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Opposed-flow flame spread across n-propanol pools
Twenty-Sixth Symposium (International) on Combustion/The Combustion Institute, 1996/pp. 1319–1325
OPPOSED-FLOW FLAME SPREAD ACROSS n-PROPANOL POOLS D. N. SCHILLER and W. A. SIRIGNANO Department of Mechanical and Aerospace Engineering University of California Irvine, CA 92697 USA
A computational study is made of the effects of forced, opposed air flow on pulsating and uniform flame spread across n-propanol pools at either normal or zero gravity conditions. The numerical model incorporates finite-rate chemical kinetics, variable properties, and an adaptive gridding scheme in the direction of flame spread. In zero gravity, the combination of forced, opposed flow and thermocapillary-driven concurrent flow can cause a gas-phase recirculation cell to form ahead of the flame and thus lead to flame pulsations. The pulsation mechanism, in this case, is essentially the same as that previously detailed for pulsating flame spread in normal gravity without forced flow [1]. In either normal gravity or zero gravity, increasing the opposed air speed (Uopp) causes a transition from uniform flame spread to pulsating flame spread. The value of Uopp that corresponds to this transition increases with initial pool temperature (To). Unlike with normal gravity flame spread, the mean flame spread rate (U¯fl) in zero gravity decreases significantly because of this transition for To , 218C. As Uopp is increased further, both U¯fl and the pulsation frequency increase in zero gravity but change only slightly in normal gravity. For high values of Uopp, U¯fl varies little with gravity level. For low values of Uopp, the flame spread rate in zero gravity is sensitive to the initial profile of fuel vapor in the gas phase.
Introduction This computational study examines the effects of forced, opposed air flow on flame spread across liquid fuel pools initially below the flash point temperature. When a liquid fuel is initially below its flash point temperature, it must be heated sufficiently to create a combustible mixture of fuel vapor before ignition and flame spread can occur. Forced, opposed flow as well as buoyancy and thermocapillary forces affect the heating of the liquid pool and the mixing of fuel vapor and air ahead of the flame. In normal gravity without forced flow, when the initial pool temperature To is in a small range below the flash point temperature, the flame periodically accelerates and decelerates (pulsates) as it propagates across the fuel surface. Flame spread is uniform for higher values of To. The transition temperature between pulsating and uniform spread in 1-g0 without forced flow is approximately 178C for n-propanol [1,2]. A review of flame spread studies can be found in Refs. 3–5. A recent numerical study [1] addressed pulsating and uniform flame spread mechanisms and the effects of gravity level, pool depth, fluid properties, and chemical kinetic coefficients on flame spread across liquid fuel pools without forced flow. Pulsating flame spread was related to a gas-phase recirculation cell that forms just ahead of the flame leading edge because of the combination of buoyancy-driven opposed flow and thermo-
capillary-driven concurrent flow [1,2,6]. One question that arises is whether flame pulsations will occur without buoyancy if forced, opposed flow is provided. Although there have been several theoretical and numerical studies of opposed-flow flame spread across solid fuels [5,7,8], this is the first numerical study of unsteady, forced, opposed-flow flame spread across liquid fuels. Liquid motion and differences in fuel volatility typically cause flame spread rates across liquid fuels to be more than an order of magnitude greater than those across solids such as PMMA or paper. A two-dimensional, transient, numerical model of flame spread across open pools of alcohol fuels without forced flow [1,2] has been modified to study flame propagation across n-propanol with forced, opposed air flow. Ross and Miller [9] compare experimental results to the model’s results for opposed flow flame spread across deep nbutanol pools in normal and reduced gravity. Physical Description of the Problem The geometry of the planar model and select gasphase boundary conditions are shown in Fig. 1. A shallow pool (Hl 4 2 mm) is used because this case agrees well with 1-g0 experiments without forced flow [1,2]. The igniter (size 4 0.4 2 0.09 mm) is modeled as a hot pocket of gas with a temperature that increases linearly from To to 1700 K in 0.1 s.
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Fig. 1. Geometry and select gas-phase boundary conditions for the two-dimensional problem (infinitely wide pool); f 4 u, v, Yi, or T.
The igniter temperature remains constant until the flame reaches xfl 4 2 cm, after which time the temperature of the igniter is no longer held fixed. The initial temperature of the liquid and gas phases is To. The initial conditions for ug, vg, and YF are assumed to be given by a Blasius profile for flow across a flat plate with zero blowing velocity and unity Schmidt number [10]. For simplicity, the initial condition for the fuel vapor concentration along y 4 0 is assumed to be vF,o 4 Psat (To)/Pa, where Pa 4 1 atm is the ambient pressure. This assumption yields a nonsensical initial YF profile for x , 0, but this region is used only for numerical convenience to make the boundary conditions ]f/]x 4 0 along the left-hand side of the gas-phase computational domain more accurate. The initial fuel vapor for x , 0 typically convects out of the computational domain without affecting the flame spread rate. Assumptions and Governing Equations The mathematical formulation of the problem and the assumptions are similar to those of Schiller et al. [1,2]. The main assumptions are: (1) the liquid surface does not recede and remains flat and horizontal; (2) solutocapillary forces are neglected; (3) the binary diffusion coefficient is the same for each species; (4) one-step, second-order, finite-rate chemical kinetics are assumed; (5) gas-phase radiation and radiation from the igniter are neglected; and (6) the effects of concentration gradients on the diffusion of energy are neglected. Other minor assumptions used to simplify the energy equation and to split the pressure in the momentum equation are given by Schiller et al. [1,2]. Laboratory coordinates are used in the solution of the governing equations and in the presentation of results. Variable density and thermophysical properties are included in the analysis. Curve fits of properties, values for the preexponential constant and activation energy in the reaction rate expression, and the unsteady, nondimensional conservation equations of mass, species, momentum, and energy are the same as in Schiller et al. [1,2].
The boundary conditions at the gas/liquid interface follow from a heat balance and balance of the shear stresses with surface tension; a continuity of the temperature and tangential velocity; no dissolution of air or combustion products into the liquid; vF,lg 4 Psat (Tlg)/Pa; and no recession of the liquid surface [1,2]. Temperature gradients along the liquid surface create thermocapillary stresses that are typically larger than gas-phase shear in the following balance: (l]u/]y)l 4 (l]u/]y)g ` rT(dT/dx)lg. Zero normal gradient boundary conditions are used in the gas phase along x 4 110 cm and y 4 Hg 4 6 cm (Fig. 1). At x 4 20 cm in the gas, u 4 1Uopp, v 4 0, YO2 4 0.233, YP 4 YF 4 0, and T 4 To. Along the exit plate (x # 0, yg 4 0), u 4 v 4 ]Yi/]y 4 0 and T 4 To. In the liquid, the side wall at x 4 20 cm is assumed adiabatic, whereas the other two walls are assumed isothermal (T 4 To).
Numerical Method The numerical method [1,2] uses the SIMPLE algorithm [11] with the SIMPLEC modification [12] and the hybrid-differencing scheme. A uniform mesh of Dxfl 4 0.4 mm is used in a 6-mm region around the flame. Behind and ahead of this region, the mesh size increases by geometric progressions. An adaptive gridding scheme is used in the x direction to provide fine resolution of the reaction zone throughout the simulation. The grid in the x direction is changed each time xfl changes by at least 1 mm since the last grid alteration. Linear interpolation is used to map each dependent variable from the old to the new grid. A 132 2 72 nonuniform grid was used for the gas phase, whereas a 132 2 32 nonuniform grid was used for the liquid. The time step was 0.4 ms. The numerical results do not change appreciably if (1) the mesh size or the time step is decreased by a factor of 2 or (2) Hg is increased to 12 cm. For each time step, good convergence and satisfaction of global continuity were attained in two to three iterations.
Results and Discussion Herein, the horizontal location of the flame (xfl) is defined as the point of maximum fuel consumption rate at a particular value of yg (41 mm for most simulations) that intersects the flame leading edge region. The overall length of liquid-phase flow ahead of the flame, dflow, is defined arbitrarily as the difference between the largest value of x where Ulg . 0.5 cm/s and xfl. The threshold value of 0.5 cm/s was selected because it is typically about 3–5% of the value of Ulg at x 4 xfl.
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opposed flow. Figure 2 shows the qualitative flow pattern and temperature contours in the gas phase for two 1-g0 simulations with different values of Uopp. For these cases, numerical error introduced by forcing zero normal-gradient boundary conditions along the truncated exit planes of the domain appears to be confined above the exit plate, several centimeters behind the flame leading edge. Even in 1-g0, the majority of the far-field, gas-phase flow is in the x direction. Contours of maximum reaction rate correspond approximately with the contours of maximum temperature. The flame leading edge region is typically 0.05–0.3 cm above the liquid surface and well within the viscous boundary layer caused by the forced flow. Hence, the speed of the local opposed flow ahead of the flame leading edge is significantly less than Uopp. For example, the Blasius profile for air at 300 K with Uopp 4 30 cm/s gives 1u 4 10.6 cm/s at y 4 0.3 cm and 15 cm from the inlet (i.e., x 4 5 cm in Fig. 1). Fig. 2. Temperature contours (starting at 300 K with a 200-K interval) and qualitative flow pattern in the gas phase for two normal gravity simulations with To 4 198C (t 4 2 s). Isotherms for a 0-g0 simulation with To 4 198C and Uopp 4 50 cm/s are very similar to (b).
Fig. 3. Mean flame spread rate and character of spread (uniform versus pulsating) as a function of gravity level, initial pool temperature, and forced, opposed air speed.
Gas-Phase Profiles Without forced gas-phase flow, numerical error occurs near the open boundary at y 4 Hg because of the blockage of buoyancy-advected vorticity by the reflective boundary condition ]v/]y 4 0 [1]. For the present problem, this error is greatly reduced by downstream advection of the plume by the forced,
Flame Spread and Extinction at Low Uopp Figure 3 shows the predicted mean flame spread rate (U¯fl) and character of flame spread (uniform versus pulsating) for 1-g0 and 0-g0 simulations at different values of To and Uopp. Values of U¯fl calculated with Uopp 4 0 in the present model (initially, vF 4 Psat(To)/Pa throughout the gas phase) differ by less than 10% from U¯fl calculated by a previous model without forced flow [1]. For very low opposed air speeds in 0-g0, the uniform flame spread rate decreases strongly with an increase in Uopp (Fig. 3) due to less fuel vapor initially in the gas phase. For all cases shown in Fig. 3, YF,o (x, y) is determined by a Blasius profile with freestream opposed air velocity U` 4 Uopp. The flame spread rate changes significantly if YF,o is artificially determined by a value of U` not equal to Uopp. For example, with To 4 158C, the flame is predicted to extinguish for Uopp 4 3 cm/s but propagate at 6.2 cm/s for Uopp 4 1 cm/ s. For Uopp 4 3 cm/s, if YF,o is determined by U` 4 1 cm/s (leading to more fuel vapor initially in the gas phase), the flame does not extinguish but rather spreads uniformly at 5.5 cm/s. Similarly, the flame extinguishes for Uopp 4 1 cm/s if YF,o is determined by U` 4 3 cm/s. U¯fl is less sensitive to the assumed profile of YF,o for higher values of Uopp in 0-g0 and for all values of Uopp in 1-g0 because the opposed flow caused by forced and/or natural convection in these cases quickly changes the distribution of fuel vapor. As shown by Bhattacharjee and Altenkirch [13] for flame spread across solid fuels, gas-phase radiation also adds to the effects of convective heat losses that lead to extinction in quiescent environments. Hence, radiation should be included in future analyses of flame spread across liquid pools for low Uopp.
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Uniform Flame Spread Flame spread is uniform when no gas-phase recirculation cell is formed ahead of the flame leading edge. Concurrent gas-phase flow ahead of the flame always exists, at least near the liquid surface, due to both the thermocapillary-driven liquid surface motion and hot gas expansion. The gas-phase recirculation cell does not form if the buoyancy-driven and/ or forced, opposed flow is not strong enough to overcome part of this concurrent motion. For uniform spread in either 1-g0 or 0-g0 (e.g., Fig. 4), hot gas expansion, together with the gas-phase shear from the concurrent liquid surface motion, forces incoming gas-phase flow away from the flame leading edge. Behind the flame, the flow accelerates due to buoyancy and/or hot gas expansion to a speed of 30–35 cm/s in 1-g0 and to 10–15 cm/s in 0-g0. Pulsating Flame Spread Fig. 4. Stream function contours in the liquid and fuel consumption rate contours and flow pattern in the gas phase near the flame leading edge for uniform flame spread in zero gravity (To 4 198C, Uopp 4 20 cm/s, t 4 1 s).
In either 1-g0 (for To . 178C) or 0-g0, flame spread transitions from uniform to pulsating as the value of Uopp is increased. The mechanism of pulsating flame spread in 0-g0 with forced, opposed flow is essentially the same as that of pulsating flame spread in 1-g0 without forced flow. This mechanism was described previously [1,2,6] for 1-g0 flame spread without forced flow, and it is summarized here for 0-g0 flame spread with forced, opposed flow. Figure 5 shows details near the flame leading edge during the slow-spread portion of a 0-g0 pulsation cycle with Uopp 4 50 cm/s and To 4 198C. The liquid surface
Fig. 5. Contours in the gas phase, liquid surface temperature and velocity profiles, and stream function contours in the liquid near the flame leading edge during the slow-spread portion of a 0-g0 flame pulsation cycle (To 4 198C, Uopp 4 50 cm/s, t 4 0.95 s). The contour interval for YF and YO2 is 0.02.
OPPOSED-FLOW FLAME SPREAD ACROSS n-PROPANOL
Fig. 6. Variation of the maximum fuel consumption rate, extent of liquid flow ahead of the flame, maximum liquid surface velocity, flame position, and instantaneous flame speed over several pulsation periods for To 4 198C, Uopp 4 50 cm/s, and 0-g0.
is heated over a distance of approximately dflow ahead of the flame leading edge. The liquid surface velocity near the flame front is due to thermocapillary convection. A gas-phase recirculation cell forms ahead of the flame because of the combination of the concurrent liquid surface flow and the opposed gasphase flow (due to the forced flow at the inlet in this case or due also, in part, to buoyancy in 1-g0). The gas-phase recirculation cell entrains fuel vapor and plays a critical role in the flame pulsation process. During this portion of the cycle, a premixed region just above the lean flammability limit (YF . 0.045) [14] forms in the downstream side of the recirculation cell. When a sufficient quantity of fuel vapor has accumulated ahead of the flame because of the gasphase recirculation cell, the maximum fuel consumption rate increases markedly, causing the flame to accelerate through the premixed region to a maximum speed of approximately 15 cm/s. Simultaneously, the increased rate of hot gas expansion destroys the recirculation cell structure ahead of the flame by forcing the flow away from the flame leading edge. After the flame accelerates through the premixed region that was formed in the recirculation cell, the maximum reaction rate and the rate of hot gas expansion decrease. The forced flow overcomes the hot gas expansion and once again creates opposed
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flow just ahead of the flame leading edge, thus reforming a small gas-phase recirculation cell. The size of the recirculation cell then increases to approximately the same length as in the beginning of the cycle. As shown by Fig. 6, the maximum fuel consumption rate, the extent of liquid flow ahead of the flame, the maximum liquid surface velocity (which occurs approximately under the flame leading edge), and the instantaneous flame speed (Ufl) oscillate in phase with each other during pulsating spread. The maximum reaction rate occurs before the end of the fast-spread portion of the cycle, thus creating a phase lag between dflow and (1rF )max. Note that dflow . 0 does not imply that there is liquid-phase convection ahead of the flame in a flame-fixed reference frame. Figure 6 shows that Ufl equals or exceeds Ulg,max during roughly the first half of the fast-spread portion of the cycle. During the slow-spread portion of a pulsation cycle in 1-g0 (regardless of To or Uopp) or in 0-g0 with relatively high values of Uopp or To, the flame propagates forward at a speed of 1–3 cm/s. For lower values of Uopp or To in 0-g0, however, the pulsating flame recedes slightly during the slow-spread portion of the cycle (Fig. 6). These 0-g0 cases have lower fuel consumption rates and flame temperatures than their 1-g0 counterparts because of lower transport rates of fuel vapor and oxygen into the reaction zone. Figure 6 shows that the maximum fuel consumption rate is near its minimum value when the flame is receding. Therefore, perhaps the flame recedes to a region of higher YF concentration to sustain propagation. As shown previously for 1-g0 flame pulsations without forced flow [1], the rate of diffusion of fuel vapor into the reaction zone from below the flame increases during the fast-spread portion of the cycle. Therefore, although the abrupt acceleration of the flame requires the formation of a premixed region ahead of the flame, combustion in the flame leading edge region is of a mixed mode throughout the pulsation cycle. Other features of the 0-g0 flame structure and liquid-phase flow pattern without forced, opposed flow are the same as in 1-g0. In both cases, the flame structure just behind the flame leading edge has similar character to a classic triple flame. The mixture in the upper trailing portion of the flame appears to have a very lean, premixed character from the initial mixture of fuel vapor and oxygen because it exists in a region of relatively low YF but high YO2. This portion of the flame extends less than 1 cm behind the flame tip. The lower portion of the trailing flame is clearly in a region of diffusive burning, as evident from the separation of YF and YO2 contours shown in Fig. 5. It extends all the way to the edge of the pool (x 4 0) for low values of Uopp and past the edge of the pool (x , 0) for larger values of Uopp. The premixed, rich part of the triple flame is absent because it is quenched by the rela-
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Fig. 7. Flame position versus time for different values of Uopp (To 4 218C, 0-g0). The extent of liquid flow ahead of the flame is also shown for Uopp 4 70 cm/s.
tively cold pool surface. For the shallow pools studied previously, the flow pattern in the liquid varies little between 0-g0 and 1-g0. During the slow-spread portion of a pulsation cycle, one large vortex in the liquid rotates clockwise about two vortex centers. The height of the vortex centers is yl ' 2Hl/3, which is typical for a shallow pool [1]. During the fastspread portion of a cycle, only one vortex center appears in the liquid near the flame leading edge. Both the mean flame spread rate and the pulsation frequency ( f ) generally increase with Uopp in 0-g0. For example, U¯fl and f increase by 30% and 100%, respectively, for To 4 158C when Uopp is varied from 30 to 70 cm/s. A higher opposed velocity results in a smaller gas-phase recirculation cell, which results in a higher pulsation frequency (less time is required to accumulate fuel vapor ahead of the flame). The increase in U¯fl with Uopp could be due to a greater burning rate caused by the increased rate of oxygen transport to the reaction zone. In 1-g0, U¯fl decreases slightly with Uopp in the pulsating regime. For example, for To 4 158C, U¯fl and f in 1-g0 decrease by only 10% as Uopp is varied from 30 to 70 cm/s. For a given To and Uopp, the pulsation frequency typically differs by less than a factor of 2 between 1-g0 and 0g0. Figure 3 shows that U¯fl is fairly insensitive to gravity level for high values of Uopp. Very high values of Uopp that could lead to a reduction in U¯fl and ultimately extinction were not investigated in this study. Transition of Flame Spread Mode As To increases, there is initially more fuel vapor in the gas phase, which leads to higher rates of fuel consumption and hot gas expansion. As the rate of
hot gas expansion increases, a higher opposed velocity at the inlet is required to create a gas-phase recirculation cell ahead of the flame. Therefore, the value of Uopp that corresponds to the transition between uniform and pulsating flame spread increases with To (Fig. 3). For example, the transition in 0-g0 occurs at Uopp ' 25 cm/s for To 4 198C and Uopp ' 70 cm/s for To 4 218C. The thickness of the viscous boundary layer above the liquid surface caused by the inlet flow increases with the square root of the distance from the inlet. Thus, in some cases, flame spread can transition from uniform to pulsating as it propagates closer to the inlet because of an increase in the local opposed velocity ahead of the flame leading edge. Figure 7 shows xfl as a function of time for different values of Uopp in 0-g0 (To 4 218C). In each case, dflow 4 0.5–1.0 cm before the igniter is deactivated (e.g., at t 4 0.27 s for Uopp 4 50 cm/s and at t 4 0.47 s for Uopp 4 70 cm/s). Subsequently, flame spread is uniform and dflow ' 0.3 cm for Uopp 4 50 cm/s until the flame reaches x 4 13 cm. For Uopp 4 70 cm/s, the variation of xfl and dflow with t is very similar to that for Uopp 4 50 cm/s (except for a longer ignition transient) until the flame reaches x ' 9 cm, after which the flame pulsates. For a pool 4 cm shorter, the flame transitions to pulsating at x ' 5 cm. This transition also occurs during propagation for Uopp 4 80 cm/s. When flame spread transitions from uniform to pulsating, U¯fl decreases by nearly a factor of 2 in 0-g0 for To 4 158C or To 4 198C, but not nearly as much for To 4 218C or for any of the initial temperatures investigated for 1-g0. This sharp decrease in the mean flame spread rate is due to the recession of the flame during the slow-spread portion of the pulsation cycle. The size of the gas-phase recirculation cell that forms at the transition to pulsating spread is also typically much larger in 0-g0 than in 1-g0. For uniform spread in 0-g0 at moderate or low values of Uopp, gas-phase flow ahead of the flame caused by hot gas expansion remains concurrent over a 1-cm distance ahead of the flame, even though dflow is less than 0.5 cm (Fig. 4). Hence, when flame spread transitions from uniform to pulsating because of an increase in Uopp, the length of the gas-phase recirculation cell in this case is 1–2 cm instead of approximately 2 mm as in 1-g0 flame spread.
Conclusions In zero gravity, the combination of forced, opposed flow and thermocapillary-driven concurrent flow can cause a gas-phase recirculation cell to form ahead of the flame and, thus, lead to flame pulsations. The pulsation mechanism in this case is essentially the same as that previously detailed for pulsating flame spread in normal gravity without forced
OPPOSED-FLOW FLAME SPREAD ACROSS n-PROPANOL
flow [1]. In either 1-g0 or 0-g0, increasing the opposed air speed causes a transition from uniform flame spread to pulsating flame spread when the opposed velocity overwhelms hot gas expansion to form a gas-phase recirculation cell ahead of the flame. The value of Uopp that corresponds to this transition increases with initial pool temperature. Unlike 1-g0 flame spread, U¯fl in 0-g0 decreases significantly due to this transition for To , 218C. Because of the variation of the viscous boundary layer thickness (created by the inlet flow) above the liquid surface, the flame can transition from uniform spread to pulsating spread as it propagates across the pool. Both U¯fl and the pulsation frequency generally increase with Uopp in 0-g0 but change only slightly in 1-g0. For a given To and Uopp, the pulsation frequency typically differs by less than a factor of 2 between 1-g0 and 0-g0. The mean flame spread rate varies little with gravity level for high values of Uopp. For low values of Uopp in 0-g0, flame spread is uniform and U¯fl decreases with increasing Uopp due to less fuel vapor initially in the gas phase. Nomenclature g0
magnitude of gravitational acceleration on Earth H height Psat saturated vapor pressure 3 1rF consumption rate of fuel vapor [kg/(m • s)] t time T temperature U¯fl mean horizontal flame speed, time-averaged after the igniter is deactivated Uopp forced, opposed air velocity along inlet u x component of velocity v y component of velocity Vg,lg vertical gas velocity at liquid surface x horizontal coordinate y vertical coordinate Y mass fraction of species Greek symbols dflow extent of liquid flow ahead of the flame rT ]r/]T, temperature coefficient of surface tension at 300 K vF mole fraction of fuel vapor Subscripts fl g i l
flame location gas phase species index [e.g., F (fuel vapor)] liquid phase
lg o
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liquid surface initial condition Standard symbols are used for fluid properties. Variables not defined in the given nomenclature are defined when first used in the text. Acknowledgments
Research at UCI was conducted in support of NASA Grant No. NAG 3-627 under the technical monitoring of Dr. Howard Ross. We thank Dr. Ross and Dr. F. Miller (NASA-LeRC) for their experimental work and many helpful technical discussions. The computational research was supported in part by the San Diego Supercomputer Center, the NASA Center for Computational Sciences, and the UCI Office of Academic Computing through allocations of computer time.
REFERENCES 1. Schiller, D. N., Ross, H. D., and Sirignano, W. A., “Computational Analysis of Flame Spread Across Alcohol Pools,” Comb. Sci. Technol., 118:205–258 (1996). 2. Schiller, D. N., Ross, H. D., and Sirignano, W. A., Thirty-First AIAA Aerospace Sciences Meeting, Reno, Nevada, Paper No. 93-0825, 1993. 3. Glassman, I. and Dryer, F., Fire Safety J. 3:123–138 (1980/1981). 4. Ross, H. D., Prog. Energy Combust. Sci. 20:17–63 (1994). 5. Sirignano, W. A. and Schiller, D. N., Physical and Chemical Aspects of Combustion: A Tribute to Irvin Glassman (Sawyer, R. F. and Dryer, F. L., Eds.), Gordon and Breach, New Jersey, 1996, submitted. 6. Schiller, D. N., Ph.D. Dissertation, U. C. Irvine, 1991. 7. Fernandez-Pello, A. C., Combust. Sci. Technol. 39:119–134 (1984). 8. Wichman, I. S., Prog. Energy Combust. Sci. 18:553– 593 (1992). 9. Ross, H. D. and Miller, F. J., Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1327–1334. 10. White, F. M., Viscous Fluid Flow, McGraw Hill, New York, 1974, pp. 261–265. 11. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, McGraw Hill, New York, 1980. 12. Van Doormaal, J. P. and Raithby, G. D., Numerical Heat Transfer 7:147–163 (1984). 13. Bhattacharjee, S. and Altenkirch, R. A., Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1991, pp. 1627–1633. 14. Glassman, I., Combustion, 2d ed., Academic Press, San Diego, CA, 1987.
OPPOSED-FLOW FLAME SPREAD ACROSS n-PROPANOL POOLS D. N. SCHILLER and W. A. SIRIGNANO Department of Mechanical and Aerospace Engineering University of California Irvine, CA 92697 USA
A computational study is made of the effects of forced, opposed air flow on pulsating and uniform flame spread across n-propanol pools at either normal or zero gravity conditions. The numerical model incorporates finite-rate chemical kinetics, variable properties, and an adaptive gridding scheme in the direction of flame spread. In zero gravity, the combination of forced, opposed flow and thermocapillary-driven concurrent flow can cause a gas-phase recirculation cell to form ahead of the flame and thus lead to flame pulsations. The pulsation mechanism, in this case, is essentially the same as that previously detailed for pulsating flame spread in normal gravity without forced flow [1]. In either normal gravity or zero gravity, increasing the opposed air speed (Uopp) causes a transition from uniform flame spread to pulsating flame spread. The value of Uopp that corresponds to this transition increases with initial pool temperature (To). Unlike with normal gravity flame spread, the mean flame spread rate (U¯fl) in zero gravity decreases significantly because of this transition for To , 218C. As Uopp is increased further, both U¯fl and the pulsation frequency increase in zero gravity but change only slightly in normal gravity. For high values of Uopp, U¯fl varies little with gravity level. For low values of Uopp, the flame spread rate in zero gravity is sensitive to the initial profile of fuel vapor in the gas phase.
Introduction This computational study examines the effects of forced, opposed air flow on flame spread across liquid fuel pools initially below the flash point temperature. When a liquid fuel is initially below its flash point temperature, it must be heated sufficiently to create a combustible mixture of fuel vapor before ignition and flame spread can occur. Forced, opposed flow as well as buoyancy and thermocapillary forces affect the heating of the liquid pool and the mixing of fuel vapor and air ahead of the flame. In normal gravity without forced flow, when the initial pool temperature To is in a small range below the flash point temperature, the flame periodically accelerates and decelerates (pulsates) as it propagates across the fuel surface. Flame spread is uniform for higher values of To. The transition temperature between pulsating and uniform spread in 1-g0 without forced flow is approximately 178C for n-propanol [1,2]. A review of flame spread studies can be found in Refs. 3–5. A recent numerical study [1] addressed pulsating and uniform flame spread mechanisms and the effects of gravity level, pool depth, fluid properties, and chemical kinetic coefficients on flame spread across liquid fuel pools without forced flow. Pulsating flame spread was related to a gas-phase recirculation cell that forms just ahead of the flame leading edge because of the combination of buoyancy-driven opposed flow and thermo-
capillary-driven concurrent flow [1,2,6]. One question that arises is whether flame pulsations will occur without buoyancy if forced, opposed flow is provided. Although there have been several theoretical and numerical studies of opposed-flow flame spread across solid fuels [5,7,8], this is the first numerical study of unsteady, forced, opposed-flow flame spread across liquid fuels. Liquid motion and differences in fuel volatility typically cause flame spread rates across liquid fuels to be more than an order of magnitude greater than those across solids such as PMMA or paper. A two-dimensional, transient, numerical model of flame spread across open pools of alcohol fuels without forced flow [1,2] has been modified to study flame propagation across n-propanol with forced, opposed air flow. Ross and Miller [9] compare experimental results to the model’s results for opposed flow flame spread across deep nbutanol pools in normal and reduced gravity. Physical Description of the Problem The geometry of the planar model and select gasphase boundary conditions are shown in Fig. 1. A shallow pool (Hl 4 2 mm) is used because this case agrees well with 1-g0 experiments without forced flow [1,2]. The igniter (size 4 0.4 2 0.09 mm) is modeled as a hot pocket of gas with a temperature that increases linearly from To to 1700 K in 0.1 s.
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Fig. 1. Geometry and select gas-phase boundary conditions for the two-dimensional problem (infinitely wide pool); f 4 u, v, Yi, or T.
The igniter temperature remains constant until the flame reaches xfl 4 2 cm, after which time the temperature of the igniter is no longer held fixed. The initial temperature of the liquid and gas phases is To. The initial conditions for ug, vg, and YF are assumed to be given by a Blasius profile for flow across a flat plate with zero blowing velocity and unity Schmidt number [10]. For simplicity, the initial condition for the fuel vapor concentration along y 4 0 is assumed to be vF,o 4 Psat (To)/Pa, where Pa 4 1 atm is the ambient pressure. This assumption yields a nonsensical initial YF profile for x , 0, but this region is used only for numerical convenience to make the boundary conditions ]f/]x 4 0 along the left-hand side of the gas-phase computational domain more accurate. The initial fuel vapor for x , 0 typically convects out of the computational domain without affecting the flame spread rate. Assumptions and Governing Equations The mathematical formulation of the problem and the assumptions are similar to those of Schiller et al. [1,2]. The main assumptions are: (1) the liquid surface does not recede and remains flat and horizontal; (2) solutocapillary forces are neglected; (3) the binary diffusion coefficient is the same for each species; (4) one-step, second-order, finite-rate chemical kinetics are assumed; (5) gas-phase radiation and radiation from the igniter are neglected; and (6) the effects of concentration gradients on the diffusion of energy are neglected. Other minor assumptions used to simplify the energy equation and to split the pressure in the momentum equation are given by Schiller et al. [1,2]. Laboratory coordinates are used in the solution of the governing equations and in the presentation of results. Variable density and thermophysical properties are included in the analysis. Curve fits of properties, values for the preexponential constant and activation energy in the reaction rate expression, and the unsteady, nondimensional conservation equations of mass, species, momentum, and energy are the same as in Schiller et al. [1,2].
The boundary conditions at the gas/liquid interface follow from a heat balance and balance of the shear stresses with surface tension; a continuity of the temperature and tangential velocity; no dissolution of air or combustion products into the liquid; vF,lg 4 Psat (Tlg)/Pa; and no recession of the liquid surface [1,2]. Temperature gradients along the liquid surface create thermocapillary stresses that are typically larger than gas-phase shear in the following balance: (l]u/]y)l 4 (l]u/]y)g ` rT(dT/dx)lg. Zero normal gradient boundary conditions are used in the gas phase along x 4 110 cm and y 4 Hg 4 6 cm (Fig. 1). At x 4 20 cm in the gas, u 4 1Uopp, v 4 0, YO2 4 0.233, YP 4 YF 4 0, and T 4 To. Along the exit plate (x # 0, yg 4 0), u 4 v 4 ]Yi/]y 4 0 and T 4 To. In the liquid, the side wall at x 4 20 cm is assumed adiabatic, whereas the other two walls are assumed isothermal (T 4 To).
Numerical Method The numerical method [1,2] uses the SIMPLE algorithm [11] with the SIMPLEC modification [12] and the hybrid-differencing scheme. A uniform mesh of Dxfl 4 0.4 mm is used in a 6-mm region around the flame. Behind and ahead of this region, the mesh size increases by geometric progressions. An adaptive gridding scheme is used in the x direction to provide fine resolution of the reaction zone throughout the simulation. The grid in the x direction is changed each time xfl changes by at least 1 mm since the last grid alteration. Linear interpolation is used to map each dependent variable from the old to the new grid. A 132 2 72 nonuniform grid was used for the gas phase, whereas a 132 2 32 nonuniform grid was used for the liquid. The time step was 0.4 ms. The numerical results do not change appreciably if (1) the mesh size or the time step is decreased by a factor of 2 or (2) Hg is increased to 12 cm. For each time step, good convergence and satisfaction of global continuity were attained in two to three iterations.
Results and Discussion Herein, the horizontal location of the flame (xfl) is defined as the point of maximum fuel consumption rate at a particular value of yg (41 mm for most simulations) that intersects the flame leading edge region. The overall length of liquid-phase flow ahead of the flame, dflow, is defined arbitrarily as the difference between the largest value of x where Ulg . 0.5 cm/s and xfl. The threshold value of 0.5 cm/s was selected because it is typically about 3–5% of the value of Ulg at x 4 xfl.
OPPOSED-FLOW FLAME SPREAD ACROSS n-PROPANOL
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opposed flow. Figure 2 shows the qualitative flow pattern and temperature contours in the gas phase for two 1-g0 simulations with different values of Uopp. For these cases, numerical error introduced by forcing zero normal-gradient boundary conditions along the truncated exit planes of the domain appears to be confined above the exit plate, several centimeters behind the flame leading edge. Even in 1-g0, the majority of the far-field, gas-phase flow is in the x direction. Contours of maximum reaction rate correspond approximately with the contours of maximum temperature. The flame leading edge region is typically 0.05–0.3 cm above the liquid surface and well within the viscous boundary layer caused by the forced flow. Hence, the speed of the local opposed flow ahead of the flame leading edge is significantly less than Uopp. For example, the Blasius profile for air at 300 K with Uopp 4 30 cm/s gives 1u 4 10.6 cm/s at y 4 0.3 cm and 15 cm from the inlet (i.e., x 4 5 cm in Fig. 1). Fig. 2. Temperature contours (starting at 300 K with a 200-K interval) and qualitative flow pattern in the gas phase for two normal gravity simulations with To 4 198C (t 4 2 s). Isotherms for a 0-g0 simulation with To 4 198C and Uopp 4 50 cm/s are very similar to (b).
Fig. 3. Mean flame spread rate and character of spread (uniform versus pulsating) as a function of gravity level, initial pool temperature, and forced, opposed air speed.
Gas-Phase Profiles Without forced gas-phase flow, numerical error occurs near the open boundary at y 4 Hg because of the blockage of buoyancy-advected vorticity by the reflective boundary condition ]v/]y 4 0 [1]. For the present problem, this error is greatly reduced by downstream advection of the plume by the forced,
Flame Spread and Extinction at Low Uopp Figure 3 shows the predicted mean flame spread rate (U¯fl) and character of flame spread (uniform versus pulsating) for 1-g0 and 0-g0 simulations at different values of To and Uopp. Values of U¯fl calculated with Uopp 4 0 in the present model (initially, vF 4 Psat(To)/Pa throughout the gas phase) differ by less than 10% from U¯fl calculated by a previous model without forced flow [1]. For very low opposed air speeds in 0-g0, the uniform flame spread rate decreases strongly with an increase in Uopp (Fig. 3) due to less fuel vapor initially in the gas phase. For all cases shown in Fig. 3, YF,o (x, y) is determined by a Blasius profile with freestream opposed air velocity U` 4 Uopp. The flame spread rate changes significantly if YF,o is artificially determined by a value of U` not equal to Uopp. For example, with To 4 158C, the flame is predicted to extinguish for Uopp 4 3 cm/s but propagate at 6.2 cm/s for Uopp 4 1 cm/ s. For Uopp 4 3 cm/s, if YF,o is determined by U` 4 1 cm/s (leading to more fuel vapor initially in the gas phase), the flame does not extinguish but rather spreads uniformly at 5.5 cm/s. Similarly, the flame extinguishes for Uopp 4 1 cm/s if YF,o is determined by U` 4 3 cm/s. U¯fl is less sensitive to the assumed profile of YF,o for higher values of Uopp in 0-g0 and for all values of Uopp in 1-g0 because the opposed flow caused by forced and/or natural convection in these cases quickly changes the distribution of fuel vapor. As shown by Bhattacharjee and Altenkirch [13] for flame spread across solid fuels, gas-phase radiation also adds to the effects of convective heat losses that lead to extinction in quiescent environments. Hence, radiation should be included in future analyses of flame spread across liquid pools for low Uopp.
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MICROGRAVITY COMBUSTION
Uniform Flame Spread Flame spread is uniform when no gas-phase recirculation cell is formed ahead of the flame leading edge. Concurrent gas-phase flow ahead of the flame always exists, at least near the liquid surface, due to both the thermocapillary-driven liquid surface motion and hot gas expansion. The gas-phase recirculation cell does not form if the buoyancy-driven and/ or forced, opposed flow is not strong enough to overcome part of this concurrent motion. For uniform spread in either 1-g0 or 0-g0 (e.g., Fig. 4), hot gas expansion, together with the gas-phase shear from the concurrent liquid surface motion, forces incoming gas-phase flow away from the flame leading edge. Behind the flame, the flow accelerates due to buoyancy and/or hot gas expansion to a speed of 30–35 cm/s in 1-g0 and to 10–15 cm/s in 0-g0. Pulsating Flame Spread Fig. 4. Stream function contours in the liquid and fuel consumption rate contours and flow pattern in the gas phase near the flame leading edge for uniform flame spread in zero gravity (To 4 198C, Uopp 4 20 cm/s, t 4 1 s).
In either 1-g0 (for To . 178C) or 0-g0, flame spread transitions from uniform to pulsating as the value of Uopp is increased. The mechanism of pulsating flame spread in 0-g0 with forced, opposed flow is essentially the same as that of pulsating flame spread in 1-g0 without forced flow. This mechanism was described previously [1,2,6] for 1-g0 flame spread without forced flow, and it is summarized here for 0-g0 flame spread with forced, opposed flow. Figure 5 shows details near the flame leading edge during the slow-spread portion of a 0-g0 pulsation cycle with Uopp 4 50 cm/s and To 4 198C. The liquid surface
Fig. 5. Contours in the gas phase, liquid surface temperature and velocity profiles, and stream function contours in the liquid near the flame leading edge during the slow-spread portion of a 0-g0 flame pulsation cycle (To 4 198C, Uopp 4 50 cm/s, t 4 0.95 s). The contour interval for YF and YO2 is 0.02.
OPPOSED-FLOW FLAME SPREAD ACROSS n-PROPANOL
Fig. 6. Variation of the maximum fuel consumption rate, extent of liquid flow ahead of the flame, maximum liquid surface velocity, flame position, and instantaneous flame speed over several pulsation periods for To 4 198C, Uopp 4 50 cm/s, and 0-g0.
is heated over a distance of approximately dflow ahead of the flame leading edge. The liquid surface velocity near the flame front is due to thermocapillary convection. A gas-phase recirculation cell forms ahead of the flame because of the combination of the concurrent liquid surface flow and the opposed gasphase flow (due to the forced flow at the inlet in this case or due also, in part, to buoyancy in 1-g0). The gas-phase recirculation cell entrains fuel vapor and plays a critical role in the flame pulsation process. During this portion of the cycle, a premixed region just above the lean flammability limit (YF . 0.045) [14] forms in the downstream side of the recirculation cell. When a sufficient quantity of fuel vapor has accumulated ahead of the flame because of the gasphase recirculation cell, the maximum fuel consumption rate increases markedly, causing the flame to accelerate through the premixed region to a maximum speed of approximately 15 cm/s. Simultaneously, the increased rate of hot gas expansion destroys the recirculation cell structure ahead of the flame by forcing the flow away from the flame leading edge. After the flame accelerates through the premixed region that was formed in the recirculation cell, the maximum reaction rate and the rate of hot gas expansion decrease. The forced flow overcomes the hot gas expansion and once again creates opposed
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flow just ahead of the flame leading edge, thus reforming a small gas-phase recirculation cell. The size of the recirculation cell then increases to approximately the same length as in the beginning of the cycle. As shown by Fig. 6, the maximum fuel consumption rate, the extent of liquid flow ahead of the flame, the maximum liquid surface velocity (which occurs approximately under the flame leading edge), and the instantaneous flame speed (Ufl) oscillate in phase with each other during pulsating spread. The maximum reaction rate occurs before the end of the fast-spread portion of the cycle, thus creating a phase lag between dflow and (1rF )max. Note that dflow . 0 does not imply that there is liquid-phase convection ahead of the flame in a flame-fixed reference frame. Figure 6 shows that Ufl equals or exceeds Ulg,max during roughly the first half of the fast-spread portion of the cycle. During the slow-spread portion of a pulsation cycle in 1-g0 (regardless of To or Uopp) or in 0-g0 with relatively high values of Uopp or To, the flame propagates forward at a speed of 1–3 cm/s. For lower values of Uopp or To in 0-g0, however, the pulsating flame recedes slightly during the slow-spread portion of the cycle (Fig. 6). These 0-g0 cases have lower fuel consumption rates and flame temperatures than their 1-g0 counterparts because of lower transport rates of fuel vapor and oxygen into the reaction zone. Figure 6 shows that the maximum fuel consumption rate is near its minimum value when the flame is receding. Therefore, perhaps the flame recedes to a region of higher YF concentration to sustain propagation. As shown previously for 1-g0 flame pulsations without forced flow [1], the rate of diffusion of fuel vapor into the reaction zone from below the flame increases during the fast-spread portion of the cycle. Therefore, although the abrupt acceleration of the flame requires the formation of a premixed region ahead of the flame, combustion in the flame leading edge region is of a mixed mode throughout the pulsation cycle. Other features of the 0-g0 flame structure and liquid-phase flow pattern without forced, opposed flow are the same as in 1-g0. In both cases, the flame structure just behind the flame leading edge has similar character to a classic triple flame. The mixture in the upper trailing portion of the flame appears to have a very lean, premixed character from the initial mixture of fuel vapor and oxygen because it exists in a region of relatively low YF but high YO2. This portion of the flame extends less than 1 cm behind the flame tip. The lower portion of the trailing flame is clearly in a region of diffusive burning, as evident from the separation of YF and YO2 contours shown in Fig. 5. It extends all the way to the edge of the pool (x 4 0) for low values of Uopp and past the edge of the pool (x , 0) for larger values of Uopp. The premixed, rich part of the triple flame is absent because it is quenched by the rela-
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MICROGRAVITY COMBUSTION
Fig. 7. Flame position versus time for different values of Uopp (To 4 218C, 0-g0). The extent of liquid flow ahead of the flame is also shown for Uopp 4 70 cm/s.
tively cold pool surface. For the shallow pools studied previously, the flow pattern in the liquid varies little between 0-g0 and 1-g0. During the slow-spread portion of a pulsation cycle, one large vortex in the liquid rotates clockwise about two vortex centers. The height of the vortex centers is yl ' 2Hl/3, which is typical for a shallow pool [1]. During the fastspread portion of a cycle, only one vortex center appears in the liquid near the flame leading edge. Both the mean flame spread rate and the pulsation frequency ( f ) generally increase with Uopp in 0-g0. For example, U¯fl and f increase by 30% and 100%, respectively, for To 4 158C when Uopp is varied from 30 to 70 cm/s. A higher opposed velocity results in a smaller gas-phase recirculation cell, which results in a higher pulsation frequency (less time is required to accumulate fuel vapor ahead of the flame). The increase in U¯fl with Uopp could be due to a greater burning rate caused by the increased rate of oxygen transport to the reaction zone. In 1-g0, U¯fl decreases slightly with Uopp in the pulsating regime. For example, for To 4 158C, U¯fl and f in 1-g0 decrease by only 10% as Uopp is varied from 30 to 70 cm/s. For a given To and Uopp, the pulsation frequency typically differs by less than a factor of 2 between 1-g0 and 0g0. Figure 3 shows that U¯fl is fairly insensitive to gravity level for high values of Uopp. Very high values of Uopp that could lead to a reduction in U¯fl and ultimately extinction were not investigated in this study. Transition of Flame Spread Mode As To increases, there is initially more fuel vapor in the gas phase, which leads to higher rates of fuel consumption and hot gas expansion. As the rate of
hot gas expansion increases, a higher opposed velocity at the inlet is required to create a gas-phase recirculation cell ahead of the flame. Therefore, the value of Uopp that corresponds to the transition between uniform and pulsating flame spread increases with To (Fig. 3). For example, the transition in 0-g0 occurs at Uopp ' 25 cm/s for To 4 198C and Uopp ' 70 cm/s for To 4 218C. The thickness of the viscous boundary layer above the liquid surface caused by the inlet flow increases with the square root of the distance from the inlet. Thus, in some cases, flame spread can transition from uniform to pulsating as it propagates closer to the inlet because of an increase in the local opposed velocity ahead of the flame leading edge. Figure 7 shows xfl as a function of time for different values of Uopp in 0-g0 (To 4 218C). In each case, dflow 4 0.5–1.0 cm before the igniter is deactivated (e.g., at t 4 0.27 s for Uopp 4 50 cm/s and at t 4 0.47 s for Uopp 4 70 cm/s). Subsequently, flame spread is uniform and dflow ' 0.3 cm for Uopp 4 50 cm/s until the flame reaches x 4 13 cm. For Uopp 4 70 cm/s, the variation of xfl and dflow with t is very similar to that for Uopp 4 50 cm/s (except for a longer ignition transient) until the flame reaches x ' 9 cm, after which the flame pulsates. For a pool 4 cm shorter, the flame transitions to pulsating at x ' 5 cm. This transition also occurs during propagation for Uopp 4 80 cm/s. When flame spread transitions from uniform to pulsating, U¯fl decreases by nearly a factor of 2 in 0-g0 for To 4 158C or To 4 198C, but not nearly as much for To 4 218C or for any of the initial temperatures investigated for 1-g0. This sharp decrease in the mean flame spread rate is due to the recession of the flame during the slow-spread portion of the pulsation cycle. The size of the gas-phase recirculation cell that forms at the transition to pulsating spread is also typically much larger in 0-g0 than in 1-g0. For uniform spread in 0-g0 at moderate or low values of Uopp, gas-phase flow ahead of the flame caused by hot gas expansion remains concurrent over a 1-cm distance ahead of the flame, even though dflow is less than 0.5 cm (Fig. 4). Hence, when flame spread transitions from uniform to pulsating because of an increase in Uopp, the length of the gas-phase recirculation cell in this case is 1–2 cm instead of approximately 2 mm as in 1-g0 flame spread.
Conclusions In zero gravity, the combination of forced, opposed flow and thermocapillary-driven concurrent flow can cause a gas-phase recirculation cell to form ahead of the flame and, thus, lead to flame pulsations. The pulsation mechanism in this case is essentially the same as that previously detailed for pulsating flame spread in normal gravity without forced
OPPOSED-FLOW FLAME SPREAD ACROSS n-PROPANOL
flow [1]. In either 1-g0 or 0-g0, increasing the opposed air speed causes a transition from uniform flame spread to pulsating flame spread when the opposed velocity overwhelms hot gas expansion to form a gas-phase recirculation cell ahead of the flame. The value of Uopp that corresponds to this transition increases with initial pool temperature. Unlike 1-g0 flame spread, U¯fl in 0-g0 decreases significantly due to this transition for To , 218C. Because of the variation of the viscous boundary layer thickness (created by the inlet flow) above the liquid surface, the flame can transition from uniform spread to pulsating spread as it propagates across the pool. Both U¯fl and the pulsation frequency generally increase with Uopp in 0-g0 but change only slightly in 1-g0. For a given To and Uopp, the pulsation frequency typically differs by less than a factor of 2 between 1-g0 and 0-g0. The mean flame spread rate varies little with gravity level for high values of Uopp. For low values of Uopp in 0-g0, flame spread is uniform and U¯fl decreases with increasing Uopp due to less fuel vapor initially in the gas phase. Nomenclature g0
magnitude of gravitational acceleration on Earth H height Psat saturated vapor pressure 3 1rF consumption rate of fuel vapor [kg/(m • s)] t time T temperature U¯fl mean horizontal flame speed, time-averaged after the igniter is deactivated Uopp forced, opposed air velocity along inlet u x component of velocity v y component of velocity Vg,lg vertical gas velocity at liquid surface x horizontal coordinate y vertical coordinate Y mass fraction of species Greek symbols dflow extent of liquid flow ahead of the flame rT ]r/]T, temperature coefficient of surface tension at 300 K vF mole fraction of fuel vapor Subscripts fl g i l
flame location gas phase species index [e.g., F (fuel vapor)] liquid phase
lg o
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liquid surface initial condition Standard symbols are used for fluid properties. Variables not defined in the given nomenclature are defined when first used in the text. Acknowledgments
Research at UCI was conducted in support of NASA Grant No. NAG 3-627 under the technical monitoring of Dr. Howard Ross. We thank Dr. Ross and Dr. F. Miller (NASA-LeRC) for their experimental work and many helpful technical discussions. The computational research was supported in part by the San Diego Supercomputer Center, the NASA Center for Computational Sciences, and the UCI Office of Academic Computing through allocations of computer time.
REFERENCES 1. Schiller, D. N., Ross, H. D., and Sirignano, W. A., “Computational Analysis of Flame Spread Across Alcohol Pools,” Comb. Sci. Technol., 118:205–258 (1996). 2. Schiller, D. N., Ross, H. D., and Sirignano, W. A., Thirty-First AIAA Aerospace Sciences Meeting, Reno, Nevada, Paper No. 93-0825, 1993. 3. Glassman, I. and Dryer, F., Fire Safety J. 3:123–138 (1980/1981). 4. Ross, H. D., Prog. Energy Combust. Sci. 20:17–63 (1994). 5. Sirignano, W. A. and Schiller, D. N., Physical and Chemical Aspects of Combustion: A Tribute to Irvin Glassman (Sawyer, R. F. and Dryer, F. L., Eds.), Gordon and Breach, New Jersey, 1996, submitted. 6. Schiller, D. N., Ph.D. Dissertation, U. C. Irvine, 1991. 7. Fernandez-Pello, A. C., Combust. Sci. Technol. 39:119–134 (1984). 8. Wichman, I. S., Prog. Energy Combust. Sci. 18:553– 593 (1992). 9. Ross, H. D. and Miller, F. J., Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1327–1334. 10. White, F. M., Viscous Fluid Flow, McGraw Hill, New York, 1974, pp. 261–265. 11. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, McGraw Hill, New York, 1980. 12. Van Doormaal, J. P. and Raithby, G. D., Numerical Heat Transfer 7:147–163 (1984). 13. Bhattacharjee, S. and Altenkirch, R. A., Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1991, pp. 1627–1633. 14. Glassman, I., Combustion, 2d ed., Academic Press, San Diego, CA, 1987.