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

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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 pressing of which is suggested to be radiative reabsorption effects in weakly burning flames.

Invited paper submitted to the 27th International Symposium on Combustion.

1. INTRODUCTION

It is well known that gravity influences many types of combustion processes, particularly due to the effects of buoyant convection on the rates of transport of thermal energy and reactants to/from the chemical reaction zones. To gain a better understanding of these influences, many experimental and theoretical studies of combustion at reduced gravity, or "microgravity" (µg) conditions have been conducted in recent years, and remarkable progress has been made in applying the µg environment to further our understanding of combustion processes. As indicated in recent reviews [, , , ], these studies are motivated by the need to assess the hazards associated with fires in spacecraft as well as obtaining information that enables a better understanding of combustion processes at earth gravity (1g) through the elimination of one of the major complicating factors.

The purpose of this paper is to show how new understanding of combustion processes has been obtained and can continue to be obtained through µg research, rather than to provide a comprehensive review of this rapidly growing and changing field. Toward this objective, first a comparison of time scales for various chemical and transport processes in flames, including buoyancy-induced transport, is used to motivate µg research. Next, examples from several combustion disciplines are used to illustrate the improvements in understanding already obtained. Emphasis is placed on examples where µg research has led to findings that were unexpected and not readily predicted based on the state of knowledge prior to the µg studies. These findings are then summarized and recommendations for future research directions are given.

2. COMPARISON OF TIME SCALES FOR PREMIXED-GAS COMBUSTION

To estimate under what conditions gravity can affect flames, and thus µg experiments might be enlightening, we compare the time scales for chemical reaction (tchem), buoyant convection in inviscid flow (tinv), buoyant convection in viscous flow (tvis), heat loss to walls via conduction (tcond) and radiant heat loss from the burned gases (trad). As a first example, premixed laminar flames are considered because they are probably the simplest type of flame. Subsequent sections will introduce time scales appropriate to other types of flames.

Premixed flame structure is determined by a balance between chemical reaction and diffusion of heat and reactants over a zone of thickness d, thus tchem Å d/SL (see Nomenclature). d in turn can be estimated as a/SL, thus tchem Å a/SL2. A buoyant transport time scale can be estimated as d/U, where d is a characteristic flow length scale and U is the velocity induced by buoyancy. U in turn can be estimated as (gd(Dr/r))1/2. Since Dr/r Å 1 for flames, the buoyant time scale for inviscid flow can be estimated as d/(gd)1/2 = (d/g)1/2. For flames propagating in tubes or stabilized on a burner, d would be the tube or burner rim diameter. For viscous flow, d cannot be independently specified; instead d Å n/U. Combining the relations d Å n/U and U Å (gd)1/2 leads to U Å (gn)1/3 and thus tvis Å d/U Å (n/U)/U Å (n/g2)1/3. In general a Prandtl number should appear in this estimate, but for gases Pr Å 1. The conductive loss time scale, tcond, can be estimated as the flame temperature divided by the rate of cooling due to conductive losses, i.e., tcond = Tf/(dT/dt) Å Tf/(rCPh(TfT_)). Since h Å 16l/d2, tcond Å d2/16a, where we have assumed (TfT_)/Tf Å 1, which is reasonable for practical flames. Similarly, trad for optically-thin gases can be estimated as Tf/(dT/dt) Å Tf(L/rCP), thus trad Å {g/(g1)}{P/4saP(Tf4 - T_4)}.

Two sets of time scales are now generated, one for near-stoichiometric hydrocarbon-air flames and the other for very weakly burning lean hydrocarbon-air flames near the flammability limits, both at P = 1 atm. For the former case, SL = 0.40 m/s, Tf = 2200K, a = n = 1.5 x 10-4 m2/s and aP = .56 m-1. For the latter case, SL = 0.02 m/s, Tf = 1500K, a = n = 1.0 x 10-4 m2/s and aP = .83 m-1. For both cases g = 9.8 m/s2, g = 1.35, T_ = 300K and d = 0.05 m (a typical diameter for burner or flame tube experiments.) The estimated time scales for these flames are shown in Table 1.

Several observations can be made based on these simple estimates:

(1) Buoyant convection is unimportant for near-stoichiometric flames because both tvis >> tchem and tinv >> tchem.

(2) Buoyant convection strongly influences near-limit flames at 1g because in this case both tvis ² tchem and tinv ² tchem.

(3) Radiation effects are unimportant at 1g because buoyant convection is a much faster process (tvis << trad and tinv << trad).

(4) Radiation effects will dominate the behavior of flames with very low SL since trad Å tchem for the near-limit flame, but these effects can only be observed at low gravity because of (3)

(5) For the representative conditions, the apparatus size, e.g., the tube diameter, must be larger than about 0.03 m if one is to observe radiation-induced extinction, otherwise conduction losses to the tube wall will exceed radiative losses.

(6) Many phenomena associated with radiative loss effects can be studied in drop tower experiments, with test durations of 2 to 10 s, since these times are typically larger than trad. As a result, combustion science has probably benefited more from the utilization of short-duration drop tower experiments than any other microgravity science discipline. The main reason for this is because of the large a and Dr/r of combustion processes as compared to most other fluid systems.

(7) Since tinv ~ g1/2 and tvis ~ g1/3, aircraft-based µg experiments at g Å 10-2 go, may not provide sufficient reduction in buoyancy effects to observe radiative effects. In fact, in some cases [] results of aircraft-based µg experiments have led to specification of less than optimal test parameters for space experiments.

(8) Since tvis ~ n1/3 ~ P-1/3 and trad ~ r/L ~ P1/P1 ~ P0, tvis/trad ~ P-1/3. Thus, the radiative time scale is similar at all pressures, but at higher pressures buoyancy effects interfere more strongly with radiative effects.

(9) A Reynolds number (Red) for buoyant flow can be defined as Ud/n ~ (gd)1/2d/n = (gd3/n2)1/2 = Grd1/2, where Grd is a Grashof number. For Red of the order 103 or larger, thus Grd > 106, buoyant flow at 1g will necessarily be turbulent, thus it is virtually impossible to study steady laminar flames in large systems at 1g. For the properties representative of hydrocarbons burning in air at 1 atm used in the time-scale estimates above, Grd = 106 corresponds to d Å 0.1 m.

The implications of these observations are discussed in the following sections. Essentially any combustion process where tinv or tvis is comparable to tchem or trad is likely to be affected by gravity and may be worthy of µg investigation. In particular, the importance of extinction limits due to radiative losses (tchem Å trad) in µg experiments cannot be overstressed because it leads to a new type of limit at low flow velocities and long residence times in addition to the high-velocity, short residence time limits that are well known from earth-based experiments. This dual-limit behavior permeates many of the phenomena discussed below for both premixed and non-premixed flames.

3. PREMIXED GAS FLAMES

3.1 Flammability limits

A key prediction of the previous section is that gravity effects will be significant only for mixtures with low SL. Low SL implies mixtures highly diluted with excess fuel, oxidant, or inert gas, but sufficient dilution will lead to a flammability limit. Thus, gravity effects may be expected to have a significant influence on flammability limits and near-limit behavior. It is well known [] that the limits are different for flame propagation in tubes in the upward, downward and horizontal directions, indicating buoyancy plays an important role.

For upward propagation, Levy [] showed that at the flammability limit the rise speed of the flame is the same as the rate of rise of an inviscid hot gas bubble Å 0.33(gd)1/2. This relation has been verified for a wide range of tube diameters and mixtures []. Buckmaster and Mikolaitis [] showed how this minimum rise speed leads to hydrodynamic strain at the tip of flame which causes extinguishment in mixtures with sufficiently low SL. The burning velocity corresponding to the limit mixture (SL,lim) is given by, after temperature-averaging the transport properties:

(1).

Except for the Lewis number effect, the functional form of Eq. (1), SL,lim ~ (ga2/d)1/4, can be obtained by setting tinv = tchem.

For downward propagation in tubes, high-g centrifuge experiments [] indicate that SL,lim ~ g1/3. This result is apparently independent of Le, which is reasonable since downward-propagating near-limit flames in tubes are nearly flat and unstrained. Experiments [] and numerical simulations [] suggest that extinction results from the sinking of a layer of cooling burned gas near the walls overtaking the flame front and blocking the flame front from the fresh unburned gases, although the g1/3 scaling was not tested in [11, 12]. The g1/3 scaling can be obtained by setting tchem = tvis, leading to

SL,lim ~ (ga)1/3 (2).

Experiments [8] on flames in tubes using a wide range of diluent gases and pressures have confirmed the a1/3 scaling and that the result is independent of tube size, with a best-fit proportionality constant of 1.3, which tends to support the proposed mechanism.

Both upward and downward limit mechanisms indicate that as g Æ 0, SL,lim Æ 0, implying that arbitrarily weak mixtures could burn, albeit very slowly. On the other hand, heat losses due to conduction or radiation would prevent arbitrarily weak mixtures from burning even at g = 0. Theories of flammability limits due to heat losses [, , ] indicate that a minimum Tf exists below which flame propagation cannot occur because the chemical reaction rate is generally a much stronger function of temperature than the heat loss rate, i.e., exponential vs. algebraic. Consequently, because dilution decreases Tf, dilution increases the impact of heat losses and leads to a flammability limit. For conductive losses, equating tchem and tcond leads to SL,lim ~ a/d or

Pelim _ SL,limd/a = constant (3),

with experiments [8, ] and computations [] indicating Pelim Å 40. For radiative losses an estimate of SL,lim is given as [, ]

(4),

the scaling of which can be obtained by equating tchem and trad. For the lean-limit CH4-air mixtures at 1 atm, Eq. (4) yields SL,lim Å 0.023 m/s, which is practically identical to the value predicted using detailed numerical models [, ]. Such small values of SL,lim are not usually observed at 1g because of buoyant convection (tinv < trad and tvis < trad) - Eqs. (1) and (2) yield SL,lim Å 0.033 and 0.078 m/s for upward and downward propagation, respectively. At reduced gravity, however, the predictions of Eq. (4) compare favorably to the results of µg experiments performed in a large combustion vessel [19, ] using a wide range of pressures, fuels, and inert gases. Similar results were obtained for CH4-air at 1 atm in a tube with d = 0.05 m [], though in this case conductive losses may have been significant since Eq. (3) predicts SL,lim Å 0.016 m/s which is close to the value for radiative losses. Thus radiative heat loss appears to be the cause of flammability limits when extrinsic heat losses, e.g., due to conduction, buoyant convection, etc., are eliminated. This is one instance where µg experiments have enabled observation of a phenomenon that can probably never be observed at 1g.

These estimates of radiative effects are valid only if the gases are "optically thin," that is, no reabsorption of emitted radiation. This may not be true in systems of large size, at high pressures, or in mixtures with high concentrations of strongly absorbing material. With this motivation, µg experiments [] were conducted in lean CH4-air mixtures seeded with inert solid particles to increase aP. Since solids emit and absorb as black- or gray-bodies whereas gases radiate in narrow spectral bands, particle-seeded gases can emit and absorb more radiation than a particle-free gas. Consistent with theoretical predictions [], measured propagation rates, peak pressures, maximum rates of pressure rise, and rates of thermal decay in the burned gases showed that at low particle loadings, the particles act to increase the radiative loss from the gases, whereas at higher loadings, reabsorption of emitted radiation becomes significant, which in turn acts to decrease the net radiative loss and augment conductive heat transport.

Even for gases, computations [] using a detailed statistical narrow-band radiation model show that flammability limits may be extended remarkably by considering reabsorption, for example in CH4-O2-N2 mixtures seeded with CO2 (Fig. 1). Note that the equivalence ratio at the flammability limit is 0.68 for optically-thin conditions vs. 0.44 with reabsorption. The latter value is even lower than the computed value for CH4-air despite the fact that CP is much higher for CO2 than N2, and thus the adiabatic Tf is lower with CO2. Still, there are two radiative loss mechanisms that lead to flammability limits even with reabsorption. One is due to the difference in composition between reactants and products; if a product of combustion that radiates significantly is not present in the reactants (e.g., H2O), radiation from this species that is emitted upstream cannot be reabsorbed by the reactants unless, by coincidence, its spectrum overlaps completely with a reactant. The second is that the emission spectra of most molecules are broader at flame temperatures than ambient temperature, thus some radiation emitted near the flame front cannot be absorbed by the reactants even if they are seeded with that molecule. Via both mechanisms some net upstream heat loss due to radiation will always occur, leading to extinction of sufficiently weak mixtures. These results suggest that fundamental (domain- and gravity-independent) flammability limits due to radiative heat loss may exist at µg, but these limits are strongly dependent on the emission-absorption spectra of the reactant and product gases and their temperature dependence, and cannot be predicted using gray-gas or optically-thin model parameters. In fact because of the spectral nature of gas radiation, very significant reabsorption effects were found in domains as small as 0.01 m even for mixtures with a Planck mean absorption length _ aP-1 Å 0.24 m.

4. STRETCHED AND CURVED FLAMES

Premixed flames are generally not flat and steady nor do they commonly propagate into a quiescent flow. Consequently, flames are usually subject to "flame stretch" (S) defined by [18]

(5).

Flame stretch affects flame propagation rates and extinction conditions, as discussed in numerous reviews, e.g., [18, ]. For flames at 1g, buoyancy imposes a flame stretch comparable to tinv-1 or tvis-1. In the absence of gravity, weak flame stretch effects that are insignificant at 1g may become dominant. One such example is expanding spherical flames for which, according to Eq. (5),

(6),

For mixtures with Le < 1, positive stretch increases the flame temperature because the increase in the rate of chemical enthalpy diffusion to the flame front in the form of scarce reactant is greater than the increase in the rate of thermal enthalpy loss. Since heat release reactions have high activation energies, small changes in temperature lead to large changes in reaction rate at the flame front and thus the propagation rate. Using Eq. (6), an evolution equation describing the propagation rate of an expanding spherical flame in the presence of heat loss effects was obtained []:

(7),

where S is the propagation speed divided by SL(r_/rf), R is the flame radius scaled by bdI(Le, e), I(Le, e) is a scaling function that is positive for Le < 1 and negative for Le > 1 (but of course the physical radius r* is always positive) and Q _ {bL(Tf)d2}/{l(Tf-T_)} is the scaled heat loss. The terms in Eq. (7) represent unsteadiness, heat release, curvature-induced stretch and heat loss, respectively. For steady planar flames, Eq. (7) reduces to S2lnS2 = -Q, which has a turning point at a maximum value of Q = 1/e = 0.3678... at S = e-1/2. This condition corresponds to the value of SL,lim given by Eq. (4). For Le < 1 the effect of curvature (2S/R) works opposite that of heat loss (Q), allowing mixtures that are non-flammable as plane flames (Q > 1/e) to exhibit expanding spherically flames until r* grows too large and thus the enhancement of the heat release rate due to curvature is too small. For mixtures just outside the limit, the extinction radius may be very large. Such behavior, termed "self-extinguishing flames" (SEFs), has been observed experimentally [19, 22] in mixtures near the µg flammability limit when Le < 1 (Fig. 2) but not so low that diffusive-thermal instabilities fragment the flame into cells or flame balls discussed below. Equation (7) also predicts, in accord with experimental observations [], that SEFs cannot occur for mixtures with Le > 1 (thus R < 0 in Eq. (7)) because in this case both curvature and heat loss weaken the flame.

Two features of the experimental observations that are not predicted by Eq. (7) is that a narrow range of mixtures can exhibit both SEFs and normal flames and that the energy release before extinguishment can be orders of magnitude greater than ignition energy input. Apparently the lack of buoyant convection preserves the spherical symmetry of these flames to the point that curvature effects and small variations in the spark ignition energy can influence the flame well after it has propagated away from the ignition source. Such behavior is predicted by computations [] that are not subject to the scaling limitations of the activation energy asymptotics used to derive Eq. (7). Such calculations also show that for small initial r*, all mixtures exhibit extinguishment, which corresponds to the well-known non-ignition behavior []. This indicates that in mixtures capable of exhibiting SEFs, flames extinguish at sufficiently small curvature due to high stretch rates and at large curvature due to radiative losses. This dual-limit behavior is also observed in many of the other flame phenomena described and in later sections.

The effects of hydrodynamic strain on flames has been studied for many years because of the need to understand how turbulence-induced strain affects flame fronts in practical combustion devices such as automotive engines. The most common apparatus for studying strained premixed flames is the counterflow round-jet configuration. The equilibrium location of the flame front is at the axial location where the local axial velocity U is equal to SL for the given S = dU/dy, where y is the axial distance from the stagnation plane. Thus as S increases, U increases, consequently the flame moves closer to the stagnation plane and the volume of burned gas decreases. As with curvature-induced strain, for Le less than/greater than unity, in the counterflow configuration SL is increased/decreased by moderate strain, but for all Le, sufficiently large S extinguishes the flame [9, ]. Thus, for Le < 1 the flame response to strain is non-monotonic since increasing S increases SL at low S but decreases SL at high S. The combination of the non-monotonic flame response to S at low Le, plus the reduced volume of burned gas (thus reduced radiative heat loss) at larger S may lead to a variety of extinction behaviors for counterflow flames depending on the relative magnitudes of trad and S-1. As a result, recent µg experiments [, ] on strained flames in low-Le mixtures (Fig. 3) have revealed extinction behavior somewhat reminiscent of SEFs and non-ignitions in spherical flames. For high S, the short residence time of reactants within the flame front (~S-1) causes extinguishment (S-1 Å tchem) (the "normal flame" branch, which is analogous to non-ignition behavior of highly curved flames) whereas for low S the residence time and the flame spacing are large and thus the impact of heat losses is significant (trad Å tchem) plus the increase in flame strength due to Le effects is weak, so radiant heat losses extinguish the flame (the "weak flame" branch, analogous to SEFs). The optimal S that produces the maximum increase in the flammable range (13 s-1) corresponds to a time scale of 0.08 s, which is less than tvis or tinv, 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 the loss rates are modeled well. In contrast, the computed limit composition is leaner that the experimental one, suggesting that the chemical mechanism used is not accurate for these weak mixtures.

Theory [] and numerical simulations [] show that strained premixed flames with radiative loss and Lewis number effects actually exhibit a more complex set of behaviors that those described above. For example there is another branch of solutions, called the Far-Standing Weakly Stretched Flame (FSWSF) in which the flame front is far from the stagnation plane and thus has a large burned-gas region. Only the FSWSF behavior can be extrapolated to zero stretch rate to obtain the planar unstretched flammability limit [35, 36]. Additionally, there are jump limits between different modes which are not readily explained based on the simple physical principles outlined here. It is uncertain whether any of these solutions are physically observable, since they have not been identified experimentally and stability analyses have not yet been performed.

Due to the decrease of radiant loss at moderate S, the extension of the flammability limit also occurs for mixtures with Le > 1, though for sufficiently high Le (e.g., C3H8-air) no C-shaped response or flammability limit extension occurs []. No analogous effect occurs for SEFs because for the spherically expanding flame there is no mechanism by which flame stretch would reduce the radiative loss.

3.3 Flame balls

Section 3.2 discussed examples of stretched flames, where heat and mass transport are influenced by the convective environment, and behavior resulting from differences in the convective environment between 1g and µg conditions. This section discusses a phenomenon for which convection plays no role whatsoever while the curvature of the flame front is the dominant influence.

Over 50 years ago, Zeldovich [] showed that for every combustible mixture the steady heat and mass conservation equations admit a solution corresponding to a stationary spherical flame, characterized by a flame radius (r*) just as the same governing equations in planar geometry admit a steadily propagating flame as a solution, characterized by SL. In this "flame ball" structure, fuel and oxygen diffuse from the surrounding gases inward to the reaction zone while heat and combustion products diffuse outward (Fig. 4). The fluid velocity is identically zero everywhere as required by mass conservation. The temperature and species mass fraction profiles for the solutions of Laplace's equation in spherical geometry are of the form c1 + c2/r, where c1 and c2 are constants. Corresponding solutions in planar and cylindrical geometry do not exist because the solution forms c1 + c2r and c1 + c2ln(r), respectively, are unbounded as r Æ _. By combining the temperature and scarce reactant mass fraction conservation equations, Zeldovich showed that for an adiabatic flame ball, the temperature at the flame ball surface (T*) is given by

(8),

thus c1 = T_ and c2 = (T* - T_)r*. Note that the temperature rise is proportional to Le-1, thus for mixtures with low Le, flame balls can have flame temperatures much higher than Tf, and thus can exist in mixtures that could not support plane flames; in the experiments described below mixtures with values of Tf as low as 465K have been found to exhibit flame balls. Zeldovich [38] also showed, as was confirmed much later by more rigorous analyses [, ], that flame ball solutions are unstable, and thus probably would not be physically observable, though the solutions are related to the minimum kernel size required for flame ignition [39].

Forty years after Zeldovich's analysis, seemingly stable flame balls were accidentally discovered in drop-tower experiments in H2-air mixtures [] and later confirmed in aircraft-based µg experiments employing a variety of low-Le mixtures []. The µg environment was needed to obtain spherical symmetry and to avoid buoyancy-induced extinction of the flame balls. The following sequence of phenomena was observed as the mixtures were progressively diluted. For mixtures sufficiently far from flammability limits, an expanding spherical front composed of many individual cells was observed that regularly subdivided to maintain a nearly constant cell spacing. For more dilute mixtures closer to the flammability limits, the cells formed initially did not split but instead closed up upon themselves to form stable spherical flame structures (the flame balls). For still more dilute mixtures all flame balls eventually extinguished. It was inferred that stable, stationary flame balls would probably occur in all combustible mixtures with low Le for mixtures close to the extinction limits, however, the short duration of drop tower experiments and the substantial fluctuations in the acceleration level in aircraft-based µg experiments precluded definite conclusions. Recent experiments performed on the STS-83 and STS-94 Space Shuttle missions [] (Fig. 5) confirmed that flame balls are stable for at least 500 seconds, the test duration in these cases.

Zeldovich [38] noted the possibility of heat losses stabilizing flame balls; consequently, after their experimental observation the effects of volumetric radiative losses on flame balls was analyzed [, ]. When heat losses are not too strong, two stationary flame ball radii are predicted (Fig. 6), a "large" flame ball that is strongly affected by heat loss and a "small" flame ball that is nearly adiabatic, and when the losses are sufficiently strong no solutions exist, indicating a flammability limit. (As a result, there are at least four steady solutions to the low Mach number conservation equations for non-adiabatic flames, namely the two solutions to the equation S2ln(S2) = -Q for planar flames and the two flame ball solutions.) Stability analyses [44, 45] predict that all small flame balls are unstable to radial disturbances, and large flame balls with weak heat loss effects, i.e., away from the flammability limits, are unstable to three-dimensional disturbances, which is consistent with the observation of splitting cellular flames in mixtures away from limits. A portion of the large flame branch close to the extinction limits is stable to both types of disturbances, which is consistent with the experimental observations. It has also been predicted [] that stable flame balls can only exist for mixtures with Le less than a critical value less than unity, which explains why flame balls are not observed for mixtures with Le close to unity (e.g., CH4-air) or larger than unity (e.g., C3H8-air), even for near-limit mixtures at µg.

Numerical predictions of non-adiabatic flame balls employing detailed chemistry, diffusion and radiation models [, ] are qualitatively consistent with the experimental results and theoretical predictions discussed above. Still, quantitative agreement has been elusive, for at least two reasons. First, it has been found [48] that flame ball properties are very sensitive to the rate of the 3-body recombination step H+O2+H2OÆHO2+H2O (Fig. 7) which vary by a factor of more than two between different published H2-O2 reaction mechanisms. Similar behavior has been noted with respect to the burning velocities of near-limit propagating H2-air flames []. The second reason is reabsorption of emitted radiation, which is an important effect in mixtures diluted with radiatively-active gases such as CO2 and SF6. An upper bound on reabsorption effects (aP,diluentÆ_) can be obtained by neglecting diluent radiation entirely because as aP,diluentÆ_ there is no radiative loss from the diluent and furthermore the "radiative conductivity" lR _ 16sT3/3aP is negligible, thus there is no additional heat transport due to radiative transfer. An example of this is shown in Fig. 8 for H2-O2-CO2 mixtures. The agreement between predicted and measured flame ball radii is much better when diluent radiation is neglected and the experimental flammability limit composition is bracketed by numerical results with and without diluent radiation. These observations strongly suggest that radiative reabsorption effects are needed for accurate numerical simulation in these cases.

A key factor in the differences between the behavior of propagating flames and flame balls is that propagating flames have a convective-diffusive zone where the temperature and concentration approach their ambient values in proportion to e-r/d, whereas flame balls have a purely diffusive zone where the approach is proportional to 1/r. Plane flames respond on the relatively short time scale tchem = d2/a discussed in section 2. The very gradual 1/r profile for flame balls leads to properties that are dominated by the far-field conditions on a length scale br*, where in this case b _ E/RgT*, and thus a diffusion time scale (br*)2/a [44, 45], which is typically on the order of 100 s. These time and length scales are particularly relevant to stability and extinction limits since there is a similar time scale for diffusion of radiant combustion products from the flame front to the far-field and a large volume of gas ~ (br*)3 where radiative loss can affect flame ball properties. Such large time and length scales are confirmed by space experiments [43] and numerical simulations [47, 48]. Analogous behavior will be discussed later in the context of droplet and candle flames, which are quasi-spherical and diffusion dominated as seen by the far-field.

4. GASEOUS NON-PREMIXED FLAMES

4.1 Stretched flames

Nonpremixed flames, where the fuel and oxidant are initially separated before combustion, are subject to stretch effects but with a number of differences from premixed flames. The most significant difference is that for nonpremixed flames the flame position is at the location of stoichiometric mixture fraction, which is dictated by mixing considerations, rather than being determined by the balance between SL and local flow velocity as in premixed flames. Consequently, nonpremixed flames have considerably less freedom of movement. Also, premixed flames have a characteristic thickness d ~ a/SL that is unrelated to the flow environment, whereas for nonpremixed flames the only characteristic thickness is the diffusion length scale d ~ (a/S)1/2. With a fixed flame front location and d increasing monotonically with decreasing S, nonpremixed flames with radiative loss exhibit only a simple C-shaped response to strain, with a short residence time extinction branch and a radiative loss extinction branch rather than the complicated responses found for premixed flames [37]. Recent µg experiments and computations [] have confirmed these expectations (Fig. 9) and show that the only significant difference between flame structure near the stretch extinction limit and the radiative extinction limit is the flame thickness, which is quite unlike the case for premixed flames [34, 35, 36, 37].

Note that the experimental results suggest there is a value of S below which no flames can be produced, whereas the model predicts flames can be produced 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 gas the surrounding reactant streams. This would induce a loss rate of the order tcond-1 Å 2.9 s-1, where if d is taken to be the burner nozzle spacing (25 mm), or 7.1 s-1 if d is taken to be the burner diameter (16 mm), either of which is roughly consistent with the minimum value of S seen in Fig. 9. Thus, it should be recognized that apparatuses which are large enough to study flames at 1g without substantial conductive loss, where maximum length scales are of the order (atvis)1/2, may be insufficient for the weaker flames that can be studied at µg, where length scales as large as (atrad)1/2 can be anticipated.

4.2 Laminar gas-jet flames

One of the simplest non-premixed flames to produce is a jet of fuel issuing into an oxidizing environment. Because of this simplicity, many 1g and µg experiments on gas-jet nonpremixed flames have been conducted. Analyses performed by Jost [] and later extended by Roper [] show that the flame height (Lf) and residence time from the jet exit to the flame tip (tjet) can be estimated by determining the height y at which the transverse diffusion time ~ d(y)2/D, where d(y) is the stream tube width, is equal to the convection time ~ U(y)/y, where U(y) is the axial velocity. U(y) is assumed constant and equal to the jet exit velocity (Uo) when buoyancy effects are negligible (momentum-controlled jets), whereas U(y) ~ (gy)1/2 when buoyancy effects dominate. In either case mass continuity requires d(y)2U(y) = do2Uo = constant for round jets or d(y)U(y) = doUo = constant for slot-jets, where do is the jet diameter (round jets) or width (slot jets). Equating the transverse diffusion time to the axial convection time leads to the predicted scalings for Lf and tjet given in Table 2. Transition from buoyancy-controlled to momentum-controlled conditions occurs at the value of Uo where the time scale for the former exceeds that of the latter, which corresponds to Uo > gdo2/D for either round-jets or slot-jets. Clearly for small Uo or large do buoyancy-dominated conditions apply at 1g, and under these conditions significant changes in flame structure may occur upon transition from 1g to µg. The importance of these scalings, particularly with respect to tjet, will become apparent when radiative loss effects and soot formation (section 4.4) are discussed.

The scalings for momentum-dominated flames presume that U is constant, which is reasonable for Burke-Schmann flames which have an annular co-flow of oxidizing gas with the same velocity as the fuel jet. For nonbuoyant jet flames without co-flow, the jet spreads and decelerates. Using Schlicting's [] classical boundary-layer analysis which predicts a centerline velocity profile of U = d02U02/8y, Haggard and Cochran [] showed that this results in a flame length given by

(9).

This result can be used to infer a residence time on the jet axis given by

(10),

Since Schmidt number (Sc) is close to unity for most gas mixtures, the scalings of Lf and tjet with Uo, do and D is the same for buoyant flames and nonbuoyant jet flames with or without viscous flow considerations. Fundamentally this is because of the similarity between diffusion of mass and momentum when Sc Å 1.

The early µg experiments [54, ] show that µg round-jet flame lengths are very similar to 1g flame lengths, but slightly longer at µg for fuels with larger D and cs, e.g. CH4, and slightly shorter at µg for fuels with smaller D and cs, e.g., propylene. The functional form of Eq. (9) was found to be in good agreement with the experimental observations at µg [54]. Other experiments [] report somewhat longer flame lengths at µg, even for propane fuel. In any case, the differences are not large. A summary of measurements for CH4 flames is given in Fig. 10 [], which shows Lf/do ~ Re _ Uodo/n ~ Uodo/D for both buoyant and nonbuoyant flames, as the aforementioned scalings predict.

All µg studies have shown larger flame widths (w) at µg than 1g due to the lower U and longer residence time. Of course, w depends critically on whether U is increasing (buoyant flow), constant (nonbuoyant Burke-Schumann flames) or decelerating (nonbuoyant jets). As a result it is much more difficult to predict flame width than flame length [2]. Also, the flame width will be affected by the lower temperatures of µg flames (see below) since D ~ T1.7. As expected, the difference between 1g and µg flame widths decreases as Re (thus Uo) increases (Fig. 11) [57]). The nonbuoyant flame widths are seen to increase slightly as Re increases, whereas all of the aforementioned models predict a self-similar flame shape with no effect of Re on w/do. This may be an effect of axial diffusion, not considered in these models, which increases the rate of mixing of fuel and oxygen over that with radial diffusion alone. The fact that w/do is lowest for the lowest values of Re (Å20) in Fig. 11, where axial diffusion is most likely to be significant, but asymptotes to a nearly fixed value at high Re, supports this suggestion.

The µg flames have a more reddish color compared to the yellow color of the 1g flames [54, 55, 56, 57], indicating a substantially lower blackbody temperature of the soot radiation and presumably a lower maximum flame temperature (though few actual measurements have been made.) Almost certainly this is because tjet is higher at µg and thus the impact of radiative heat loss ~ tjet/trad is greater. Recent space experiments using two-color soot pyrometry [5] indicate maximum temperatures of about 1800 - 1900K for ethylene flames with Re = 140, though no corresponding 1g measurements were reported. Consistent with the radiative loss hypothesis, the fraction of total fuel enthalpy emitted as radiative loss is far greater at µg than 1g. The measurements presented by Bahadori et al. [] indicate surprisingly consistent and remarkably high radiative loss fractions of 0.45 - 0.60 for C3H8 flames at µg at various pressures, O2 mole fractions and flow rates, as compared to 0.07 - 0.09 at 1g. Similar results were found for CH4 flames. Values for C2H4 flames in recent space experiments [5] are 0.56 and 0.60 at 0.5 atm and 1.0 atm, respectively. Thus, it is apparent that the difference in residence times at 1g and µg lead to considerably different properties even for flames of the nearly same overall length.

Flickering is a common characteristic of non-premixed gas-jet flames at 1g that is not observed at µg [54, 55, 56]. Flickering is typically thought to be due to a hydrodynamic instability associated with the growth of the thermal boundary layer of thickness dL adjacent to the flame. The corresponding flicker frequency (w) can be estimated as Lf/dL. Classical boundary-layer theory [53] predicts dL ~ LfGrLf1/4, where GrLf _ gLf3/n2, thus w ~ (g3Lf/n2)1/4, which indicates very little effect of flame height on w and of course predicts no flickering at µg. A boundary-layer analysis for a semi-infinite fuel layer adjacent to a semi-infinite oxidizing layer [] supports this scaling relation with a proportionality constant of 0.13 when n is evaluated at Tf, leading to a prediction w = 17 Hz for Lf = 50 mm, which is close to experimental observations by Durox et al. []. These authors also show that, approximately, Lf ~ g-1/3, thus according to the proposed scaling law w ~ g2/3, which is close to their experimental results w ~ g0.660.75. On the other hand, recent experiments by Maxworthy [] suggest the instability mechanism is a globally excited oscillation forced by a region of absolutely unstable flow near the base of the flame, rather than a convective instability as described above.

4.3 Turbulent flames

In the case of turbulent non-premixed jet flames, the effective turbulent diffusivity is not a fixed value, but rather proportional to the product of the turbulence intensity (u') and the integral length scale (LI). Since one can expect u' ~ Uo and LI ~ do, for round jets Lf ~ Uodo2/u'LI ~ do - i.e., Lf is independent of Uo. This prediction is supported by classical experiments at 1g [] as well as recent µg experiments [] (Fig. 12). An interesting feature of Fig. 12 is that ratio Lf(µg)/Lf(1g) stays practically constant at even after transition to turbulent conditions (high Uo and thus high Re), which is consistent with the simple scaling predictions given above. Note also that the maximum value of Uo or Re at which the flame can exist (the "blow-off" limit) is different at 1g and µg. This is somewhat surprising considering that blow-off conditions are typically thought to be controlled by behavior near the flame base [], where buoyancy effects are not expected to be significant. This suggests that blow-off of turbulent jet flames is partially affected by the velocities induced by the buoyant plume well above the jet exit, even at very high Uo. Intuitively one would expect the 1g flames to blow off at lower Uo since the buoyant flow would induce higher "effective" Uo and higher u' and indeed Fig. 12 shows that this is the case. These results show that buoyancy effects are quite ubiquitous, and indicates the value of µg studies, even under some conditions commonly thought not to be strongly affected by buoyant convection.

4.4 Soot formation processes

It is well known that under appropriate circumstances, hydrocarbon-air flames will emit solid particles, generically called soot, composed mostly of carbon. The most important practical application of studies of soot formation is to non-premixed flames in turbulent flows, for example in Diesel engines, gas turbines and industrial furnaces, but it is impractical to study such processes directly because the unsteadiness and distortion of turbulent non-premixed flames limits the available residence times and spatial resolution. Bilger [] showed that with suitable assumptions, gas-phase scalar properties (e.g., temperature, species mole fractions) within turbulent non-premixed flames can be correlated solely as functions of the local mixture fraction Z, defined as the mass fraction of material that originated in the fuel stream. It has been suggested [] that soot properties in turbulent flames may also be assessed via the "laminar flamelet" concept, and some validity of the concept has been established (Fig. 13), but buoyant laminar flames at 1g do not provide a satisfactory means to accomplish this because they do not have the same temperature, concentration and residence time characteristics as the nonbuoyant flamelets in turbulent flows (Fig. 14) []. Soot formation reactions occur mostly on the fuel side of the flame front at Zst < Z < 2Zst , where Zst is the stoichiometric mixture fraction (denoted f = 1 in the figure) due to pyrolysis of fuel molecules and subsequent nucleation of particulate carbon. For buoyant flows, soot particles initially convect toward regions of higher Z before a decrease of Z as the particles are swept toward the flame and into the air side of the flame where they may be oxidized. In contrast, for nonbuoyant flames the soot particles experience a monotonic decrease of Z throughout their life. Thus, measurements of soot properties at µg conditions could provide assessment of the laminar flamelet concept for soot formation in ways that cannot readily be done at 1g.

The early µg studies of gas-jet flames [54, 55] show that µg flames have a much greater propensity to emit soot, which indicates that the increase in tjet at µg and thus greater time for soot formation plus the broader region where composition and temperature are favorable for soot formation (Fig. 14), outweighs the lower temperatures at µg, which would act to decrease the soot formation rate []. Recent quantitative measurements [] show a peak soot volume fraction about twice as high in µg as 1g for 50% C2H2/50% N2-air flames, plus a broader region of soot formation at µg.

An interesting finding [67] of the µg gas-jet flame experiments is that these flames exhibit "smoke points," corresponding to a critical Uo below which soot is consumed within the flame, and above which soot is emitted from the top of the flame. This is expected for buoyant round-jet flames since tjet ~ Uo1/2, and thus as Uo increases, the time available for soot formation increases, leading to a greater total soot production, and at the smoke point some of the soot formed on the fuel side of the flame cannot be oxidized in the time available after passing through to the air side of the flame front before the temperature drops to values too low for significant oxidation to occur. In contrast, for nonbuoyant flames, tjet is predicted to be independent of Uo, which would suggest that no smoke point would occur. Mortazavi et al. [], using a fully elliptic numerical model, showed that at least for some cases (Fig. 15), tjet does increase monotonically with Uo, which could explain the existence of smoke points for nonbuoyant flames. This increase in tjet with Uo was suggested to be due to the effects of axial diffusion, but if axial diffusion were the dominant factor tjet should asymptote to a constant value for long flame lengths, where axial diffusion is negligible compared to radial diffusion (note that w/do asymptotes to a fixed value at high Uo, Fig. 11, but Lf/do increases linearly with Uo, Fig. 10). Also, Burke-Schumann flames, where Uo is nearly constant, seem to exhibit smoke points, at least for C3H8-air flames []. Thus, a simple description of the mechanism of smoke points at µg remains elusive. These studies suggest that residence time considerations may be misleading, and that particle temperature-composition-time history effects are important, as Fig. 14 would suggest, which may have implications for 1g smoke points as well.

Under conditions of weak convection, thermophoretic forces on soot particles, which act to move particles to regions of lower temperature, is an important effect. If the convection velocity and temperature gradient are the same direction, the convective and thermophoretic forces may balance at some location. This can lead to an accumulation region for soot. Striking examples of this have been seen in µg gas-jet flames (Fig. 16) [] where inside the flame front, the temperature gradient and outward radial flow are in the same direction, creating an annulus of soot accumulation that is convected by the axial flow through the tip of the flame. Downstream of the flame tip, for reasons not yet described, the soot annulus becomes unstable and fragments, creating a crown-like structure. When the jet velocity is very low, the temperature gradient due to the heat loss to the burner rim creates such a strong thermophoretic force that soot agglomerates actually reverse direction and collide then bounce off the rim, in a manner superficially similar to water droplets bouncing off a heated surface. Clearly such effects are only observable at µg where convection velocities can be comparable to the thermophoresis velocity, predicted [72] to be about 5 mm/s for the conditions in Fig. 16. Thus, µg conditions provide a useful means of assessing models of soot particle migration that may be applicable to 1g conditions as well.

5. CONDENSED-PHASE COMBUSTION

5.1 Droplet combustion

Probably the very first microgravity combustion experiments were conducted by Kumagai and collaborators [], who studied the burning rates of isolated spherically symmetric fuel droplets. While the combustion of isolated droplets has been studied for many years because of the obvious applications to the combustion of fuel sprays, a number of new insights have been obtained and new features identified through µg research. At 1g, experimental measurements of the burning rate are compromised by buoyant convection, which destroys the spherical symmetry of the system and induces additional heat and mass transport, both of which alter the burning rate and render modeling much more difficult. Classical theory [, ] predicts that the quasi-steady burning rate for spherically symmetric buoyancy-free combustion is given by

ddo2 - dd2 = Kt; K _ (8l/rdCP)ln(1+B) (11)

where the transfer number B is essentially the ratio of the enthalpy generated by chemical reaction to the enthalpy needed to evaporate the fuel. In experimental studies the droplet diameter dd could be fixed, for example by forcing fuel through a porous sphere at a rate that balances evaporation, leading to a steady mass burning rate () = (¹/4)rdddK, but most experiments have employed fuel droplets where dd decreases with time. Somewhat surprisingly, many experimental results using fuel droplets follow Eq. (11) reasonably well despite the effects of unsteadiness, heat losses, soot formation and water absorption discussed below.

Just as with flame balls in premixed gases, for spherical droplet flames a steady solution exists even in an infinite domain, with a flame front located at df = ddln(1+B)/ln(1+f). The structure of droplet flames is different from flame balls, however, because steady flame balls are convection-free whereas in droplet flames there is a Stefan flow due to fuel vaporization at the droplet surface. Mass conservation dictates that the Stefan velocity decays as 1/r2, but nevertheless this flow causes the temperature and concentration profiles vary with radius in proportion to (1+B)dd/2r rather than 1/r as in flame balls (though these profiles, when normalized by flame radius, are indistinguishable at large r because convection effects are negligible at large r.) Moreover, unlike flame balls, heat losses are not required for droplet flames to be stable because the nonpremixed droplet flame location cannot move away from the radius where the fuel and oxidant fluxes are in stoichiometric proportions.

The characteristic transport time scale for droplet combustion is tdrop Å df2/a. As with other types of flames discussed earlier, this leads to two types of extinction limits, one for small df where tdrop < tchem, thus fuel and oxidant cannot react within the time before they diffuse through each other, and one for large df where tdrop > trad, thus the temperature decrease due to heat loss reduces the reaction rate sufficiently to cause extinction. The former occurs when df < (atchem)1/2 for hydrocarbons in air and the latter when df > (atrad)1/2. This dual-limit behavior is fully supported by theory [, ] and is expected in view of the discussions for gaseous flames in the previous sections. Recent space experiments have enabled observation of radiative extinction of large droplets in air [] and O2-He [] atmospheres.

Even at µg, burning rates may be affected by the residual motion of the droplet caused by the droplet growth and deployment process. This has two effects: (1) it affects the heat and mass transport rates within the flame and (2) it alters the soot formation and accumulation process which in turn alters the radiative transfer environment. Still, Fig. 17 [] shows that discrepancies remain even when these influences are assessed. In Fig. 17, for the lower set of data [80] the reported relative velocity between gas and droplet is estimated by using soot particles as markers of the local gas motion, whereas for the upper set [] the relative velocity is that between the droplet and the chamber wall. It has been proposed [80] that part of the discrepancy between the two data sets may be due to effect (2), namely that even a small convection velocity (i.e., on the order of df/tdrop= a/df Å 10 mm/s for the conditions in Fig. 17) may significantly alter buildup of soot, and the gas velocity relative to the droplet provides a better assessment of such convection effects.

When the convection velocities are very low, soot particles in µg droplet flames have been found [, , ] to exhibit thermophoresis effects, leading to concentration of soot in a thin spherical shell between the droplet and the flame front (Fig. 18). This behavior is analogous to the phenomena observed in non-premixed gas-jet flames (Section 4.4) and occurs because the soot particles form between the droplet surface and the flame front then are convected towards the flame front by the Stefan flow but halted by thermophoresis. The soot particles can then attain an equilibrium position [84] where a significant amount agglomerates, forming a semi-rigid shell. In some cases this soot shell in turn can break apart suddenly, leading to multiple fragments of burning soot agglomerates. Since the velocity and temperature gradients may be readily modeled in simple geometries such as a spherical droplet flame, these experimental observations provide a means to assess the effects of thermophoresis on soot particle transport.

Radiative heat transfer seems to affect droplet burning rates at µg in somewhat unusual ways. In particular, it seems to cause K to decrease with increasing initial droplet diameter (ddo). A summary of various µg measurements of K vs. ddo for n-heptane droplets is shown in Fig. 19 []. This figure shows values of K that are fairly consistent between different investigators and exhibits a monotonic decrease in K with ddo over the range 0.1 mm < ddo < 3.3 mm. This has been suggested [] to be due to the effect of soot accumulation and the resulting decrease in net heat release and soot radiation and possibly due to the soot shell acting as an insulating barrier to heat transfer. The buildup of soot is apparently more significant for larger ddo because of the larger ratio of droplet volume to flame surface area for larger ddo. Note that the calculations [85] shown in Fig. 19, which do not consider soot formation or soot radiation but do consider gas radiation, also show a decrease in K with ddo. This indicates that even gas radiation can cause this diameter-dependent K, which is reasonable since the same volume to area consideration could apply for gaseous radiators as well as soot. An important conclusion one can draw from these results is that when radiative effects are important K is not constant; it is a function not only of the instantaneous droplet diameter (dd) but also the initial diameter (ddo). This indicates that the quasi-steady approximation leading to Eq. (11) is not valid, otherwise K should be independent of ddo, at least for a given instantaneous dd. This is probably due to the time required for diffusion of thermal energy and radiant CO2 and H2O combustion products into the far-field; this process will cause the total radiative loss to increase with time. As discussed in section 3.3, this behavior has been observed with flame ball experiments performed in space [43], where the time for radiative loss to reach steady state is typically on the order of 100 s even in flame balls with diameters much smaller than typical droplet flame diameters. Another indication that quasi-steady conditions are not achieved in droplet flames is the fact that a constant ratio df/dd predicted by the quasi-steady theory is not generally achieved, especially for the large droplets studied at µg [85, 86]. The importance of unsteadiness on df/dd and other droplet properties was recently examined by King [].

Another complicating factor that arises for fuels that are miscible in water is that water vapor from the combustion products can be absorbed by the fuel, which causes significant departure from Eq. (11) and reduces the flammability of the fuel. Evidence of this has been found [] in methanol droplet flames, where the extinction diameter depends substantially on the initial droplet diameter (Fig. 20) []. To obtain the observed agreement between model [88] and experiment [78], in the model it is necessary to assume a well-mixed droplet rather than pure spherically-symmetric diffusion-controlled mixing of water. This indicates significant internal flow in the droplet exists even at µg, presumably due to the deployment process and/or thermocapillary effects.

Finally, in some recent space experiments, flame oscillations with amplitudes comparable to the mean flame diameter have been observed []. The amplitude of these oscillations grew with time and extinction occurred after typically 8 cycles. The frequency of oscillation was relatively constant at about 1 Hz. Cheatham and Matalon [77] have predicted oscillations of roughly this frequency in spherically symmetric droplet flames of pure fuels under near-extinction conditions when the Lewis number of the reactants is sufficiently low. To date this phenomena has only been reported for methanol/dodecanol (80/20 mass fraction) bi-component droplets burning in air, and it is uncertain whether the same phenomena can occur in pure fuels as well. Moreover, µg experiments in O2-He atmospheres [79], which have much higher Le than air, did not exhibit oscillations. One possible explanation could be that a droplet support fiber was used in the experiments reported in [90] but not in [79]. The fiber would act to increase the conductive and radiative heat loss, which according to Cheatham and Matalon should encourage oscillations. Another possible explanation is that since the oscillations are predicted only near extinction conditions, depletion of oxygen (leading to extinction) was a much more significant factor in the smaller combustion chamber used in the methanol/dodecanol experiments. Such oscillations will be discussed further in the following section in the context of candle flames.

5.2. Candle flames

An excellent example of just how different 1g and µg flames can be is afforded by examining perhaps the most common and familiar of all combustion processes - a candle flame. At 1g, the candle flame is supported by a continual influx of fresh air entrained by buoyant flow which mixes with vaporized fuel from the wick. This generates a self-sustaining flame and flow configuration. Obviously this mechanism does not apply at µg. It was shown in previous sections that for spherical flame balls and droplet flames, it is theoretically possible to have steady combustion without forced convection due to the nature of the diffusion equation in spherical geometry. An interesting question then is whether candle flames, which are not strictly spherical, can behave the same way. Space experiments [, ] have shown that candle flames may be steady for long periods of time - as much as 45 minutes for candles with small diameter wicks (Å1 mm). The flame shape is typically hemispherical rather than teardrop-shaped as at 1g (Fig. 21). Eventually, the flames always extinguished, though the mechanism has not been determined. The spherical flame model would predict that the flame could be steady indefinitely.

Interestingly, before extinction, the flames were frequently observed to oscillate in that the edge of the nearly hemispherical flame would advance and retreat. The amplitude of the oscillations would increase and on one retreating cycle the entire flame would extinguish (Fig. 22). In some of the Mir experiments [92], hundreds of oscillation cycles were observed. The flames created by the larger wicks (thus larger flame diameters) would start oscillating spontaneously before extinction, whereas with smaller wicks the flame would only oscillate when a solid object was placed near it.

While no conclusive explanation of these oscillations has been presented, Cheatham and Matalon [77] have shown that near extinction, oscillatory instabilities can occur in spherically-symmetric fuel droplet flames with radiative loss at sufficiently high Lewis numbers. Their predicted oscillation frequencies (0.7 - 1.4 Hz) are close to the experimental observations. On the other hand these authors noted the differences between spherically-symmetric flames and the roughly hemispherical candle flame. Alternatively, Buckmaster [] has shown that a flame "edge" separating burning and non-burning regions of non-premixed flames such as the µg candle flame seen in Fig. 21 can exhibit oscillatory behavior at sufficiently high Le. For quasi-stationary edges the critical Lewis number is 1 + 8/b(1-(T_/Tf)) Å 2 when Le for the other reactant is unity. The Lewis number of O2 in N2 is close to unity, but the Lewis number of the fuel vapors (which is compared of molecules with much higher molecular weights and thus much lower D than O2) is probably close to 2 and thus the edge-flame instability is a potential candidate to explain the observed oscillations. The mechanism of these oscillations is analogous to the diffusive-thermal mechanism of premixed flames, for which oscillations in burning velocity are predicted [] for sufficiently high Le and have been observed experimentally at both 1g and µg []. While neither the droplet instability mechanism [77] nor the edge-flame instability mechanism [93] have been definitively linked to the candle-flame experiment, both predict a greater propensity for oscillation with greater heat losses, which is consistent with the observation that the oscillations occur near extinction conditions.

The oscillations are probably associated with near extinction conditions, but it is unclear what would cause a change in the environmental conditions over time that is more favorable for extinction. In the earlier Spacelab experiments [91], the limitation on the rate of diffusion of O2 through a protective screen was suspected to cause a decrease over time in the far-field O2 concentration seen by the flame, but this effect was thought to be greatly reduced in the later Mir experiments [92]. Perhaps in the Mir experiments, oxygen depletion still occurred but much more gradually, thereby maintain the flame in a near-extinction state for a much long period of time. In any case, these results show that even the simple candle flame experiment has led to new and as yet unexplained phenomena in a µg environment.

5.3. Flame spread over solid fuel beds

The study of flame spread over flat solid fuel beds is a useful means of understanding more complex two-phase nonpremixed flames, such as those found in fires in enclosures, e.g., manned spacecraft and terrestrial buildings. Flame spread is classified as opposed-flow, where the direction of flame propagation is opposite that of the convective flow past the flame front, or concurrent-flow flame spread. 1g downward flame spread is characterized by opposed flow since the upward buoyant flow is opposite the direction of flame spread, whereas upward flame spread is characterized as concurrent flow. At µg without a forced flow, flame spread will necessarily be of the opposed-flow variety, because the flame spreads toward the fresh oxidant with a self-induced velocity equal to the spread rate (Sf). At 1g this self-induced convection can justifiably be ignored since buoyancy-induced flows are of the order (gag)1/3 Å 0.10 m/s, whereas Sf is typically less than 0.01 m/s. In concurrent-flow flame spread, convective and diffusive transport are in the same direction and as a result the fuel surface area exposed to the hot combustion products increases with time, frequently leading to accelerating rather than steady flame spread []. Very few studies of forced concurrent-flow flame spread at µg have been conducted, e.g. [], consequently this section will focus on opposed-flow flame spread.

For opposed-flow conditions, Sf can be estimated by equating the conductive heat flux to the fuel bed (= l(dW)(Tf - Tv)/d, where in this case d = a/U is the thermal transport zone thickness, U is the opposed-flow velocity (forced, buoyant and/or self-induced), to the rate at which the fuel bed enthalpy increases (=rsCP,sts(Tv-T_)WSf). In this model, chemical reaction is assumed to occur as soon as the fuel and O2 are mixed (infinite-rate chemistry.) For thermally-thin fuels, where heat conduction through the solid fuel is negligible, this leads to the classical deRis [] approximate result

(12).

Note that Sf is independent of U and pressure. Delichatsios [] derived an "exact" result and showed that Eq. (12) is valid except that the Ã2 factor should be replaced by ¹/4. For thermally thick (essentially semi-infinite) fuels, where heat conduction through the solid fuel is dominant, ts is the depth of thermal penetration into the solid fuel. This in turn can be estimated by equating the conductive heat flux to the fuel bed to the heat flux within the solid fuel (= lsy(dW)((TvT_)/ts), where the subscript y refers to the direction normal to the fuel surface), leading to the exact solution found by deRis [98]:

(13)

Note that, unlike the thin-fuel case, for thick fuels Sf ~ U1P1.

Similar to many of the other types of flames described above, for either thin or thick fuels there are two possible extinction mechanisms. The time for thermal energy to diffuse across the convection-diffusion zone (tdiff) can be estimated as d/U = a/U2, thus a high-velocity limit occurs when tdiff > tchem or U > (a/tchem)1/2. A radiant heat loss limit when tdiff > trad or U < (a/trad)1/2, so that a radiative loss parameter can be defined as H _ tdiff/trad = a/U2trad. Such dual-limit extinction behavior has been seen in experiments at 1g and µg with and without forced flow [] (Fig. 23). From a practical point of view, it is interesting to note that as a consequence of these phenomena, the most flammable flow environment, i.e., that supporting combustion at the lowest O2 mole fraction, occurs at an intermediate value of U Å 0.1 m/s, a value that might be expected from ventilation drafts in a manned spacecraft.

The above radiation loss estimate is based on gas-phase radiation. Of course, surface radiative loss may be important as well. Rhatigan et al. [] show that both may be important for a polymethylmethacrylate (PMMA) cylinder in a stagnation-point flow. The ratio of surface loss to gas-phase loss can be estimated as (ses(Tv4T_4)dW)/(saP(Tf4-Tv4)d2W) = (Ues(Tv4-T_4))/(aaP(Tf4-Tv4)). Thus at low U (or Sf if the atmosphere is quiescent) gas-phase radiation is expected to dominate. For typical µg conditions with Sf = 5 mm/s, this estimate suggests gas-phase radiation should be stronger, but with forced flow (higher U), solid-phase radiation may be dominant. Note that the surface radiative loss is proportional to ses(Tv4T_4)dW and d ~ U-1, as with gas-phase radiation, the impact of surface radiative loss decreases with increasing U.

Since the impact of radiative loss, whether due to gas-phase or surface radiation, increases with decreasing U, Sf is generally found to be lower at µg than 1g [100]. An interesting set of experiments by Olson [] (Fig. 24) shows that at µg, an imposed forced flow increases Sf since the net U (sum of forced flow and Sf) increases and thus the impact of radiative loss decreases, whereas at sufficiently high U (whether buoyant or forced), Sf decreases as the high-U limit is approached. Note that for 21% O2 or lower, the behavior predicted by Eq. (12), where Sf is independent of U, is never achieved. Only at 30% O2 is Tf high enough that the infinite-rate chemistry result where Sf independent of U obtained for a range of U.

Since the heat loss parameter H = a/U2trad with a ~ P1 and trad ~ P0, H ~ P-1, and thus for thin fuels Sf should increase with P towards the ideal (adiabatic) value given by Eq. (12). This is confirmed by quiescent thin-fuel space experiments [, , ] which show Sf increasing from 3.2 to 4.9 to 5.9 mm/s as P is increased from 1.0 to 1.5 to 2.0 atm with fixed O2 mole fraction (0.50). For these conditions H decreases from 24 to 6.8 to 3.5, thus, even at the highest P radiative effects are still likely to be important. This is consistent with computations [103] which show that the calculated adiabatic Sf for these mixtures is about 12 mm/s (almost independent of P), which is twice the measured value even at the highest pressure tested (2 atm).

Neither N2 nor O2 emit thermal radiation and thus in the experiments cited above only the H2O and CO2 combustion products produce significant thermal radiation. A typical value of aP-1 is 1.2 m (Section 2), which is far greater than d, most of the radiation process can be considered optically thin (no reabsorption). Under conditions where the optical thickness parameter aPd approaches unity, reabsorption of emitted radiation cannot be neglected. With reabsorption, radiation emitted near the zones of peak temperature may not be lost to the surroundings and may augment conventional thermal conduction to increase Sf above that without radiative transfer. Moreover, even if reabsorption is still not important, for large aP the radiation to the fuel surface may increase Sf (see Section 6.1). Evidence of the behavior of flame spread with strongly radiating gases is seen in experiments [] using CO2 and SF6 diluents (Fig. 25), where Sf is higher and the minimum O2 concentration that will sustain flame propagation is lower at µg than for downward flame spread at 1g, whereas the opposite (conventional) behavior was found in He, Ar and N2 diluents (not shown). (A more meaningful comparison to µg results is obtained with downward rather than upward flame spread at 1g because µg flame spread and downward spread at 1g are characterized as opposed-flow. Nevertheless, the experimental results [106] show that for O2-SF6 atmospheres, the spread rates are higher at µg than for any direction of spread at 1g.)

From a practical point of view, the data in [106] show that for non-radiating diluents, e.g. He, Ar, and N2, µg is a less hazardous environment than 1g since the limit O2 mole fraction is higher at µg than 1g (0.21 vs. 0.16 for He diluent), whereas for radiant diluents (e.g., CO2 and SF6) the limit O2 mole fraction is lower at µg than 1g (0.21 vs. 0.24 for CO2 diluent). This is particularly significant in view of the fact that a CO2-based fire suppression system will be used on the International Space Station.

For the near-limit O2-CO2 and O2-SF6 mixtures, aP-1 Å 44 mm and 4 mm respectively. These are of the same order of magnitude as d for these cases (6.0 mm and 0.73 mm, respectively), and thus radiative effects should be much more important than in radiatively-inactive diluents. A detailed assessment of radiation effects, however, requires consideration of at least three factors other than the optical thickness parameter aPd: (1) as discussed in Section 3.1, the spectral absorption coefficient (k) depends strongly on temperature and wavelength - at some wavelengths k-1 is as small as 0.25 mm for CO2; (2) the absorption spectra of the radiant combustion products H2O and CO2 may or may not overlap with the diluent spectrum; and (3) the H2O and CO2 generated by combustion are present only near the flame front and not in the fresh reactants. To date, flame spread calculations have employed optically-thin radiation models with either constant aP [103, 104, 105] or variable depending on local temperature and composition [], and thus cannot assess factors (1) - (3).

All of the above flame spread discussions pertain to thin fuel beds, in which steady spread is possible at µg because the ideal Sf is independent of U. For thick fuels, Sf ~ U and thus Sf is indeterminate for quiescent µg conditions. When unsteady heat conduction to the fuel bed is considered, the thermal penetration depth ts ~ (ast)1/2 which results in Sf ~ t-1/2, where t is the time lapse from ignition [105]. Essentially this is because at µg without forced flow, d Æ _, thus ts Æ _. In a sense, then, all fuel beds are thermally-thin at quiescent µg conditions unless the radiative effects discussed in Section 6.1 are considered.

One possible concern with comparisons of µg flame spread space experiments to models is that the fuel bed width W (30 mm for thin fuels [103, 104, 105] or and 6.2 mm for thick fuels []) is comparable or smaller than d. Consequently, these experiments can hardly be considered two-dimensional. Both lateral heat losses, which would retard flame spread, and lateral O2 influx, which would enhance spread, are likely to be important in these experiments, thus their effects may cancel to some extent. Bhattacharjee et al. [103] point out that due to radiative losses, d may be much smaller than a/U = a/Sf, but the O2 diffusion zone thickness is still given by D/U, where D is the oxygen diffusivity, since there is no analog to radiative loss for the mass fraction of O2 and other chemical species. Since Le = a/D Å 1, the length scale for O2 diffusion is nearly the same as that for thermal energy in adiabatic flames. Thus, the µg flames have probably benefited substantially from lateral O2 influx, especially for the lower pressures and O2 mole fractions, where d/W ~ a/SfW is largest. In fact, it might be possible for Sf to be higher at smaller W because of the lateral O2 influx. Space experiments using cylindrical fuel rods are planned [] to examine a truly two-dimensional configuration.

An unusual observation has been reported recently by Olson et al. [] in µg flame spread experiments conducted using thin paper samples coated with potassium acetate to inhibit flaming combustion but allow smoldering propagation through the solid. Fingering fronts (Fig. 26) were observed at µg when the imposed air flow velocity U (parallel to the surface) was less than 50 mm/s whereas a simple ring-shaped smoldering front was observed in normal gravity. Olson et al. proposed that this is due to the limited O2 mass transport rate at low convection velocities, which causes the O2 consumption regions to become localized spots of reaction as opposed to a continuous front because of the scarcity of available O2. This proposition does not address the issue of why heat conduction does not smooth out the fingers. The following alternative explanation for the fingering behavior is proposed here. Heat transport can occur either in the gas phase on the length scale d ~ a/U, or through the solid fuel bed on a scale ds ~ as/us where us is the smolder front velocity and as the solid thermal diffusivity. Oxygen transport can only occur through the gas phase on a length scale D/U Å a/U ~ d. Radiative loss from the fuel surface can suppress heat transport through the gas, but no corresponding effect on O2 transport can occur. Thus at low U, the system can be thought of as a low Lewis number front where the effective Le is as/D << 1. At high U, i.e., at 1g, d is smaller, gas-phase heat transport is more dominant and radiative effects are weaker, thus the effective Le is a/D Å 1. These assertions are consistent with Olson et al.'s estimates [108] of the relative importance of gas-phase and solid-phase transport at varying U. Moreover, it is well known that premixed [15, 27, 41] and nonpremixed [, ] combustion systems with non-unity Le exhibit diffusive-thermal instabilities due to the imbalance of heat and mass transport to the flame front. At Le < 1 these instabilities manifest themselves as cellular flame fronts that produce patterns qualitatively similar to those seen in Fig. 26, whereas for Le Å 1 the fronts are stable. The explanation proposed here is also consistent with recent 1g experiments [] where a horizontal fuel bed is placed in a channel of adjustable vertical height. At low channel heights or low U, fingering very similar to that seen in Fig. 26 is observed. In this case the conductive heat loss to the ceiling of the channel causes the suppression of gas-phase heat transfer. In both cases the key mechanism seems to be the suppression of gas-phase heat transfer which would otherwise act to smooth out the front, though this factor was not mentioned in [110] or [113].

5.4 Flame spread over liquid fuel pools

The processes involved in flame spread over liquid fuels encompasses practically all of the solid-fuel flame spread phenomena mentioned above, plus the effect of liquid-phase flow. A recent review of the subject is given by Ross []. Typically Tv-T_ is much smaller for liquid fuels than solid fuel beds, thus Sf will be much higher for liquid fuels with otherwise comparable thermophysical properties, regardless of the effect of the liquid-phase flow. Also, if Tv-T_ is small there is some prevaporization of the fuel bed even at T = T_, thus premixed-gas combustion phenomena may be present. Perhaps the most important consequence of the liquid-phase flow is that, due to the temperature gradient on the fuel surface upstream of the flame front, a surface tension gradient is produced which causes the surface layer of fuel to move in the upstream direction (away from the flame front), which tends to increase Sf. At 1g, in addition to the effects of buoyancy on gas-phase transport, this heated layer of liquid fuel must lie near the fuel surface, whereas at µg no such limitation exists.

1g experiments by many investigators, summarized in [114], have shown that for low fuel temperatures, the mean Sf is small (typically a few tens of mm/s) and the spread process is pulsating, alternating between a fast "jump" velocity and a slow "crawl" velocity, whereas at higher fuel temperatures, Sf is faster and steady. For the conditions exhibiting unsteady flame spread at 1g, µg flame spread cannot be maintained (somewhat like thick-fuel flame spread over solid fuels) whereas at the conditions of higher fuel temperature that exhibit uniform spread at 1g, steady spread is also exhibited at µg []. No definitive explanation for the correspondence between the 1g and µg behavior has been advanced. For the conditions exhibiting pulsating spread at 1g and no spread at µg, when a forced flow comparable to that induced by buoyancy at 1g (0.30 m/s) is imposed, flame spread is still different at 1g and µg (Fig. 27) []. In particular, 1g spread is not affected substantially by the imposed flow and it still exhibits pulsations but µg spread is steady, not pulsating as it is for exactly the same conditions at 1g. In fact, in no instance has pulsating spread been observed experimentally at µg. Also, the steady spread rate at µg (15 mm/s) is lower than either the jump velocity (about 10 mm/s in this example) or crawl velocity (2.2 mm/s) at 1g.

A detailed numerical model of liquid-fuel flame spread [] predicts pulsating spread at µg for the conditions shown in Fig. 27 and a value of Sf much closer to the 1g Sf. Remarkably, if thermal expansion is artificially suppressed, very good agreement between model and experiment is found. It is proposed [116] that the agreement results from three-dimensional effects; specifically, it is hypothesized that in the experiment, flow induced by thermal expansion can be relaxed in the lateral dimension, whereas the two-dimensional model does not permit this. That three-dimensional effects might be dominant is somewhat surprising considering that for U = 0.30 m/s, d/W ~ a/UW Å 0.017 for this flame and thus the flame thickness d is much smaller than the fuel bed width W. Also, this hypothesis does not explain why pulsating flame spread is observed at 1g over a pool of the same width. As discussed by Ross [114], numerous models of pulsating spread have been proposed, but current consensus focuses on the role of a gas-phase recirculation cell ahead of the flame front which is driven from above by the forced or buoyant opposed flow and driven from below by the surface-tension driven flow which is in the direction of spread. This cell appears to periodically develop and transport premixed reactants toward the flame front which subsequently flash over, causing a period of high Sf, at which point the cell is destroyed, the spread process reverts to a slower mode more similar to solid-fuel spread, after which the cycle repeats. Still, an understanding of the differences between 1g and µg flame spread has not been obtained. In the case of liquid-fuel flame spread, µg experiments have pointed to limitations in our current state of understanding of combustion processes at 1g.

6. RECOMMENDATIONS FOR FUTURE STUDIES

6.1 Reabsorption effects

The µg studies described here have pointed to new unresolved issues and opportunities for further improvements in our understanding. Perhaps the most pressing and wide-ranging issue requiring further µg research are the effects of reabsorption of emitted radiation, including both reabsorption by the gas itself and in the case of two-phase combustion, absorption by the condensed phase, e.g., the solid fuel bed or fuel droplet. All of the radiative effects discussed above are critically dependent on the degree of reabsorption. To study reabsorption effects requires radiatively-active diluents such as CO2 and SF6 (which have small n), high pressures (which also reduces n) and/or large systems. All of these conditions lead to high Grd at 1g and thus turbulent flow. Hence, µg experiments provide an excellent opportunity to study reabsorption effects on combustion processes without the additional complication of turbulent flow.

Reabsorption effects are a subject of importance not only to µg studies, but also for combustion at high pressures and in large furnaces. For example, at 40 atm, a typical pressure for premixed-charge internal combustion engines, aP Å 18 m-1, thus aP-1 = 0.045 m, for the products of stoichiometric combustion. This length scale is comparable to the cylinder radius, thus reabsorption effects within the gas cannot readily be neglected. Simple estimates [] indicate radiative loss may influence flame quenching by turbulence. Similarly, reabsorption cannot be neglected in atmospheric-pressure furnaces larger than aP-1 Å 2.2 m. Moreover, many combustion devices employ exhaust-gas or flue-gas recirculation; for such devices significant amounts of absorbing CO2 and H2O will be present in the unburned mixtures.

While reabsorption could affect practically all types of flames reviewed here, the only cases to date where such effects have been studied through the use of high pressures and radiatively-active diluent gases are propagating premixed-gas flames [19, 24, 26], flame balls [48] and flame spread over thermally-thin fuels [106]. As discussed above, all of these have shown substantially different behavior from optically-thin conditions. Two examples of effects expected for other types of flames are given below.

In the case of droplet combustion the impact of radiation could be even more important than was demonstrated for flame spread over solid fuel beds because the Stefan flow severely limits the ability of conduction to heat the droplet. This is because greater conductive flux to the droplet leads to more fuel vaporization, thus a faster radial outward flow, making conduction to the droplet more difficult. This is why the heat release factor B affects the burning rate only weakly (logarithmically). Of course, radiative transfer is not subject to such limitations. The classical d2-law is readily extended to include a radiative flux qr to the droplet surface, leading to a transcendental relation for the dimensionless burning rate W (neglecting reabsorption within the gas):

(14),

The predictions of Eq. (14) are shown in Fig. 28 for B = 3 and 8.5, characteristic of methanol and heptane, respectively, in air. While the author is not aware of Eq. (14) being presented previously, the effect of radiative transfer to the droplet surface has been studied numerically [, ] with qualitatively similar predictions. In general, the absorption spectrum of the liquid fuel droplet must also be considered, but typical values of k-1 over the relevant range of wavelengths are on the order of 1 mm, and thus droplets with dd > 1 mm can be expected to absorb the incident radiation almost completely.

For a spherical shell of radiant combustion products of thickness d which is much smaller than the flame standoff distance df, qr Å Ld/4. Then for typical values B = 8.5, L = 2 x 106 W/m3, d = 10 mm, dd = 5 mm, CP = 1400 J/kgK, l = 0.07 W/mK (temperature averaged) and Lv = 400 kJ/kg, we obtain R = 0.63 and W/WR=0 = 1.11, thus a moderate effect of radiant heat transfer can be expected for large droplets. For droplets in radiatively-active diluents such as CO2, the effect could be much stronger, perhaps dominant because of the absence of non-radiant N2 and because the concentration of radiators does not decrease away from the flame front. Using the P1 approximation, solutions of the radiative transfer equations for a sphere of unit emissivity in an infinite gray gas [] show that qr = [4/(2+3aPdd/2)]s(Tg4 T_4), where Tg is the gas temperature. Using volume-average properties Tg = 1000K and aP = 20 m-1 for CO2, we obtain R = 18, thus W/WR=0 = 8.0, indicating radiation is the dominant mechanism of heat transport to the droplet. As discussed in the following section, radiative effects can be dominant even in air atmospheres for high-pressure droplet flames.

In the case of high molecular weight diluents such as CO2 or SF6 another phenomenon that may arise near extinction is low Le diffusive-thermal instabilities, which are normally associated with premixed-gas flames but have also been observed in gas-jet diffusion flames [111] and flame spread at 1g [] and µg [106]. In the case of droplets this may lead to non-spherical flame shapes and/or non-uniform burning intensities on the flame sheet [112].

As discussed in Section 5.3, flame spread over thermally-thick fuel beds in quiescent atmospheres at µg is thought [108] to be inherently unsteady, however, this is probably not true when radiative transfer from the gas to the fuel bed is considered. If the flame front is modeled as an isothermal volume of gas at constant temperature with dimensions d by d by W, the radiative emission causes heat transfer to the fuel bed proportional to Ldg2W = L(ag2/Sf2)W in addition to the conductive flux l(dW)(Tf - Tv)/d. Equating the total heat transfer to rsCP,sts(Tv-T_)WSf leads to, for thick fuel beds,

(15),

which vanishes in the absence of gas radiation (L = 0). Space experiments of the type reported by West et al. [105] in O2-CO2 or O2-SF6 atmospheres to test for the existence of steady spread over thick fuel beds at µg and the qualitative features of Eq. (15) could be quite instructive.

While modeling of radiation effects in flames under optically thin conditions [20, 21, 36, 37, 48, 50, 103, 104, 105, 107] is reasonably straightforward, modeling of spectrally dependent emission and absorption is a challenging task because local fluxes depend on the entire radiation field, not just local scalar properties and gradients. Some studies using gray-gas models have been reported [88, 89] but recent studies [26, ] have shown that the accuracy of these methods for assessing reabsorption effects is doubtful because of the wide variation in spectral absorption coefficient with temperature and species. A useful comparison of various approximate radiative treatments for a nonpremixed PMMA-air flame in a stagnation flow is given by Bedir et al. [123]. Comparisons for larger, multi-dimensional systems would be worthwhile. Of course, quasi-blackbody radiation from soot particles is another complication to assessing radiative effects in µg flames, though many µg flames are non-sooting under conditions where the same flame is sooting at 1g. Moreover, recent studies of soot formation in µg flames [67, 69, 72] are starting to provide an assessment of soot radiation under conditions where µg flames are sooting.

6.2 High pressure combustion

All practical combustion engines operate at pressures much higher than atmospheric. The impact of buoyancy for premixed flames scales as tchem/tvis ~ (ga/SL3)2/3 ~ Pn-4/3, where n is the overall order of reaction (SL ~ Pn/2-1). Since typically n < 4/3 for weak mixtures [] where buoyancy effects are likely to be important, the impact of buoyancy generally increases with pressure. Also, as discussed in Section 1, the effects of radiative transport are more difficult to assess at higher pressure due to increased interference from buoyant transport. While these observations apply to all types of combustion processes, very few high-pressure µg combustion experiments have been performed. Vielle et al. [] examined high-pressure droplet combustion at µg and found a substantial increase in K with P at both 1g and µg, but the amount of increase was different at 1g and µg. While these authors did not discuss radiative effects, it would appear that such effects could be quite important. In Eq. (14), the only factor that depends on P is qr ~ L ~ P1, thus the horizontal axis in Fig. 28 is proportional to P. (For most types of flames d would decrease with increasing P, but for droplets the scalar profiles are determined only by stoichiometry [75, 76] and thus d is independent of P.) Consequently, radiative effects should be much more important at higher pressures in droplet flames. Further assessment of radiative effects in high-pressure droplet combustion and probably other types of flames appears warranted.

6.3 Three-dimensional effects

In Sections 5.3 and 5.4, the effects of lateral heat and mass transport on flame spread over relatively narrow solid and liquid fuel beds was discussed. µg experiments with varying fuel bed width W should be performed to determine the effect of three-dimensionality. Also, three-dimensional modeling of flame spread over finite-width samples using a code such as that developed by the NIST group [], extended to include gas-phase radiation, would be instructive. An approximate but much less computationally intensive approach would be to add an approximate volumetric heat loss to account for lateral heat losses. This term is equal to 6l(T(x,y)-T_)/W2, where x and y are the coordinates parallel to and perpendicular to the fuel bed, respectively. (Note that this term is half of the usual value because the surrounding gas will not remain at fixed temperature as isothermal walls would; in this case the surrounding gas will increase in temperature as much as the combustion gases will decrease in temperature.) Of course, similar terms of the form 6rDi(Yi(x,y)-Yi,_)/W2 are needed to account for lateral diffusion of each species i.

Another three-dimensional effect is found in the development of flame balls from an ignition kernel. To date, it is known that large flame ball branch is linearly unstable to three-dimensional disturbances if the scaled heat loss is sufficiently small (Fig. 6), which will occur mixtures sufficiently far from the extinction limit or if the local enthalpy is increased sufficiently from an ignition source. The transition from splitting flame balls to stable flames has not been analyzed to date nor is there any method to predict the number of flame balls produced from a given ignition source. Modeling using a code such as that developed by the NRL group [] could shed some light on this subject.

6.4 Gas-jet flames

Comparisons of flame length and residence time for buoyant and nonbuoyant round jet and slot jet flames were presented in Section 4.2. The predicted scalings are different for round-jets and slot-jets, and despite numerous studies of round jet flames at µg, somewhat surprisingly there have been no studies of nonbuoyant slot-jet flames at µg to test these predictions. It would be interesting to see if the conclusions reached based on the round-jet studies also apply to slot-jets. For example, do slot-jet flames at µg exhibit smoke points? Can this information be used to explain the existence of smoke points in round-jet flames at µg? Also, the flame length depends on g for buoyant slot-jet flames but not round-jet flames, thus flame heights should be quite different at 1g and µg for slot jets even though they are not for round-jets (Fig. 10). Residual accelerations in aircraft experiments could be even more problematic for slot-jet flames than round-jet flames because tjet ~ g-1/2 for buoyant round jets whereas tjet ~ g-1/3 for buoyant slot jets.

There has been little investigation of the blow-off behavior of laminar µg gas-jet flames. Gas-jet flames is one of few types of flames where µg experiments have not revealed dual-limit behavior, but the requisite experiments have not been performed. Such behavior has been observed at 1g [111] with small residence time extinction at high Uo and conductive loss (to the burner rim) extinction at low Uo. It could be anticipated that dual-limit behavior would occur for gas-jet flames of fixed fuel flow rate but varying do, leading to small residence time extinction at small do and radiative extinction at large do.

6.5 Quasi-steady spherical diffusion flames

In section 5.1 the effects of radiative loss on the extinction of droplet flames was discussed. An issue that arises in comparing the predicted radiative extinction limits of droplet flames to experiments is whether quasi-steady conditions can be obtained before extinguishment, because extinction occurs for sufficiently large droplet and flame diameters but the droplet diameter decreases throughout its life. Evidence of the lack of quasi-steadiness in prior µg droplet investigations was presented in section 5.1. Of course, numerical models can account for transient effects, but it is difficult to model the multi-dimensional ignition process in a quantitative way. It would be interesting to compare droplet experiments and computations to results obtained using a fuel-wetted porous sphere through which the liquid fuel is forced at a slowly increasing rate until extinction occurs when df > (atrad)1/2. In this case a true quasi-steady extinction might be obtained, though a long duration of µg or long computation time is required because of the need to establish the steady diffusion-dominated far-field temperature and composition profiles (thus the total radiative loss). The candle flames discussed in section 5.2 are quite similar to the wetted porous sphere, though the porous sphere experiment has the advantages of being able to characterize the diameter at the fuel surface more precisely as well as minimizing the departure from spherical symmetry inherent in candle flames.

A simpler model experiment which possesses some of the features of droplet or candle flames is porous sphere through which a gaseous fuel is forced. In this case the burning rate is a prescribed quantity and the steady flame the flame radius is given by CP/2¹lln(1+f). Some initial results in short-duration drop-tower experiments have been reported [, ], though steady flame structures have not been obtained due to the short µg duration. It appears unlikely that steady structures would be possible given the fuel flow rates (thus air consumption rates) and chamber sizes that have been employed to date; oxygen depletion even at the chamber walls would probably be significant on the time scale for the establishment of steady-state conditions, thought to be at least 10 s [129]. Steady structures might be obtained through the use of smaller porous spheres that would allow smaller flow rates, as well as diluted fuel with enriched oxygen atmospheres to increase f and thus decrease df and the characteristic time scale df2/a.

6.6 Catalytic combustion

The potential benefits of catalytic combustion for reduced emissions and improved fuel efficiency in many combustion systems are well known [, , ] Since catalysis occurs at surfaces, catalysis is inherently a multi-dimensional and/or unsteady process requiring transport of reactants to the surface and heat and products away from the surface. While boundary layer approximations can be incorporated into models of reaction at catalytic processes, the only truly one-dimensional steady catalytic configuration would be a spherical surface immersed in a nonbuoyant quiescent gas - i.e., a "catalytic flame ball." In this case the radius r* is fixed but the surface temperature T* and fuel concentration Y* are unknown. T* and Y* can be related through energy conservation (including surface radiation) and the diffusion equations, leading to the following expression for the surface reaction rate in moles per second (Q):

(15)

where T* the measured surface temperature and the subscript * refers to properties evaluated at the temperature T*. Through varying r*, Y_, pressure, and diluent gas, Q can be determined for a range of surface fuel concentrations and temperatures and compared to models. Of course, the conditions must be unfavorable for the initiation of a propagating flame or a flame ball that stands off from the surface for the catalysis process to be examined.

6.7 Chemical models

One of the most important contributions of µg combustion experiments has been an improved understanding of extinction processes. Of course, extinction processes are inherently related to finite-rate chemistry effects. Thus, to obtain closure between µg experiments and model predictions, accurate chemical models are needed. To date, in practically all comparisons for lean premixed hydrocarbon-air flames the models [21, 26, 33] predict higher SL and leaner flammability limits than the experimental observations [22, 23, 33, ]. The discrepancy seems to be more than experimental uncertainty or unaccounted heat losses could explain. In contrast, for flame balls in H2-air mixtures [48] and strained premixed H2-air flames at 1g [], the same chemical reaction mechanisms predict smaller flame balls, lower SL and richer flammability limits than the experimental observations. All of these chemical models predict the burning velocities of flames in mixtures away from extinction limits very faithfully. A substantial part of the discrepancy seems to be due to differences in the 3-body recombination rates for the H + O2 + M reactions, and in particular the third-body efficiency of various species [48]. These reactions are extremely important in near-limit flames, but of much lesser importance in mixtures away from limits, because of the competition between chain-branching and chain-inhibiting steps near limits [21]. Further consideration of the proper rates of these reactions in the intermediate temperature range (1100-1400K in most cases) would be most welcomed.

7. CONCLUSIONS

µg experiments have helped broaden our understanding of combustion fundamentals into regimes not explored in the past. In particular they have helped to integrate radiation into flame theory. Although flame radiation has long been recognized as an important heat transfer mechanism in large fires [96], 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, both premixed and non-premixed flames have exhibited dual-limit extinction behavior, with a residence time limited extinction at high strain or curvature and a radiative loss induced extinction at low strain or curvature. The high-strain limit is readily observed at 1g, and in fact when forced flow is absent, buoyant flow causes the strain that leads to extinction. For weak mixtures these limits converge, but the convergence and the entire low-strain extinction branch can only be seen at µg (cf. Fig. 23). This dual-limit behavior has been 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 also led to observation and clarification of a number of other combustion phenomena, for example thermophoresis effects in soot formation, spherically symmetric droplet burning rates, diffusion-controlled premixed flames (flame balls) and flame instabilities in droplets and candle flames. Considering the rapid progress made in just the past few years, further advances are certain to occur in the next few years. Hopefully this report on the current state of understanding can help motivate and inspire such advances.

ACKNOWLEDGMENTS

The author would like to express his deepest gratitude to the NASA Lewis Research Center for having supported his work on µg combustion for over 10 years. Discussions with Tom Avedisian, Yousef Bahadori, John Buckmaster, 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 others have been invaluable in extending the author's knowledge of this subject and in the preparation of this manuscript. The author would also like to thank numerous others who offered useful suggestions that could not be accommodated because of space limitations.

NOMENCLATURE

aP Planck mean absorption coefficient

A flame surface area

B transfer number

cs stoichiometric molar ratio of fuel to air

CP constant-pressure heat capacity

d characteristic flow length scale

d tube diameter

df droplet flame diameter

dd droplet diameter

ddo droplet initial diameter

do jet exit diameter

D fuel mass diffusivity

E overall activation energy of the heat-release reactions

f stoichiometric fuel to air mass ratio

g acceleration of gravity

go earth gravity

Grd Grashof number based on characteristic length scale (d) = gd3/n2

GrLf Grashof number based on flame length = gLf3/n2

h heat transfer coefficient in a cylindrical tube = 16l/d2

H radiative loss parameter for flame spread = tdiff/trad = a/U2trad

K droplet burning rate constant

Lf Flame length for gas-jet flame

LI turbulence integral scale

Lv latent heat of vaporization of liquid fuel

Le Lewis number (mixture thermal diffusivity / scarce reactant mass diffusivity)

Mass burning rate

M fuel molecular weight

P pressure

Pr Prandtl number

Q scaled heat loss ~ tchem/trad (Eq. (7))

r radial coordinate

r* flame radius for spherical flames

R scaled flame radius (Eq. (7))

R radiation parameter (Eq. (14))

Rg gas constant

Re jet Reynolds number = Uodo/n

S scaled flame speed = SL/SLo

Sf flame spread rate over solid fuel bed

SL premixed laminar burning velocity

SLo burning velocity of planar unstretched flame

SL,lim burning velocity at the flammability limit

Sc Schmidt number = n/D

tchem chemical time scale

tdiff thermal diffusion time scale = a/U2

tdrop droplet flame time scale = df2/a

tinv inviscid buoyant transport time scale

tjet residence time of nonpremixed jet flame

tad radiative loss time scale

tvis viscous buoyant transport time scale

T temperature

Tf adiabatic flame temperature

u' turbulence intensity

U convection velocity

U local axial velocity in counterflow configuration

Uo jet exit velocity

V local radial velocity in counterflow configuration

w flame width

W solid fuel bed width

y axial coordinate

Y fuel mass faction

Z mixture fraction

Zst stoichiometric mixture fraction

Greek symbols

a thermal diffusivity

b non-dimensional activation energy = E/RgTf

g gas specific heat ratio

d flame thickness

Dr density change across the flame front

e surface emissivity

k spectral absorption coefficient

l thermal conductivity

lR radiative conductivity = 16sT3/3aP

L radiative heat loss per unit volume = 4saP(Tf4 - T_4)

n kinematic viscosity

Q reaction rate (Eq. (16))

r density

ts fuel bed half-thickness

s Stefan-Boltzman constant

S flame stretch rate

w flame flicker frequency

Subscripts

f flame front condition

s solid fuel bed

v solid fuel vaporization condition

_ ambient conditions

REFERENCES

       Time scale          Stoichiometric flame   Near-limit flame   



Chemistry (tchem)                      0.00094 s              0.25 s 



Buoyant, inviscid (tinv)                 0.071 s             0.071 s 



Buoyant, viscous (tvis)                  0.012 s             0.010 s 



Conduction to tube wall                   0.95 s               1.4 s 

(tcond)                                                              



Radiation (trad)                          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           Lf             tjet       



 Round-jet         Momentum           Uodo2/D           do2/D      



 Round-jet         Buoyant            Uodo2/D       (Uodo2/gD)1/2  



  Slot-jet         Momentum           Uodo2/D           do2/D      



  Slot-jet         Buoyant        (Uo4do4/D2g)1/3  (Uo2do2/g2D)1/3 

Table 2. Predicted scalings of flame heights (Lf) and residence times (tjet) for nonpremixed round-jet and slot-jet flames under momentum-dominated and buoyancy-dominated conditions.

FIGURE CAPTIONS

Figure 1. Predicted values of burning velocity and peak flame temperature in CH4 - (0.21 O2 + 0.49 N2 + 0.30 CO2) mixtures under adiabatic conditions, with optically-thin radiative losses, and including reabsorption effects [26].

Figure 2. Characteristics of Self-Extinguishing Flames in CH4-air mixtures at 1 atm for various mole percent CH4 and spark ignition energies [133]. The "x" symbols denotes extinction.

Figure 3. Measured and predicted extinction strain rates for strained premixed CH4-air flames at µg [34] 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. Image of flame balls in a 7.0% H2 - 14.0% O2 - 79.0% SF6 mixture (Le Å 0.06) obtained during the STS-94 space shuttle mission, taken 400 seconds after ignition. The radius of the balls is about 4 mm (they appear to have different sizes because of varying distances from the camera).

Figure 6. Predicted effect of heat loss on flame ball radius and stability properties [44] showing radially unstable (small) flame ball solution, radially stable (large) flame ball solution, and 3-d instability for large flame balls.

Figure 7. Comparison of computed flame ball radii as a function of H2 mole fraction in H2-air mixtures for 3 different H2-O2 chemical mechanisms, along with preliminary results from the STS-83 and STS-94 space experiments [48].

Figure 8. Computed flame ball radius as a function of the H2 mole fraction for steady flame balls in H2-O2-CO2 mixtures with H2:O2 = 1:2, for optically-thin CO2 radiation and with CO2 radiation artificially suppressed (optically-thick limit for CO2 radiation.) Preliminary experimental results from the STS-94 mission are also shown (filled circles) [48].

Figure 9. Measured and predicted extinction strain rates for strained nonpremixed N2-diluted CH4 vs. air counterflow flames at µg [50] showing dual-limit response analogous to premixed flames (Fig. 3).

Figure 10. Measured flame lengths, normalized by jet diameter, as a function of the jet Reynolds number for nonpremixed CH4-air jet flames at 1g and µg. Data is taken from a variety of sources and compiled in [57]. 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 11. Measured flame widths, normalized by jet diameter, as a function of the jet Reynolds number for nonpremixed CH4-air jet flames at 1g and µg. Data is taken from a variety of sources and compiled in [57]. 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 12. Measured flame heights for nonpremixed C3H8-air jet flames at 1g and µg [58] showing transition to turbulence. Note 1g flame lengths are shorter than µg flame lengths, even at very high Reynolds numbers.

Figure 13. Evidence of the validity of flamelet concept, which presumes properties of nonpremixed flames depend only on local mixture fraction (Z), for soot volume fraction [66].

Figure 14. Schematic comparison of structure of non-premixed buoyant (left) and non-buoyant (right) nonpremixed gas-jet flames [67], showing differences between temporal history of temperature and mixture fraction for incipient soot particles at 1g and µg.

Figure 15. Computed residence times as a function of flame length for nonbuoyant flames [70], suggesting that, in contrast to analytical approximations (Eq. (10)), residence time does not asymptote to a constant value at large Re (thus long flame lengths).

Figure 16. Direct photographs of sooting n-C4H10 non-premixed gas-jet flames at 1g (left) and µg (right) at Re Å 42, jet diameter 10 mm, showing evidence of thermophoresis-induced agglomeration at µg. Photographs courtesy of Prof. O. Fujita.

Figure 17. Comparison of burning rate constants (K) as a function of estimated relative velocity between droplet and gas for heptane droplets burning in air for different µg experiments [80, 81]. In the NASA experiments [80], the relative velocity was inferred from soot particles that were assumed to follow the flow, whereas in the Kumagai experiments [81], the velocity of the droplet in the (falling) laboratory frame of reference is shown.

Figure 18. 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) [86].

Figure 19. Effect of initial droplet diameter (ddo) on quasi-steady burning rate (K) for heptane droplets burning in air and an O2-He atmosphere at µg, showing that K decreases with increasing ddo, apparently due to effect of increase accumulation of soot and gas-phase radiant species for larger ddo [85].

Figure 20. Droplet extinction diameter as a function of initial droplet diameter for methanol-water mixtures [89]. Lines are calculations, symbols are experimental results [78]. The observed and predicted dependence of the extinction diameter on the initial diameter is evidence of the effects of water absorption by the methanol fuel.

Figure 21. Direct photographs of candle flames at 1g and µg, showing impact of buoyant flow on flame shape [91].

Figure 22. Flame height vs. time for a candle flame at µg, showing evidence of oscillations just before extinction. Adapted from [91].

Figure 23. Minimum mole percent O2 in N2 supporting flame spread over a thin solid fuel bed, as a function of the opposed flow velocity (U) [100], showing dual-limit behavior, i.e., residence-time limited extinction at high U and radiative loss extinction at low U.

Figure 24. 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 O2 [102], showing dual-limit response. Note that the infinite-rate kinetics prediction [98, 99] that the spread rate is independent of U, is only satisfied at O2 mole fractions higher than that in air.

Figure 25. Flame spread rate over a thin solid fuel bed at 1g (downward spread) and µg as a function of O2 mole fraction [106], showing atypical behavior where the spread rates are higher and the minimum O2 model fraction supporting combustion is lower at µg.

a) Carbon dioxide diluent

b) Sulfur hexafluoride diluent

Figure 26. Fingering patterns observed in smoldering flame spread over a thin paper fuel sample in µg [110]. Flaming combustion was inhibited by soaking the fuel sample in potassium acetate. An imposed convective velocity of 0.05 m/s flows from right to left. Grid pattern scale is 10 mm by 10 mm. Photograph courtesy of Dr. S. Olson.

Figure 27. 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 [116]. 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 (HGE). 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 28. 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.

Figure 1. Predicted values of burning velocity and peak flame temperature in CH4 - (0.21 O2 + 0.49 N2 + 0.30 CO2) mixtures under adiabatic conditions, with optically-thin radiative losses, and including reabsorption effects [26].

Figure 2. Characteristics of Self-Extinguishing Flames in CH4-air mixtures at 1 atm for various mole percent CH4 and spark ignition energies [133]. The "x" symbols denotes extinction.

Figure 3. Measured and predicted extinction strain rates for strained premixed CH4-air flames at µg [34] 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. Image of flame balls in a 7.0% H2 - 14.0% O2 - 79.0% SF6 mixture (Le Å 0.06) obtained during the STS-94 space shuttle mission, taken 400 seconds after ignition. The radius of the balls is 4 to 5 mm (they appear to have different sizes because of varying distances from the camera).

Figure 6. Predicted effect of heat loss on flame ball radius and stability properties [44] showing radially unstable (small) flame ball solution, radially stable (large) flame ball solution, and 3-d instability for large flame balls.

Figure 7. Comparison of computed flame ball radii as a function of H2 mole fraction in H2-air mixtures for 3 different H2-O2 chemical mechanisms, along with preliminary results from the STS-83 and STS-94 space experiments [48].

Figure 8. Computed flame ball radius as a function of the H2 mole fraction for steady flame balls in H2-O2-CO2 mixtures with H2:O2 = 1:2, for optically-thin CO2 radiation and with CO2 radiation artificially suppressed (optically-thick limit for CO2 radiation.) Preliminary experimental results from the STS-94 mission are also shown (filled circles) [48].

Figure 9. Measured and predicted extinction strain rates for strained nonpremixed N2-diluted CH4 vs. air counterflow flames at µg [50] showing dual-limit response analogous to premixed flames (Fig. 3).

Figure 10. Measured flame lengths, normalized by jet diameter, as a function of the jet Reynolds number for nonpremixed CH4-air jet flames at 1g and µg. Data is taken from a variety of sources and compiled in [57]. 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 11. Measured flame widths, normalized by jet diameter, as a function of the jet Reynolds number for nonpremixed CH4-air jet flames at 1g and µg. Data is taken from a variety of sources and compiled in [57]. 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 12. Measured flame heights for nonpremixed C3H8-air jet flames at 1g and µg [58] showing transition to turbulence. Note 1g flame lengths are shorter than µg flame lengths, even at very high Reynolds numbers.

Figure 13. Evidence of the validity of flamelet concept, which presumes properties of nonpremixed flames depend only on local mixture fraction (Z), for soot volume fraction [66]. QR is the fraction of heat release assumed to be lost due to thermal radiation.

Figure 14. Schematic comparison of structure of non-premixed buoyant (left) and non-buoyant (right) nonpremixed gas-jet flames [67], showing differences between temporal history of temperature and mixture fraction for incipient soot particles at 1g and µg.

Figure 15. Computed residence times as a function of flame length for nonbuoyant flames [70], suggesting that, in contrast to analytical approximations (Eq. (10)), residence time does not asymptote to a constant value at large Re (thus long flame lengths).

Figure 16. Direct photographs of sooting n-C4H10 non-premixed gas-jet flames at 1g (left) and µg (right) at Re Å 42, jet diameter 10 mm, showing evidence of thermophoresis-induced agglomeration at µg. Photographs courtesy of Prof. O. Fujita.

Figure 17. Comparison of burning rate constants (K) as a function of estimated relative velocity between droplet and gas for heptane droplets burning in air for different µg experiments [80, 81]. In the NASA experiments [80], the relative velocity was inferred from soot particles that were assumed to follow the flow, whereas in the Kumagai experiments [81], the velocity of the droplet in the (falling) laboratory frame of reference is shown.

Figure 18. 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) [86].

Figure 19. Effect of initial droplet diameter (ddo) on quasi-steady burning rate (K) for heptane droplets burning in air and an O2-He atmosphere at µg, showing that K decreases with increasing ddo, apparently due to effect of increase accumulation of soot and gas-phase radiant species for larger ddo [85].

Figure 20. Droplet extinction diameter as a function of initial droplet diameter for methanol-water mixtures [89]. Lines are calculations, symbols are experimental results [78]. The observed and predicted dependence of the extinction diameter on the initial diameter is evidence of the effects of water absorption by the methanol fuel.

                                                          





            1g                           µg               

Figure 21. Direct photographs of candle flames at 1g and µg, showing impact of buoyant flow on flame shape [91].

Figure 22. Flame height vs. time for a candle flame at µg, showing evidence of oscillations just before extinction. Adapted from [91].

Figure 23. Minimum mole percent O2 in N2 supporting flame spread over a thin solid fuel bed, as a function of the opposed flow velocity (U) [100], showing dual-limit behavior, i.e., residence-time limited extinction at high U and radiative loss extinction at low U.

Figure 24. 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 O2 [102], showing dual-limit response. Note that the infinite-rate kinetics prediction [98, 99] that the spread rate is independent of U, is only satisfied at O2 mole fractions higher than that in air.

Figure 25. Flame spread rate over a thin solid fuel bed at 1g (downward spread) and µg as a function of O2 mole fraction [106], showing atypical behavior where the spread rates are higher and the minimum O2 model fraction supporting combustion is lower at µg. a) Carbon dioxide diluent

Figure 25. Flame spread rate over a thin solid fuel bed at 1g (downward spread) and µg as a function of O2 mole fraction [106], showing atypical behavior where the spread rates are higher and the minimum O2 model fraction supporting combustion is lower at µg. b) Sulfur hexafluoride diluent

Figure 26. Fingering patterns observed in smoldering flame spread over a thin paper fuel sample in µg [110]. Flaming combustion was inhibited by soaking the fuel sample in potassium acetate. An imposed convective velocity of 0.05 m/s flows from right to left. Grid pattern scale is 10 mm by 10 mm. Photograph courtesy of Dr. S. Olson.

Figure 27. 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 [116]. 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 (HGE). 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 28. 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.