Numerical Modeling of a Two Dimensional

Proceedings NHTC’01 Proceedings of ASMEof NHTC’01 th 35th NationalHeat HeatTransfer TransferConference Conference 35 National June 10-12, 2001, Anaheim, California Anaheim, California, June 10-12, 2001

NHTC2001-20126 NHTC01-11419 NUMERICAL MODELING OF A TWO DIMENSIONAL AXISYMMETRIC LAMINAR ETHYLENE-AIR DIFFUSION FLAME Hongsheng Guo*, Fengshan Liu, Gregory J. Smallwood, and Ömer L. Gülder Combustion Research Group Institute for Chemical Process and Environmental Technology National Research Council Canada Building M-9, Montreal Road, Ottawa, K1A 0R6 Email: [email protected]

ABSTRACT A numerical modeling of soot formation has been conducted for an axisymmetric, laminar, coflow diffusion ethylene-air flame at atmospheric pressure. A detailed reaction mechanism, which consists of 36 species and 219 reactions, and complex thermal and transport properties are used. The fully coupled elliptic equations are solved. A simple two-equation soot model is coupled with the detailed gas-phase chemistry. The interactions between the soot and gas-phase chemistry are taken into account. The radiation heat loss is obtained by the discrete ordinate method coupled to a SNBCK based wide band model for radiative properties of CO, CO2, H2O and soot. The predicted flame temperature and soot volume fraction are compared with the experimental results. Although we observe a good qualitative agreement between the numerical and experimental results, predicted flame temperature and soot volume fraction are lower than the experimental values in the centerline region. The numerical results indicate that the radiative heat loss due to the presence of soot is significant in this type of flames. KEYWORDS: combustion, soot, modeling, laminar INTRODUCTION Soot is a by-product of incomplete hydrocarbon combustion and is generated in regions of the flame where not enough oxygen is present to yield a complete conversion of fuel *

into carbon dioxide and water. The detrimental effect of soot on human health is a current concern and various restrictions are being placed on soot emissions from various sources. From an operational point of view, soot formation is not desired in most combustion devices. Therefore quantitative understanding of the soot inception, growth and oxidation mechanisms and the ability to model these processes are critical to the development of strategies to control soot emissions. Modeling soot formation in practical combustion systems is an extremely challenging problem. Although Frenklach and his co-workers have derived some detailed kinetic models of soot inception, growth and oxidation [1-3], it is currently still very difficult and time consuming to implement this kind of models to the simulations of multidimensional combustion systems. Smooke et al. [4-5] numerically modeled two slightly sooting, coflow, laminar diffusion flames using the sectional soot model. In addition to the momentum, energy and gas species conservation equations, several soot section equations (usually more than 10) need to be solved. Based on some semiempirical assumptions, Leung et al. [6] conducted the simulations of soot formation for counterflow ethylene-air and propane-air diffusion flames by a simplified two-equation soot model. Only two additional equations need to be solved in this model. Fairweather et al. [7] further used this two-equation model to simulate the soot formation in a turbulent reacting flow. A reduced ten-step gas phase chemistry and flamelet

Corresponding author.

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model were used to model the gas-phase combustion process. Moss et al. [8] conducted a simulation of soot formation and burnout in a high temperature laminar diffusion flame by a similar, but a little different, two-equation soot model. The simulation was performed by a parabolic code based on the boundary layer approximation and radiation heat transfer was neglected. Kennedy et al. [9] simulated soot formation in a coflow, laminar ethylene-air diffusion flame by the two-equation model of Leung et al. [6]. However they also used a parabolic code based on the boundary layer approximation and neglected the axial direction diffusion terms in all gas species and soot mass governing equations. In the present paper, we use the two-equation model of Leung et al. [6] to simulate a coflow, laminar diffusion C2H4-air flame. Computationally we employ the primitive variable method in which the fully elliptic governing equations are solved with detailed gas-phase chemistry and complex thermal and transport properties. The effects of soot inception, growth and oxidation on gas-phase chemistry are accounted for. The radiation heat loss is accounted for by the discrete ordinate method coupled to a SNBCK-based wide band model for radiative properties of CO, CO2, H2O and soot. The results are compared to those obtained in our laboratory by Gülder et al. [10, 11]. GOVERNING EQUATIONS AND NUMERICAL METHOD Gas-Phase Equations The investigated flame is a coflow laminar diffusion flame in which a cylindrical fuel stream is surrounded by a coflowing air jet. In cylindrical coordinates (r, z), the governing equations for the gas-phase are [12] Continuity:

∂ (rρv ) + ∂ (rρu ) = 0 ∂r ∂z

(1)

∂u ∂u ∂p 1 ∂  ∂u  + ρu =− +  rµ  ∂r ∂z ∂z r ∂r  ∂r  ∂  ∂u  2 ∂  µ ∂ (rv ) + 2 µ  −  ∂z  ∂z  3 ∂z  r ∂r  ρv

2 ∂  ∂u  1 ∂  ∂v  µ  +  rµ  + ρg z 3 ∂z  ∂z  r ∂r  ∂z  (2)

Radial momentum:

(3) Energy:

∂T  1 ∂  ∂T   ∂T c p  ρv + ρu =  rλ + ∂z  r ∂r  ∂r   ∂r ∂  ∂T  KK +1  ∂T    ∂T + VkZ λ  − ∑  ρc pk Yk  Vkr  ∂z  ∂z  k =1  ∂z    ∂r −

KK +1

∑h W ω k =1

k

k

k

+ qr (4)

Gas species:

ρv

∂Yk ∂Y 1 ∂ + ρu k = − (rρYkVkr ) − ∂r ∂z r ∂r

∂ ( ρYkVkz ) + Wk ωk ∂z k=1, 2, …, KK

(5)

where u and v are the velocities in axial (z) and radial (r) directions, respectively; T the temperature of the mixture; ρ the density of the mixture (soot and gas); Wk the molecular weight of the k th gas species; λ the mixture thermal conductivity; cp the specific heat of the mixture under constant pressure; cpk the specific heat of the k th gas species under constant pressure; ω k the mole production rate of the k th gas species per unit

Axial momentum:

−

∂v ∂v ∂p ∂  ∂v  + ρu =− + µ  ∂r ∂z ∂r ∂z  ∂z  2 ∂  ∂v  2 1 ∂  ∂  +  rµ  −  µ (rv ) r ∂r  ∂r  3 r ∂r  ∂r  2 1 ∂  ∂u  ∂  ∂u  −  rµ  +  µ  3 r ∂r  ∂z  ∂z  ∂r  2µ v 2 µ ∂ (rv ) + 2 µ ∂u − 2 + 2 r 3 r ∂r 3 r ∂z ρv

volume. It should be pointed out that the production rates of gas species include the contribution due to the soot inception, surface growth and oxidation (see the next section). Quantity h k denotes the specific enthalpy of the k th gas species; g z the gravitational acceleration in the z direction; µ the viscosity of the mixture; Yk the mass fraction of the k th gas species; Vkr and Vkz the diffusion velocities of the k th gas species in r and z directions; and KK the total gas phase species number. The quantities with subscript KK+1 correspond to those of soot. As an approximation, the thermal properties of graphite are used to represent those of soot. The last term on the right hand side of Eq. 4 is the source term due to radiation heat transfer. It is obtained by the discrete

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ordinate method coupled to a SNBCK-based wide band model for properties of CO, CO2, H2O and soot [13]. The spectral absorption coefficient of soot was assumed to be 5.5 fν / λ .

VT , xi = −0.55

The diffusion velocities are written as:

Vkr = −

1 ∂Y Dk k + VTkr + Vcr Yk ∂r

k=1,2,…,KK (6)

1 ∂Y Vkz = − Dk k + VTkz + Vcz Yk ∂z

Quantities VT,r and VT,z are the particle thermophoretic velocities in r and z directions, respectively. They are obtained by

µ ∂T , ρT ∂xi

xi =r, z

(11)

The source term S m in Eq. 9 accounts for the contributions of soot nucleation (ω n ), surface growth (ω g ) and oxidation ( ω O ). Therefore

k=1,2,…,KK

S m = ω n + ω g − ωO

(12)

(7) where VTkr and VTkz are the thermal diffusion velocities in r and z directions for the k th gas species; Vcr and Vcz the correction diffusion velocities, which are used to ensure that the net diffusive flux of all gas species and soot is zero [14], in r and z directions. In the current simulation, only the thermal diffusion velocities of species H 2 and H are accounted for by the method given in reference 14. The thermal diffusion velocities of all other gas species are set as zero. Quantity Dk is related to the binary diffusion coefficients through the relation

Dk =

1− Xk , Xj ∑ D jk j≠ k KK

k=1,2,…,KK

(8) where Xk is the mole fraction of the k th species, and Djk is the binary diffusion coefficient. Soot Model A semi-empirical two-equation soot model [6, 7] is used to obtain the soot behavior in the flame. The transport equations for the soot mass fraction and number density are given as

ρv

∂Ys ∂Y 1 ∂ + ρu s = − (rρVT ,rYs ) − ∂r ∂z r ∂r

∂ (ρVT , zYs ) + S m ∂z

(9)

The model developed by Leung et al. [6] is used to obtain the three terms in Eq. 12. The model assumes the chemical reactions for nucleation and surface growth respectively as: C2H2 → 2C(S) + H2 C2H2 + nC(S) → (n+2)C(S)+H2

(R1) (R2)

with the reaction rates given as:

r1 = k1 (T )[C2 H 2 ] r2 = k 2 (T ) f ( As )[C2 H 2 ]

(13) (14)

where f(As) denotes the function dependence on soot surface area per unit volume. Following Fairwhether et al. [7] and Kronenburg et al. [15], we simply assume that the function dependence is linear, i.e. f(As)=As . Neoh et al. [16] investigated the soot oxidation process in flames, and found that the oxidation due to both O2 and OH is important, depending on the local equivalence ratio. The radical O contributes to soot oxidation too in some regions. We, therefore, account for the soot oxidation by O2, OH and O, and assume the following reactions: O2 + 0.5C(S) → CO OH + C(S) → CO + H O + C(S) → CO

(R3) (R4) (R5)

The reaction rates for these three reactions are obtained by: 1

∂N ∂N 1 ∂ ρv + ρu =− (rρVT , r N ) − ∂r ∂z r ∂r ∂ (ρVT ,z N ) + S N ∂z

r3 = k3 (T )T 2 As [O2 ] (10)

where Ys is the soot mass fraction, and N is the soot number density defined as the particle number per unit mass of mixture.

r4 = ϕ OH k 4 (T )T r5 = ϕO k 5 (T )T

−1

−1

2

2

As X OH

As X O

(15) (16) (17)

where XOH and XO denote the mole fractions of OH and O, and ϕ OH and ϕO are the collision efficiencies for OH and O attacking on soot particles. The collision efficiency of OH is

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treated as described by Kennedy et al. [9], who accounted for the variation of the collision efficiency of OH with time by assuming a linear relation between the collision efficiency and a dimensionless distance from the fuel nozzle exit. A collision efficiency of 0.5 for radical O attacking on the particles is used [17]. Table 1 Rate constants as Ki K1 K2 K3 K4 K5

−E

Ae

RT

(units are kg, m, s, kcal, kmol and K) A E Reference 1.35E+06 41 [7] 5.00E+02 24 [7] 1.78E+04 39 [7] 1.27E+03 0 [8] 6.65E+02 0 [17]

All the reaction rate constants, k i (i=1,…,5), are given in Table 1. The source term S N in Eq. 10 accounts for the soot nucleation and agglomeration, and is calculated as:

SN =

2 N R − Cmin A 1

 6M C(S ) 2Ca   πρC ( S )

   

1

6

 6κT   ρ C(S ) 

   

1

2

(18)

[C(s )] 6 [ρN ] 1

11 6

where NA is Avogadro’s number (6.022 x 1026 particles/kmol), Cmin is the number of carbon atoms in the incipient carbon particle (9x104), κ is the Boltzman constant (1.38x10-23 J/K), ρ C ( S ) is the soot density (1800kg/m3), and Ca is the agglomeration rate constant for which a value of 3.0 [7] is used, although values 9.0 or 0.0 (ignoring the agglomeration) have been used by other investigators [6, 18]. Numerical Model The equations 1-5, 9 and 10 are closed with the equation of state and appropriate boundary conditions on each side of the computational domain. The SIMPLE numerical scheme [19] is used to solve the governing equations. The diffusion terms in the conservation equations are discretized by the central difference method and the convective terms are discretized by the upwind difference method. The governing equations of gas species, soot mass fraction and soot number density are solved in a fully coupled fashion on every grid to speed up the convergence process [20]. The chemical reaction mechanism used is basically from GRIMech3.0 [21], with the exception being the removal of all the reactions and species related to NOX formation. The revised reaction scheme consists of 36 species and 219 reactions. All the thermal and transport properties are obtained by using the

database of GRIMech3.0 and the algorithms given in references 14 and 22. RESULTS AND DISCUSSIONS An atmospheric pressure, laminar C2H4-air laminar diffusion flame studied in our laboratory previously [10, 11] is modeled. The flame was generated with a burner in which the pure fuel (C2H4) flows from an uncooled 10.9mm inner diameter vertical steel tube, and the air flows from the annular region between the fuel tube and a 100 mm diameter concentric tube. The wall thickness of the fuel tube is 0.95 mm. The volume flow rate of fuel is 194 ml/min and that of air is 284 l/min at 294K. The computational domain covers an area from 0 to 3.0 cm in the radial direction and 0 to 12.0 cm in the axial direction. The inflow boundary (z=0) corresponds to the region immediately above the fuel nozzle. A parabolic velocity profile (with the mean value of 4.01 cm/s) is used for the region from r=0 to 0.545 cm (fuel nozzle exit) at z=0, and an uniform velocity of 70.86 cm/s is used for the region from r=0.64 to 3.0 cm (air nozzle exit) at z=0. For the region from r=0.545 to 0.64 cm at z=0, a solid wall boundary condition is used. Symmetric and free-slip boundary conditions are used for r=0 and r=3.0, respectively. At z=12.0, a zero-gradient outlet boundary condition is employed. The experimental results showed that the fuel nozzle outer surface temperature was about 100K higher than the room temperature due to the flame heating [11], which indicates that the inlet fuel and air are preheated by flame. In order to consider the influence of this preheating, the fuel and air inlet temperatures are increased to 340 K in the simulation. It should be noted that this temperature increase is only an approximation. The exact inlet temperature and velocity profiles should be obtained by experiment. Flame Temperature Figures 1 and 2 show the predicted and measured [10] flame temperatures, respectively. They indicate that the flame height from the numerical result is similar to that from the experiment, and the predicted temperature distribution throughout the flame envelope is reasonable. For both the experiment and simulation, results show that the temperature profiles have a maximum in the annular region of the lower part of the flame, and these maximum temperature contours do not converge to the axis in the upper part of the flame. This is due to the radiation cooling by soot, since significant amount of soot exists in the upper part of flame (see the result below). The numerical result without soot does not present this feature, as shown in figure 3. Therefore we can conclude that the radiative heat loss due to the presence of soot is significant for this type of flames, especially for the upper part. It is noted that the flame temperature values with soot are underpredicted, especially in the centerline region. The maximum flame temperature (2020K) is about 130K lower than

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300 1224 2148

300 1224 2148

6

6

5

5

4

4

3

3

2

2

1

1

-0.5 0

-0.5 0

0.5

0.5

r, cm

r, cm

Fig.1 Predicted flame temperature, K

Fig. 3 Predicted flame temperature, K (without soot)

300 1224 2148

6

5

4

3

2

1

-0.5 0

0.5

r, cm Fig.2 Measured flame temperature, K

the measured value (2150K), although both occur in the similar annular region. Two factors may result in the predicted temperature lower than the real value. First, the inlet fuel and air temperatures in the model are lower than the real case, although the inlet temperatures were increased by 46K. As shown in Figs.1and 2, the measured temperature on the centerline at 2mm above the nozzle exit (the lowest position for which the measured data is available) is about 200K higher than the predicted value at the same position. Therefore it is necessary to provide more precise inlet condition in the future model. Secondly, the soot model (reactions R1-R5) may be oversimplified (see the discussion below), which may lead to the incorrect gas phase composition and heat release rate. Soot Volume Fraction Figures 4 and 5 show the predicted and measured [11] soot volume fractions, respectively. It can be found that the simulation captures the general features of soot in the flame. The peak soot volume fraction from simulation is quite close to that from the experiment, and the predicted soot concentration distribution is similar to that from the experiment. The soot volume fraction maximum occurs at radial positions slightly inside those corresponding to the maximum flame temperature (see figures 1 and 2), and therefore most soot is formed on the fuel rich side of the flame reaction zone.

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0.0 4.0 8.0

0.0 3.3 6.6

6

6

5

5

4

4

3

3

2

2

1

1

-0.5

0

0.5

-0.5

r, cm Fig. 4 Predicted soot volume fraction, ppm

0.5

Fig. 6 C2H2 mass fraction, %

0.0 4.0 8.0

0.00 0.22 0.44

6

6

5

5

4

4

3

3

2

2

1

1

-0.5

0

r, cm

0

0.5

-0.5

r, cm

0

0.5

r, cm Fig. 7 Inception rate x103, g/cm3-s

Fig. 5 Measured soot volume fraction, ppm

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0.00 0.50 1.00

0.00 2.59 5.18

6

6

5

5

4

4

3

3

2

2

1

1

-0.5

0

-0.5

0.5

r, cm Fig. 8 Surface growth rate x103, g/cm3-s

0.00 0.10 0.20

6

6

5

5

4

4

3

3

2

2

1

1

0

0.5

Fig. 10 Rate of oxidation by OH x103, g/cm3-s

0.00 0.95 1.90

-0.5

0

r, cm

-0.5

0.5

0

0.5

r, cm

r, cm

Fig. 11 Rate of oxidation by O x103, g/cm3-s

Fig. 9 Rate of oxidation by O2 x103, g/cm3-s

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However, the centerline region soot volume fractions are underpredicted, and the predicted annular region with higher soot volume fraction value is wider than that from the experiment. The discrepancies may be due to the assumed soot model. As indicated by Sunderland et al. [23], the soot nucleation is a complex process involving polycyclic aromatic hydrocarbons (PAH) that eventually become visible soot particles, although it can be empirically correlated with C2H2 concentration alone, as expressed by R1. Moreover, not only C2H2 but also some other species may contribute to the soot surface growth process, which can be described by the effects of various species on active sites and parallel channels [23-25]. This means that the reaction R2 is not adequate to describe the soot surface growth process. These simplifications in the assumed soot model contribute to the discrepancies between the simulation and experimental results and need to be improved in the future model. Lower predicted flame temperature, especially in the centerline region, may contribute to the discrepancies too. Soot Inception and Surface Growth Acetylene (C2H2) is the most important precursor of soot, as the rates of soot inception and surface growth are directly related to its concentration in the present model. Figure 6 gives the mass fraction of C2H2. Figures 7 and 8 contain the soot inception and surface growth rates, respectively. From figures 1 and 6-8, we observe that the predicted inception and surface growth rates strongly depend on the flame temperature and the concentration of C2H2. The maximum rates of inception and growth occur in the annular region between the zones with the peak flame temperature and C2H2 concentration. The inception rate is much lower than that of the surface growth (please note the order of magnitude difference in the scales for figures 7 and 8), and thus most of the soot in the flame is produced by surface growth process. This is in agreement with the conclusion indicated by reference 25 for heavily sooting flames. Therefore the improvement of soot surface growth model is relatively more important than that of the inception model for improving the prediction. Soot Oxidation Figures 9-11 illustrate the predicted oxidation rates of soot by O2, OH and O, respectively. It is observed that the oxidation by O2 is significant in the fuel lean region, while that by OH is more important in the fuel rich region. However, the rate of oxidation by O2 is higher than that by OH in most of the soot region, and thus O2 dominates the soot oxidation process in the present model. This result is a little different from those by Smooke et al. [4] and Kennedy et al. [9]. It possible that the present model overpredicted the rate of oxidation by O2, because the O2 oxidation model used here is from Fairwhether et al. [7], who derived the model constant by comparing the model predictions and experimental data when the soot oxidation by OH was ignored.

Oxygen atom (O) also contributes to the soot oxidation in similar region to OH, although its rate is much lower than that by OH and O2. This soot oxidation process is qualitatively consistent with the findings of Neoh et al. [16]. CONCLUSIONS The simulation of a coflow laminar diffusion C2H4-air flame by a detailed gas phase reaction model, complex transport and thermal properties and a simplified two equation soot model was conducted. The simulation results were compared to those from experiments. The results show that the simulation captured the main features of the flame. The predicted flame height and temperature distribution shape are similar to those from experiment. The radiative heat loss due to the presence of soot is significant in this type of flames. The maximum value and the concentration distribution of soot from simulation are quite close to those from experiment. However, both the temperature and soot volume fraction values in the centerline region were underpredicted. The oversimplifications in the assumed soot model and the imprecise inlet condition may be responsible for the discrepancies. The simulation also shows that the oxidation by O2 is significant in the fuel lean region, while the oxidation by OH is more important in the fuel rich region. Oxygen atom also contributes to the soot oxidation in similar region to OH, although its rate is much lower than those by O 2 and OH.

REFERENCES 1. Frenklach, M., Clary, D.W., Gardiner, W.C. and Stein, S.E., Twentieth Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, 1984, pp.887-901. 2. Frenklach, M. and Wang, H., Twenty-Third Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, 1990, pp.1559-1566. 3. M. Frenklach and H. Wang, in Soot Formation in Combustion: Mechanisms and Models (H. Bockhorn, Ed.), Springer Series in Chemical Physics, Vol. 59, SpringerVerlag, Berlin, 1994, pp. 162-190. 4. Smooke, M.D., Mcenally, C.S., Pfefferle, L.D., Hall, R.J. and Colket, M.B., Combust. Flame 117: 117-139 (1999). 5. McEnally, C.S., Schaffer, A.M., Long, M.B., Pfefferle, L.D., Smooke, M.D., Colket, M.B. and Hall, R.J., Twenty-Seventh Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, 1998, pp.1497-1505. 6. Leung, K.M., Lindstedt, R.P. and Jones, W.P., Combust. Flame 87:289-305 (1991). 7. Fairwhether, M., Jones, W.P. and Lindstedt, Combust. Flame 89:45-63 (1992). 8. Moss, J.B., Stewart, C.D. and Young, K.J., Combust. Flame 101: 491-500 (1995). 9. Kennedy, I.M, Yam, C., Rapp, D.C, and Santoro, R.J., Combust. Flame 107: 368-382 (1996).

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10. Gülder, Ö.L., Snelling, D.R., and Sawchuk, R.A., TwentySixth Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp.2351-2357. 11. Gülder, Ö. L., Thomson, K.A., and Snelling, D.R., Influence of the Fuel Nozzle Material on Soot Formation and Temperature Field in Coflow Laminar Diffusion Flames, Combustion Institute/Canadian Section, 2000. 12. Kuo, K.K., Principles of Combustion, John Wiley & Sons, 1986. 13. Liu, F., Smallwood, G. J., and Gülder, Ö.L., AIAA paper 993679. 14. Kee., R.J., Warnatz, J., and Miller, J.A., Sandia Report, SAND 83-8209. 15. Kronenburg, A., Bilger, R.W. and Kent, J.H, Combust. & Flame 121:24-40 (2000). 16. Neoh, K.G., Howard, J.B. and Sarofim, A.F., in Particulate Carbon: Formation During Combustion (Siegla, D.C. and Smith, G.W., Eds.), Plenum, New York, 1981, P.261. 17. Bradley, D., Dixon-Lewis, G., Habik, S.E., and Mushi, E.M., Twentieth Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, 1984, pp.931-940. 18. Ezekoye, O.A. and Zhang, Z., Combust. Flame 110: 127-139 (1997). 19. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980 20. Liu, Z., Liao, C., Liu, C. and McCormick, S., AIAA 95-0205. 21. Gregory P. Smith, David M. Golden, Michael Frenklach, Nigel W. Moriarty, Boris Eiteneer, Mikhail Goldenberg, C. Thomas Bowman, Ronald K. Hanson, Soonho Song, William C. Gardiner, Jr., Vitali V. Lissianski, and Zhiwei Qin http://www.me.berkeley.edu/gri_mech/. 22. Kee., R.J., Miller, J.A., and Jefferson, T.H., Sandia Report, SAND 80-8003. 23. Sunderland, P.B. and Faeth, G.M., Combust. Flame 105:132146 (1996). 24. Mauss, F., Schafer, T., and Bockhorn, H., Combust. Flame 99:697-705 (1994). 25. Kennedy, I.M., Prog. Energy Combust. Sci. 23: 95-132 (1997)..

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