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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E 47th St., New York, N.Y The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Satiety or of its Divisions or Sections, or printed in its publications Discussion is printed only if the paper Is published In an ASME Journal. Authorization to photocopy material for internal or personal use under circumstance not falling within the fair use provisions of the Copyright Act is granted by ASME to libraries and other users registered with the Copyright Clearance Comer (CCC) Transactional Reporting Service provided that the base tee of $0.30 per page Is paid directly to the CCC, 27 Congress street Salem MA Requests for special permission or tad( reproduction should be addressed to the ASME Techrdcal Publishing Department 95-GT-115 Copyright by ASME AS Rights Reserved Primed in U.S.A. STUDIES ON LEAN-BLOWOUT CHARACTERISTICS OF A PREMIXED JET FLAME Viswanath R. Katta Systems Research Laboratories, Inc. A Division of Calspan Corporation Dayton, Ohio W. M. Roquemore Wright Laboratory Aero Propulsion and Power Directorate Wright-Patterson Air Force Base, Ohio ,1111) ABSTRACT The stability characteristics of a fuel-lean premixed jet flame are investigated using a time-dependent, axisymmetric numerical model with a detailed-chemical-kinetics mechanism for H2-02-N2 combustion. Temperature- and speciesdependent transport properties are incorporated. The mathematical model is validated by computing the burning velocities for different equivalence ratios and comparing them with experimentally measured values. Premixed flames that are stably attached to the fuel tube are obtained over a wide range of equivalence ratios. The lower limit for equivalence ratio at which the flame lifts-off from the fuel tube is found to be 0.4. Calculations have also correctly predicted the following scenario: when a premixed flame lifts-off at the base, it becomes unstable and is eventually blown out of the computational domain. The blow-out process is studied by analyzing the stable flame at 0 = 0.4 and the unstable flame at 0 = Entrainment of ambient air by the fuel jet upstream of the lifted-flame base reduces the local equivalence ratio which, in turn, is found to be responsible for the blow-out of the flame. INTRODUCTION Combustion efficiency, lean blowout, high-altitude relight, and emissions are some of the important characteristics considered in the design of gas turbine combustors. In developing accurate models for combustor design, a thorough understanding of these characteristics in simple systems such as jet flames is required. Gaseous jet flames fall into two general categories. In the first, the fuel bums as it is brought into contact with the air. The combustion processes in this case are mainly determined by the rate of diffusion of the oxidizer and fuel; hence, these are commonly referred to as diffusion flames. In the second, the fuel and oxidizer are well mixed prior to burning, and the combustion processes depend on the burning velocity of the fuel/oxidizer er mixture; hence, these are referred to as premixed flames. The physical mechanisms governing the stability, lift-off, and blowout of a jet flame are not fully understood. The stabilization mechanism for a jet diffusion flame has been investigated by Vanquickenbore and van Tiggelen (1966), Gollahalli et al. (1985), Pitts (1988, 1990), and Takahashi et al. (1984, 1992). When a diffusion flame is lifted-off, the fuel and oxidizer in the region upstream of the flame base mix rapidly and create a partially premixed condition. Therefore, the flame stability which is also closely related to the lift-off and blowout characteristics of the flame is believed to be controlled by the balance between the approach velocity and the turbulent burning velocity of the mixture (Vanquickenbore and van Tiggelen, 1966). However, Takahashi et al. (1984) argued that the local velocity in the vicinity of an attached flame is laminar; hence, the laminar burning velocity of the mixture should be taken into account in studies of flame stability. These issues may be addressed systematically by performing detailed numerical simulations of both the attached and lifted flames. Laminar jet flames of both diffusion and premixed types are often investigated theoretically and experimentally to elucidate the chemical and physical processes of combustion. By taking advantage of the fundamental differences between diffusion and premixed flames, investigators have developed separate mathematical models for simulating these flames. For example, for the study of jet diffusion flames, multidimensional-flow models have been used which incorporate simple chemical kinetics such Presented at the International Gas Turbine and Aeroengine Congress and Exposition Houston, Texas - June 5-8, 1993

2 Downloaded From: on 07/10/ Terms of Use: as mixture fraction (Mitchell et al., 1980) and one-step global chemistry (Ellzey at al., 1990). Premixed flame characteristics, on the other hand, are generally studied using one-dimensional flow and detailed chemical-kinetics models (Dixon-Lewis and Islam, 1983). Obviously, these approaches have limitations with respect to understanding the chemistry and fluid dynamics aspects of jet flames, e. g., stability and blowout. With the continuous advancements in computer-hardware technology and the need to improve the understanding of combustion phenomena under complex conditions, modelers are incorporating detailed chemistry into diffusion-flame models and multidimensionality into premixed-flame calculations. As a result, models are becoming somewhat general in treating different types of combustion systems simultaneously. In the present study, a time-dependent, axisrmnetric computational-fluid-dynamics with finite-rate chemistry (CFDC) model is used to investigate the blowout process of a H2/air premixed jet flame. A wide spectrum of jet flames from fuel-lean premixed (near-lean-blowout condition) to fuel-rich diffusion was simulated using this model. MODELING Time-dependent, axisynunetric Navier-Stokes equations written in the cylindrical-coordinate system are solved along with species- and energy-conservation equations (Katta et al., 1994a). The body-force term resulting from the gravitational field is included in the axial-momentum equation. A detailed-chemical-kinetics model has been used to describe hydrogen-air combustion. This model consists of eleven species; namely, H 2, 02, H, 0, OH, 1110, H02, H20 2, N, NO, and N2. The chemical kinetics used among the constituent species is reported in ICatta et al. (1994b). The rate constants for the H reaction system are taken from Cowart et al. (1989) and those for the N 2 reactions are taken from Miller and Bowman (1989) and Hanson and Salimian (1984). Temperature- and species-dependent property calculations are incorporated in the model. The enthalpy of each species is calculated from polynomial curve fits (Burcat, 1984). The viscosity and the thermal conductivity of individual species are estimated from the Chapman-Enskog collision theory, and the mixture properties are obtained employing Wilke's semi-empirical formulae. The effective binary-diffusion coefficient (Williams, 1985) of the individual species in the local mixture is calculated using molecular dynamics and the Lennard-Jones potentials. The governing equations are integrated on a nonuniform staggered-grid system. An orthogonal grid having rapidly expanding cell sizes in both the axial and the radial direction is employed. The finite-difference forms of the momentum equations are obtained using an implicit QUICKEST scheme (Leonard, 1979; Katta et al., 1994a) and those of the species and energy equations are obtained using a hybrid scheme of upwind and central differencing (Spalding, 1972). At every time-step, the pressure field is calculated by solving the pressure Poisson equations simultaneously and utilizing the LU (Lower and Upper diagonal) matrix-decomposition technique. Simple laminar jet flames are considered in this study. Fuel in the form of a H 2/air mixture is emerging from a tube that is surrounded by a very low speed co-annular flow of ambient air. The different burning patterns for this jet flame are dependent on the H 2-to-air ratio of the fuel. The fuel and oxidizer in the fuel stream burn along the conical surface (premixed-flame cone) that is attached to the fuel tube. In the fuel-rich conditions, the excess fuel diffuses in a radially outward direction and establishes a diffusion flame when it encounters ambient oxygen that is diffusing in a radially inward direction into the combustion products. As the fuel content in the central stream increases, the combustion along the premixed cone becomes weak and the flame transforms into a pure diffusion type. Initial and Boundary Conditions A vertically mounted premixed jet flame is simulated in this study. The burner assembly consists of a central fuel tube and a coannular air duct. The computational domain is bounded by the axis of symmetry and an outflow boundary in the radial direction and by the inflow and another outflow boundary in the axial direction. The outer boundaries in the z and r directions are located sufficiently far from the nozzle exit and the symmetry axis, respectively, that the propagation of boundary-induced disturbances into the region of interest is minimized. Since most of the chemical reactions associated with the premixed fuelhir mixture occur within a narrow zone surrounding the burning cone, grid points are highly clustered in this region. A prescribed velocity profile is used at the fuel-inflow boundary, while a flat velocity profile is imposed at the air-inflow boundary. An extrapolation procedure with weighted zero- and first-order terms is employed to estimate the flow variables at the outflow boundary. The burner consists of a 7.2-mm-diameter central tube and a large coannular duct. A mixture of H N 2 was used as the fuel. A parabolic velocity having an average of 8 in/s was prescribed at the central-tube exit. In order to reduce the influence of the boundary on the flame, a low uniform shroud flow of ambient air at 0.15 m/s was used in the annular duct. Axisymmetric calculations were made on a physical domain of 200 x 150 mm utilizing a 201 x 71 nonuniform grid system. Grid points are clustered in such a way that the maximum grid spacing in the region containing the reaction cone of the premixed flame is 0.2 mm. For the given flow conditions, a steady-state solution was obtained by neglecting the unsteady terms in the governing equations. The flame in these calculations was ignited by temporarily increasing the temperature (to

3 kr K) at a few grid points near the fuel-tube exit during the first 10 time-steps. The dynamic-flame simulations were performed using the previously obtained steady-state solution as the initial flow condition. RESULTS AND DISCUSSION The procedure described in the previous section was used to make calculations for H 2/air premixed flames having different equivalence ratios (ratio of the mole fraction of H2 with respect to air to the mole fraction of the stoichiometric H 2/air mixture). The results were used for evaluating the mathematical model and investigating the blowout characteristics of the premixed flame. Validation of the Model An important parameter in premixed-type combustion is the burning velocity (or flame speed) which is defined as the velocity of the unburned gases normal to the inner cone as they move into the combustion zone. This quantity is often used to validate the physical and chemical models for laminar premixed flames. Calculations for 11 2/air premixed flames having different equivalence ratios were made using the numerical procedure described in the Modeling section. The predicted burning velocities (S0 for different flames are compared in Fig. 1 with the experimental data obtained by Takahashi et al. (1983) who used shadowgraphs and Laser Doppler Velocimetry (LDV) for locating the inner cone and for measuring the local unburned-gas velocity, respectively. Experimental data on burning velocity compiled by Takahashi et al. (1983) from different sources show considerable scatter, with upper and lower limits being represented in Fig. 1 by plus and minus symbols, respectively. The scatter is primarily caused by the variations in the experimental techniques used for 3.5 identifying the inner cone. The computed results shown in Fig. 1 are obtained from the inner cone that is identified from a low-value (104 of the maximum) iso-heat-releaserate contour. As expected, the burning velocity increased initially with equivalence ratio (0) and then decreased. The predicted maximum burning velocity and the corresponding CD are 3.16 rn/s and 1.8, respectively, showing good agreement with the data of Takahashi et al. (1983). Overall, the agreement between the computed and measured burning velocities is very good for the fuel-lean and slightly fuel-rich cases, and the model seems to predict reasonably well for the fuel-rich conditions also. Fuel-lean Flames Flame locations and temperatures obtained for stoichiometric and three other fuel-lean conditions are shown in Fig. 2. The volumetric flow rate and, hence, the velocity (V, = 8 m/s) at the burner exit were maintained constant in all calculations. It should be noted that changes in the equivalence ratio are associated with changes in the mass-flow rate due to different densities of the fuel/air mixtures. The flame location (rf) at a given axial distance in Fig. 2(a) is identified by scanning radially for the peak temperature a Calculations } Exp. (Takahashi et al) Equivalence Ratio 0 Fig. 1. Comparison of predicted and measured burning velocities at different equivalence ratios for H 2/air system T (K) 2500 (a) Fig. 2. Properties of flame surface for different fuel-lean conditions, (a) Location of flame surface obtained by radial scanning for peak temperature. Variation of temperature along flame surface. 3

4 which is shown in Fig. 2. Flames for all equivalence ratios except for 4) = 0.2, are anchored at the tube exit in the shear layer formed between the fuel/air mixture and the annulus air flow. As mentioned earlier, a parabolic velocity profile representing fully developed tube flow was used at the fuel-tube exit. The lower fuel velocity near the tube walls aids in stabilizing the highly fuel-lean flames. As a result, stable burning was obtained for equivalence ratios up to 0.4. When to -was fiirther reduced to 0.2, the flame lifted off the burner and slowly blew out of the computational domain. Note that the data shown in Fig. 2 for 4) = 0.2 were obtained before the solution had reached a converged value. Calculations for the 4) = 0.2 case were initiated by making use of the converged flame for 4) = 0.4 as an initial solution. As the calculations advanced in time, the lean-fuel mixture (4) = 0.2) and flame quenching were convected downstream by replacing the previous 4) = 0.4 solution. In general, the temperature along the flame surface was found to decrease with height (or axial distance) as a result of the less-than-unity Lewis number of the stoichiometric and fuel-lean hydrogen/air flames. However, as expected, the average flame temperature decreased from 2500 to 1500 K when the equivalence ratio was varied from 1.0 to 0.4. When the equivalence ratio was reduced to 0.2, the flame temperature decreased to about 1000 K and the flame was quenched. Attached vs Lifted Flame For understanding the lift-off mechanism of premixed hydrogen flames, additional calculations were made in the neighborhood of 4) = 0.4. Interestingly, for the chosen velocity conditions, the flame was found to lift-off when the equivalence ratio was slightly lower than 0.4. The flowfields near the bases of the 4) = 0.4 and 4) = flames are shown in Figs. 3 and 4, respectively. The velocity field near the base of the attached flame (4) = 0.4) is superimposed on the temperature field in Fig. 3(a). The temperature corresponding to each contour line is shown in the table at the top of the figure. The fuel/air mixture emanating from the fuel tube with a parabolic velocity profile is burning along the outer edge of the jet. As seen in Fig. 3 the intense combustion of the premixed fuel and air has significantly increased the velocity in the flame zone, and the flame is stably attached to the fuel tube. Entrainment of outer air into this attached flame is shown in Fig. 3. Here, the iso-mole-fraction contours of oxygen are plotted with thick solid lines. Note that since this is a fuel-lean flame, oxygen is not completely consumed in the flame zone (minimum oxygen mole fraction is 0.105). Streamlines (thin solid lines) and isofuel mole-fraction contours (broken lines) are superimposed on the oxygen-concentration field in Fig. 3. Both the inner and the outer cones of the premixed flame can be T (K) X, i Fig. 3. Near-base structure of attached fuel-lean flame at (I) = 0.4. (a) Velocity superimposed over iso-temperature contours. Streamlines (thin solid lines) are superimposed over iso-mole-fraction contours of fuel (broken lines) and oxygen (thick solid lines). Contours tables are given at the top of each plot. identified from this plot. Burning of the incoming fuel/air mixture deflects the streamlines along the inner cone, and the entrainment of annulus air by the higher-velocity combustion products causes the streamlines originating on the annulus-air side to converge toward the outer cone. The flowfield near the base of the lifted flame is shown in Fig. 4. The equivalence ratio used for this flame was 0.395, and the calculations were initiated using the solution shown in Fig. 3 O. e., = 0.4 flame). Figure 4 represents the solution obtained ms after the start of the calculation. During this period the flame base has moved from its attached location (i. e., z = 0) to about 4 mm above the tube exit. On the other hand, since the velocity of the fuel/air mixture at the jet center is 16 m/s, the 4) = mixture has replaced the 4) = 0.4 mixture up to z = 300 mm during the same time period. Since the fuel-jet velocity at other radial locations is lower than 16 m/s and higher than 1 m/s (note that a parabolic distribution was used), the fresh mixture (4) = 0.395) in the outer core of the fuel jet has propagated downstream at least 20 mm. Since the velocity of the combustion products in the flame zone is much higher than 1 mis, local quenching caused by the fuel-lean mixture conditions would have convected the combustion products of the I 1 I Downloaded From: on 07/10/ Terms of Use:

5 : *; T (K) 1750 (a) I XI 0.24 Fig. 4. Instantaneous-near-base Structure of lifted fuel-lean flame captured 12.5 ms after the initiation of calculations. (a) Velocity superimposed over iso-temperature contours. Streamlines (thin solid lines) are superimposed over iso-mole-fraction contours of fuel (broken lines) and oxygen (thick solid lines). initial solution beyond z = 20 mm, leaving a cold flow in the region depicted in Fig. 4. Therefore, the flame seen in Fig. 4 between z = 4 and 20 mm should be supported by the burning of the fuel/air mixture having an equivalence ratio of The reason for the lift-off of the premixed flame is evident from Fig. 4 where the mole fraction of oxygen (thick solid lines), streamlines (thin solid lines), and the mole fraction of the fuel (broken lines) are plotted. When the flame is lifted, fuel in the center jet diffuses radially outward into the annulus-air stream; at the same time oxygen in the air stream diffuses into the fuel jet. This results in a mixture in the fuel-jet shear layer whose equivalence ratio is < Entrainment of air by the rapidly moving fuel, in fact, brings oxygen to the fuel side of the flame near the base [cf. note the wrapping of isomolar Contour No. 8 of oxygen around the flame base in Fig. 4]. The leanness of the mixture in the fuel-jet shear layer is also evident from the streamline pattern and temperature distribution in the neighborhood of the flame base. Comparison of Figures 3 and 4 indicates that the fourth streamline from the center which is entering the flame zone is deflected identically by the premixed combustion in the two cases. However, this is not the case near the flame base. When the flame is attached to the fuel tube [Fig. 3], the streamlines (e.g., fifth from the centerline) are deflected more than those of the lifted flame [e.g., fifth and sixth from centerline in Fig. 4]. In the latter case the streamlines remain almost straight while entering the combustion zone. The differences in the streamline patterns suggest that the burning velocity at the flame base is decreased; in other words, the mixture becomes more fuel lean when the flame is lifted. The lower temperatures near the flame base in Fig. 4(a) as compared to those in Fig. 3(a) also indicate a decrease in the local equivalence ratio. As mentioned earlier, the flame shown in Fig. 4 represents the solution for the 0 = flame at ms after initiation of the calculations. As the solution is further advanced in time, the flame slowly moves downstream and eventually blows out of the computational domain. Figure 5 shows the solution obtained after 90 ms. A trace of hot gases can be identified at the top in Fig. 5(a) from the iso-temperature contour. Laminar mixing between the fuel and the ambient oxygen is evident in Fig. 5. Flow structures at an axial distance of 2 mm above the fuel tube for the attached (0 = 0.4) and lifted (0 = 0.395) flames are shown in Figs. 6, 7, and 8. Legends "Lifted-a" and "Lifted-b" in these figures represent the flame (a) Fig. 5. Instantaneous-near-base Structure of lifted flame at t= ms. (a) Velocity superimposed over isotemperature contours. Streamlines (thin solid lines) are superimposed over iso-mole-fraction contours of fuel (broken lines) and oxygen (thick solid lines). 5

6 moo a Attached a Lifted-a 0--- Lifted (a) e-- Attached e-- Lifted-a --e Lifted-13 Fig. 6. Radial distributions of (a) temperature, and heat-release rate across the attached and lifted flames at an axial location 2 mm above the fuel tube. solutions obtained at times 12.5 and ins after initiation of the calculations. The maximum temperature of the attached flame [Fig. 6(a)] is 1506 K which matches closely the adiabatic flame temperature of 1465 K. When the flame is lifted, the maximum temperature at this axial location (z = 2 mm) decreases to 1235 after 12.5 ms of real time and then to 350 K after ms. The equivalence ratios corresponding to these adiabatic flame temperatures are 0.31 and 0.02, respectively. The heat-release rates across the flames for the three cases considered are given in Fig. 6. The temperature and flowfield around the base of the lifted premixed flame shown in Fig. 4 suggest that the increased leanness of the local mixture first quenched the flame and then gradually blew it out at downstream locations. It is known that local extinction can also occur if the residence time resulting from the flow velocity is much higher than the reaction time of the chemical kinetics. However, this study indicates that blow-out of the premixed jet flame does not occur as a result of the local-extinction phenomenon. This observation is evident from the axial and radial velocity plots in Fig. 7. Note that the parabolic axial-velocity profiles used at the exit of Fig. 7. Radial distributions of (a) axial and radial velocity across the attached and lifted flames at an axial location 2 mm above the fuel tube >c Fig. 8. Species structure of the attached and lifted flames at an axial location 2 mm above the fuel tube. 6

7 the fuel tube are identical for the attached (d) = 0.4) and lifted ( 1) = 0.395) flames. The axial velocity [Fig. 7(a)] at the flame surface ( between r = 3 and 4 mm) is highest for the attached flame and lowest for the "Lifted-b" flame. Since the equivalence ratios used for the attached and lifted flames are very similar, one might expect only a small difference in the reaction times for the two cases. This means that if the flame is attached for the "Attached" case, it should not be extinguished for the "Lifted-a and -b" cases because of the residence time (or velocity). The negative radial velocity (r > 4 mm) in Fig. 7 which indicates the entrainment of ambient air into the flame zone has increased with lift-off of the flame. This additional air mixes with the fuel and makes the local mixture dilute with respect to the fuel which, in turn, leads to flame-quenching. The molar-concentration distributions of fuel (H 2) and oxygen across the flame for both the attached and the lifted condition are plotted in Fig. 8. The slight difference in the mole fraction of H2 in the fuel jet for the attached and lifted flames (at r < 3 mm in Fig. 8) is due to the difference in the equivalence ratios used. For the attached flame at r = 3.8 mm (flame location), the fuel (H 2) has been consumed completely and the oxygen concentration has reached a minimum value. This structure in species profiles is typical of a fuel-lean premixed flame. Interestingly, the species structure is similar for the "Lifted-a" flame also. That is, at the flame location (r = 4 mm), the fuel is totally consumed, and oxygen has reached a minimum value. However, the minimum oxygen concentration is higher for the "Lifted-a" flame than for the "Attached" flame, suggesting a leaner condition for the former case. When the flame is blown out farther downstream ("Lifted-b" case) because of the leanness of the local mixture, the species structure at z = 2 mm approaches that of a typical cold jet. That is, the zero-fuel location matches with maximum oxygen-concentration location. CONCLUSIONS A time-dependent, axisymmetric computational-fluiddynamics code with a detailed H N 2 chemistry model is used to simulate vertically mounted H 2/air jet flames. A simple combustion system is investigated in which a central fuel jet issues into a low-speed coannulus air flow at an average velocity of 8 m/s. The entire spectrum of premixed jet flames from fuel-lean (near-lean-blowout) to fuel-rich are computed using this code. The predicted burning velocities for different equivalence ratios are compared with experimental data. The code has correctly predicted the maximum burning velocity of H 2/air jet flames which occurs at an equivalence ratio of 1.8. Lifting characteristics of the fuel-lean flame are investigated by gradually reducing the fueuair ratio (while maintaining the volumetric flow rate constant) of the stoichiometric flame. For the velocity conditions considered in this study, it is found that a stable flame that is attached to the fuel tube establishes for equivalence ratios up to 0.4. When the equivalence ratio is reduced further, the flame lifts-off from the base of the fuel tube. The lifted flame is found to be unstable and is eventually blown out of the computational domain. This numerical prediction of the lean-blow-out process agrees qualitatively with that observed in the experiments. The blow-out process is studied by analyzing the attached and lifted flames computed for equivalence ratios of 0.4 and 0.395, respectively. It is found that blow-off in premixed flames occurs as a result of the entrainment of ambient air into the fuel jet which decreases the local equivalence ratio of the mixture around the lifted-flame base further. Velocity (or flow residence time) seems to play only a secondary role in the blow-out characteristics of a premixed jet flame. On the other hand, it is known that velocity affects the lift-off and blow-out processes of a diffusion flame. Since the CFD model used in this study is capable of simulating the entire range of laminar jet flames from the fuel-lean premixed type to the fuel-rich diffusion type, a logical extension of this work is to investigate the blow-out characteristics of a jet diffusion flame. ACKNOWLEDGMENTS This work was supported, in part, by Air Force Contract F C-2033 and the Air Force Office of Scientific Research. The authors would like to thank Dr. Fumiaki Takahashi for helpful discussions and Mrs. Marian Whitaker for help in preparing the manuscript. 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8 Katta, V. R., Goss, L. P., and Roquemore, W. M., 1994a, "Numerical Investigations of Transitional H2/N2 Jet Diffusion Flames," AIAA Journal, Vol. 32, No. 1, p. 84. Katta, V. R., and Roquemore, W. M., 1994b, "Numerical Studies on Multidimensional 112/021N2 Premixed Jet Flame," AIAA Paper No , 32nd Aerospace Sciences Meeting and Exhibit, Jan , Reno, Nevada. Leonard, B. P., 1979, "A Stable and Accurate Convective Modeling Procedure Based on Quadratic Upstream Interpolation," Computational Methods in Applied Mechanics and Engineering, Vol. 19, p. 59. Miller, J. D., and Bowman, C. T., 1989, "Mechanism and Modeling of Nitrogen Chemistry in Combustion," Progress in Energy and Combustion Science, Vol. 15, p Mitchell, R. E., Sarofim, A. F., and Clomburg, L. A., 1980, "Experimental and Numerical Investigation of Confined Laminar Diffusion Flames," Combustion and Flame, Vol. 37, Nos. 1 and 2, p Pitts, W. P., 1988, "Assessment of Theories of the Behavior and Blowout of Lifted Turbulent Jet Diffusion Flames," Twenty -Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, p Pitts, W. P., 1990, "Large-Scale Turbulent Structures and the Stabilization of Lifted Turbulent Jet Diffusion Flames," Pitts, W. P., Twenty - Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, p Spalding, D. B., 1972, "A Novel Finite Difference Formulation for Difference Expressions Involving Both First and Second Derivatives," International Journal for Numerical Methods in Engineering, Vol. 4, p Takahashi, F., Mizomoto, M., and Ikai, S., 1983, Alternative Energy Sources III (T. Nejat Veziroglu, Ed.), Vol. 5 Nuclear Energy/Synthetic Fuels, Hemisphere Publishing Co., WA, p Takahashi, F., Mizomoto, M., Ikai, S., and Fukati, N., 1984, "Lifting Mechanism of Free Jet Diffusion Flames," Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, p Takahashi, F., Schmoll, W. J., and Vangsness, M. D., 1990, "Effects of Swirl on the Stability and Turbulent structure of Jet Diffusion Flames," AIAA , 28th Aerospace Sciences Meeting and Exhibit, Jan. 8-11, Reno, Nevada. Vanquickenborne, L., and van Tiggelen, 1996, "The Stabilization Mechanism of Lifted Diffusion Flames," Combustion and Flame, Vol. 10, p. 59. Williams, F. A., 1985, Combustion Theory - The Fundamentals of Chemically Reacting Flow Systems, Addison-Wesley Publishing Co., Reading, MA. 8