DETAILED NUMERICAL SIMULATION

OF FLAME BALL STRUCTURE AND DYNAMICS

byMing-Shin Wu and Paul D. Ronney

Department of Mechanical Engineering

University of Southern California, Los Angeles, CA 90089 USA

Renato O. Colantonio

NASA Lewis Research Center, Cleveland, OH 44135 USA

and

David M. VanZandt

ADF Corp., Brookpark, OH 44142 USA

ABSTRACT

A numerical study was conducted to examine the structure and dynamics of steady, source-free spherical premixed flames ("flame balls") which have been observed in microgravity experiments. A time-dependent spherically symmetric code was employed with detailed chemical, transport, and radiation sub-models. Steady properties, stability limits and dynamics of flame balls are computed for H2-air mixtures. The chemical and radiation models employed were found to affect flame ball properties substantially. The special and unusual role of thermal radiation from the combustion products is described. In particular the far-field radiative loss is found to affect the behavior of flame balls in a manner very different from propagating planar flames. One new feature was identified: mixtures capable of exhibiting both stable flame balls and steadily propagating flames depending on the initial condition. Numerical results are compared to theoretical predictions, prior steady-state numerical calculations and prior experimental results on flame ball size. Additional experiments were performed to measure flame ball radiant emission. Qualitative agreement with theory and experiment is found, however, quantitative agreement with experiment is only fair, indicating the need for improved µg conditions, e.g., in orbiting spacecraft.

1. INTRODUCTION

Over 50 years ago, Zeldovich [1] showed the possibility of stationary spherical flames existing in premixed gases in an infinitely large domain. These structures, termed "flame balls," were predicted for every combustible mixture, just as in planar geometry steadily propagating flames are possible for every combustible mixture. The solution eigenvalue for flame balls is the steady flame radius (r*) whereas for plane flames the eigenvalue is the burning velocity. Analogous to heat conduction from a sphere in an infinite medium, the temperature decays as 1/r, where r is the radial coordinate, for r > r*. Corresponding solutions do not exist in cylindrical or planar geometries since, except in spherical geometry, the steady heat conduction equation cannot be satisfied with the required far-field boundary condition that the temperature remain finite as r Æ ∞.

Zeldovich also predicted that adiabatic flame balls are unstable and thus not physically observable, just as plane flames are frequently subject to instabilities which render them unsteady and/or non-planar. This prediction was later verified using the method of activation energy asymptotics [2, 3]. However, recent experiments [4, 5] have reported apparently stable flame balls with typical radii 0.5 cm in a "microgravity" (µg) environment obtained in drop towers or aircraft flying parabolic trajectories when two conditions were satisfied: (1) the mixture was diluted with excess air, inert gas or chemical inhibitor to near the flammability limit and (2) the Lewis number (Le), defined as the mixture thermal diffusivity (a) to the mass diffusivity of the stoichiometrically scarce reactant, was less than about 0.5. µg conditions were needed to observe flame balls because buoyant convection distorts their spherical shape and/or extinguishes them at earth gravity (go). The limited µg duration in the drop tower (2.2 seconds) precluded definite conclusions concerning flame ball stability [4]. The aircraft experiments [5] provided longer µg durations but exhibited higher gravity levels (typically 10-2go) which caused significant flame ball motion. Consequently, experiments are being conducted on Space Shuttle mission MSL-1 (April 1997) [6] to study flame balls in long-duration, high-quality (≈ 10-6go) µg environments.

These apparently stable flame balls have been observed only at µg near flammability limits. Under such conditions, radiant heat losses are known to be significant in propagating flames in mixtures with higher Le [7]. In fact, Zeldovich [1] had suggested that radiant loss might stabilize flame balls. With these motivations, theories of non-adiabatic flame balls were developed for Le < 1 [8 - 10]. It was predicted that for sufficiently strong volumetric heat losses, no steady solutions exist, indicating a flammability limit, whereas for weaker losses, two values of r* are possible. As the turning-point limit (static extinction limit) is approached, the difference between the radii of the larger and smaller balls (denoted “cold giant” (CG) and “hot dwarf” (HD) [11]) decreases to zero. All HD/CG flames are predicted to be unstable/stable to radial, one-dimensional perturbations. For sufficiently weak losses, CG flames are unstable to three-dimensional perturbations, thus only at intermediate heat loss are any flames stable to all perturbations. This is consistent with experimental observations that sufficiently weak mixtures do not burn whereas mixtures far from the lean flammability limit exhibit continuously splitting cellular flames rather than steady flame balls.

These analytical theories assume highly simplified chemical, thermodynamic and transport properties. Buckmaster, Smooke and Giovangigli [11] (hereafter BSG) and Smooke and Ern [12] (hereafter SE) conducted detailed numerical simulations of steady non-adiabatic flame balls in H2-air mixtures and found behavior qualitatively consistent with the analytical theories. However, no transient properties were studied, hence stability limits, equilibrium overshoots, etc. could not be studied and compared to theoretical predictions or experiments. Also their computed r* at the flammability limit is about half the experimental observation and the computed variation in r* with composition is much wider than experimental observations. It would be of interest to determine how the chemistry and radiation sub-models affects these results. An initial study of the role of these sub-models was performed by SE, however, but only to the extent of showing that these sub-models do have at least some effect. It is not certain based on currently available information whether they have a significant effect on flame ball properties at the experimentally observable near-limit conditions. Also, it is of interest to examine the effect of computational domain size, which may be important since the slow 1/r decay of the thermal field outside the flame ball indicates that a large domain may be needed to obtain domain-independent results.

Consequently, the goal of this study is to model flame ball properties, including the effects chemical reaction models, radiation models, domain size and unsteadiness, and provide predictions which can be compared to the space flight experiments. The numerical model is described in section 2, followed by the numerical results in section 3. Preliminary measurements of flame ball radiation at µg are described in section 4 and compared to the numerical predictions. Concluding remarks are given in section 5.

2. NUMERICAL MODEL

A one dimensional, time-dependent flame code employing detailed chemical and transport sub-models, developed by Rogg [13, 14], was employed. The usual nonsteady equations of energy and species conservation along with the equation of state of ideal gas were solved in spherical geometry at constant pressure. The equations are:

,

(1)

,

.

Here r is the density, Cp the bulk specific heat at constant pressure of mixture, T the temperature, t the time, u the radial convection velocity, r the radius, l the thermal conductivity, hi and wi the specific enthalpy and the mass rate of production of specie i, respectively, N the number of the gaseous species, Cpi and Ji the specific heat at constant pressure and the mass diffusion flux of specie i, respectively, qrad the differential radiative heat loss, Yi the mass fraction of specie i, p the pressure, R the universal gas constant, and the mass averaged molecular weight of the mixture. Because of the constant pressure assumption, the system expands/shrinks due to thermal expansion, though the change in system size and the resulting flow velocities are very small because the expansion occurs at a very small radius relative to the outer boundary radius. Optically-thin radiation was assumed with heat loss per unit volume qrad = 4sap(T4-To4), where s is the Stefan-Boltzman constant, ap the Planck mean absorption coefficient, T the local temperature and To the ambient temperature. Data on ap were taken from Hubbard and Tien [15] (hereafter HT). At the outer boundary, T was maintained at To and the composition was maintained at the unburned mixture composition. Zero-gradient conditions were enforced at r = 0. Unless otherwise noted, an outer boundary radius (rb) of 100 cm was employed. 191 computational grid points were employed with dynamically-adaptive re-gridding. Once steady solution was obtained for one equivalence ratio, the far-field composition was modified slightly with the previous steady state solution as the initial condition and the calculation re-started to obtain the solution for the new equivalence ratio. Near the lean and rich dynamic stability limits, the equivalence ratio was changed in increments as small as 0.0001 to ensure accurate determination of these limits. The initial time step was 1 µsec, and as numerical accuracy considerations allowed was then increased to as much as 1000 seconds as solutions approached steady state. Calculations were run until either the flame extinguished, an expanding, steadily propagating flame developed or a steady flame ball evolved. In the first and last cases all convective velocities decay to zero as required by mass conservation.

3. NUMERICAL RESULTS

Steady properties

Figure 1a shows predicted steady temperature, heat release and radiative heat loss profiles for a CG flame at equivalence ratio (f) = 0.10 (4.03% H2) using three different chemical mechanisms [16 - 18]. Figure 1b shows species concentration profiles. Figure 1a shows that the heat release rate peaks away from r = 0 and drops to zero rapidly at larger r. H2 is mostly consumed near this peak but some H2 leaks through to the center. The temperature and H2O concentrations decrease slowly toward their ambient values as r increases. According to theoretical predictions [1 - 3] these profiles decay as r-1 for adiabatic flame balls with constant transport properties. A comparison of Figs. 1a and 1b shows that for flame balls with radiative heat loss the decay in the temperature profile is steeper than the H2O concentration profile, which is expected because radiation is a sink for thermal energy whereas there is no sink of H2O in the far-field.

Figures 1a and 1b show that the profiles vary considerably for the different chemical mechanisms. This is illustrated further in Table 1, which shows that the radii at peak OH concentration (rOH) (for f = 0.1) differ markedly even though all mechanisms have been calibrated to obtain similar laminar burning velocities for mixtures away from flammability limits, as shown in Table 1 for f = 0.6. These discrepancies between models may occur because the reaction rate parameters of the H2 - O2 system are still not well known at low flame temperature; similar difficulties have been found in studies of lean propagating planar H2-air flames [19].

Figure 2 shows comparisons of profiles obtained with a detailed chemical mechanism [17], a reduced mechanism [17] with the same starting mechanism, and the detailed mechanism with different radiation models [11, 15]. Figure 2 shows that the radiation model affects the profiles considerably. The Hubbard and Tien (HT) [15] and BSG [11] radiation models are similar at higher temperatures (above about 950K) but BSG radiation is weaker than HT radiation at lower temperatures. This can account for the differences in the calculated flame balls properties since weaker radiation leads to larger flame balls [8 - 10]. We shall show that low-temperature radiation, although generally unimportant in conventional propagating flames, is important to flame ball behavior. Figure 2 also shows that the reduced mechanism tested performs reasonably well on flame balls, especially considering that the reduction scheme was tailored to obtain optimal results for propagating flames having much higher peak temperatures than flame balls.

Table 1 provides a summary of comparisons of various flame ball properties using 5 chemical and 2 radiation models. Table 1 includes the total heat release integrated over the entire computational domain (QH) and total radiative heat loss (QL), the latter being an experimentally measurable quantity. In addition to the previously noted effects of chemical and radiation properties, Table 1 shows that results obtained by us and BSG, even using the same chemical mechanism, radiation model and rb, are slightly different, perhaps due to differences in the transport models employed; SE [12] noted that the transport model employed does in fact affect the flame ball properties slightly.

For brevity, in the computations described below we employ only the GRI chemical model [16] and the HT radiation model [15] because these seem to be the most widely employed in recent literature.

It is of interest to determine the energy budget of flame balls, that is, to determine how the heat generated by chemical reaction is distributed. The usual steady energy conservation equation can be integrated from r = 0 to r = rb to obtain

(2)

Note that in Eq. 1, the convective term is excluded because the only way to satisfy the steady continuity equation in spherical geometry without sources or sinks is to have the convection velocity identically zero everywhere. The four terms on the right hand side of Eq. 1 represent (1) the thermal conduction flux evaluated at r = 0 and r = rb; the former is zero because of the zero-slope boundary condition at r = 0 and the latter may or may not be zero depending on rb as discussed below, (2) the differential enthalpy diffusion, i.e. the transport of energy by species diffusion, (3) the total heat generated by chemical reaction (QH) and (4) the total heat loss by radiation (QL).

With this motivation, Figure 3 shows, for a steady flame ball at f = 0.1, profiles of temperature, heat release plus the differential enthalpy diffusion (integrated starting from r = 0), integrated radiative heat loss (ditto) and differential enthalpy diffusion by itself (ditto). Because chemical reaction rates increase rapidly with temperature, heat release occurs at small r where temperature is highest; half the heat release occurs at r < 0.45 cm. In contrast, because the dependence of radiative loss on temperature is weaker, coupled with the r2dr volume effect of spherical geometry, less than 1% of the integrated radiation occurs at r < 0.45 cm. The radius inside which half the radiative loss occurs (rrad,1/2) is 3.5 cm. Even this powerful effect is in a sense under-emphasized for H2-air mixtures because there is no radiator in the unburned mixture; the only radiator is the product H2O which must diffuse from the reaction zone into the far-field. If N2 were replaced with a radiating gas such as CO2, an even stronger effect of far-field radiation might be expected.

To assess how significant the far-field radiation is compared to the near-field radiation, one must consult the theoretical models of non-adiabatic flame balls [8 - 11]. This is an important issue because the effects of near-field and far-field loss are qualitatively different [9]. In particular, only with far-field losses are oscillatory solutions and a dynamic stability limit predicted in addition to the turning-point limit. Also, the three-dimensional instability of flame balls depends only on the magnitude of near-field loss [9]. The near-field is defined as the region where r/r* is O(1), whereas the far-field is defined as the region where r/r* is O(b), where b = E/RT*, E is the overall activation energy, R the gas constant and T* the peak temperature. The theoretical models examine flame structure in the asymptotic limit b Æ ∞. While there is no unique way to distinguish the near-field region from the far-field region for finite b, a reasonable approach would be to define r/r* = b1/2 as the transition point. If we then assume E for lean H2-air mixtures to be 27 kcal/mole as Mitani and Williams [20] estimated and use the calculated T* = 1140K, then b ≈ 12 and the threshold would be 121/2 * 0.375 cm ≈ 1.3 cm. Figure 3 shows that about 0.45 Watts of radiation out of the total 2.88 Watts, or 16%, is emitted from r < 1.3 cm. To assess which has a greater impact on flame ball structure, the 16% near-field or the 84% far-field radiation, we again consult the asymptotic analyses. The strength of the near-field radiation per unit volume is given by k(T - To), where k is O(1/b) and (T - To) is O(1). (The fact that radiative heat loss is in reality not a linear function of T is irrelevant since only radiation inside the flame ball, where T varies only by an O(1/b) amount from T*, was considered in [8].) The total volume is O(1) since r/r* is O(1). Hence, the total radiation is O(1/b)·O(1)·O(1) = O(1/b). The strength of the far-field radiation per unit volume is also given by k(T - To), but here k is O(1/b2) and (T - To) is O(1/b). (Here the linear relation between heat loss and T is approviate since the deviation of T from To is small.) The total volume is O(b3) since r/r* is O(b). Hence, the total radiation is O(1/b2)·O(1/b)·O(b3) = O(1). Therefore, the total far-field radiation must be O(b) larger than the near-field radiation to have the same impact. For the f = 0.1 mixture the ratio of far-field to near-field radiation is 2.43/0.45 = 5.4 ≈ .45b. Thus, both near-field and far-field radiation are probably important in this case, with near-field radiation perhaps slightly more important according to this rough estimate. Again for CO2-diluted mixtures, the balance might shift toward a greater impact of far-field losses.

Figure 3 also shows that the integrated heat release plus differential enthalpy diffusion is balanced by the integrated radiative heat loss (QL) for flame ball at steady state, as Eq. 1 would predict if the conductive flux at the (isothermal) outer boundary were zero. This energy balance will not be satisfied if the computational domain is not large enough so that the volume of radiating gas is sufficient to remove all of the heat generated plus the differential enthalpy diffusion. In this case, the only way to establish a steady state would be to have a non-zero conductive flux at r = rb, which leads to domain-dependent predictions of flame ball properties. Figure 4 illustrates this point by showing predicted flame ball properties for various rb. For this particular mixture, a boundary radius of about 50 cm is required to obtain practically domain-independent results, however, the effect is modest above rb = 20 cm, which corresponds to the value employed by BSG. Notice also that the radiative heat loss , an easily measurable quantity, is a rather robust property in that it is affected only slightly by rb.

Figures 5a and 5b shows how the equivalence ratio (f) affects steady flame ball properties. Figure 5a shows values of rOH predicted by us and by BSG. Figure 5b shows our calculated values of T*, adiabatic flame temperature for the homogeneous mixture (Tad), QH and the differential enthalpy diffusion = QH - QL. Figure 5a shows that both numerical models predict rOH increases as f increases. This is expected since for the stable CG branch, as f increases the available fuel increases, thus the flame ball must grow larger in order to radiate away energy in proportion to the increased heat release - note that rOH increases almost linearly with f. BSG reports two values of rOH for each f, whereas we report only one. This is because we employed a time-dependent code, thus the only steady solutions that our code can converge to are stable solutions, which occur only on the CG branch and only for a portion of this branch, whereas BSG employed a steady code with arc-length continuation methods enabling them to find, in addition, the steady but unstable portion of the CG solution branch as well as the unstable HD solutions. For reference, experimentally observed flame ball radii [5] are also shown, though quantitative comparison with the numerical results is subject to a number of limitations, as discussed at the end of section 3.

At our computed lean dynamic stability limit of f = 0.0847, d(rOH)/d(f) is large but finite, thus a turning-point limit has not been reached. This is consistent with theoretical predictions [9] that when far-field losses are significant (as was asserted above) the dynamic stability limit is slightly above the turning-point limit. By comparison, BSG predicted a turning-point limit of f = 0.0866. While both types of limits are interesting to study, for comparison with experiments it is preferable to consult the dynamic stability limit, since steady but unstable solutions would not be experimentally observable. For comparison, the experimental lean stability limit for flame balls at µg [5] is f = 0.0825 ± 0.0013, which is within 5% of both predictions. All of these values are much leaner than the leanest H2-air mixtures that can be burned at earth gravity, which is f = 0.097 - 0.104 for upward flame propagation in a 5 cm diameter tube [21].

The kink in the rOH curve at f ≈ 0.255, which was also noted by BSG, corresponds to a transition from O2 to H2 leaking through the reaction zone to the flame ball center. Thus at f > 0.255, O2 is the “deficient” reactant at the reaction zone and the relevant Le is LeO2 ≈ 1.19 rather than LeH2 ≈ 0.30. Joulin [22] predicted this transition occurs at f = fc ∫ LeH2/LeO2 ≈ 0.25. Experimental evidence of this transition at f ≈ 0.27 is reported in [5]. Consequently, theory, computation and experiment all concur on this unusual property of flame balls, namely the shift from lean burning to rich burning at the reaction zone at an overall mixture equivalence ratio very different from unity.

Figure 5b shows that Tad becomes larger than T* when f > 0.255 ≈ fc. This is to be expected because for adiabatic flame balls, T* = To + (Tad - To)/Le, and thus only when the effective Le is less than unity, which corresponds to f < fc, can T* be greater than Tad even in the presence of heat losses.

The transition from lean to rich burning at f ≈ fc leads to a rich stability limit because at Le close to or larger than unity, even CG flames are unstable to radial perturbations, regardless of the magnitude of heat loss [10]. Consistent with this prediction, no stable flame balls were predicted by us beyond f ≈ 0.2853 > fc. This Le effect also explains why stable flame balls have never been seen experimentally in mixtures with Le 1.

Figures 5a and 5b shows that as f increases, rOH increases while T* decreases. These results are somewhat counter-intuitive because for propagating flames in lean mixtures, as f increases, flame thickness decreases and peak temperature increases. Nevertheless, these results are in accord with BSG’s theoretical predictions for CG flames. The more intuitive behavior is observed for HD flames, however, HD flames are apparently always unstable.

Dynamical properties

To assess effects of initial conditions on flame ball dynamics, calculated steady temperature and composition profiles were stretched or shrunk by mapping the grid locations for steady solutions (ri) to new locations (rj) according to

(3)

where c = ro/r* is the stretching/shrinking parameter (here r* is the steady value of rOH), ro is the modified initial flame ball radius and x = ri/rb (0 ≤ x ≤ 1). This stretching/shrinking scheme was chosen because it allows linear stretching/shrinking (rj/rb = cx) at small x without changing rb and because it provides symmetric mapping for stretching vs. shrinking, that is, the function for c > 1 is obtained from the function for c < 1 by exchanging ri and rj and replacing c by 1/c.

Figure 6 shows an example of flame ball evolution to steady-state conditions. About 100 sec is required for the flame ball to evolve to within 10% of its steady value. This is comparable the theoretically-derived [8, 9] evolution time scale b2r*2/a, estimated as 70 sec for these conditions. This rather long time scale is a consequence of the need for heat conduction and H2O diffusion to the far-field (whose radius is of the order br*) where much of the radiative loss occurs, so that the balance between heat generation and heat loss (and therefore rOH) can be established.

Figure 7 shows how ro affects flame ball evolution. Sufficiently large or small ro leads to extinguishment whereas intermediate ro leads to stable balls. That large ro should extinguish whereas smaller ro do not is again counter-intuitive yet consistent with theoretical predictions [9]. Physically, it occurs because heat release is proportional to area whereas heat loss is proportional to volume, thus the heat loss to heat generation ratio is greater, and flames are weaker, for larger ro. Overshoot and slight oscillatory behavior is seen in Figs. 6 and 7, also in accord with theory [9]. Oscillations are predicted [9] only when far-field losses dominate near-field losses, which is consistent with our earlier assertion that far-field losses are significant in the mixtures under study.

Figure 8a shows how initial conditions affect the fate of flame balls as a function of mole percent H2 (XH2). For compositions having steady solutions, initial radii close to r* eventually evolve to r* whereas for larger or smaller ro the flames eventually quench. As the lean and rich stability limits are approached, the range of ro leading to steady flames narrows to zero. Consistent with theoretical predictions [2, 3], it was not possible to obtain any stable HD flames.

None of the CG-like initial conditions led to steadily propagating flames. Obviously, sufficiently rich mixtures do exhibit propagating flames. Planar calculations for H2-air premixed flames with GRI chemistry and HT radiation were carried out to determine the lean extinction limit for H2-air mixtures. It is found that the equivalence ratio for lean limit is 0.296 (XH2 = 11.1) at a burning velocity of 1.61 cm/sec, which is higher than the rich stability limit (f = 0.285, XH2 = 10.7) for flame balls, indicating that stable flame ball solutions and planar flame solutions do not co-exist for any value of f. Interestingly, for 0.285 < f < 0.296, there are no stable flames of any kind.

As discussed earlier, the rich stability limit for flame balls is a consequence of the transition from lean to rich burning at the flame front and the corresponding shift in the effective Lewis number even though the overall equivalence ratio is considerably less than unity. In contrast, the lean planar flammability limit has no such strong dependence on Le but instead is primarily dependent on Tad [7]. This suggests that for mixture with different O2/N2 ratios the rich stability limit for flame balls would be significantly different (in terms of the mole percent H2 at the limit) whereas the planar flammability limit would not change much since values of Cp for O2 and N2 are similar and thus Tad would depend mostly on the H2 mole fraction and not the O2/N2 ratio. To test this hypothesis, planar flame and flame ball calculations were performed for H2 - O2 mixtures with no N2. Figure 8b shows the stability map for H2 - O2 mixtures, analogous to that for H2 -air mixtures in Fig. 8a. The lean flammability limit for planar flames shifted only slightly, from 11.1% H2 in air to 11.9% H2 in O2 (f = 0.0677). whereas the rich stability limit for flame balls shifted from 10.7% H2 in air to 28.6% H2 in O2 (f = 0.200). Thus, in H2 - O2 mixtures with 0.0677 ≤ f ≤ 0.200, both steadily propagating flames and stable flame balls are possible whereas for H2-air mixtures there is no such overlap.

Comparison with experiment

Figure 5a shows comparisons of predicted (by us and BSG) and observed [5] flame ball or cell radii. Only semi-quantitative comparisons can be made for at least two reasons: (1) the limited duration of µg in the aircraft experiments (<20 sec) may not have allowed achievement of steady-state conditions in the experiments (see Fig. 6); (2) the acceleration levels on the aircraft (≈ 10-2 go) caused significant motion of flame balls and their thermal plumes, which alters the radiant emission and causes some convective transport in addition to the diffusive transport. Only cases with nearly stationary flame balls near the center of the chamber were analyzed; consequently, many experimental data had to be discarded from consideration.

Given these caveats, Fig. 5 shows predicted CG radii comparable to observations except at near-limit conditions where experimental radii are considerably larger. (Experimental radii are based on H2O emission at 800-900 nm, not OH emission, however, [5] shows that these indicators yield similar radii).

The aircraft µg experiments suggest a rich stability limit of only f ≈ 0.1, which is much leaner than our numerical rich stability limit. This discrepancy occurs because three-dimensional instabilities, which cannot be examined with our model or BSG’s model, occur for CG flames of sufficiently large radius [8] (thus sufficiently large f) and hence only a portion of the CG branch is stable to all disturbances.

4. FLAME BALL RADIATION MEASUREMENTS

Since no prior data on flame ball radiation were available, additional aircraft µg experiments were performed to obtain preliminary data on flame ball radiation. The apparatus was similar to that used in [5] but with a larger chamber (32 cm diameter) into which two Oriel 7109 radiometers were installed diametrically opposed. Radiometers were calibrated using blackbody sources of known intensity. Radiometer signals were amplified and recorded by an A/D converter and microcomputer.

The measured radiation from f = 0.0928 mixtures (average over 3 experiments with a total of 8 flame balls) was 0.85W per ball with an rms deviation of 0.82W (due to two anomalously high readings) and was 0.29W (average of 2 flame balls in 1 experiment) for f = 0.0851 mixtures. Our corresponding predictions are 2.16W and 1.19W, respectively, which are considerably higher than measurements. This may be due to the differences between experimental conditions and model assumptions noted above. For example, the short µg duration may not allow the far-field to fill with H2O vapor allowing full radiation to be exhibited (Fig. 6). Another possibility is that the flame ball motion in the aircraft experiments disturbs the thermal field in such a way as to reduce the measured radiation, though it is not currently known whether this motion should increase or decrease radiation. Preliminary results from the space flight experiments [23] suggest that both of these effects are of critical importance.

5. CONCLUSIONS

Numerical studies of flame balls in H2-air mixtures were conducted to examine stability properties, extend prior steady-state calculations and compare predictions with experiments. Several chemical and radiation models were tested and were found to affect flame ball properties substantially. Thus, results of flame ball experiments in space may provide a new basis for testing these models at low flame temperatures (<1200K). Radiation profiles, steady properties, stability limits and dynamical response all suggest that far-field radiative loss affects flame ball behavior substantially and in ways very different from propagating planar flames. Many phenomena were predicted that are counter-intuitive yet qualitatively consistent with flame ball theories, e.g. flame radii, temperatures, dynamic response, effects of f, etc. One new feature identified: mixtures capable of exhibiting both stable flame balls and propagating flames. It is also found that rather large domain sizes (typically 100 times the flame ball radius) are needed to obtain domain-independent solutions due to the long 1/r “tail” of the far-field thermal and composition profiles. When this condition is satisfied, the conductive flux at the outer boundary is zero and the radiative loss is balanced by the sum of the heat release and differential enthalpy diffusion.

Reasonable quantitative agreement of predicted extinction limits with experiment is found, however, agreement for flame radii and radiant emission is not very satisfactory, indicating the need for improved chemical mechanisms and radiation models at low flame temperature. However, to obtain reliable data for comparison to these models requires much longer durations and much better quality of µg, e.g., in orbiting spacecraft. Such experiments are in progress and will be reported in future works.

ACKNOWLEDGMENTS

This work was supported by NASA-Lewis under Grant NAG3-1523. We thank Mr. Quin Blackburn for assistance with data analysis, Prof. John Buckmaster for theoretical discussions and Prof. Fokion Egolfopoulos for his computational expertise.

REFERENCES

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9. Buckmaster, J. D., Joulin, G. and Ronney, P. D., Combust. Flame 84, 411-422 (1991).

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TABLES

Chemistry model	Radiation model	Steady rOH, cm	rrad,1/2, cm	T*, K	QH, watts	QL, watts	SL (f = 0.6), cm/s
GRI [16]	HT [15]	0.375	3.48	1140	3.05	2.88	93.1
Yetter [18]	HT [15]	0.509	3.27	1070	3.70	3.54	99.4
Peters [17]	HT [15]	0.284	3.84	1202	2.47	2.36	78.6
Peters [17], 2-step	HT [15]	0.262	4.28	1227	2.00	1.89	85.7
BSG [11]	HT [15]	0.371	3.54	1145	3.00	2.86	86.9
GRI [16]	BSG [11]	0.494	4.92	1123	3.94	3.66	93.1
BSG [11]	BSG [11]	0.569	2.86	1121	4.67	3.20	86.9
BSG [11]	BSG [11]	0.487 [11]	NR	1128 [11]	NR	NR	NR

Table 1. Characteristics of steady CG flame balls at f = 0.1 computed with different chemical and radiation models. rrad,1/2 is the radius inside which half of the radiative loss occurs. The last row shows computed results reported by BSG [11]. The last two rows were computed using rb = 20 cm; for all other cases rb = 100 cm. NR = not computed by BSG.

Figure 1a. Profiles of temperature, heat release per unit volume and heat loss per unit volume for a steady flame ball at f = 0.1 (4.03% H2 in air) for 3 different H2-O2 chemical mechanisms [16 - 18]. Note logarithmic horizontal axis.

Figure 1b. Species mass fraction profiles for a steady flame ball at f = 0.1 (4.03% H2 in air) for 3 different H2-O2 chemical mechanisms [16 - 18].

Figure 2. Performance of a detailed chemical mechanism [17], a reduced chemical mechanism with the same starting mechanism [17] and of HT [15] and BSG [11] radiation models for a steady flame ball at f = 0.1.

Figure 3. Integrated (starting from zero radius) heat release plus differential enthalpy diffusion and heat loss profiles, along with temperature and H2O mass fraction, for a steady flame ball at f = 0.1.

Figure 4. Effect of far-field boundary radius (rb) on the maximum temperature, integrated heat release and integrated heat loss on a steady flame ball at f = 0.1.

Figure 5a. Radius at peak OH concentration (rOH) for steady flames computed by us and BSG compared to experimentally observed radius [5] as a function of f for flame balls in H2-air mixtures. For experimental data, at f > 0.1 flame balls were not stable due to three-dimensional instabilities; in these cases values of r shown correspond to those just before the beginning of the splitting process.

Figure 5b. Computed radius of peak OH concentration, maximum temperature, integrated heat release, differential enthalpy diffusion and adiabatic flame temperature as a function of f for steady flame balls in H2-air mixtures.

Figure 6. Evolution of a flame ball to steady-state, starting with ro/r* = 0.5, where ro is the initial radius and r* is the steady value of rOH, in mixtures with f = 0.1.

Figure 7. Effect of initial condition on evolution of flame balls in mixtures with f = 0.1.

Figure 8a. Eventual fate of flame balls as a function of ro and XH2 for H2-air mixtures.

Figure 8b. Eventual fate of flame balls as a function of ro and XH2 for H2-O2 mixtures.