cao, S.B. Pope, G.M. Goldin, Combust. Theor. Model 8 (1) (24) [7] R. Cao, S.B. Pope, A.R. Masri, Combust. Flame 142 (4) (25) [8] R. Cabra, J.Y. Chen, R.W. Dibble, A.N. Karpetis, R.S. Barlow, Combust. Flame 143 (4) (25) [9] Q. Tang, J. Xu, S.B. Pope, Proc. Combust. Inst. 28 () [1] R.P. Lindstedt, S.A. Louloudi, E.M. Vaos, Proc. Combust. Inst. 28 () [11] R.P. Lindstedt, S.A. Louloudi, Proc. Combust. Inst. 29 (22) [12] J. Janicka, W. Kolbe, W.J. Kollmann, Non-Equil. Thermodyn. 4 (1979) [13] R. Cao, S.B. Pope, Combust. Flame 143 (4) (25) [14] C.G. Speziale, S. Sarkar, T.B. Gatski, J. Fluid Mech. 227 (1991) [15] R.P. Lindstedt, M.P. Meyer, Proc. Combust. Inst. 29 (22) [16] K.A. Rizos, Detailed Chemical Kinetic Modelling of Homogeneous Systems, PhD thesis, Imperial College London, October 23. [17] 26 (to be uploaded). [18] W. Grosshandler, RADCAL: A Narrow-Band Model for Radiation Calculations in a Combustion Environment NIST Technical Note 142, [19] R.P. Lindstedt, S.A. Louloudi, Proc. Combust. Inst. 3 (25) [2] R.P. Lindstedt, H.C. Ozarovsky, Combust. Flame 143 (4) (25) [21] S.B. Pope, Prog. Energ. Combust. Sci. 11 (1985) [22] J.P. Hessler, J. Phys. Chem. A 12 (24) (1998) [23] K.M. Leung, R.P. Lindstedt, Combust. Flame 12 (1 2) (1995) [24] D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, G. Hayman, Th. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys. Chem. Ref. Data 23 (1994) [25] J. Troe, Proc. Combust. Inst. 28 (21) [26] C.C. Hsu, A.M. Mebel, M.C. Lin, J. Chem. Phys. 15 (6) (1996) [27] R.S. Timonen, E. Ratajczak, D. Gutman, A.F. Wagner, J. Phys. Chem. 91 (1987) [28] D.L. Baulch, C.J. Cobos, R.A. Cox, C. Esser, P. Frank, Th. Just, J.A. Kerr, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys. Chem. Ref. Data 21 (1992) [29] H.H. Grotheer, G. Riekert, D. Walter, T. Just, J. Phys. Chem. 92 (14) (1988)