Journal of Thermal Science and Technology

Transcription

1 Science and Technology Vol. 5, No., Development o an Ethanol Reduced Kinetic Mechanism Based on the Quasi-Steady State Assumption and Feasiility Evaluation or Multi-Dimensional Flame Analysis * Masaki OKUYAMA **, Shinichiro HIRANO **, Yasuhiro OGAMI **, Hisashi NAKAMURA **, Yiguang JU *** and Hideaki KOBAYASHI ** ** Institute o Fluid Science, Tohoku University, -- Katahira, Aoa-ku, Sendai, Miyagi , Japan okuyama@lame.is.tohoku.ac.jp *** Department o Mechanical and Aerospace Engineering, Princeton University, D5 E-Quad, Olden Street, Princeton, NJ 8544, United States Astract A -step reduced kinetic mechanism o ethanol, a potential sustainale energy source as a iouel, was developed ased on the detailed reaction mechanism proposed y Saxena and Williams using the Computational Singular Perturation (CSP) method ased on the Quasi-steady State Assumption (QSSA). Feasiility evaluation o the reduced kinetic mechanism or multi-dimensional lame analysis, i.e., the dierence in numerical results and convergence time etween the detailed reaction mechanism and the reduced kinetic mechanism, was also perormed to investigate the applicaility o the ethanol reduced kinetic mechanism to the development o practical comustors. To consider urther industrial applications, the reduced kinetic mechanism was incorporated into the commercial computational luid dynamics (CFD) code FLUENT using the User Deined Function (UDF) code developed in the present study. Numerical results calculated with the detailed reaction mechanism and the reduced kinetic mechanism, i.e., temperature proiles, chemical species proiles and laminar urning velocities, were in good agreement or oth two-dimensional premixed and non-premixed lame calculations. Convergence time using the reduced kinetic mechanism was consideraly reduced compared to that using the detailed reaction mechanism, indicating the applicaility and advantage o a reduced kinetic mechanism ased on QSSA or multi-dimensional lame analysis. An additional reduction o the computational time was achieved y using oth the reduced kinetic mechanism and In Situ Adaptive Taulation (ISAT) solver y Pope et al. Key words: Reduced Kinetic Mechanism, Quasi-Steady State Assumption, Ethanol, Multi-Dimensional Flame Analysis, Computational Time. Introduction *Received Apr., (No. -39) [DOI:.99/jtst.5.89] Copyright y JSME Biomass ethanol, which is a potential sustainale energy source, is expected as an alternative uel o practical comustors such as internal comustion engines due to its low CO emissions rom the viewpoint o Caron neutral. It is also expected as a hydrogen carrier or urther applications. These acts have motivated experimental and numerical studies on the chemical kinetics o ethanol oxidation. In previous works, several detailed reaction mechanisms o ethanol oxidation have een proposed ()-(5), e.g., a 383-step reaction 89

2 Vol. 5, No., mechanism including 57 species y Marinov (3) and a 35-step reaction mechanism including 46 species y Saxena and Williams (5), the validity o which was also tested against the experimental data. However, despite the availaility o detailed ethanol reaction mechanisms, application o these mechanisms to multi-dimensional lame analysis or industrial purposes is diicult ecause the present perormance o computers is limited. Namely, recently availale detailed reaction mechanisms o ethanol oxidation consist o tens o chemical species and hundreds o elementary reactions, which require a large memory and convergence time, thus making it quite diicult to perorm multi-dimensional numerical simulations with detailed reaction mechanisms. Hence, the overall reaction mechanism which consists o a single reaction or two reactions is currently used in many cases (6) ; however, the accurate prediction o the lame temperature and gas emissions, which is essential or practical comustor design, is diicult when the overall reaction mechanism is used. Furthermore, the overall reaction mechanism is not capale o numerical simulations under a wide range o operating conditions (7), i.e., amient pressures, equivalence ratios and initial gas temperatures. The reduced kinetic mechanism ased on the Quasi-steady State Assumption (QSSA) has een a promising solution to this prolem. The reduced kinetic mechanism decreases the calculation cost y reducing the numer o chemical species which are considered and simultaneously presents etter numerical results, i.e., the accurate lame temperature and major species mole raction proiles, under a relatively wide range o conditions when the Quasi-steady State (QSS) species are selected appropriately or the target condition (7)-(9). In previous studies, a large numer o reduced kinetic mechanisms were proposed or methane and other major uels (8) ; however, to the est o our knowledge, little research on the reduced kinetic mechanism o ethanol comustion has een conducted. Moreover, the easiility o reduced kinetic mechanisms proposed in the previous research (8, 9) was mostly evaluated ased on ignition delay calculations and one-dimensional laminar lame calculations, ew o which were applied to the multi-dimensional practical numerical simulations. Saito et al. (7) pointed out that numerical instaility was caused y the molecular diusion o the H radical into the cold area in the case o two-dimensional diusion lame calculations using the 4-step CH 4 /air reduced kinetic mechanism y Peters (8). This indicates that the easiility o the reduced kinetic mechanism or such things as the choice o QSS species should e tested on multi-dimensional numerical simulations eore eing applied to highly time-consuming industrial lame analysis. Although some reduced kinetic mechanisms have een successully implemented into the Computational Fluid Dynamics (CFD) code (,), the comparison o the total convergence time and numerical results etween the detailed reaction mechanism and reduced kinetic mechanism has not een suicient. The purpose o the present study, thereore, was to develop a reduced kinetic mechanism o ethanol oxidation ased on QSSA and to evaluate the easiility o that mechanism or multi-dimensional numerical simulations. Firstly, several possile ethanol-reduced kinetic mechanisms were developed and each o them was tested on one-dimensional premixed lame calculation, which is more suitale or the evaluation o mechanism validity and computational cost ecause the sensitivity to the reaction kinetics is signiicant. Then the astest and stale one was applied to two-dimensional premixed Bunsen lame calculations and diusion lame calculations to elucidate the applicaility o the reduced kinetic mechanism ased on QSSA to multi-dimensional numerical simulations.. Numerical methods. Development o an ethanol reduced kinetic mechanism In this study, a reduced kinetic mechanism was developed y using the method o 9

3 Vol. 5, No., Computational Singular Perturation (CSP) y Lu et al. () ased on the QSSA. In this method, to speciy the QSS species, reaction time scale analysis was conducted on the result o the zero-dimensional calculation using the Perect Stirred Reactor (PSR) (3) model. This was perormed over a wide range o equivalence ratio, φ, o.5 to.6 and amient pressure, P, o. to 5. MPa. Then the QSS species was speciied when its time scale is less than the critical time scale ased on the extinction residence time and the threshold actor (). As a starting mechanism, the detailed reaction mechanism o ethanol oxidation proposed y Saxena and Williams (5), which consists o 46 chemical species and 35 reactions, was chosen. In this strategy, various reduced mechanisms, i.e., 9- to 6-step reduced kinetic mechanisms, were generated. The procedure o generation and evaluation o reduced kinetic mechanisms are descried in the results and discussion section. The gloal reactions, considered species and QSS species o the -step ethanol reduced kinetic mechanism developed in the present study are shown in Tale. Here the considered species are the chemical species used in the calculation with the reduced kinetic mechanism. The -step reduced kinetic mechanism o the present study includes important species or the comustion analysis, e.g., CO, H and OH, this analysis eing expected to calculate the lame structure properly under a wide range o conditions. Tale. Gloal reactions o -step reduced kinetic mechanism developed in the present study. Considered species CH5OH, O, HO, CO, CO, H, H, OH, O, HO, HO, CHO, CH4, CH3, CH4, CH6,CH, CHCO, HCCO, CH3OH, CH3CHO,C3H3,C3H6, Ar, He,N QSS species CH4O, CH3, C3H8, C3H4, CHOH, C3H5, CH3CHOH, CH, T-CH, CH, I-C3H7, CH5, HCO, N-C3H7, CH3O, CH3CHO, CHCHO, S-CH, CH3CO, CHCHOH No Gloal reactions R H + O OH + O R O + H H + OH R3 OH + H H + HO R4 H H R5 OH + O HO R6 OH HO R7 OH + CO H + CO R8 H + CH3 CH4 R9 H + CO CHO R O + CH3 H + CHO R O + CH4 H + CHCO R OH + CH3 H + HO + CH4 R3 CH3 CH6 R4 OH + CH3 H + HO + CH R5 O + CH H + HCCO R6 H + CH3OH HO + CH3 R7 H + HO + CHCO O + CH3CHO R8 H + CH5OH OH + CH3 R9 H + O + C3H3 CO + CH4 R O + C3H6 H + CO + CH4. One-dimensional premixed lame analysis The reduced kinetic mechanisms were irstly applied to the one-dimensional premixed lame calculations using the PREMIX (4) code, and their perormance were evaluated preliminary to the two-dimensional lame simulations. The continuity, momentum, energy and species equations or a steady laminar low were solved with the damped Newton method (4). Coeicients or viscosity, thermal conductivity and mass diusion were 9

4 Vol. 5, No., calculated using the thermodynamic and transport dataase y Saxena and Williams (5) and transport model y CHEMKIN-II (5). To enale the numerical simulation with the reduced mechanism, the suroutine CKWYP in the PREMIX code, which calculates the rate o each elementary reaction, was replaced y a suroutine developed y the authors. Numerical simulations were conducted under various conditions, i.e., equivalence ratios, φ, o.5 to.5 and amient pressures, P, o. to 3. MPa. The temperature o the premixed gas, T, was constant at 98 K. The length o the calculation domain was X = cm in this study. Zero-gradient oundary condition or all physical quantities was speciied at the end o calculation domain. CPU time was measured to evaluate the convergence time using a typical personal computer (CPU: Pentium4-3. GHz, memory:. GB)..3 Two-dimensional premixed and non-premixed lame analysis A reduced kinetic mechanism was applied to the -D premixed Bunsen lame calculations and -D diusion lame calculations using the commercial code FLUENT (6) or urther practical calculations. The continuity, momentum, energy and species equations or a laminar steady low were solved with the SIMPLE algorithm. Eects o uoyancy and radiation were not considered in this study. Coeicients or viscosity, thermal conductivity and mass diusion were calculated using the thermodynamic and transport dataase y Saxena and Williams (5) and the transport model y FLUENT. Figure shows the computational domains o the -D numerical simulations in this study. The domain sizes were mm x 4 mm and 5 mm x mm or -D Bunsen lame calculations and -D diusion lame calculations, respectively. A structured grid was employed or oth calculation domains. The grid interval was 5 μm and μm or each calculation, and the total numers o grid points were 3 and 5, respectively. The let-hand sides o oth domains were symmetric planes. The constant velocities, temperature, chemical species mole ractions were given at the urner inlet or oth calculations. As or -D Bunsen lame calculations, the velocity proile o the laminar plug low was employed at the low inlet oundary. As or -D diusion lame calculations, the gas phase C H 5 OH jet normal to the oundary with a uniorm velocity o. m/s lowed into the air co-low with a uniorm velocity o. m/s. The gauge pressure was set to zero, and the zero-gradient condition normal to the oundary was adopted or other variales at the urner outlet. The right-hand sides o oth calculation domains and the urner lip were considered as adiaatic walls. The initial temperature o the uel and the oxidizer, T, was constant at 98 K. a Figure. Computational domains o the two-dimensional lame calculations o the present study: (a) two-dimensional Bunsen lame calculations; () two-dimensional diusion lame calculations. 9

5 Vol. 5, No., To enale calculation with the reduced kinetic mechanism, the suroutine developed y the authors was implemented as a User Deined Function (UDF) to the solver o FLUENT. The UDF suroutine is called just once per iteration at each grid point, and it creates a reerence tale o reaction rates or all gloal reactions o the reduced kinetic mechanism in the computer memory. The convergence o the calculation was judged when residuals o the aove-mentioned equations ecame steady. Two-dimensional numerical simulations were perormed using the Altix37 supercomputer system o the Institute o Fluid Science, Tohoku University (CPU: 56 CPU x 4, perormance:.64 TFLOPS x 4, memory: 3 TB x 4). To evaluate the convergence time, the CPU time was measured. 3. Results and Discussion 3. Development and selection o an ethanol reduced kinetic mechanism ased on -D laminar lame analysis Figure shows the relationship etween the convergence time and the numer o species considered in the reduced kinetic mechanisms developed in the present study, including 5 to 3 species. The convergence time using the detailed reaction mechanism is also plotted in Fig.. Overall, the convergence time was reduced linearly as the numer o tracked species decreased, indicating that the numer o considered species was dominant in the calculation cost o one-dimensional numerical simulations o the present study. Adaptive mesh reinement was perormed several times during the calculation; however, the computational time or the mesh reinement was negligily small compared with the total CPU time. The astest tested mechanisms is already shown in Tale, i.e., the -step reduced kinetic mechanism including 6 chemical species (Tale ) that converged 4 times aster than the detailed mechanism, indicating the advantage o the ethanol reduced kinetic mechanism in the present study. Locally, the convergence time using the 9-step reduced kinetic mechanism, in which C H 6 was eliminated as the QSS species rom the -step reduced kinetic mechanism, was slightly larger than that using the -step reduced kinetic mechanism. It was presumed that the decrease o the numer o the considered species does not always contriute to the reduction o the convergence time, indicating that the choice o steady-state species in the reduced kinetic mechanism, e.g., C H 6 mentioned aove, aects the calculation staility even i the total numer o considered species and reaction steps are the same. Convergence time [s] step detailed mechanism Reduced mechanisms o the present study 9-step reduced mechanism -step reduced mechanism (Tale. ) Numer o considered species Figure. Relationship etween numer o species and convergence time (C H 5OH/air -D lame calculation, φ =., P =. MPa, T = 98 K). 93

6 Vol. 5, No., Figure 3 shows the species mole raction proiles and temperature proiles in the lame zone calculated with the detailed reaction mechanism and the -step reduced mechanism under the condition o φ =. and P =. MPa. Both results, including the intermediate species mole raction, were in good agreement. The application range o the -step reduced kinetic mechanism was susequently tested at various equivalence ratios and amient pressures, as shown in Fig. 4. Experimental data y Gülder (7) and Egolopoulos (8) et al. at atmospheric pressure are also plotted in Fig. 4. Generally, the laminar urning velocities calculated using the detailed reaction mechanism and the -step reduced kinetic mechanism showed good agreement at various amient pressures up to 3. MPa. They also agreed well with the experimental data. Although Schwer et al. (9) pointed out the prolem o limited application range o the reduced kinetic mechanism, the -step reduced kinetic mechanism developed in the present study (Tale ) includes a suicient numers o chemical species and reactions to perorm industrial lame analysis under a certain range o conditions. Thus, the -step reduced kinetic mechanism o the ethanol comustion was chosen and applied in the two-dimensional numerical simulations detailed in the next section. a.4 O Temperature 5 7.E-3 OH.9 H O 6.E-3 C H 4 Mole raction C H 5OH H CO CO 5 5 Temperature [K] Mole raction 5.E-3 4.E-3 3.E-3.E-3.E-3 CH 3CHO CH O CH 3 H -.. X [cm].e X [cm] Figure 3. Proiles o species mole raction and temperature with the detailed reaction mechanism (Lines) and the -step reduced kinetic mechanism (Symols): (a) major species; () intermediate species (C H 5OH/air -D lame calculation, φ =., P =. MPa, T = 98 K). Laminar urning velocity [cm/s] Experimental data + : Guelder * : Egolopoulos -step reduced mechanism Detailed reaction mechanism. MPa 3. MPa Equivalence ratio, φ. MPa Figure 4. Laminar urning velocities calculated with the detailed reaction mechanism (Lines) and the -step reduced kinetic mechanism (Symols) (C H 5OH/air -D lame calculation, T = 98 K). 94

7 Vol. 5, No., 3. Application o the ethanol reduced mechanism or two-dimensional lame analysis Convergence o the two-dimensional lame analysis with the -step ethanol reduced kinetic mechanism (Tale ) was achieved ater the ollowing improvements. In the reduced kinetic mechanism ased on QSSA, the mole ractions o QSS species were calculated internally using an algeraic division equation derived rom the steady-state relation (8) in the UDF code developed in the present study. For example, the mole raction o the CH radical, which is one o the QSS species, is calculated as ollows: [CH] = ABV DEN ABV = R (59) [T - CH ] [H] + R (6) [T - CH ] [OH] + R + R (69) [CH O] [H] + R (7) [HCO] [CO] + R (8) [C H] [O] + R () [C H] [O ] DEN = R (59) [H ] + R (6) [H O] + R (67) [O] + R (68) [O ] + R (69) [H O] + R (7) [CO ] + R (67) [CO] [H] + R (68) [HCO] [O] (8) [CO] + R () [CO ], () where R and R are the orward and ackward rate constants o elementary reactions in the detailed reaction mechanism y Saxena and Williams. The numer in parenthesis is the numer o the elementary reaction in the detailed reaction mechanism y Saxena and Williams. It is considered that this expression tends to cause numerical instaility ecause species mole ractions in the equation are near zero at the initial step o the calculation even when the under-relaxation actors are suiciently small. In the present study, thereore, the initial temperature and species proile were used to avoid this prolem. The initial solution was calculated using the ollowing overall one-step reaction o the FLUENT dataase (6) : C H 5OH + 3O CO + 3H O () E ω.5.6 C H 5 OH = A exp [C H5OH] [O ]. (3) RT Here the pre-exponential actor, A, is x 9 m.5 /mol.75 s and the activation energy, E, is.56 x 8 J/kmol. It is considered that the chemical species included in Eq. () prevented the calculation instaility caused y the aove mentioned algeraic equations. This method could also lead to the reduction o the total convergence time. Figure 5 shows the numerical results o the -D premixed Bunsen lame calculated with the -step reduced kinetic mechanism (Tale ) and the detailed reaction mechanism at φ =. and P =. MPa. As or the result using -step reduced kinetic mechanism, the numerical instaility o the reduced kinetic mechanism caused y the intermediate species diusion reported y Saito et al. (7) was successully avoided, indicating that QSS species were appropriately selected in the -step reduced kinetic mechanism o the present study. The calculation results with the detailed reaction mechanism shown in Fig. 5 were otained using the In Situ Adaptive Taulation (ISAT) () solver module o FLUENT ased on the in-situ generation o look-up tales ecause those without ISAT could not converge as mentioned later. The proiles o temperature, major chemical species, e.g., H O and CO, and intermediate species, e.g., C H 4, were in good agreement, meaning that the -step reduced kinetic mechanism ased on QSSA is applicale to two-dimensional premixed lame calculations. 95

8 Vol. 5, No., a.4.4 c.4 d (K) (K) e X (cm) Figure 5. Temperature and species mole raction proiles calculated with the -step reduced kinetic mechanism: (a) Temperature; () CO mole ractions; (c) C H 4 mole ractionss, and those with the detailed reaction mechanism: (d) Temperature; (e) CO mole ractions; () C H 4 mole ractions (C H 5OH/air -D premixed lame calculation, φ =., P =. MPa, T = 98 K) a c X (cm) X (cm) E E (K) d e (K) Figure 6. Temperature and species mole raction proiles calculated with the -step reduced kinetic mechanism: (a) Temperature; () CO mole ractions; (c) OH mole ractions, and those with the detailed reaction mechanism: (d) Temperature; (e) CO mole ractions; () OH mole ractions (C H 5OH/air -D diusion lame calculation, P =. MPa, T = 98 K). 96

9 Vol. 5, No., Note that the eect o mesh reinement on these calculation results was small in the present study. It should also e noted that the temperature and species distriution normal to the lame ront in Fig. 5 calculated with FLUENT agreed well with those calculated with the PREMIX code. The temperature and species mole raction proiles o the -D diusion lame calculated using the -step reduced kinetic mechanism and the detailed reaction mechanism are also shown in Fig. 6. Numerical results using the -step reduced kinetic mechanism were in good agreement with the results using the detailed reaction mechanism with the ISAT solver, meaning that the reduced kinetic mechanism developed in the present study is also applicale to -D non-premixed lame calculations. Figure 7 shows the relationship etween the numer o iterations and CPU time or -D Bunsen lame and diusion lame calculations. These calculations were all conducted under conditions o same amient pressure, equivalence ratio, under-relaxation actor or every variale and the initial proiles o temperature and chemical species or the comparison o the computational time. As or the -D Bunsen lame calculation, the calculation with the detailed mechanism and the normal SIMPLE solver module o FLUENT were unale to converge within 35 hours, and the CPU time per iteration was more than 6 times longer than that with the -step reduced mechanism, showing the advantage o the multi-dimensional analysis using the reduced mechanism. It can also e considered that the computational time o the algeraic calculation or QSS species is suiciently small compared with the total calculation cost. The calculation time with the detailed reaction mechanism using the ISAT solver was also plotted. The calculation with detailed reaction mechanism with ISAT could iterate 5 times aster than that without ISAT, i.e., ISAT speed-up actor (9) o 5, showing that the ISAT is also eective or -D calculations in this study. The CPU time per iteration with the -step reduced mechanism without ISAT was 5 times longer than that using the detailed reaction mechanism using the ISAT solver; however, as or the total computational time, the calculation with the -step reduced mechanism converged.5 times aster than that using detailed reaction mechanism and ISAT solver. It can e concluded that the eect o the decrease o the calculation time y reducing the considered species is relatively larger than the speed-up eect y using ISAT in this case. As or the -D diusion lame calculation, on the other hand, the calculation using the detailed reaction mechanism with the ISAT solver could converge aster than that with the -step reduced kinetic mechanism without ISAT, which indicates that the speed-up eect o ISAT is relatively larger in the case o diusion lame calculations than the case o premixed lame calculations. a CPU time [hour] Not -step reduced mechanism -step reduced mechanism with ISAT Detailed mechanism with ISAT Detailed mechanism CPU Time [hour] step reduced mechanism -step reduced mechanism with ISAT Detailed mechanism with ISAT Numer o iterations 5 Numer o iterations Figure 7. Relationship etween the numer o iterations and the CPU time: (a) -D Bunsen lame calculations, φ =., P =. MPa; () -D diusion lame calculations, P =. MPa. 97

10 E-5 6E-5 E-7.4E-7 Journal o Thermal Vol. 5, No., Furthermore, as in the study y Montgomery et al. () the computational time when oth the -step reduced kinetic mechanism and ISAT solver were simultaneously used was consideraly less than that using only the detailed reaction mechanism or -D premixed and non-premixed lame calculations. It should e noted here that the accuracy o the inal solution and the total computational time strongly depend on the error tolerance parameter when the ISAT solver is used (, 6, ). At the same time, unrealistic proiles o some intermediate species such as C 3 H 3 with split peaks could e seen depending on the error tolerance (6), as shown in Fig. 8, in the present -D diusion lame calculations using the ISAT solver o FLUENT, showing the advantage o the reduced kinetic mechanism ased on QSSA over the ISAT solver in terms o the prediction o the chemical species proiles including intermediate species with low calculation cost. a e-7 3E-7 3E-7 E E E-5 E E-7.8E-7.4E-7 6E-8 E-8 3.4E-7 3E-7.6E-7.E-7.8E-7.4E-7 E-7 6E-8 E-8 Figure 8. C 3H 3 mole raction proiles: (a) numerical result calculated with the -step reduced kinetic mechanism o the present study; () numerical result calculated with the detailed reaction mechanism and ISAT solver o FLUENT (-D diusion lame calculation, P =. MPa). 4. Conclusions A -step reduced kinetic mechanism or ethanol comustion ased on QSSA was developed in this study and the easiility o that mechanism or multi-dimensional numerical simulation was evaluated. The validity o the ethanol reduced kinetic mechanism was irstly tested on -D premixed lame calculations, and it was conirmed that the numerical results with a reduced kinetic mechanism and a detailed reaction mechanism showed etter agreement at various equivalence ratios and amient pressures. The developed ethanol reduced mechanism was then applied to two-dimensional premixed and non-premixed lame calculations. The -D lame calculations using the ethanol reduced kinetic mechanism successully converged ater improvement o initial temperature and species proiles. The numerical results agreed well with those with the detailed reaction mechanism, and the convergence time was suiciently reduced y using the reduced kinetic mechanism or -D numerical simulations, indicating that the reduced kinetic mechanism ased on QSSA can e applied to the multi-dimensional industrial lame analysis with excellent results. Moreover, computational time was consideraly decreased when the reduced kinetic mechanism was used with ISAT, although depending on its error tolerance parameter, some prolems were seen on the prediction o intermediate species proiles when the ISAT solver o FLUENT was used. 98

11 Vol. 5, No., Reerences () Norton, T.S. and Dryer, F.L., International Journal o Chemical Kinetics, Vol. 4 (99), pp () Dagaut, P., Boettner, J.C., Cathonnet, M., Journal o Chemical Physics, Vol. 89 (99), pp (3) Marinov, N.M., International Journal o Chemical Kinetics, Vol. 3 (999), pp (4) Liu, J., Kazakov, A., Chaos, M., and Dryer, F.L., 5 th US Comustion Meeting 7, C6. (5) Saxena, P., Williams, F.A., Proceedings o the Comustion Institute, Vol. 3 (7), pp (6) Lacaze, G., Richardson, E., Poinsot, T., Comustion and Flame, Vol. 56 (9), pp (7) Saito, H., Ogami, Y., Koayashi, H., Niioka, T., Mohri, T., Hozumi, Y., Shiozaki, T., Journal o the Comustion Society o Japan, Vol. 4 (5), pp. 9-8 (in Japanese). (8) Peters, N., in Reduced Kinetic Mechanisms and Asymptotic Approximations or Methane-Air Flames, Springer, Berlin, (99), pp (9) Lu, T., Law, C. K., Comustion and Flame, Vol. 54 (8), pp () Cremer, M.A., Montgomery, C.J., Wang, D.H., Heap, M.P., and Chen, J.Y., Proceedings o the Comustion Institute, Vol. 8 (), pp () Montgomery, C.J., Zhao, W., Eklund, D.R., and Chen, J.Y., AIAA Paper , AIAA Computational Fluid Dynamics Conerence, Orlando, FL, June 3-6, 3. () Lu, T., Ju, Y., Law, C.K., Comustion and Flame, Vol. 6 (), pp (3) Glarorg, P., Kee, R. J., Grcar, J. F., Miller, J. A., Sandia Report, SAND86-89, 988. (4) Kee, R.J., Grcar, J.F., Smook, M.D., Miller, J.A., Sandia Report, SAND 85-84, 985. (5) Kee, R.J., Rupley, F.M., Miller, J. A., Sandia Report, SAND 89-89, 989. (6) FLUENT Inc., FLUENT User's Guide Ver (8). (7) Gülder, O.L., Proceedings o the Comustion Institute, Vol. 9 (98), pp (8) Egolopoulos, F.N., Du, D.X., Law, C.K., Proceedings o the Comustion Institute, Vol. 4 (99), pp (9) Schwer, D., Lu, P., Green, H., Comustion and Flame, Vol. 6 (), pp () Pope, S.B., Comustion Theory and Modeling, Vol. (997), pp () Singer, M.A. and Pope, S.B., Comustion Theory and Modeling, Vol. 8 (4), pp