Understanding
Combustion Processes
Through Microgravity Research
Paul D. Ronney
Department of Aerospace and Mechanical Engineering
University of Southern California, Los Angeles, CA 90089-1453 USA
ABSTRACT
A review of research on the effects of gravity on combustion processes is presented, with an emphasis on a discussion of the ways in which reduced-gravity experiments and modeling has led to new understanding. Comparison of time scales shows that the removal of buoyancy-induced convection leads to manifestations of other transport mechanisms, notably radiative heat transfer and diffusional processes such as Lewis number effects. Examples from premixed-gas combustion, non-premixed gas-jet flames, droplet combustion, flame spread over solid and liquid fuels and other fields are presented. Promising directions for new research are outlined, the most important of which is suggested to be radiative reabsorption effects in weakly burning flames.
Invited paper presented at the 27th International Symposium on Combustion
Boulder, CO, August 5, 1998.
INTRODUCTION
Gravity influences many combustion processes, particularly due to buoyant convection which affects transport of thermal energy and reactants to/from the chemical reaction zones. Recently many experimental and theoretical studies of combustion at "microgravity" (µg) conditions have been conducted. These studies are motivated by the need to assess fire hazards in spacecraft and to enable better understanding of combustion processes at earth gravity (1g) through elimination of buoyancy [, , , ].
This paper discusses how new understanding of combustion processes has been obtained through µg research, rather than providing a comprehensive review of this rapidly changing field. First, comparisons of time scales for various chemical and transport processes in flames, including buoyancy-induced transport, are given. Next, examples of unexpected results and new understandings obtained through µg research are discussed. These findings are then summarized and future research directions are suggested.
COMPARISON OF TIME SCALES FOR PREMIXED-GAS COMBUSTION
To determine the conditions where gravity can affect flames, we compare estimated time scales for chemical reaction (t_{chem}), inviscid buoyant convection (t_{inv}), viscous buoyant convection (t_{vis}), conductive heat loss to walls (t_{cond}) and radiant heat loss (t_{rad}). Premixed laminar flames are considered first because of their simplicity. Subsequent sections introduce time scales for other flames.
The chemical time scale is (see Nomenclature) t_{chemš}d/S_{L} where d=a/S_{L}, thus t_{chemš}a/S_{L}^{2}. The convective transport time scale is d/U, where d is a characteristic flow length scale, Uš(gd(Dr/r))^{1/2} is the buoyant convection velocity and Dr the density change across the flame. Since Dr/rš1 for flames, t_{invš}d/(gd)^{1/2}=(d/g)^{1/2}. For inviscid flow d is determined by the apparatus dimensions, for example the burner or tube diameter. For viscous flow, d cannot be specified independently; instead dšn/U, thus Uš(gn)^{1/3} and t_{visš}d/Uš(n/U)/Uš(n/g^{2})^{1/3}. t_{cond} is the flame front temperature (T_{f}) divided by the rate of temperature decrease due to conductive loss, thus t_{cond}=T_{f}/(dT/dt)šT_{f}/(rC_{P}h(T_{f}-T)), thus t_{condš}d^{2}/16a. Similarly, for optically-thin radiation t_{rad}šT_{f}/(L/rC_{P})š{g/(g-1)}{P/4sa_{P}(T_{f}^{4}-T^{4})}.
Two sets of time scales are shown in Table 1, one for near-stoichiometric hydrocarbon-air flames and one for near-limit flames, both at P=1 atm. For near-stoichiometric flames S_{Lš}0.40 m/s, T_{fš}2200K, a=n=1.5x10^{-4} m^{2}/s and a_{Pš}0.56 m^{-1}. For near-limit flames, S_{Lš}0.02 m/s, T_{fš}1500K, a=n=1.0x10^{-4} m^{2}/s and a_{Pš}0.83 m^{-1}. For both cases gš9.8 m/s^{2}, g=1.35, T_{š}300K and dš0.05 m (a typical apparatus dimension.)
Several observations can be made based on these simple estimates:
(1) Buoyant convection is unimportant for near-stoichiometric flames because t_{vis}ªt_{chem} and t_{inv}ªt_{chem}
(2) Buoyant convection strongly influences near-limit flames at 1g because t_{vis}<t_{chem} and t_{inv}<t_{chem}.
(3) Radiation effects are unimportant at 1g compared to buoyant convection since t_{vis}>t_{rad} and t_{inv}>t_{rad}.
(4) Radiation effects dominate near-limit flames since t_{radš}t_{chem}, but these effects are only observable at reduced gravity because of (3)
(5) The apparatus size (d) must be larger than about 0.03 m to observe radiation-induced extinction, otherwise conduction losses exceed radiative losses (t_{cond}<t_{rad}).
(6) Many radiative loss effects can be studied in drop-towers (test duration 2-10 s), since these times are typically larger than t_{rad}.
(7) Since t_{inv}~g^{1/2} and t_{vis}~g^{1/3}, aircraft-based µg experiments at gš10^{-2} g_{o}, may not provide sufficiently reduced buoyancy to observe radiative effects.
(8) Since t_{vis}~n^{1/3}~P^{-1/3} and t_{rad}~r/L~P^{1}/P^{1}~P^{0}, t_{vis}/t_{rad}~P^{-1/3}. Thus, t_{rad} is independent of P, but at higher P buoyancy effects interfere more strongly with radiative effects.
(9) A Reynolds number Re_{d}ŽUd/n for buoyant flow is estimated as (gd)^{1/2}d/n^{2}=Gr_{d}^{1/2}, where Gr_{d}Žgd^{3}/n^{2} is a Grashof number. For Re_{d}=10^{3}, thus Gr_{d}=10^{6}, buoyant flow at 1g is generally turbulent, thus it difficult to obtain steady laminar flames in large systems at 1g.
The implications of these observations are discussed in the following sections. Essentially any combustion process where t_{chem} or t_{rad} exceeds t_{inv} or t_{vis} may be affected by gravity and is worthy of µg investigation.
PREMIXED-GAS FLAMES
Flammability limits
The previous section showed that gravity effects are significant only for mixtures with low S_{L}, implying mixtures highly diluted with excess fuel, oxidant, or inert gas, but sufficient dilution causes flammability limits. Thus, gravity should have significant influences on near-limit behavior, which is supported by the fact that limits are different for upward, downward and horizontal propagation [].
Practically all flammability limit studies show that burning velocity at the flammability limit (S_{L,lim}) is nonzero. Giovangigli and Smooke [] have shown that there is no purely chemical flammability limit criterion for planar unstretched flames; without losses S_{L} decreases asymptotically to zero as dilution increases. Consequently, loss mechanisms such as those discussed below are needed to explain limit mechanisms. The resulting predictions of S_{L,lim} indicate that usually S_{L,lim} depends only weakly on chemical reaction rate parameters. Thus, limit mechanisms may be inferred by comparing predicted and measured S_{L,lim} without detailed chemical knowledge. The mixture composition at the limit affects S_{L,lim} only weakly through T_{f}, thus, comparing predicted and measured limit compositions is not especially enlightening; comparisons of S_{L,lim} values is much more useful. Consequently, this discussion will emphasize comparisons of predicted and measured values of S_{L,lim}.
For upward propagation, Levy [] showed that the flame rise speed at the limit (š0.33(gd)^{1/2}) is identical to that of an inviscid hot gas bubble. This relation was later verified for many tube diameters and mixtures []. Buckmaster and Mikolaitis [] showed how this minimum rise speed causes hydrodynamic strain at the flame tip, which causes extinguishment for sufficiently low S_{L}. The predicted burning velocity of the limit mixture (S_{L,lim}) is, after temperature-averaging transport properties:
(1).
The form of Eq. (1), S_{L,lim}~(ga^{2}/d)^{1/4}, can be obtained by setting t_{inv}=t_{chem}.
For downward propagation in tubes, centrifuge experiments [] indicate S_{L,lim}~g^{1/3}, independent of Le, which is reasonable since downward-propagating near-limit flames are nearly flat and unstrained. Experiments [] and numerical simulations [] suggest extinction results from sinking regions of cooling burned gas near the walls overtaking the flame and blocking it from fresh reactants, although the g^{1/3} scaling was not tested in [, ]. The g^{1/3} scaling can be obtained by setting t_{chem}=t_{vis}, thus
S_{L,lim}~(ga)^{1/3} (2).
Experiments [] employing varying diluent gases and pressures confirm the a^{1/3} scaling and lack of dependence on tube size, which supports the proposed mechanism.
Both upward and downward limit mechanisms indicate that as gÆ0, S_{L,lim}Æ0, implying arbitrarily weak mixtures could burn very slowly. However, conductive or radiative losses prevent arbitrarily weak mixtures from burning even at g=0. Theories which relate flammability limit to heat losses [, , ] predict a minimum T_{f} below which flame propagation cannot occur because chemical reaction rates are much stronger functions of temperature than heat loss rates (exponential vs. algebraic). Consequently, because dilution decreases T_{f}, dilution increases the impact of losses, leading to flammability limits. For conductive losses, setting t_{chem}=t_{cond} leads to
Pe_{lim}_{Ž}S_{L,lim}d/a=constant (3),
with experiments [, ] and computations [] indicating Pe_{limš}40. For radiative losses, setting t_{chem}=t_{rad} leads to [, ]
(4),
For lean-limit CH_{4}-air mixtures at 1 atm, Eq. (4) yields S_{L,limš}0.023 m/s, similar to detailed numerical model predictions [, ]. Such small S_{L,lim} are not observed at 1g because of buoyant convection (t_{inv}<t_{rad} and t_{vis}<t_{rad}); Eqs. (1) and (2) yield S_{L,limš}0.033 and 0.078 m/s for upward and downward propagation, respectively. At µg, however, predictions of Eq. (4) compare favorably to experiments in large combustion vessels [, ] using varying pressures, fuels, and inert gases. Also, similar results were obtained for CH_{4}-air mixtures at 1 atm in tubes with d=0.05 m [], suggesting these limits are apparatus-independent. Thus, radiative losses may cause flammability limits when extrinsic losses (conduction, buoyant convection, etc.) are eliminated. In this instance µg experiments enabled observation of phenomena not observable at 1g.
These radiative effects apply only for "optically thin" gases (no reabsorption of emitted radiation) which is inappropriate for large systems, high pressures, or mixtures with strongly absorbing material. With this motivation, µg experiments [] were conducted using lean CH_{4}-air mixtures seeded with SiC particles. Since solids emit/absorb as black- or gray-bodies whereas gases radiate in narrow spectral bands, particle-seeded gases emit/absorb more radiation than particle-free gases. Measurements of propagation rates, pressures and post-flame thermal decay showed that, consistent with theoretical predictions [], at low particle loadings, the particles increase radiative loss (optically-thin conditions), whereas at higher loadings, reabsorption of emitted radiation becomes significant, which decreases net radiative loss and augments conductive heat transport.
Even for gases, computations [] using detailed statistical narrow-band radiation models show that flammability limits are extended considerably with reabsorption (Fig. 1). With gases, however, there are two mechanisms leading to flammability limits even with reabsorption. One is the difference in composition between reactants and products; if H_{2}O or radiatively-active combustion products are absent from the reactants, radiation from these species that is emitted upstream cannot be reabsorbed by the reactants. The second mechanism is that emission spectra are broader at T_{f} than T, thus some radiation emitted near the flame front cannot be absorbed by the reactants. Via both mechanisms upstream loss occurs, leading to extinction of weak mixtures. These results suggest fundamental (domain- and gravity-independent) flammability limits due to radiative losses may exist at µg, but these limits are strongly dependent on emission/absorption spectra of reactant and product gases and their temperature dependence, and cannot be predicted using gray-gas or optically-thin model parameters.
Stretched flames
Premixed-gas flames generally are not flat and steady nor propagate into quiescent flows. Consequently, flames are subject to "flame stretch" SŽ(1/A)(dA/dt) [], which affects S_{L} and extinction conditions [, ]. At 1g, buoyancy imposes flame stretch comparable to t_{inv}^{-1} or t_{vis}^{-1}. At µg, weak flame stretch effects that are insignificant at 1g may dominate. One example is expanding spherical flames for which
(5).
For Le<1, positive stretch increases T_{f} because the increased chemical enthalpy diffusion to the flame front in the form of scarce reactant exceeds the increased thermal enthalpy loss. Since heat release reactions have high activation energies, small T_{f} changes cause large changes in reaction rate and thus S_{L}. An evolution equation for non-adiabatic expanding spherical flames is given by []:
(6),
where SŽ(dr_{f}/dt)/(S_{L}(r/r_{f})), RŽr_{f}/(bdI(Le, e)), I(Le, e) is a scaling function (I>0 for Le<1 and I<0 for Le>1) and QŽ{bL(T_{f})d^{2}}/{l(T_{f}-T)}. The terms in Eq. (6) represent unsteadiness, heat release, curvature-induced stretch and heat loss, respectively. For steady planar flames, Eq. (6) becomes S^{2}lnS^{2}=-Q, which exhibits a maximum Q=1/e=0.3678... at S=e^{-1/2}, which corresponds to S_{L,lim} from Eq. (4). For Le<1 the curvature effect (2S/R) opposes heat loss (Q), allowing mixtures that are non-flammable as plane flames (Q>1/e) to exhibit expanding spherical flames until r_{f} grows too large and thus the curvature benefit is too small. For mixtures just outside the limit, the extinction radius may be very large. Such behavior, termed "self-extinguishing flames" (SEFs) is observed experimentally [, ] (Fig. 2) at µg in near-limit mixtures with Le slightly less than unity. (Mixtures with lower Le exhibit diffusive-thermal instabilities or flame balls discussed below.) Equation (6) also predicts, consistent with experimental observations [], that SEFs cannot occur for Le>1 (thus R<0) because both curvature and heat loss weaken the flame.
Two experimental observations not predicted by Eq. (6) are that narrow mixture ranges exhibit both SEFs and normal flames and that the energy release before extinguishment can be orders of magnitude greater than the ignition energy. Such behavior is predicted by computations [] not subject to scaling limitations of activation energy asymptotics used to derive Eq. (6). These calculations also show that for small initial r_{f}, all mixtures exhibit extinguishment, corresponding to non-ignition behavior []. Thus, in mixtures exhibiting SEFs, flames extinguish at large curvature (small r_{f}) due to large S and at small curvature due to radiative losses. This dual-limit behavior is also exhibited by many other types of flames described later.
Flames in hydrodynamic strain induced by counterflowing round-jets are frequently employed to model turbulence-induced flame stretch effects. At steady-state, the flame resides at the axial location (y) where the axial velocity (U_{y}) equals S_{L} for the given S=dU_{y}/dy. As S increases, U_{y} increases, thus the flame moves toward the stagnation plane (smaller y) and the burned gas volume (thus radiative loss) decreases. As with curvature-induced stretch, for Le less than/greater than unity, moderate hydrodynamic strain increases/decreases S_{L}, but for all Le, large strain extinguishes the flame []. Consequently, µg experiments [] in low-Le mixtures (Fig. 3) reveal extinction behavior analogous to spherical flames. For large S, the short residence time (~S^{-1}) causes extinguishment (S^{-1š}t_{chem}) (the "normal flame" branch, analogous to non-ignition behavior of spherical flames). In contrast, for low S the residence time and burned gas volume are large, thus radiative loss is significant (t_{radš}t_{chem}), so radiative loss extinguishes the flame (the "weak flame" branch, analogous to SEFs). The optimal S (š13 s^{-1}) producing the minimum flammable fuel concentration corresponds to S^{-1}=0.08 s, which is less than t_{vis} or t_{inv}, thus the C-shaped response and the entire weak-flame branch cannot be observed at 1g. The optimal S is nearly the same for model and experiment, indicating that loss rates are modeled well, but the computed limit composition is leaner than the experimental limit, suggesting that the chemical mechanism used is inaccurate for weak mixtures. Due to the radiant loss decrease at moderate S, the flammability limit extension also occurs for Le>1, though for sufficiently high Le no C-shaped response or flammability limit extension occurs []. (For spherically expanding flames no flammability extension occurs for Le>1 because in this case there is no mechanism to reduce radiative loss by flame stretch.)
The combination of non-monotonic response to S plus the reduced radiative loss at larger S causes several new extinction branches depending on t_{chem}, t_{rad}, S^{-1} and Le [, ]. It is uncertain whether these branches are physically observable since they have not been identified experimentally and stability analyses have not been performed.
Flame balls
Over 50 years ago, Zeldovich [] showed that the steady mass, energy and species conservation equations admit solutions corresponding to stationary spherical flames, characterized by a flame radius (r_{f}). Fuel and oxygen diffuse from the ambient mixture inward to the reaction zone while heat and combustion products diffuse outward (Fig. 4). Mass conservation requires that the fluid velocity be zero everywhere. The temperature and species mass fraction profiles have the form c_{1}+c_{2}/r, where c_{1} and c_{2} are constants. Corresponding solutions in planar and cylindrical geometry cannot exist because the solution forms c_{1}+c_{2}r and c_{1}+c_{2}ln(r), respectively, are unbounded as rÆ. Zeldovich [] and others [, ] also showed that flame ball solutions are unstable, thus probably not physically observable, though these solutions are related to flame ignition [].
Forty years after Zeldovich [], apparently stable flame balls were accidentally discovered in drop-tower experiments using H_{2}-air mixtures [] and aircraft-based µg experiments using various low-Le mixtures []. The µg environment facilitated spherical symmetry and prevented buoyancy-induced extinction. For mixtures sufficiently far from flammability limits, expanding spherical fronts composed of many individual cells were observed, whereas for more dilute mixtures cells that formed initially did not split and instead closed up upon themselves to form flame balls. (For still more dilute mixtures all flames eventually extinguished.) It was inferred that flame balls can occur in all near-limit low-Le mixtures, however, the short duration of drop-tower experiments and substantial g fluctuations in the aircraft-based µg experiments precluded definite conclusions. Recent Space Shuttle experiments [] confirmed that flame balls can exist for >500 seconds (the entire experiment duration.)
Zeldovich [] noted that radiative losses might stabilize flame balls; consequently, after their experimental observation, radiative loss effects on flame balls were analyzed []. For moderate loss, two solution branches are predicted (Fig. 5), a strongly non-adiabatic large-radius branch and a nearly-adiabatic small-radius branch. For sufficiently strong losses no solutions exist, indicating extinction limits. Stability analyses [] predict that all small flames are unstable to radial disturbances, and large flames with weak loss (far from flammability limits) are unstable to three-dimensional disturbances. Close to extinction limits the large-radius branch is stable to both disturbances. These predictions are consistent with the observed splitting cellular flames away from limits and stable balls close to limits. For Le close to or larger than unity, all flame balls are unstable for any loss magnitude [] explaining why they are never observed in (for example) CH_{4}-air mixtures (Leš0.9) or C_{3}H_{8}-air mixtures (Leš1.7).
Numerical predictions of non-adiabatic flame balls employing detailed chemistry, diffusion and radiation models [, ] are qualitatively consistent with these experimental and theoretical results. Still, quantitative agreement has been elusive (Fig. 6) for at least two reasons. First, flame ball properties are very sensitive to the 3-body recombination step H+O_{2}+H_{2}OÆHO_{2}+H_{2}O [] whose rate varies widely between different published H_{2}-O_{2} reaction mechanisms. The second reason is reabsorption of emitted radiation in mixtures diluted with radiatively-active CO_{2} or SF_{6}. An upper bound on self-absorption of diluent radiation (a_{P}Æ) is assessed by neglecting diluent radiation entirely because as a_{P}Æ radiative loss from the diluent vanishes and no additional heat transport occurs due to radiation. Agreement between predicted and measured flame radii is much better in this case (Fig. 7), strongly suggesting radiation modeling including reabsorption is needed for accurate predictions in these cases.
A key difference between propagating flames and flame balls is that propagating flames have convective-diffusive zones where temperature and concentration approach their ambient values in proportion to e^{-r/}^{d}, whereas flame balls have purely diffusive zones where the approach is proportional to 1/r. Plane flames respond on short time scales t_{chem}=d^{2}/a, whereas the gradual 1/r flame ball profiles produce properties dominated by the far-field length scale br_{f}, and thus diffusion time scales (br_{f})^{2}/a [], typically 100 s. These scales are relevant to stability and extinction limits since they affect the times for radiant combustion products to diffuse to the far-field and indicate large volumes of gas (~b^{3}r_{f}^{3}) where radiative loss affects flame balls. Such large scales are confirmed by space experiments [] and numerical simulations [, ]. Droplet and candle flames (discussed later), which have quasi-spherical, diffusion-dominated far-fields, exhibit analogous behavior.
GASEOUS NON-PREMIXED FLAMES
Stretched flames
Nonpremixed flames, where fuel and oxidant are separated before combustion, are affected by stretch differently than are premixed flames. The most significant difference is that the flame position is the location of stoichiometric mixture fraction, dictated by mixing considerations, rather than being determined by balances between S_{L} and U as in premixed flames. Consequently, nonpremixed flames have considerably less freedom of movement. Also, premixed flames have characteristic thicknesses d~a/S_{L} unrelated to the flow environment, whereas nonpremixed flames have only the diffusion length scale d~(a/S)^{1/2}. With fixed flame location and d increasing monotonically with decreasing S, nonpremixed flames with radiative loss exhibit only simple C-shaped responses to strain (Fig. 8) [], with a short residence time extinction branch (t_{chem}>S^{-1}) and a radiative loss extinction branch (t_{rad}<S^{-1}) rather than the complicated responses found for premixed flames []. The only significant difference in flame structures near the two limits is d [], a situation quite unlike premixed flames.
For the radiative extinction branch, t_{chem} is still a factor because t_{radš}S^{-1} results in order unity decreases in flame temperature, thus exponentially large decreases in t_{chem}. Thus, even conditions far from extinction at 1g may exhibit radiative extinction at µg due to much larger residence times. This mechanism also applies to radiative extinction of other types of nonpremixed flames discussed later. It is also somewhat analogous to the lowest branch of strained premixed flames (Fig. 3) and the large-radius branch of flame balls (Fig. 5).
Figure 8 suggests that no flames exist below some value of S, whereas the model predicts flames at arbitrarily low S. Similar behavior was seen for premixed flames (Fig. 3). This suggests an additional loss mechanism not considered by the model, probably axial conductive heat losses to the jets or radial conductive loss to inert gases surrounding the reactant streams. This would induce t_{cond}^{-1š}2.9 s^{-1} if d is the jet spacing (25 mm), or 7.1 s^{-1} if d is the jet diameter (16 mm). Either of these are roughly consistent with the minimum S in Fig. 8. Thus, apparatuses large enough to study flames at 1g without substantial conductive loss, where maximum length scales are about (at_{vis})^{1/2}, are insufficient for the weaker flames attainable at µg, where maximum length scales are about (at_{rad})^{1/2}.
Laminar gas-jet flames
A fuel jet issuing into an oxidizing environment is one of the simplest types of flames. Jost [] and Roper [] estimated the flame height (L_{f}) and residence time from jet exit to flame tip (t_{jet}) by determining the height (y) where the transverse diffusion time (d(y)^{2}/D, where d(y) is the stream tube width), equals the convection time (U(y)/y, where U(y) is the axial velocity.) When buoyancy and viscosity effects are negligible (momentum-controlled jets), U(y) is constant and equals the jet exit velocity (U_{o}) whereas when buoyancy effects dominate U(y)~(gy)^{1/2}. In either case mass conservation requires d(y)^{2}U(y)=d_{o}^{2}U_{o}=constant for round jets or d(y)U(y)=d_{o}U_{o}=constant for slot-jets. The resulting estimated scalings for L_{f} and t_{jet} are given in Table 2. Transition from buoyancy-controlled to momentum-controlled conditions occurs where the time scale for the former exceeds the latter, which corresponds to U_{o}>gd_{o}^{2}/D for either round-jets or slot-jets. The scalings for momentum-dominated flames presume constant U, which is reasonable for co-flowing Burke-Schumann flames, but for nonbuoyant jet flames without co-flow, the jet spreads and decelerates. For this situation []
Þ (7).
Since Scš1, the scalings of L_{f} and t_{jet} are similar with or without viscosity. Figure 9 shows measurements of L_{f} for CH_{4} flames []. Note that, as the scalings predict, L_{f}/d_{o}~ReŽU_{o}d_{o}/n at both 1g and µg and only small differences exist between 1g and µg flame lengths.
All µg studies show larger flame widths (w) at µg than 1g due to lower U and longer t_{jet} []. Also, w is larger at µg because the temperatures are lower (see below) and D~T^{1.75}. Because w depends on whether U is accelerating (buoyant jets), constant (nonbuoyant Burke-Schumann flames) or decelerating (nonbuoyant jets), w is more difficult to predict than L_{f} []. The difference between 1g and µg widths decreases as Re (thus U_{o}) increases (Fig. 10). The nonbuoyant widths increase slightly with Re, whereas all aforementioned models predict self-similar flame shapes with no effect of Re. This may be due to axial diffusion, not considered in these models, which increases mixing over that with radial diffusion alone. That w/d_{o} is lowest for lowest Re (š20), where axial diffusion is most significant, but asymptotes to fixed values at high Re, supports this suggestion.
µg gas-jet flames are redder compared to yellow 1g flames [, ], indicating lower temperatures in the soot production regions and presumably lower maximum flame temperatures. This occurs because t_{jet} is larger at µg and thus radiative loss effects (~t_{jet}/t_{rad}) are greater. Drop-tower [] and space [] experiments indicate surprisingly large and consistent radiative loss fractions (0.45-0.60) at µg compared to 0.07-0.09 at 1g, for various fuels, pressures, O_{2} mole fractions and flow rates. Thus, differences in t_{jet} at 1g and µg result in widely varying characteristics even for flames having the same L_{f}.
Turbulent flames
In turbulent non-premixed jet flames, D is not fixed but rather is nearly proportional to u’L_{I}. Since approximately u’~U_{o} and L_{I}~d_{o}, for round jets L_{f}~U_{o}d_{o}^{2}/u’L_{I}~d_{o}, thus L_{f} is independent of U_{o}. This prediction is supported by classical 1g experiments [] as well as recent µg experiments [] (Fig. 11). Note that L_{f}(µg)/L_{f}(1g) is practically constant even beyond the transition to turbulent conditions (high U_{o} and Re). Note also that the maximum Re where the flame exists (the "blow-off" limit) is different at 1g and µg. This is surprising since blow-off conditions are typically controlled by behavior near the flame base [], where buoyancy effects are often considered insignificant. This suggests that blow-off is partially affected by convection induced by the buoyant plume far above the jet exit, even at very high U_{o}. Intuitively the 1g flames should blow-off at lower U_{o} since buoyant flow would induce higher "effective" U_{o}, which is consistent with Fig. 11. This shows that buoyancy effects are quite ubiquitous even under conditions commonly thought to be unaffected by buoyancy.
Soot formation processes
µg gas-jet flames have much greater tendencies to emit soot than 1g flames [, ], indicating increases in t_{jet} (thus greater time for soot formation) plus broader regions where composition and temperature are favorable for soot formation [, ], outweigh lower temperatures at µg, which decreases soot formation []. Recent quantitative measurements [] show peak soot volume fractions about twice as high at µg than 1g for 50% C_{2}H_{2}/50% N_{2}-air flames.
Surprisingly, µg gas-jet flames exhibit "smoke points," corresponding to critical U_{o} below which soot is consumed within the flame, and above which soot is emitted from the flame []. Smoke points are expected for buoyant round-jet flames since t_{jet}~U_{o}^{1/2}, thus increasing U_{o} increases the time available for soot formation, but for nonbuoyant flames, t_{jet}~U_{o}^{0}, suggesting no smoke point should exist. Fully elliptic numerical computations [] show that for some circumstances, t_{jet} does increase monotonically with U_{o}, which could explain smoke points for nonbuoyant flames. This behavior was suggested to result from axial diffusion effects [], but in this case t_{jet} should asymptote to constant values for large L_{f}, where axial diffusion is negligible. Thus, simple explanations of µg smoke points remain elusive. Residence time considerations alone may be misleading; soot precursor temperature-composition-time history effects may also be important.
With weak convection, thermophoretic forces, which move particles towards lower temperatures is an important effect. If convection and temperature gradient are in the same direction, the convective and thermophoretic forces may balance at some location. This leads to soot accumulation inside the flame front (Fig. 12) [], where an annulus of soot accumulation is convected by axial flow through the flame tip. Downstream, for reasons not yet explained, the soot annulus fragments, creating crown-like structures. These effects are only observable at µg where convection velocities are comparable to thermophoresis velocities, (š5 mm/s for the conditions of Fig. 12 [].)
CONDENSED-PHASE COMBUSTION
Droplet combustion
The first microgravity combustion experiments were isolated fuel-droplet tests conducted by Kumagai []. At 1g, experimental measurements are compromised by buoyant convection, destroying the spherical symmetry and inducing additional heat and mass transport, which alters burning rates and complicates modeling. Classical theory [, ] predicts burning rates for spherically-symmetric buoyancy-free droplet burning are given by
d_{do}^{2} - d_{d}^{2}=Kt; K Ž (8l/r_{d}C_{P})ln(1+B) (8).
In experimental studies the droplet diameter d_{d} could be fixed by forcing fuel through a porous sphere at rates that balances evaporation, leading to a steady mass burning rate ()=(¼/4)r_{d}d_{d}K, but most experiments employ fuel droplets where d_{d} decreases with time. Somewhat surprisingly, many fuel-droplet results follow Eq. (8) well despite unsteadiness, heat losses, soot formation and water absorption effects discussed below.
As with flame balls, for spherical droplet flames steady solutions exist even in infinite domains, with the flame front located at d_{f}=d_{d}ln(1+B)/ln(1+f). While flame balls are convection-free, droplet flames exhibit Stefan flow due to fuel vaporization at the droplet surface. Mass conservation dictates that the Stefan velocity decays as 1/r^{2}, causing temperature and concentration profiles to vary with radius in proportion to (1+B)^{-d}^{d}^{/2r} rather than 1/r as in flame balls, though these profiles, when normalized by flame radius, are indistinguishable at large r. Unlike flame balls, heat losses are not required for stable droplet flames because the flame location cannot move away from the stoichiometric contour.
The characteristic time scale for droplet combustion is t_{dropš}d_{f}^{2}/a. This leads to two extinction limits [, ], one for small d_{f }where t_{drop}<t_{chem}, thus fuel and oxidant cannot react before interdiffusing, and one for large d_{f} where t_{drop}>t_{rad}, thus the temperature decrease from radiative loss reduces the reaction rate sufficiently to cause extinction. The former occurs when d_{f}<(at_{chem})^{1/2} and the latter when d_{f} >(at_{rad})^{1/2}. Recent space experiments have reported radiative extinction of large droplets in air [] and O_{2}-He [] atmospheres. Even when radiation does not cause extinguishment, it causes a decrease in K, especially when soot formation is significant [, ].
In experiments, K is not constant as the quasi-steady model (Eq. 8) predicts, but rather decreases with increasing d_{do} (Fig. 13) []. This suggests that non-steady effects, specifically the diffusion of thermal energy and radiant combustion products into the far-field, may be significant; this would cause changing radiative loss over time. Analogous behavior occurs in flame balls [], where radiative loss requires š100 s to reach steady-state even in flame balls much smaller than typical droplet flames. Another indication of unsteadiness in droplet flames is that constant d_{f}/d_{d} values are not generally achieved, especially for large droplets at µg [, ], in contrast to quasi-steady theory. Unsteadiness effects in droplet flames are analyzed by King []. The effect of d_{do} on K has also been proposed [] to result from soot accumulation, which is more significant for larger d_{do} and acts to decrease net heat release and increase radiation.
As with non-premixed gas-jet flames, soot particles in µg droplet flames exhibit thermophoresis effects [, ], leading to soot agglomeration between the droplet and flame front (Fig. 14). The agglomerates may break apart suddenly, leading to multiple burning fragments. Since velocity and temperature gradients are readily modeled in spherical droplet flames, such experimental observations enable assessment of thermophoresis effects on soot particle transport.
Another complicating factor arises in fuels that are miscible in water. The fuel may absorb water vapor from the combustion products, causing significant departure from Eq. (8) and reducing flammability. Such behavior is found [, ] in methanol flames, where extinction diameters depend substantially on d_{do} due to water absorption during the burn.
Finally, in recent space experiments, flame oscillations with amplitudes comparable to the mean flame diameter have been observed []. The oscillation amplitude grew with time with extinction occurring after typically 8 cycles. The oscillation frequency was š1 Hz. Cheatham and Matalon [] predicted oscillations of roughly the correct frequency in droplet flames of pure fuels under near-extinction conditions when the reactant Lewis numbers are sufficiently high. To date oscillations have been reported for methanol/dodecanol (80/20 mass fraction) bi-component droplets burning in air; it is uncertain whether oscillations also occur in pure fuels. Moreover, µg experiments in O_{2}-He atmospheres [], with much higher Le, did not exhibit oscillations. One possible explanation is that droplet support fibers were used in [] but not []. The fiber could increase conductive and radiative losses, which encourage oscillations []. Another explanation is that since oscillations occur only near extinction conditions, depletion of oxygen (leading to extinction) was much more significant in the smaller combustion chamber used in the methanol/dodecanol experiments. Oscillatory instabilities are discussed further in the following section.
Candle flames
An excellent example of the differences between 1g and µg flames is seen in perhaps the most common and familiar of all combustion processes - candle flames. At 1g, candle flames are supported by air entrained via buoyant flow, which generates self-sustaining flames and flow configurations. Obviously this mechanism cannot apply at µg. The spherical diffusion equation admits steady solutions for flame balls and droplet flames without forced convection. An interesting question is whether candle flames, which are not strictly spherical, can behave similarly. Space experiments [, ] indicate that candle flames can be steady for >45 min, with flame shapes typically hemispherical (Fig. 15). Eventually, the flames always extinguished, whereas the spherical flame model predicts that the flame would burn indefinitely. Protective screens used in the experiments may have limited the O_{2} supply, eventually allowing sufficient O_{2} depletion to cause extinguishment.
Before extinction, the candle flame edge frequently advanced and retreated periodically. The oscillation amplitude increased over time and on one retreating cycle the entire flame extinguished. With larger wicks (thus larger flame diameters) oscillations started spontaneously, whereas with smaller wicks oscillations occurred only when solid objects were placed near the flame. Only a few cycles before extinction were observed in the Spacelab experiments [] whereas hundreds of cycles were observed in the Mir experiments []. Probably this is because the protective screen was much more permeable in the Mir experiments, thereby decreasing the O_{2} depletion rate and maintaining the flame at near-extinction conditions much longer.
At least two possible explanations for these oscillations have been advanced. Cheatham and Matalon [] showed that near extinction, oscillatory instabilities occur in spherically-symmetric droplet flames with radiative loss at sufficiently high Lewis numbers. Their predicted oscillation frequencies (0.7-1.4 Hz) are comparable to experimental observations, however, the differences between spherically-symmetric droplet flames and roughly hemispherical candle flames were noted []. Alternatively, Buckmaster [] showed that the flame "edge" separating burning and non-burning regions of non-premixed flames exhibit oscillatory behavior at (for quasi-stationary edges) Le>1+8/b(1-(T/T_{f}))š2 when Le for the other reactant is unity. Leš1 for O_{2} in N_{2}, but Le for fuel vapors is probably closer to 2, thus edge-flame instabilities could explain the observed oscillations. While neither instability mechanism has been definitively linked to the candle-flame experiments, both predict greater propensities for oscillation with greater heat losses, consistent with the observation that oscillations occur near extinction.
Flame spread over solid fuel beds
Flame spread over solid fuel beds is typically classified as opposed-flow, where convection opposes flame propagation, or concurrent-flow. 1g downward flame spread is opposed-flow since upward buoyant flow opposes flame spread, whereas upward flame spread is concurrent-flow. At µg without forced flow, flame spread is always opposed-flow since the flame spreads toward the fresh oxidant with a self-induced velocity equal to the spread rate (S_{f}). At 1g self-induced convection is negligible since buoyancy-induced flows are typically (ga_{g})^{1/3š}0.10 m/sªS_{f}. Very few concurrent-flow flame spread have been conducted at µg [], consequently this section focuses on opposed-flow spread.
S_{f} is estimated by equating the conductive heat flux to the fuel bed (=l(dW)(T_{f}-T_{v})/d, where d=a/U is the thermal transport zone thickness and U is the opposed-flow velocity (forced, buoyant and/or self-induced)), to rate of fuel bed enthalpy increase (=r_{s}C_{P,s}t_{s}(T_{v}-T)WS_{f}). Assuming mixing-limited reaction (infinite-rate chemistry), for thermally-thin fuels, where heat conduction through the solid is negligible, S_{f} is predicted as [, ]
(9).
Note S_{f} is independent of U and P. For thermally thick fuels, where heat conduction through the solid fuel dominates, t_{s} is the thermal penetration depth into the solid, estimated by equating the conductive heat flux to the fuel bed to the heat flux through the fuel (=l_{sy}(dW)((T_{v}-T)/t_{s}), where the subscript y refers to the direction normal to the fuel surface). This leads to the exact solution []
(10)
Note that, unlike the thin-fuel case, for thick fuels S_{f}~U^{1}P^{1}.
Dual-limit extinction behavior is observed in µg flame spread experiments (Fig. 16) []. The time for thermal energy to diffuse across the convection-diffusion zone (t_{diff}) is d/U=a/U^{2}, thus high-U extinction occurs when t_{diff}>t_{chem} or U>(a/t_{chem})^{1/2} and radiative extinction occurs when t_{diff}>t_{rad} or U<(a/t_{rad})^{1/2}. (Surface radiative loss may also be important, particularly at moderate and higher U [].) Interestingly, the minimum O_{2} concentration supporting combustion (c_{O2,lim}), and thus the greatest hazard, corresponds to Uš0.1 m/s, which is lower than buoyant convection at 1g, and might correspond to ventilation drafts in manned spacecraft.
A radiative loss parameter can be defined as HŽt_{diff}/t_{rad}=a/U^{2}t_{rad}. Since H~U^{-2}, S_{f} is lower at µg where U (thus H) is lower. Experiments [] (Fig. 17) show that an imposed forced flow at µg increases S_{f} since U (sum of forced flow and S_{f}) increases, thus H decreases, whereas at higher U (whether buoyant or forced), S_{f} decreases as the high-U limit is approached. For 21% O_{2} or lower, the infinite-rate chemistry prediction of Eq. (9), S_{f}~U^{0}, is never achieved. Only at 30% O_{2} is T_{f} high enough that this condition is achieved.
Since a~P^{-1} and t_{rad}~P^{0}, H~P^{-1}, thus for thin fuels S_{f} should increase with P towards the ideal (adiabatic) value (Eq. (9)). This is confirmed by quiescent thin-fuel space experiments [, ] which show S_{f} increasing from 3.2 to 5.9 mm/s as P increases from 1.0 to 2.0 atm with fixed O_{2} mole fraction (0.50). For these conditions H decreases from 24 to 3.5, thus, even at the highest P radiative effects are probably still important. This is consistent with computations [] which predict S_{f}=12 mm/s (almost independent of P) for adiabatic conditions for this fuel/atmosphere combination.
Neither N_{2} nor O_{2} emit thermal radiation, thus, for flames in O_{2}-N_{2} atmospheres only H_{2}O and CO_{2} combustion products radiate significantly. For these cases typically a_{P}^{-1š}1.2mªd, thus radiative transfer is optically-thin (negligible reabsorption). When a_{P}d³1, reabsorption effects cannot be neglected. With reabsorption, some radiation is not lost and may augment conduction to increase S_{f} above that without radiation. This behavior is seen experimentally [] using strongly emitting/absorbing CO_{2} and SF_{6 }diluents (Fig. 18), where S_{f} is higher and c_{O2,lim} is lower at µg than 1g, whereas the opposite (conventional) behavior is found in non-radiant diluents (not shown). These data indicate that for non-radiating diluents µg is less hazardous since c_{O2,lim} is higher at µg than 1g (0.21 vs. 0.16 for He), whereas for radiant diluents, µg is more hazardous (0.21 vs. 0.24 for CO_{2}). This is particularly significant considering that CO_{2}-based fire suppression systems will be used on the International Space Station. To date, flame spread calculations have employed optically-thin radiation models with constant a_{P} [, ] or variable depending on local temperature and composition [], and thus cannot assess reabsorption effects.
These discussions pertain to thin fuels, for which steady µg spread is possible because theoretically S_{f}~U^{0}. For thick fuels, S_{f}~U^{1}, thus S_{f} is indeterminate for quiescent µg conditions (U=S_{f}). When unsteady solid-phase conduction is considered, t_{s}~(a_{s}t)^{1/2}, which results in S_{f}~t^{-1/2} []. Consequently, all fuel beds at quiescent µg conditions eventually become thermally-thin (penetration depth greater than the bed thickness) unless the radiative effects discussed later are considered. Of course, flames may extinguish due to large d (thus large radiative loss) before reaching steady-state, thermally-thin conditions.
A difficulty in comparing space experiments to two-dimensional model predictions is that the fuel bed width W (30 mm for thin fuels [, ] and 6.2 mm for thick fuels []) is smaller than the thermal transport zone thickness (d). Consequently, these experiments can hardly be considered two-dimensional. Both lateral heat loss, which retards spread, and lateral O_{2} influx, which enhances spread, are probably important, thus their effects may partially cancel. Some authors [] suggest that radiative losses decrease d to values much smaller than a/U=a/S_{f}, but the oxygen transport zone thickness (d_{O2}) is still D_{O2}/U, since no analog to radiative loss exists for O_{2} transport. Since Le=a/D_{O2š}1, d_{O2} is nearly the same as d in adiabatic flames. Thus, µg flames have probably benefited substantially from lateral O_{2} influx, especially for lower pressures and O_{2} mole fractions, where d/W~a/S_{f}W is largest. In fact, S_{f} might be higher at smaller W due to lateral O_{2} influx. Space experiments using cylindrical fuel rods are planned [] to examine truly two-dimensional spread.
A surprising observation of fingering fronts was found in space experiments using paper samples treated to inhibit flaming combustion but allow smoldering propagation (Fig. 19) []. Fingering was observed at µg when U<50 mm/s whereas smooth fronts were observed at 1g. This was proposed [] to result from limited O_{2} mass transport at µg with low U, which caused the O_{2} consumption regions to become localized spots instead of continuous fronts. This proposition does not explain why heat conduction does not smooth out potential fingers as it does (for example) in premixed-gas flames with Le>1. The following alternative explanation is proposed here. Gas-phase heat transport occurs on the length scale d~a/U, and solid-phase transport occurs on the scale d_{s}~a_{s}/u_{s} where u_{s} is the smolder front velocity and a_{s} the solid thermal diffusivity. Oxygen transport occurs only through the gas phase on the scale D_{O2}/Uša/U~d. Radiative loss can suppress heat transport through the gas, but no corresponding effect on O_{2} transport can occur. Thus at low U, the effective Le is a_{s}/D_{O2}´1. At higher U or at 1g, d is smaller, gas-phase heat transport dominates and radiative effects are weaker, thus the effective Le is a/D_{O2š}1. These assertions are consistent with estimates [] of the relative importance of gas-phase and solid-phase transport. Both premixed [, ] and nonpremixed [, ] flames with effective Le<1 exhibit diffusive-thermal instabilities that cause fingering patterns, whereas for Le³1 the fronts are stable. This explanation is also consistent with 1g experiments [] on horizontal fuel beds burning in oxidant channels of adjustable vertical height. At small heights or low U, fingering similar to Fig. 19 was observed. In this case conductive loss to the channel ceiling causes suppression of gas-phase heat transfer. Apparently in both cases the key factor is suppression of gas-phase heat transfer while allowing solid-phase heat transfer, which reduces the effective Le (though this factor was not mentioned in [] or [].)
Flame spread over liquid fuel pools
Flame spread over liquid fuels encompasses practically all solid-fuel flame spread phenomena discussed above, plus liquid-phase flow effects. Typically T_{v}-T is smaller for liquid than solid fuels, thus S_{f} is higher. Also, if T_{v}-T is small some fuel prevaporization occurs even at T=T, thus partially-premixed gas-phase combustion phenomena may occur. Because of the fuel surface temperature gradient upstream of the flame, surface tension gradients are produced that cause the surface layer to move upstream (away from the flame), which increases S_{f}. At 1g, this heated liquid layer must lie near the surface, whereas at µg no limitation exists. 1g experiments, summarized in [], show that at low fuel temperatures, the average S_{f} is small (typically 10 mm/s) and the spread alternates between a fast "jump" velocity and a slow "crawl" velocity. At higher fuel temperatures, S_{f} is faster and steady. For the conditions exhibiting pulsating spread at 1g, µg flame spread cannot be maintained, whereas for the conditions exhibiting uniform spread at 1g, steady spread is also exhibited at µg []. Pulsating spread has never been observed at µg. No definitive explanation for these observations has been advanced. For the conditions exhibiting pulsating spread at 1g and no spread at µg, flame spread is still different at 1g and µg when forced flows comparable to buoyancy-induced convection (U=0.30 m/s) are imposed (Fig. 20) []. Specifically, 1g spread is almost unaffected by the imposed flow but µg spread is steady with S_{f} being lower (š15 mm/s) than either the 1g jump velocity (š100 mm/s) or crawl velocity (š22 mm/s). Detailed numerical modeling [] predicts pulsating spread at µg for the conditions of Fig. 20 and values of S_{f} much closer to the measured 1g S_{f}. Remarkably, if thermal expansion is artificially suppressed, good agreement between the model and µg experiments is found. It is proposed [] that this agreement results from three-dimensional effects, specifically, in the experiment, flow induced by thermal expansion is relaxed in the lateral dimension, whereas the two-dimensional model does not permit this. That three-dimensional effects might dominate is surprising considering that for this flame d/W~a/UWš0.02, thus d>W. Also, this hypothesis does not explain why pulsating flame spread is observed at 1g but not µg. In liquid-fuel flame spread, µg experiments have identified limitations in our current understanding of combustion processes at 1g.
RECOMMENDATIONS FOR FUTURE STUDIES
Reabsorption effects
The µg studies described here suggest new unresolved issues and opportunities for further improvements in understanding. Perhaps the most important is the effects of reabsorption of emitted radiation, including both reabsorption by the emitting gas and in two-phase combustion, absorption by the condensed phase. All radiative effects discussed above are critically dependent on the degree of reabsorption. To study reabsorption effects requires radiatively-active diluents (CO_{2}, SF_{6}), high pressures and/or large systems. All of these conditions lead to higher Gr_{d} at 1g and thus turbulent flow. Hence, µg experiments enable study of reabsorption effects without the additional complications due to turbulence. Reabsorption effects are important not only to µg studies, but also to combustion at high pressures and in large combustors. For example, at 40 atm, typical of premixed-charge internal combustion engines, a_{Pš}18 m^{-1}, thus a_{P}^{-1}=0.045 m, for stoichiometric combustion products. This length scale is comparable to cylinder radii, thus reabsorption effects within the gas cannot readily be neglected. Simple estimates [] indicate radiative loss may influence flame quenching by turbulence in lean mixtures. Similarly, reabsorption cannot be neglected in atmospheric-pressure furnaces larger than a_{P}^{-1š}2.2 m. Moreover, many combustion devices employ exhaust-gas or flue-gas recirculation; for such devices the unburned mixtures contain significant amounts of absorbing CO_{2} and H_{2}O. While reabsorption could affect practically all types of flames reviewed here, to date reabsorption effects have been studied only for propagating premixed-gas flames [, ], flame balls [] and flame spread over thermally-thin fuels []. All have shown substantial differences from optically-thin behavior. Two examples of effects expected for other flames are given below. For droplet combustion, reabsorption effects could be substantially more important than for flame spread over solid fuels because for droplets the Stefan flow severely limits heat conduction to the droplet surface. This is why heat release (B) affects burning rates (K) only weakly (logarithmically) (Eq. 8). Radiative transfer is unaffected by the Stefan flow. Equation 8 is readily extended to include surface radiative flux (q_{r}):
(11),
Figure 21 shows the predictions of Eq. (11). (While apparently Eq. (11) has not been presented previously, numerical studies [, ] have shown qualitatively similar predictions. Moreover, these studies show that typical radiative absorption lengths for liquid fuels at relevant wavelengths are on the order of 1 mm, thus large droplets could absorb most incident radiation.) For spherical shells of radiant combustion products having thickness d´d_{f}, q_{rš}Ld/4, then for typical values B=8.5, L=2 x 10^{6} W/m^{3}, d=10 mm, d_{d}=5 mm, C_{P}=1400 J/kgK, l=0.07 W/mK and L_{v}=400 kJ/kg, Eq. (11) predicts R=0.63 and W/W_{R=0}=1.11, thus moderate effects of radiative transfer are expected. For droplets in radiatively-active diluents such as CO_{2}, the effect could be much stronger. Using the P1 approximation, for a sphere of unit emissivity in an infinite gray gas, q_{r}=[4/(2+3a_{P}d_{d}/2)]s(T^{4}-T^{4}) []. Using volume-averaged properties T=1000K and a_{P,CO2}=20 m^{-1}, R=18 and thus W/W_{R=0}=8.0, indicating radiation completely dominates heat transport. As discussed later, at high pressures radiative effects may prevail even in O_{2}-N_{2} atmospheres. Flame spread over thermally-thick fuel beds in quiescent atmospheres at µg is typically considered inherently unsteady [], however, radiative transfer to the fuel bed could enable steady spread. If the flame is modeled by an isothermal volume with dimensions dxdxW, radiation induces a radiative flux Ld^{2}W=L(a^{2}/S_{f}^{2})W which augments the conductive flux l(dW)(T_{f}-T_{v})/d. Equating this total heat transfer to r_{s}C_{P,s}t_{s}(T_{v}-T)WS_{f} yields
(12),
which vanishes without gas radiation (L=0). Thick-fuel space experiments in O_{2}-CO_{2} or O_{2}-SF_{6} atmospheres could be employed to check for steady spread and test the accuracy of Eq. (12). While optically-thin radiation modeling is reasonably straightforward, modeling of spectrally-dependent emission and absorption is challenging because local fluxes depend on the entire radiation field, not just local scalar properties and gradients. Some relevant computations have employed gray-gas models [] but recent studies [, ] show that these methods are probably inaccurate because of the wide variation in spectral absorption coefficient with temperature, species and wavelength. Comparisons of various radiative treatments for small one-dimensional nonpremixed flames have been made []. Comparisons for larger, multi-dimensional systems would be valuable. Moreover, recent studies of µg soot formation [, , ] may enable improved modeling of soot radiation at µg.
High pressure combustion
All practical combustion engines operate at pressures much higher than atmospheric. The impact of buoyancy for premixed flames scales as t_{chem}/t_{vis}~(ga/S_{L}^{3})^{2/3}~P^{n-4/3}, where n is the overall reaction order (S_{L}~P^{n/2-1}). Since typically n<4/3 for weak mixtures [], where buoyancy effects are most important, the impact of buoyancy increases with pressure. Also, as discussed earlier, radiation effects are more difficult to assess at higher pressure due to increased interference from buoyant transport. Nevertheless, few high-pressure µg combustion experiments have been performed. High-pressure droplet combustion experiments [] revealed substantial but different increases in K with P at 1g and µg. Radiative effects were not discussed, but could have been important since in Eq. (11), the only pressure-dependent factor is q_{r}~L~P^{1}, thus in Fig. 21, R~P. (For most flames length scales decrease with increasing P, which would decrease radiative effects, but for droplet flames d_{f} depends only on stoichiometry [75, 76].) Further assessment of radiative effects in high-pressure droplet combustion and other flames appears warranted.
Three-dimensional effects
In earlier sections, effects of lateral heat and mass transport on flame spread were discussed. µg experiments with varying fuel bed width (W) are needed to assess three-dimensional effects. Complementary three-dimensional modeling using codes such as those developed by NIST [], extended to include gas-phase radiation, would be instructive. An approximate but much less expensive approach would be to incorporate volumetric terms 6l(T(x,y)-T)/W^{2} and 6rD_{i}(Y_{i}(x,y)-Y_{i,})/W^{2}, where x and y are the coordinates parallel and perpendicular to the fuel bed, into the two-dimensional model to account for lateral heat losses and lateral diffusion of each species i. Another three-dimensional effect is found in the development of flame balls from ignition kernels. Currently, it is known that large flame balls are linearly unstable to three-dimensional disturbances for weak loss (Fig. 5), but the transition from splitting flame balls to stable flames is not well understood nor can the number of flame balls produced from an ignition source be predicted. Modeling using three-dimensional premixed flame codes [] is needed.
Gas-jet flames
Table 2 shows predicted scalings of flame lengths (L_{f}) and residence times (t_{jet}) for buoyant and nonbuoyant round-jet and slot-jet flames. Despite numerous investigations of round-jet flames at µg, no µg slot-jet results are available to test those predictions. Currently it is unknown whether slot-jet flames at µg would exhibit smoke points, or whether this information could be used to explain smoke points in µg round-jet flames. Since L_{f} depends on g for buoyant slot-jet but not round-jet flames, L_{f} should be quite different at 1g and µg for slot jets but not round-jets. Residual accelerations in aircraft µg experiments will be more problematic for slot-jet than round-jet flames because t_{jet}~g^{-1/2} for buoyant round-jet flames whereas t_{jet}~g^{-1/3} for slot-jet flames. There has been little investigation of blow-off behavior of laminar gas-jet flames at µg. Dual-limit behavior might occur for flames of fixed mass flow rate but varying d_{o}, with short residence time extinction at small d_{o} (t_{jet}~d_{o}^{2}/U_{o}) and radiative extinction at large d_{o} (thus large t_{jet}) Experiments should be conducted by diluting the fuel rather than increasing U_{o} to obtain blow-off without transition to turbulence. In this way dual-limit behavior was observed at 1g [] with short residence time and conductive loss (to the burner rim) extinction branches.
Quasi-steady spherical diffusion flames
As discussed earlier, comparing predicted radiative extinction limits of droplet flames to experiments is problematic because quasi-steady conditions may not be obtained, since extinction occurs for sufficiently large droplet and flame diameters but the droplet diameter decreases throughout its life. Numerical models can account for transient effects, but the multi-dimensional ignition process is difficult to model quantitatively. Comparisons of droplet experiments and computations to corresponding results obtained using fuel-wetted porous spheres would be most interesting. The fuel should be forced through the porous sphere at slowly increasing rates until extinction occurs when d_{f} >(at_{rad})^{1/2}, thus obtaining truly quasi-steady extinction. Long µg durations would be required to establish steady diffusion-dominated far-field temperature and composition profiles (thus steady radiative loss). Candle flames are similar to wetted porous spheres, though without true spherical symmetry nor any means to control or measure the instantaneous d_{d}. A simpler related experiment employs porous spheres through which gaseous fuel is forced at prescribed , resulting in a flame diameter d_{fš}C_{P}/2¹lln(1+f). Some drop-tower experiments using this configuration have been reported [, ], though steady-state conditions were not obtained due to short µg durations. Steady-state behavior appears unlikely given the (thus air consumption rates) and chamber sizes employed to date; near-wall oxygen depletion would be significant over the times required to reach steady-state (>10 s [].) Steady-state behavior might be obtained in drop-towers by using smaller and diluted fuel with enriched oxygen atmospheres to increase f, thus decrease d_{f} and t_{drop}~d_{f}^{2}/a.
Catalytic combustion
Catalytic combustion holds promise for reduced emissions and improved fuel efficiency in many combustion systems [, ]. Since catalysis occurs at surfaces, catalysis is inherently multi-dimensional and/or unsteady, requiring reactant transport to the surface and heat and products transport from the surface. While boundary-layer approximations can be invoked, probably the only truly one-dimensional steady catalytic configuration is a spherical surface immersed in nonbuoyant quiescent premixed gas - a "catalytic flame ball." In this case r_{f} is fixed but the surface temperature T_{s} and fuel concentration Y_{s} are unknown. These are related through the energy conservation (including surface radiation) and diffusion equations to obtain the surface reaction rate in moles per second (Q):
(13)
where e_{s} is the surface emissivity and properties with the subscript s are gas-phase properties evaluated at T=T_{s}. By varying r_{s}, Y, pressure and diluent gas, Q(T,Y) can be inferred from Eq. (13) and the measured T_{s}. Of course, conditions must be unfavorable for initiation of propagating flame or flame balls that stand off from the surface.
Chemical models
An important contribution of µg combustion experiments has been improved understanding of extinction processes, which are inherently related to finite-rate chemistry. To obtain closure between experiments and computations, accurate chemical models are needed. For lean premixed hydrocarbon-air flames, most models [, ] predict higher S_{L} and leaner flammability limits than experimental observations [, , ]. The discrepancy seems larger than experimental uncertainty or unaccounted heat losses could explain. In contrast, for H_{2}-air flame balls [] and 1g strained premixed H_{2}-air flames [], these chemical models predict smaller balls, lower S_{L} and richer flammability limits than experiments. These chemical models predict S_{L} in mixtures away from extinction limits very faithfully. The discrepancies result largely from differences in rates for H+O_{2}+M reactions, particularly the Clapeyron efficiencies of various M species []. These reactions are extremely important in near-limit flames due to competition between chain-branching and chain-inhibiting steps near limits [], but are much less important away from limits. Further scrutiny of the rates for these reactions at intermediate temperatures (1100-1400K) would be welcomed.
CONCLUSIONS
µg experiments have broadened our understanding of combustion fundamentals into regimes not previously explored. In particular, they have helped integrate radiation into flame theory. Although flame radiation has long been recognized as an important heat transfer mechanism in large fires [], its treatment has largely been ad hoc because of the difficulty of predicting soot formation. Also, large-scale fires at 1g are inevitably turbulent, leading to complicated flame-flow interactions. Small-scale µg flames are laminar, often soot-free and have significant influences of radiation. As a result of radiation effects, both premixed and non-premixed flames have exhibited dual-limit extinction behavior, with residence time limited extinction at high strain or curvature and radiative loss induced extinction at low strain or curvature. The high-strain limit is readily observed at 1g; when forced flow is absent, buoyant flow causes this strain. For weak mixtures these limits converge, but the convergence and the entire low-strain extinction branch is only seen at µg. This dual-limit behavior is observed for stretched and curved premixed-gas flames, strained non-premixed flames, isolated fuel droplets and flame spread over solid fuels. Besides radiative effects, µg studies have enabled observation and clarification of numerous other phenomena, for example thermophoresis effects in soot formation, spherically-symmetric droplet burning, diffusion-controlled premixed flames (flame balls) and flame instabilities in droplets and candle flames. Considering the rapid progress made recently, further advances are certain to occur. Hopefully this report on the current state of understanding can help motivate and inspire such advances.
ACKNOWLEDGMENTS
The author expresses his deepest gratitude to the NASA-Lewis Research Center for supporting his µg combustion work for over 10 years. Comments from Tom Avedisian, Yousef Bahadori, John Buckmaster, Mun Choi, Dan Dietrich, Fred Dryer, Gerard Faeth, Guy Joulin, Yiguang Ju, Kaoru Maruta, Vedha Nayagam, Takashi Niioka, Sandra Olson, Howard Ross, Kurt Sacksteder, Dennis Stocker, Peter Sunderland, Gregory Sivashinsky, James Tien, Karen Weiland, Forman Williams and anonymous reviewers have been invaluable in preparing this manuscript. The author also thanks numerous others who offered useful suggestions that could not be accommodated because of space limitations.
NOMENCLATURE
a_{P} Planck mean absorption
coefficient
A flame surface area
B transfer number (chemical enthalpy generation / enthalpy needed
for fuel evaporation)
c_{s} stoichiometric molar ratio of fuel to air
C_{P} constant-pressure heat capacity
d characteristic flow length scale or tube diameter
d_{f} droplet flame diameterd_{d} droplet
diameter
d_{do} droplet initial diameter
d_{o} jet exit diameter (round jets) or slot width (slot
jets)
D mass diffusivity
E overall activation energy of the heat-release reactions
f stoichiometric fuel to air mass ratio
g acceleration of gravity
g_{o} earth gravity
Gr_{d} Grashof number based on characteristic length
scale (d)=gd^{3}/n^{2}
h heat transfer coefficient in a cylindrical tube=16l/d^{2}
H radiative loss parameter for flame spread=t_{diff}/t_{rad}=a/U^{2}t_{rad
}K droplet burning rate constant
L_{f} flame length for gas-jet flame
L_{I} turbulence integral scale
L_{v} latent heat of vaporization of liquid fuel
Le Lewis number (a/D = thermal diffusivity / reactant mass
diffusivity)
mass burning rate
M fuel molecular weight
P pressurer radial coordinate
r_{f} flame radius
R scaled flame radius (Eq. (6)); radiation parameter (Eq. (11))
R_{g} gas constant
Re jet Reynolds number=U_{o}d_{o}/n
S_{f} flame spread rate over solid fuel bed
S_{L} premixed laminar burning velocity
S_{L,lim} burning velocity at the flammability limit
Sc Schmidt number=n/D
t_{chem} chemical time scale
t_{diff} thermal diffusion time scale=a/U^{2
}t_{drop} droplet flame time scale=d_{f}^{2}/a
t_{inv} inviscid buoyant transport time scale
t_{jet} residence time of nonpremixed jet flame
t_{rad} radiative loss time scale
t_{vis} viscous buoyant transport time scale
T temperature
T_{ad} adiabatic flame temperature
u’ turbulence intensity
U convection velocity
U_{y} local axial velocity in counterflow configuration
U_{o} jet exit velocityw gas-jet flame width
W solid fuel bed widthy axial coordinate
Y fuel mass faction
a thermal
diffusivity
b non-dimensional
activation energy=E/R_{g}T_{f
}c_{O2,lim}
minimum oxygen concentration supporting combustion
d flame thickness
g gas specific heat ratio
l thermal
conductivity
L radiative
heat loss per unit volume=4sa_{P}(T_{f}^{4} - T^{4})
n kinematic
viscosityr
density
t_{s} fuel bed half-thickness
(thin fuel) or thermal penetration depth (thick fuel)
s Stefan-Boltzman constant
S flame stretch rate
Subscripts
d droplet surface condition
f flame front condition
s solid fuel or solid surface condition
v solid or liquid fuel vaporization condition
ambient conditions
Time scale |
Stoichiometric flame |
Near-limit flame |
Chemistry (t_{chem}) | 0.00094 s |
0.25 s |
Buoyant, inviscid (t_{inv}) | 0.071 s |
0.071 s |
Buoyant, viscous (t_{vis}) | 0.012 s |
0.010 s |
Conduction to tube wall (t_{cond}) | 0.95 s |
1.4 s |
Radiation (t_{rad}) | 0.13 s |
0.41 s |
Table 1. Estimates of time scales for stoichiometric and near-limit hydrocarbon-air flames at 1 atm pressure.
Geometry |
Flow mechanism |
L_{f} |
t_{jet} |
Round-jet |
Momentum |
U_{o}d_{o}^{2}/D |
d_{o}^{2}/D |
Round-jet |
Buoyant |
U_{o}d_{o}^{2}/D |
(U_{o}d_{o}^{2}/gD)^{1/2} |
Slot-jet |
Momentum |
U_{o}d_{o}^{2}/D |
d_{o}^{2}/D |
Slot-jet |
Buoyant |
(U_{o}^{4}d_{o}^{4}/D^{2}g)^{1/3} |
(U_{o}^{2}d_{o}^{2}/g^{2}D)^{1/3} |
Table 2. Predicted scalings of flame heights (L_{f}) and residence times (t_{jet}) for nonpremixed round-jet and slot-jet flames under momentum-dominated and buoyancy-dominated conditions.
Figure 1. Predicted values of burning velocity and peak flame temperature in CH_{4 }- (0.21 O_{2 }+ 0.49 N_{2 }+ 0.30 CO_{2}) mixtures under adiabatic conditions, with optically-thin radiative losses, and including reabsorption effects [].
Figure 2. Characteristics of Self-Extinguishing Flames in CH_{4}-air mixtures at 1 atm for various mole percent CH_{4} and spark ignition energies []. The "x" symbols denotes extinction.
Figure 3. Measured and predicted extinction strain rates for strained premixed CH_{4}-air flames at µg [] showing dual-limit behavior, i.e., residence-time limited extinction at high strain rates (upper branch, "strong flames") and radiative loss extinction at low strain rates (lower branch, "weak flames").
Figure 4. Schematic diagram of a flame ball, illustrated for the case of fuel-limited combustion at the reaction zone. The oxygen profile is similar to the fuel profile except its concentration is non-zero in the interior of the ball. The combustion product profile is identical to the temperature profile except for a scale factor.
Figure 5. Predicted effect of heat loss on flame ball radius and stability properties [] showing radially unstable (small) flame ball solution, radially stable (large) flame ball solution, and three-dimensional instability for large flame balls.
Figure 6. Comparison of computed flame ball radii as a function of H_{2} mole fraction in H_{2}-air mixtures for 3 different H_{2}-O_{2} chemical mechanisms, along with preliminary results from the STS-83 and STS-94 space experiments [].
Figure 7. Computed flame ball radius as a function of the H_{2} mole fraction for steady flame balls in H_{2}-O_{2}-CO_{2} mixtures with H_{2}:O_{2}=1:2, for optically-thin CO_{2} radiation and with CO_{2} radiation artificially suppressed (optically-thick limit for CO_{2} radiation.) Preliminary experimental results from the STS-94 mission are also shown (filled circles) [].
Figure 8. Measured and predicted extinction strain rates for strained nonpremixed N_{2}-diluted CH_{4} vs. air counterflow flames at µg [] showing dual-limit response analogous to premixed flames (Fig. 3).
Figure 9. Measured flame lengths, normalized by jet diameter, as a function of the jet Reynolds number for nonpremixed CH_{4}-air jet flames at 1g and µg. Data is taken from a variety of sources and compiled in []. The data show a nearly linear relationship between flame length and Reynolds number, with generally longer flame lengths at µg, due to the differences between residence times under buoyancy-driven vs. momentum-driven residence times.
Figure 10. Measured flame widths, normalized by jet diameter, as a function of the jet Reynolds number for nonpremixed CH_{4}-air jet flames at 1g and µg. Data is taken from a variety of sources and compiled in []. The data show larger flame widths at µg due to the differences between accelerating flow at 1g vs. decelerating flow at µg. The data also show that, consistent with theoretical predictions, the width is nearly independent of Reynolds number for nonbuoyant conditions, except at low Re where boundary-layer approximations are invalid.
Figure 11. Measured flame heights for nonpremixed C_{3}H_{8}-air jet flames at 1g and µg [] showing transition to turbulence. Nozzle diameter is 0.8 mm. Note 1g flame lengths are shorter than µg flame lengths, even at very high Reynolds numbers.
Figure 12. Direct photographs of sooting n-C_{4}H_{10} non-premixed gas-jet flames at 1g (left) and µg (right) at Reš42, jet diameter 10 mm, showing evidence of thermophoresis-induced agglomeration at µg. Photographs courtesy of Prof. O. Fujita.
Figure 13. Effect of initial droplet diameter (d_{do}) on quasi-steady burning rate (K) for heptane droplets burning in air and an O_{2}-He atmosphere at µg, showing that K decreases with increasing d_{do}, apparently due to effect of increase accumulation of soot and gas-phase radiant species for larger d_{do} [].
Figure 14. Direct photographs of heptane droplets burning in air at µg showing spherically-symmetric combustion (left) and a soot "tail" formed by weak convection effects (right) []. Photographs courtesy of Prof. T. Avedisian.
1g |
µg |
Figure 15. Direct photographs of candle flames at 1g and µg, showing impact of buoyant flow on flame shape [].
Figure 16. Minimum mole percent O_{2} in N_{2} supporting flame spread over a thin solid fuel bed, as a function of the opposed flow velocity (U) [], showing dual-limit behavior, i.e., residence-time limited extinction at high U and radiative loss extinction at low U.
Figure 17. Flame spread rate over a thin solid fuel bed as a function of the opposed flow velocity (U), for three values of the mole percent O_{2} [], showing dual-limit response. Note that the infinite-rate kinetics prediction [, ] that the spread rate is independent of U, is only satisfied at O_{2} mole fractions higher than that in air.
Figure 18. Flame spread rate over a thin solid fuel bed at 1g (downward spread) and µg as a function of O_{2} mole fraction [], showing atypical behavior in O_{2}-CO_{2} and O_{2}-SF_{6} atmospheres where the spread rates are higher and the minimum O_{2} model fraction supporting combustion is lower at µg.
Figure 19. Fingering patterns observed in smoldering flame spread over a thin paper fuel sample in µg []. Flaming combustion was inhibited by soaking the fuel sample in potassium acetate. An imposed convective velocity of 0.065 m/s flows from right to left. Grid pattern scale is 10 mm by 10 mm. Photograph courtesy of Dr. S. Olson.
Figure 20. Measured (thick lines) and computed (thin lines) flame position vs. time for flame spread over a 1-butanol pool 20 mm wide and 25 mm deep []. The 10, 20 and 30 notations refer to the opposed flow velocities (U) in cm/s. Both computed results are for µg conditions, U=30 cm/s, either with or without hot gas expansion. The comparison of predicted and measured results suggest a very strong influence of expansion which is much less effective in the experiment because of the relaxation of expansion in the transverse dimension, a factor not captured within the two-dimensional model.
Figure 21. Predicted effect of radiative heat transport coefficient (R) on droplet burning rate constant referenced to the value without radiative transport, showing importance of absorption of radiation at the droplet surface on the resulting burning rate. B=3 and B=8.5 are characteristic of methanol and heptane, respectively, burning in air.