1335

Twenty-Sixth Symposium (International) on Combustion/The Combustion Institute, 1996/pp. 1335–1343

QUIESCENT FLAME SPREAD OVER THICK FUELS IN MICROGRAVITY

JEFF WEST,1 LIN TANG,2 ROBERT A. ALTENKIRCH,2 SUBRATA BHATTACHARJEE,1 KURT SACKSTEDER3

and MICHAEL A. DELICHATSIOS4

1Department of Mechanical EngineeringSan Diego State University, San Diego, CA 92182, USA

2School of Mechanical and Materials EngineeringWashington State University, Pullman, WA 99164, USA

3NASA Lewis Research CenterCleveland, OH 99135, USA

4Factory Mutual ResearchNorwood, MA 02062, USA

Experimental results for flame spread over thick PMMA in microgravity are reviewed. The results wereobtained aboard three different space shuttle missions, STS-54, STS-63, and STS-64. For the three con-ditions, 50% O2 in N2 at 1 atm, 50% O2 at 2 atm, and 70% O2 at 1 atm, the flame-spread rate slowlydecreases with time, which varied from about 50 s to over 300 s. Computational modeling that includesthe effects of radiation captures the essential features of the flame position versus time trajectory. Whencomputations are carried out past the experimental time, the flames eventually retreat and then extinguishafter spread times of about 450–600 s.

With respect to the flame, the flow velocity into the flame is the spread rate. Absent any additional flowto press the flame close to the surface to provide a heat flux that allows the heated layer in the solid todevelop, the process remains unsteady. The thermal and mass diffusion scales each are approximately thethermal diffusivity of the gas divided by the spread rate. The computed temperature and oxygen fieldsshow that the distances over which temperature changes take place are small compared to those overwhich oxygen diffuses. This effect is due to the radiation causing a reduction in the length scale charac-teristic of the temperature field compared to the mass diffusion scale. The mismatch in the actual thermalscale and the mass diffusion scale grows with time until the oxygen diffusion rate to the flame is unable tosustain it. For fuels with thickness below some critical value, the fuel thickness is heated fast enough andthe spread rate is high enough that the mismatch in the thermal and the mass diffusion scales is unim-portant, and the spread rate is steady.

Introduction

It is well known that the propagation speed of aflame, Vf, spreading over a thick fuel bed in an op-posing flow environment depends on the strengthand character of the opposing flow [1–4]. Buildingon the understanding of the spread of a flame overa thin fuel, in which forward heat transfer throughthe solid is suppressed, we investigate here the morepractical thick fuel configuration for quiescent flamespread in microgravity. This configuration, importantwith respect to fire safety in spacecraft, has not beeninvestigated to any great extent, particularly experi-mentally, because the slow spread rates, muchslower than those for a thin fuel, result in a need forsubstantial experimental time, which is difficult toobtain in Earth-bound facilities such as drop towers.

Here we describe the results of experiments forspread over polymethylmethacrylate (PMMA) sam-

ples in the microgravity environment of the spaceshuttle. The results are coupled with modeling in aneffort to describe the physics of the spread processfor thick fuels in a quiescent, microgravity environ-ment and uncover differences between thin andthick fuels. A quenching phenomenon not presentfor thin fuels is delineated, namely, the fact that forthick fuels the possibility exists that, absent anopposing flow of sufficient strength to press theflame close enough to the fuel surface to allow theheated layer in the solid to develop, the heated layerfails to become “fully developed.” The result is thatthe flame slows, which, in turn, causes an increasein the relative radiative loss from the flame leadingeventually to extinction. This potential inability of athick fuel to develop a steady spread rate is not pres-ent for a thin fuel because the heated layer is thefuel thickness, which reaches a uniform temperatureacross the thickness relatively rapidly.

1336 MICROGRAVITY COMBUSTION

TABLE 1Description of the generic terms of Eq. (1)

Equation f Cf-˙sf

Continuity 1 0 0

x momentum u l

]P1

]x

y momentum v l

]P1

]y

Species: fuel yF k/cg

Ta,c21B q y y exp 1c g F 0 5 6TSpecies: oxygen y0 k/cg

-ssF

Species: nitrogen yN k/cg 0

Energy: gas phase T k/cg

10 4 4{Dh s- ` q- 1 4a (T 1 T )}c F ign P,GB `cg

Energy: Solid Phase Ts ks /cs

q-ign

cs

Experiment

The experimental apparatus was largely the sameas used for previous cellulosic sample experiments[5,6]: a 39-L chamber filled before flight with thetest atmosphere, two cine cameras providing top andside views of the PMMA samples, and a data acqui-sition and control computer. Three shuttle missions,space transportation system (STS) flights, were con-ducted with quiescent test atmosphere mixtures ofO2 in N2 at 70% O2/1 atm pressure (STS-54), 50%O2/1 atm (STS-63), and 50% O2/2 atm (STS-64). Ineach flight, two PMMA samples, 25.4 mm long 26.35 mm wide 2 3.18 mm physical thickness (thehalf-thickness in the model formulation), were em-bedded in an aluminum structure with one large faceexposed to the chamber atmosphere in the plane ofthe structure surface and the bottom and side facesinsulated from the structure. A resistively heatedKanthol wire for ignition was embedded 1.6 mmfrom one end. Once ignited, the flame spreads alongthe long dimension of the sample.

Three type-R thermocouples were installed foreach sample: in the gas phase, on the surface, andembedded in the fuel. Differing heights and depthsof the nonsurface thermocouples between the twosamples were to provide a reconstructed five-pointtemperature profile. In 70% O2, 0.127-mm-diame-ter thermocouple wire was used throughout. For the50% O2 tests, 0.076-mm wire was used for the sur-face and embedded thermocouples and 0.025-mmwire in the gas.

A sliding plate mechanism for flame quenchingwas deployed under computer control to preservethe fuel surface shape following flame spreading andto minimize oxygen consumption so that the oxygencontent of the environment could be consideredfixed. In STS-54, the flame arrival time at the sample

end was estimated by the rise time to a thresholdtemperature at the thermocouple locations. In STS-64, the first sample quench time was computed usinga higher threshold temperature, and the secondsample was allowed to burn until the cine film wasconsumed. In STS-63, both samples were allowed toburn until the cine film was consumed, but the cam-eras were started 60 s after ignition for the secondsample.

Model

The mathematical model employed has been re-ported elsewhere [6–8], so only a brief descriptionis presented here. Dimensionless conservation equa-tions can be expressed in a generic format as follows:

](qf) ](quf) ](qmf) ] ]f` ` 4 Cf5 6]t ]x ]y ]x ]x

] ]f` C ` s- (1)f f5 6]y ]y

where the different terms are explained in Table 1.The momentum source terms are written in incom-pressible form as a simplification because the addi-tional viscous terms present, in principle, in com-pressible flows were found earlier to be ratherunimportant in these slow flows with surface blowing[9]. The ignition source term applies only to theshaded ignition zone of the computational domainshown in Fig. 1. The Planck-mean absorption coef-ficient, aP,GB, calculated from the method of globalenergy balance [8,10], considering CO2, H2O, meth-ylmethacrylate vapor, and the temperature distri-bution in the flame accounts for reabsorption of ra-diation despite the apparent thin optical limit of theradiation source term. Values of aP,GB were found to

QUIESCENT FLAME SPREAD IN MICROGRAVITY 1337

Fig. 1. Computational domain and boundary conditions for unsteady computation in laboratory-fixed coordinates.

4 m9 for fuel and 4 0 for others.˙ ˙J9 J9i i

vary between 2 and 4 m11. Feedback of gas radiationto the fuel surface as well as surface reradiation areincluded.

In addition to the conservation equations, theequation of state for density, a square-root depen-dence of viscosity and thermal conductivity in thegas on temperature, and a pyrolysis formula basedon negligible surface regression, constant solid den-sity and first-order kinetics [11] are used to completethe formulation.

P M l k T` `q 4 ; 4 4¯ !RT l k Tr r r

1/22q T B ks s p sm9 4 5 60T [3.615Dh ` 4.605c (T 1 T )]a,p v s s `

Ta,pexp 1 (2)1 22Ts

The properties used are: s 4 1.92; 4 25.90DhcMJ/kgfuel; Bc 4 5.928 2 109 m3/kg • s; Ta,c 4 10,700K; cg 4 1.183 kJ/kg •K; kr 4 0.086 W/m •K; lr 40.675 2 1013 N • s/m2; Tr 4 (T` ` Tad)/2 4 1958K; M` 4 30 kg/kmol; cs 4 1.465 kJ/kg •K; 40Dhv0.941 MJ/kgfuel; ks 4 0.188 W/m •K; qs 4 1190 kg/m3; Bp 4 2.282 2 109 s11; and Ta,p 4 15,600 K.Boundary conditions are depicted in Fig. 1.

The set of equations are solved numerically usingthe SIMPLER algorithm [12]. A 25.4-mm-long sam-ple of PMMA, with the same width and half-thick-ness dimensions as the experiment, is embeddedflush with an inert surface, with the exception of the70% O2 environment for which the computational

sample was 10 mm longer, for reasons that will beapparent below. The inert surface is used to directthe gas flow toward the reaction zone and to providesuitable boundary conditions for analysis. The down-stream inert surface, behind the ignition end, is 4cm long, and the upstream inert surface, towardwhich the flame is spreading, is 23.46 cm long (22.46cm for 70% O2). The computation is performed overa 30-cm-long (xmax 1 xmin in Fig. 1) 2 20-cm-high(ymax 1 ymin) domain. The ignition power input perunit volume in the region indicated in Fig. 1 is 2.22 108 W/m3, which when multiplied by the ignitionvolume results in 2.4 W of ignition power, whichcorresponds approximately to the experimental in-put. This power input is continued until flame igni-tion occurs, which is usually at approximately 1.8 s,at which time it is shut off.

Computations on several grids were performed toestablish the level of refinement required for essen-tially grid-independent results, the grid chosen being140 (x nodes) 2 52 (y nodes) with 8 y-directionnodes in the solid. Along the y direction, the gridstep size is 0.625 mm from the fuel surface into thegas to y 4 10 mm. After 10 mm, a power-law vari-ation for grid distribution to the top of the domainis used. The grid distribution along the x directionconsists of a step size of 1.74 mm from x 4 0 to x4 4 cm (downstream inert surface), followed by astep size of 0.283 mm to x 4 6.54 cm (fuel sample),and then followed by a power-law variation to theright-hand side of the domain (upstream inert sur-face). The time step ranged from 0.05 to 0.5 s, and

1338 MICROGRAVITY COMBUSTION

Fig. 2. The progress of spreading-flame position overburning PMMA samples. The flame in 70% O2, 1 atm (cir-cles) appears retarded by the thermocouples (0.127-mmwire). The position of the 50% O2, 2-atm flame (squares)is obscured by the glowing thermocouple at about 50 s afterignition (0.025-mm wire), but flame progress is unaffected.Data from the other 50% O2, 2-atm flame (not shown) isindistinguishable from the first until it is quenched by thecomputer at about 70 s after ignition. Two flames in 50%O2, 1 atm (triangles) show distinguishable thermocoupleinfluence: The flame encountering the thermocouple 3 mmabove the fuel surface (sample 1) follows a trajectory de-layed but otherwise similar to its companion encounteringthe thermocouple 1 mm above the surface.

Fig. 3. Computed and experimental (from Fig. 2) flameposition as a function of time.

the relative convergence criterion for all field vari-ables was 0.005.

Results and Discussion

Experimental Results

For brevity, only flame position as a function oftime data are presented here. These data allow adescription of the phenomena involved to be devel-oped. In Fig. 2, position of the spreading flames,measured using image analysis software [13], withtime is shown. The flattened portions of these plotsare associated with the flames encountering the gas-phase thermocouples, which glow brightly and partlyobscure the dim flame images. The size of the gas-phase thermocouples was initially chosen to matchthat of earlier experiments with thin, cellulosic fuels[6]. Following the first experiment at 70% O2 whenit was found that thermocouples of that size influ-ence the spread process for a portion of the experi-ment, presumably because of the low flame temper-

atures due to substantial radiative heat loss from theflame, computed to be as much as 60% of the heatreleased, Earth-bound near-extinction experimentswere conducted to determine if the use of smallerthermocouples would be less intrusive, which it wasfound to be. As a result, for subsequent flight ex-periments, the smallest thermocouples that could bephysically installed were used. Once the flames passthe thermocouple, they resume a trajectory of grad-ually decreasing slope, that is, a decreasing spreadrate. The flame in 70% O2, 1 atm was beginning toescape the thermocouple when it was quenched bythe experiment’s computer.

Modeling Results

In Fig. 3, the experimental results are presentedagain for comparison to the computational results.The computations track the flame progression rela-tively well as the flame spreads from time zero andcaptures the fact that the spread rate decreases withtime. As the computations are extended beyond theexperimental time, the flame continues to spread ata decreasing rate until it reaches a maximum dis-tance of progression, at which time it retreats andthen extinguishes. This behavior was found for allthree environments considered. For 70% O2, theprogression was beyond the 25.4-mm experimentalsample length, and so computationally, the samplewas extended 10 mm as mentioned above.

QUIESCENT FLAME SPREAD IN MICROGRAVITY 1339

Fig. 4. Computed spread rate, normalized with the max-imum computed spread rate, as a function of time for fourdifferent oxygen concentrations.

Discussion

Previous results for thin cellulosic fuels in a qui-escent, microgravity environment showed that theflame-spread rate following ignition quickly adjustedto a fixed value [6,10]. Here we find, however, thatfor the thick fuels, the spread rate continually de-creases with time until eventually, computationally,the flame extinguishes. This behavior is somewhatunderstandable, as described below.

Consider a region near the leading edge of theflame in which the thermal diffusion length is Lg 4ag/Vf, Vf being the velocity scale. Actually, the lengthover which heat is transferred is less than ag/Vf be-cause of radiation [14], the consequences of whichwill be evident later, but initially in the spread pro-cess, radiation is unimportant until the flame evolvesto sufficient size. Within the solid, there is a y-di-rection thermal scale for the heated layer depth of

, ag/ being the time the solid is2 2d 4 a a /V V!y s g f fheated from T` to Tv and an x-direction scale of dx4 as/Vf.

At the flame leading edge, an approximate steady-state energy balance on the solid gives

q c (T 1 T ) V ds s v ` f y

0 4 4' k (B 1 r) Dh /c 1 e r(T 1 T ) d (3)g v g s v ` x

where the first term on the right-hand side is theheat conduction from the flame to the fuel upstreamthrough the gas and solid [15,16], and the secondterm is the radiative loss from the surface in whichthe x-direction length in the solid has been taken tobe larger than that in the gas for now. For a thin fuel,in which dy is s, dy/dx is small such that upstreamconduction in the solid is negligible, and dx is Lgbecause the heat conduction is through the gas, weget a steady solution for Vf as long as the heat con-ducted from the gas exceeds the heat lost by radia-tion; that is,

0k (B 1 r) Dhg v2V 'f c q c (T 1 T ) sg s s v `

20 4 4k (B 1 r) Dh 4e r(T 1 T ) ag v s v ` g` 11 2! c q c (T 1 T ) s q c (T 1 T ) sg s s v ` s s v `

(4)

When radiation is neglected, Eq. (4) gives the deRis–Delichatsios formula [1,15,16], except for a fac-tor of p/4. As mentioned above, experiments for thincellulosic samples in a quiescent, microgravity en-vironment yield a steady spread rate for those envi-ronments in which the flame spreads.

Assuming slow variations from one steady state toanother, the unsteady counterpart of the above en-ergy balance, Eq. (3), is obtained by adding the term(d/dt) qscs(Tv 1 T`)dydx to the left-hand side inwhich we use the same steady-state length scales asan approximation. For the thin fuel again, with dx 4Lg and dy 4 s, the unsteady term divided by theconvective term gives (1/Vf)d/dt ag/Vf so that thetimescale becomes ag/ , which is of the order of 1–2Vf10 s for the spread rates measured [6,10]. For thethick fuel, with dx 4 as/Vf, the timescale is as/ .2VfWith as ù 1013 ag and the spread rate approxi-mately two orders of magnitude, or more, smallerthan for the thin fuel, the timescale can be more thanan order of magnitude larger than that for a thin fuel,large to the extent that the flame is unable to adjustto a steady state as the in-depth heated layer in thesolid near the leading edge continues to grow withtime. From the approximate energy balance for thethick fuel, neglecting the radiative loss, we get, earlyin the spread process, that Vf } 1/ , where the pro-t!portionality factor is inversely proportional to B.That is, for higher oxygen concentrations, Vf de-creases initially more rapidly. Once this decreasetakes place, the flame seeks to adjust to a steady statesuch that the Vf dependence on t changes and is afunction of the environment.

In Fig. 4, we show computational spread rates forseveral O2 in N2 environments normalized by themaximum spread rate for the particular condition.The higher the oxygen concentration, the faster thespread rate decreases initially as the flame seeks asteady state. However, for the environmentalconditions of Fig. 4, the flames extinguished with thetime to extinction being longer for the higher oxygenconcentrations. After the initial adjustment in spreadrate, the higher the oxygen concentration, the moregradually the spread rate decreases to extinction.

Steady spread over PMMA in a quiescent, micro-gravity environment should exist if the fuel, orheated layer in the solid, is thin enough as it existsfor thin cellulosic samples. Steady-state computa-tions, in which the unsteady term in the computa-tional model is dropped and the steady computationscarried out in flame-fixed coordinates, as described

1340 MICROGRAVITY COMBUSTION

Fig. 5. Computed steady spread rates, normalized withthe maximum spread rate computed for the same thin fuelthickness (obtained under conditions of infinitely fast, gas-phase chemistry [17] with the maximum possible heat con-duction forward from the flame) as a function of fuel thick-ness normalized with the critical fuel thickness from Eq.(5). Steady spread rates are not found using finite-ratechemistry for thicknesses above those shown.

elsewhere [8,10] (except that the solid-phase modelfor PMMA used here is employed), show that forthicknesses of and below 0.05 mm for 50% O2/1 atmand 0.2 mm for 100% O2/2 atm, steady spread forPMMA of the same width as the experimental sam-ples is obtained. For these computations, a domainapproximately 111 cm long and 89 cm high with a90 2 50 nonuniform grid in the gas and 10 grid linesin the solid was used. The relatively large domain isneeded to allow the oxygen field to develop. Thecomputed spread rates for these limiting thicknessesare approximately 2 and 1 mm/s, respectively. Thesethicknesses are comparable to the thickness of 0.083mm for which cellulosic samples gave steady spreadrates in an O2 percentage as low as 35% at 1 atm [6].

In summary, steady spread rates of a few mm/sover fuels of a thickness of about 0.1 mm or less canbe obtained. Thicker fuels and lower spread ratesexhibit unsteady spreading, the transition betweensteady and unsteady spreading being, of course, en-vironment dependent. This “critical” thickness canbe qualitatively derived from Eq. (4) when the twoterms under the radical sign balance one another.

In Fig. 5, we show computed spread rates fromthe steady-state computations mentioned above.The spread rates are normalized with a theoreticalspread rate for infinitely fast, gas-phase chemistryderived from an extension of de Ris’s Oseen-flow-approximation theory [1]. The expression forVf,thin,EST [17] takes into account details of the hy-drodynamic flow field and flame lift, neither of whichare present in Oseen flow. The critical thickness de-rives from setting the terms under the radical signin Eq. (4) equal to zero to give

2kg 0(B 1 r)Dhv1 2cgs 4 (5)crit 4 4q c (T 1 T )4e r(T 1 T )as s v ` s v ` g

The computations in Fig. 5 are for “wide” samplesin order to minimize the effects of radiative lossesfrom the sides of the flame. Under this approxima-tion, radiative and surface losses have similar effects,and the fact that the arguments leading up to Fig. 5contain only surface losses becomes less importantthan for the narrower experimental samples whosewidth was chosen in part to minimize oxygen con-sumption.

The results of Fig. 5 show a qualitative correlationbetween spread rate and thickness. Steady spreadingis obtained up to thicknesses of about 3–4, beyondwhich no steady spread is found. The depression ofthe spread rate below unity for the thin samples isdue to the effects of radiation. Without radiation, Vf/Vf,thin,EST is unity for s/scrit less than unity. For Fig.5, the same value of the Planck-mean absorption co-efficient was used for all three conditions, whichturns out to be a reasonable approximation for thewide samples. For narrow samples, when side lossesare accounted for in the radiation modeling, resultspresented as in Fig. 5 do not correlate as well, butthe result that there is some thickness above whichsteady spreading is not obtained remains.

The mechanism causing the unsteady spread andeventual extinction for the thick fuel can be ex-plained by comparing computed temperature andoxygen contours at several times during the unsteadyspread process, as shown from computations in Fig.6. The thermal diffusion scale in the gas is ag/Vf, asmentioned above, and for unit Lewis number, thisis the mass diffusion scale as well. However, the ac-tual thermal length scale in the gas over which tem-perature changes occur is reduced because of theeffects of radiation [14]. The reduction increases asthe spread rate decreases, and the effects of radia-tion become more pronounced while the mass dif-fusion scale increases. As a result, as the flameevolves over time for the thick fuel, the spread ratedecreases with time, and the mismatch between thethermal length scale and the mass diffusion scalegrows. Eventually, the diffusion rate of oxygen to thehigh-temperature flame, proportional to yox,`/Lg,is too small to sustain the flame, and the flameextinguishes. This behavior is evident in Fig. 6.While there are oxygen concentrations below whichthin fuels are unable to support spreading flames,and this inability has been identified with the effectsof radiation on the gas-phase chemical kinetics [8]and oxygen diffusion [18], apparently the spread ratefor thin fuels is high enough that the flame spreadsat a steady rate.

QUIESCENT FLAME SPREAD IN MICROGRAVITY 1341

Fig. 6. Computed temperature (left-hand side) and oxygen (right-hand side) contours for 50% O2/2 atm (y0` 4 0.533)for three different times during the spread process. The outermost oxygen contour is for y0` 4 0.45, and the temperaturecontours are from 450–1350 K for 48 s to 1050 K for 261 and 450 s. The thickness of the region over which oxygendiffuses grows faster with time than the thickness of the thermal layer surrounding the flame such that the high-tem-perature contours find themselves in a region in which the flux of oxygen is decreasing with time.

Conclusions

Experiments for flame spread over flat surfaces ofthick fuels in microgravity show that the flame-spread rate slowly decreases with time. Because thefuel is thick, the heated layer in the solid evolves intime. Spread rates are low enough that radiative ef-fects are important, and they cause the distance overwhich temperature changes occur in the gas to bereduced over what it would be in the absence ofradiation when the conduction and mass diffusionscales are comparable. In contrast, the distance overwhich oxygen must diffuse to reach the flame is un-affected by radiation. As a result, this distance growsin time as the flame slows, which results in a reduc-tion of the diffusion rate to the flame to lead even-tually to flame extinction. There appears to be a crit-ical thickness below which steady spreading isobtained in microgravity. Below this thickness, thefuel is heated rapidly, and the spread rate remainshigh enough that any difference in the distance over

which temperature changes and mass diffusion oc-cur is unimportant in comparison to the thick fuels.

Acknowledgments

This work was supported by NASA through ContractNAS3-23901. We thank Sandra Olson for serving as a con-tract monitor during a period of the project and Prof. S. V.Patankar for providing us with an initial version of the soft-ware. We gratefully acknowledge the contributions ofRalph Zavesky, John Koudelka, and the SSCE flight hard-ware team at the NASA Lewis Research Center and theprogram of support of NASA Headquarters.

Nomenclature

B transfer number [17]Bc pre-exponential factor of the gas-phase re-

action, 5.928 2 109 m3/kg • sBp pre-exponential factor the pyrolysis reac-

tion, 2.282 2 109 s11

1342 MICROGRAVITY COMBUSTION

cg specific heat of the gas, 1.183 kJ/kg •Kcs specific heat of the PMMA, 1.465 kJ/kg •KEg activation energy for the gas-phase reac-

tion, 88,950 kJ/kmolEign total energy added by the ignitor during

ignition, JEs activation energy for the pyrolysis reaction,

129,800 kJ/kmolLg gas-phase thermal and mass diffusion

scale, ag/Vf, mLp length of the pyrolysis zone, mm9 mass flux, kg/m2 • sM` molecular weightP pressure, atmq9ign ignitor heat flux, W/m2

r stoichiometric fuel–air ratios stochiometric air–fuel ratio, M0/( Mf)v8 v80 fs-f source term in the conservation equations

(see Table 1)t timeT temperature of the gas, KTa,c activation temperature of the gas-phase re-

action, KTa,p activation temperature of the pyrolysis re-

action, KTs temperature of the solid fuel, KT` temperature of the ambient gas and fuel,

Ku gas velocity in the x direction, m/sv gas velocity in the y direction, m/sVf spread rate, m/sVf,thin,EST limiting spread rate for thin fuels for infi-

nite-rate chemistry [17]x coordinate parallel to the fuel surface, my coordinate perpendicular to the fuel sur-

face, myo` oxygen mass fraction in the environment

Greek Symbols

as thermal diffusivity of the gas, m2/sas thermal diffusivity of the condensed phase,

m2/sdx, dy x- and y-direction lengths, m

0Dhc heat of combustion for the gas-phase re-action, 25,900 kJ/kg

0Dhv heat of evaporation for the pyrolysis reac-tion, 941.08 kJ/kg

f generic dependent variable, Eq. (1)C diffusion coefficient, Eq. (1)es radiative emittance of the fuel surface,

unitykg thermal conductivity of the gas, W/m •Kks thermal conductivity of the solid phase, W/

m •Kl absolute viscosity of the gas, N • s/m2

q density of the gas, kg/m3

qs density of the solid fuel, 1190 kg/m3

s fuel half-thickness, mscrit critical fuel half-thickness, Eq. (5), m

Subscripts

ad adiabaticcrit criticalf flameg gas phaseign ignition` ambient conditionsmax maximummin minimump pyrolysisr references solid phasev vaporization

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