Flame spread along horizontal solid fuel cylinders

Proceedings of the Combustion Institute, Volume 29, 2002/pp. 211–217

FLAME SPREAD ALONG HORIZONTAL SOLID FUEL CYLINDERS F. J. HIGUERA ETS Ingenieros Aerona´uticos Pza. Cardenal Cisneros 3 28040 Madrid, Spain

The spread of a flame along a thermally thick horizontal solid fuel cylinder in a quiescent atmosphere is analyzed modeling the gas-phase kinetics by means of a single-step irreversible Arrhenius reaction and assuming that vaporization of the solid occurs at a constant temperature. The natural convection flow of the warm gas surrounding the flame is mostly normal to the axis of the cylinder, but a weak flow is also induced along the axis by the axial gradient of the pressure variation that gravity generates in the warm gas. This latter flow aids flame propagation in the lower part of the cylinder. The effects of finite rate kinetics and of radiative losses from the surface of the solid are studied, computing the spread rate as a function of the Damko¨hler number, the oxygen concentration in the atmosphere and the strength of the radiative losses for a set of material properties chosen to mimic flame spread on PMMA. The effect of a small inclination of the cylinder is also discussed, showing that a continuous transition from upward to downward flame spread occurs when the tilt angle is small of the order of the inverse of the power 1/8 of the Grashof number based on the radius of the cylinder.

Introduction The characteristics of the spread of a flame along a cylinder of solid fuel depend on the inclination of the cylinder to the vertical or to the direction of the forced stream acting on the cylinder. Upward and wind-aided flame spread along vertical or streamaligned cylinders are accelerative processes, as they are for flat surfaces, in which the streamwise extent of the region of warm gas above or downstream of the vaporization front increases continuously with time, at least insofar as radiative losses can be neglected. But, contrary to the case of flat surfaces, flame spread along cylinders ceases to be accelerative when the cylinder is at an angle to the vertical or to the incoming stream. This is because the natural or forced convection around an inclined cylinder develops a component perpendicular to the axis of the cylinder that sweeps the warm gas and prevents heat from reaching distances above or downstream of the vaporization front that are large compared with the radius of the cylinder. Tewarson and Khan [1] studied experimentally the upward spread of a flame along the coating of vertical electric cables and pine samples, while Sibulkin and Lee [2], Bakhman et al. [3,4], and Weber and de Mestre [5] investigated the effect of the inclination of the sample. Tizo´n [6] and Tizo´n et al. [7] showed that the analysis of these flame-spread problems can be reduced to the analysis of the flow around the lowest or windward generatrix of the cylinder. Tizo´n’s analysis is extended here to the case of horizontal cylinders in

the absence of forced flow, for which the axial buoyancy-induced flow is similar in character to the natural convection flow around a heat source in a horizontal ceiling [8]. An analysis of the problem for infinitely fast gas-phase reactions has been presented elsewhere [9].

Order of Magnitude Estimates The scales of the natural convection flow accompanying the spread of a flame along a horizontal solid fuel cylinder in a quiescent oxidizing atmosphere have been discussed by Weber and de Mestre [5]. These scales can be worked out as follows. Assume that the Grashof number Gr ⳱ q20ga3/l20 k 1, where q0 and l0 are the density and viscosity of the ambient gas, g is the acceleration of gravity, and a is the radius of the cylinder, as in the sketch in Fig. 1. Then the flow and the flame are confined to a natural convection boundary layer surrounding the warm part of the cylinder and to a plume above the cylinder. The order-of-magnitude balance of convection, buoyancy and viscous forces in the boundary layer, assumed to be laminar, is q0w2c /a ⳱ q0g ⳱ l0wc/d2g, where wc and dg are the characteristic values of the velocity of the gas and the boundary layer thickness, respectively, and the density variation due to the heat released by the flame has been taken to be of the order of q0. These conditions yield the well-known results wc ⳱ g1/2a1/2 and dg ⳱ a/Gr1/4. The heat flux reaching the solid surface is of order qg ⳱ k0DTg/dg, where k0 is the thermal conductivity of the ambient

211

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FIRE—Ignition and Flame Spread

so that ds K lc and conduction is essentially radial, this condition reads qscsUDTs/lc ⳱ ksDTs/d2s in order of magnitude, where qs and cs are the density and specific heat of the solid, and ds is the thickness of the heated layer estimated before. From this balance [5], U ⬃ Gr3/8N2 Fig. 1. Definition sketch.

gas and DTg is the characteristic gas-phase temperature variation, equal to the difference between the adiabatic flame temperature and the ambient temperature T0. In the solid, the heat flux is of order qs ⳱ ksDTs/ds, where ks, DTs, and ds are the thermal conductivity of the solid, its temperature variation, equal to the difference between the vaporization temperature and the ambient temperature, and the characteristic thickness of the heated layer. The condition that qg and qs be of the same order determines the characteristic thickness ds ⳱ dg/N, with N ⳱ k0DTg/ksDTs. This thickness is small compared with the radius of the cylinder if NGr1/4 k 1. The natural convection boundary layer develops only on the burning surface and does not contribute directly to the spread of the flame, because the flow in it is perpendicular to the axis of the cylinder and thus does not transfer heat to the nonvaporizing region. Its existence, however, sets the stage for an indirect driving mechanism which is the cause of the natural convection flows around heated horizontal surfaces [8]. This is because the component of the buoyancy normal to the surface originates a hydrostatic pressure variation of order Dp ⳱ q0gdg in the boundary layer, which is an overpressure in the lower part of the cylinder and a depression in the upper part, relative to the gas at rest ahead of the flame front. The axial gradient of this hydrostatic pressure pushes the gas from the warm to the cold region in the lower half of the cylinder surface, aiding flame spread, and from the cold to the warm region in the upper half of the cylinder surface, opposing flame spread (dashed arrows in Fig. 1). The axial extent of the region around the flame front that is affected by this indirect driving mechanism (lc), and the order the axial velocity it induces (uc), follow from an order-of-magnitude balance of axial convection, pressure force, and viscous force: q0uc2/lc ⳱ Dp/lc ⳱ l0uc/d2g, with Dp given by the estimate above. Thus lc ⳱ dgGr1/8 ⳱ a/Gr1/8 and uc ⳱ wc/ Gr1/8 ⳱ vcGr1/8, where vc is the characteristic velocity normal to the cylinder surface. The order of the spread rate, U, can be obtained from the energy equation in the solid, where axial convection in a reference frame moving with the front and conduction balance each other. Assuming that NGr1/8 k 1,

ks qscsa

(1)

If, on the contrary, NGr1/8 K 1, so that ds k lc, the energy equation in the solid requires qscsUDTs/ds ⳱ ksDTs/ds2 in orders of magnitude, and U ⬃ Gr1/4N

ks qscsa

Formulation The solid is assumed to vaporize at a constant temperature Tv, and the fuel vapor reacts with the oxygen of the air in a single irreversible Arrhenius reaction releasing an energy Q per unit mass of fuel consumed. The oxygen-to-fuel mass stoichiometric ratio is s0. The Lewis numbers of both reactants are taken to be equal to unity. The Prandtl number and the specific heat of the gas (Pr and cp) are constant, as well as the density, specific heat, and conductivity of the solid. Radiation losses from the surface are taken into account, though radiative transfer in the gas is left out. The Schvab-Zeldovich variables sYF ⳮ YO Ⳮ 1 1Ⳮ s and H ⳱ T ⳮ 1 Ⳮ c(YF Ⳮ YO ⳮ 1) Z⳱

(2)

will be used in the problem formulation instead of the fuel vapor and oxygen mass fractions. Here T is the gas temperature scaled with the ambient temperature T0, YF is the mass fraction of fuel vapor, YO is the mass fraction of oxygen scaled with its value YO⬁ in the ambient atmosphere, s ⳱ s0/YO⬁ is the air-to-fuel mass stoichiometric ratio, and c ⳱ (Q/ cpT0)/(1 Ⳮ s). The mixture fraction Z and the excess of enthalpy H are transported by the flow as passive scalars. The fuel consumption rate per unit volume and time is W ⳱ qYO⬁BYFYO exp(ⳮTa/T)

(3)

where B and Ta are a frequency factor and an activation temperature, and the factor YO⬁ appears due to the scaling of YO. Let y denote the distance to the cylinder and x and z the distances on the cylinder surface along and normal to its axis, respectively, measured from the lowest point of the vaporization front (thick dashed curve in Fig. 1), which moves along the bottom of

FLAME SPREAD ALONG HORIZONTAL SOLID FUEL CYLINDERS

the cylinder with a speed U to be determined. In this moving reference frame, and on the lowest generatrix of the cylinder, the vaporizing region where T ⳱ Tv extends to x ⬍ 0 and T ⬍ Tv in the nonvaporizing region x ⬎ 0. It was shown by Tizo´n [6] and Tizo´n et al. [7] for the related case of upward flame spread along an inclined cylinder, that the spread rate U can be determined from the analysis of the flow around the bottom, for z K a, where every variable can be expanded in the form f (x, y, z) ⳱ f1(x, y) Ⳮ O(z/a)2, except for the z-component of the velocity, which is of the form w(x, y, z) ⳱ (z/a)w1(x, y) Ⳮ O(z/a)3. Carrying these series expansions into the boundary-layer equations governing the flow in the gas phase, we are led, at leading order in z/a, to ⳵qu ⳵qv Ⳮ Ⳮ qw ⳱ 0 ⳵x ⳵y

冢

q u

(4)

⳵u ⳵u ⳵p Ⳮv ⳱ ⳮ Ⳮ (1 ⳮ q)gx ⳵x ⳵y ⳵x ⳵ ⳵u Ⳮ Pr k ⳵y ⳵y

冣

冢 冣

(5)

冢

冣

Ⳮ Pr

冢

⳵ ⳵w k ⳵y ⳵y

冢

冣

冢

q u

冢 冣

⳵T ⳵T ⳵ ⳵T Ⳮv ⳱ k ⳵x ⳵y ⳵y ⳵y Ⳮ qDYFYO exp(ⳮTa/T )

冣

(6) (7)

⳵ ⳵ ⳵ ⳵ Ⳮv (Z, H) ⳱ k (Z, H) ⳵x ⳵y ⳵y ⳵y

冣

(8)

冢 冣

qT ⳱ 1

ⳮU

⳵Ts c ⳵T ⳵2Ts Ⳮ s m s⳱ ⳵x cp ⳵ys ⳵ys2

(9) (10)

where the subscript 1 has been suppressed and the gas-phase variables (x, y, u, v, w, p, T, q) have been scaled with the factors (lc/Pr1/4, dg/Pr1/2, uc/Pr1/4, vc/Pr1/2, wc, Dp/Pr1/2, T0, q0), respectively. The thermal conductivity, k, which is scaled with its ambient gas value, k0, will be taken to be of the form k ⳱ T 1/2 is what follows. The Damko¨hler number in the reaction term of equation 9 is D ⳱ (1 Ⳮ s)cYO⬁B/冪g/a, and the activation temperature is scaled with T0. Finally, gx in equation 5 is the negative of the axial component of the acceleration of gravity scaled with gdg/lcPr1/4. This term appears when the cylinder is slightly inclined; then the axial component of the buoyancy, which may aid or oppose flame spread, becomes of the order of the axial pressure gradient when the angle of inclination of the cylinder to the horizontal becomes of the order of dg/lc K 1.

(11)

where the temperature of the solid Ts is scaled with T0, the distance normal to the surface ys (negative in the solid) is scaled with (ks/k0)dg/Pr1/2, the vaporization mass flux m in the second term of the lefthand side, which represents the apparent convection due to the recession of the surface, is scaled with q0vc/Pr1/2, and the spread rate U is scaled with (k0/ks)2Gr3/8Pr3/4ks/qscsa. The boundary conditions at the surface ( y ⳱ ys ⳱ 0) are

冧

(12a)

冧

(12b)

⳵Z ⳱ 0, T ⳱ Ts ⬍ Tv ⳵y ⳵H ⳵T ⳵T k ⳱k ⳱ SR(T 4 ⳮ 1) Ⳮ s ⳵y ⳵y ⳵ys

u⳱v⳱w⳱

for x ⬎ 0, and

mZ ⳮ k

⳵p ⳱ qⳮ1 ⳵y q u

The energy equation for a thermally thick solid in the conditions leading to equation 1 is

u ⳱ w ⳱ 0, qv ⳱ m

⳵w ⳵w q u Ⳮv Ⳮ w2 ⳱ 1 ⳮ q ⳵x ⳵y

213

⳵Z ⳱ m, T ⳱ Ts ⳱ Tv ⳵y

⳵H ⳵T ⳱ m(T ⳮ 1) ⳮ k ⳵y ⳵y ⳵Ts 4 ⳱ ⳮSR(T ⳮ 1) ⳮ mL ⳮ ⳵ys

mH ⳮ k

for x ⬍ 0. Here L is the effective heat of vaporization scaled with cpT0 and SR ⳱ ⑀rT 30dg/Pr1/2k0, with ⑀ and r denoting the emissivity of the surface and the Stefan-Boltzmann constant. Far from the surface and ahead of the flame u⳱ w⳱p⳱Z⳱H⳱Tⳮ1 ⳱0 for y r ⬁ and x r ⬁ Ts ⳱ 1 for ys r ⳮ⬁ and x r ⬁

(13) (14)

while, insofar as the recession of the surface due to fuel consumption can be ignored, the solution takes a well-known asymptotic form for x r ⳮ⬁ in which all the x-derivatives become zero [10–13]. The solution of equations 2–14 should determine u, v, w, p, Z, H, T, and q in the gas, Ts in the solid, m in the vaporizing region of the surface x ⬍ 0, and the spread rate U. The solution depends on the nondimensional parameters c, s, D, Ta, Tv, L, Pr, SR, gx, and cs/cp. Results and Discussion Problem 2–14 was solved numerically using finite differences and a pseudo-transient iteration, which is appropriate because the presence of ⳵p/⳵x in

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FIRE—Ignition and Flame Spread

Fig. 2. Spread rate (solid) and flame tip (dashed, righthand side scale) as functions of YO⬁ for SR ⳱ 0, 0.1, and 0.2, increasing from top to bottom.

equation 5 with p given by equation 7 makes the problem for the gas elliptic despite the boundary layer approximation (see Ref. [14]). A set of parameter values was chosen to mimic flame spread along a cylinder of PMMA in ambient temperature atmospheres with different oxygen mass fractions. In these conditions s0 ⳱ 1.92 and Q/(cpT0) ⳱ 80.1, leading to [7,15]

冧

1.92 80.1 ,c⳱ , Ta ⳱ 75.2 YO⬁ 1Ⳮ s c Tv ⳱ 2.3, L ⳱ 3.6, s ⳱ 1.667, Pr ⳱ 0.72 cp (15) s⳱

Solutions were computed for different values of the oxygen mass fraction YO⬁, the Damko¨hler number D (which gauges the radius of the cylinder), and SR. Infinite Fast Reaction Consider first the solution in the Burke-Schumann limit D r ⬁, in which the energy equation (equation 9) simplifies to YFYO ⳱ 0 outside the flame, which is an infinitely this sheet. The reactants do not coexist and their mass fractions on different sides of the flame are given in terms of the mixture fraction by (YF, YO) ⳱ [0, 1 ⳮ Z/Zs] for Z ⬍ Zs and (YF, YO) ⳱ [(Z ⳮ Zs)/(1 ⳮ Zs), 0] for Z ⬎ Zs, where Zs ⳱ 1/(1 Ⳮ s) is the value of Z at the flame. The nondimensional spread rate, denoted by UBS, is given in Fig. 2 as a function of YO⬁ for three different values of the parameters SR measuring the radiation heat losses from the surface. Also plotted in Fig. 2 is the distance from the vaporization front to the foremost point of the flame, which is at the surface ahead of the front in the Burke-Schumann limit. As can be

seen, the spread rate decreases monotonically with decreasing YO⬁ and no solution exists below a minimum YO⬁ which increases with SR. The distance from the vaporization front to the nose of the flame first increases and then decreases with decreasing YO⬁, at least for small values of SR. This is because the air-to-fuel stoichiometric ratio s is inversely proportional to YO⬁ and the flame requires more oxygen and shifts away from the surface when YO⬁ begins to decrease, until this tendency is offset by the fall of the vaporization flux and fuel mass fraction when YO⬁ decreases further. In computations carried out varying c at constant s, the flame was seen to move monotonically toward the surface with decreasing c. In any case, the value of YO⬁ or c for which a solution ceases to exists, and the corresponding value of the spread rate and the distribution of vaporization flux, are bounded away from zero. Extinction in the Burke-Schumann limit considered here reflects the condition that the heat released by the flame, which does not tend to zero, ceases to sufficient to heat up the solid under the vaporizing surface and make up for the heat of vaporization (see Sibulkin et al. [16]). Finite Rate Effects The flame quenches before reaching the cold surface when the Damko¨hler number is finite. Inspection of the numerical solutions reveals that the two reactants coexist in a quench layer, with the natural convection flow making the leakage of fuel into the region ahead of the flame more prominent than the leakage of oxygen into the region between the flame and the surface. The thickness of the flame increases and its front recedes away from the surface when the Damko¨hler number decreases, until the flame extinguishes at a value of the Damko¨hler number that is at or very close to the extinction value for the uniform burning state at x r ⳮ⬁ [10,11]. The spread rate U scaled with its Burke-Schumann value UBS is given in Fig. 3 as a function of the Damko¨hler number for YO⬁ ⳱ 0.23 and SR ⳱ 0. The modified Damko¨hler number used in this fig¯ ⳱ s(1 Ⳮ s)2(T f4/cT a3)D exp(ⳮTa/Tf), where ure is D Tf ⳱ 1 Ⳮ c ⳮ Zs[L Ⳮ (cs/cp)(Tv ⳮ 1)] is the temperature of the flame for x r ⳮ⬁ in the BurkeSchumann limit. This modified Damko¨hler number is the parameter that would appear in a high-activation energy asymptotic analysis (Lin˜a´n [17]); it is a more accurate measure than D of the relative importance of chemical reaction to conduction in equation 9, because it accounts for the small values of the mass fractions of the reactants in the reaction region and the small thickness of this region compared with the boundary layer thickness. Values of the spread rate computed for other values of YO⬁ and of the activation temperature fall close to the curve of Fig. 3, in line with the results of Tizo´n et al. [7].

FLAME SPREAD ALONG HORIZONTAL SOLID FUEL CYLINDERS

Fig. 3. Spread rate scaled with its Burke-Schumann value as a function of the modified Damko¨hler number defined in the text, for YO⬁ ⳱ 0.23 (Tf ⳱ 8.95).

Fig. 4. Spread rate as a function of gx for YO⬁ ⳱ 0.18, 0.23, and 0.33, increasing from bottom to top. Dashed lines give the asymptotic behavior for gx r Ⳳ⬁.

Slightly Tilted Cylinders The spread rate as a function of gx, which measures the inclination of the cylinder, is given in Fig. 4 in the Burke-Schumann limit for SR ⳱ 0, three values of YO⬁, and the values of the other parameters used before. Positive values of gx correspond to upward spread, with the axial component of buoyancy aiding the pressure gradient to push the gas in the direction of flame spread. Negative values of gx correspond to downward spread. In this case, the axial velocity is negative (upward) far downstream of the flame front, but the pressure gradient is sufficiently

215

strong to generate positive axial velocities (downward) around the flame front, where the thickness of the boundary layer and the overpressure generated in the warm gas via equation 7 decrease with increasing x. The pressure gradient can be neglected in equation 5 compared with the axial buoyancy force when gx k 1. The balance of convection, buoyancy, and viscous force in the boundary layer where y ⳱ O(1) then requires u2/x ⳱ gx ⳱ u in orders of magnitude, leaving out order unity factors of q, 1 ⳮ q, Pr, and k. Thus, u ⬃ x ⬃ gx in the region of buoyancy-driven axial flow around the flame front. The heat flux reaching the surface in this region is ⳵T/⳵y ⳱ O(1), and therefore ys ⳱ O(1) in the thermal layer of the solid in order to cope with this flux. The balance of axial convection and radial conduction in the energy equation 11 then gives U ⳱ O(gx). The limiting problem that is obtained rescaling x, u, and U with gx and then letting gx r ⬁ coincides with the small inclination limit of the analysis of Ref. [6] for upward propagation with non-small inclination angles. Its solution determines the asymptotic behavior leading to the dashed curves at the right of Fig. 4. In particular, U/gx ⬇ 29.49 for the set of parameter values in equation 15 with Yo⬁ ⳱ 0.23. When (ⳮgx) k 1, on the other hand, the warm gas in pushed upward and a boundary layer of thickness y ⳱ O(1) develops only at distances of O(ⳮgx) downstream of the front. The characteristic values of the variables in the much thinner boundary layer that exists in the region around the front where the axial pressure gradient matters, and the axial extent of this region, are determined by the order-ofmagnitude balances u2/x ⳱ p/x ⳱ gx ⳱ u/y2 and p/y ⳱ 1 (from equations 5 and 7), which give the scales x ⳱ (ⳮgx)ⳮ5/3, y ⳱ (ⳮgx)ⳮ2/3, and u ⳱ (ⳮgx)ⳮ1/3. The scales of the thickness of the thermal layer in the solid and the spread rate are ys ⳱ (ⳮgx)ⳮ2/3 and U ⳱ (ⳮgx)ⳮ1/3, from the conditions that all or part of the heat flux reaching the surface, ⳵T/⳵y ⳱ O[(ⳮgx)2/3], should enter the solid and from the balance of axial convection and radial conduction in the solid, respectively. A limiting problem can be set up rescaling the variables with their characteristic values and then letting gx r ⳮ⬁; see Ref. [9]. From the solution of this problem, (ⳮgx)1/3U ⬇ 9.71 for Yo⬁ ⳱ 0.23 (dashed curve at the left of Fig. 4). The region around the front ceases to be slender, and the boundary layer approximation breaks down, when y/x ⳱ ⳮgx becomes of the order of lc/dg ⳱ Gr1/8, which happens when the angle of the cylinder to the horizontal ceases to be small, because gx is just this angle scaled with Grⳮ1/8 (up to a power of Pr). By then U ⳱ O(Grⳮ1/24), which coincides with the order of the rate of downward spread (see, e.g., Ref. [18]). The present analysis bridges the gap between upward spreading with non-small inclination

216

FIRE—Ignition and Flame Spread

angles (gx ⳱ O(Gr1/8) k 1; see Ref. [6]) and downward spreading. Comparison with Experiments Experiments dealing with flame spread along solid fuel cylinders at different orientations have been cited in the introduction [2–5]. In these and related experiments with fuel slabs, the spread rate is always found to increase when the direction of flame propagation changes from downward through horizontal to upward, a result discussed by Weber and de Mestre [5] in terms of the variation of the dominant mechanism of heat transfer with the inclination of the sample. On the basis of their experimental results with pine needles and those of Sibulkin and Lee [2] with PMMA, Weber and de Mestre [5] proposed a number of correlations for the spread rate as a function of the inclination, ranging from extensions of equation 1 for inclined samples to curve fitting. The results of the preceding section show the relatively narrow range of inclinations of the sample to the horizontal in which conduction, which is the dominant mechanism of heat transfer for downward propagation, and high Gr natural convection, which is the dominant mechanism for upward propagation, are equally important The spread rate computed for horizontal cylinders of PMMA can be compared with the experimental results of Sibulkin and Lee [2]. These authors report a spread rate of about 2.3 ⳯ 10ⳮ4 m/s for a rod of cast PMMA of radius a ⳱ 6.35 ⳯ 10ⳮ3 m, which is the largest radius used in their experiments. The Grashof number is then Gr ⳱ 11152. Evaluating the scaling factor for the spread rate in the paragraph following equation 11 with qs ⳱ 1200 kg/m3, cs ⳱ 1875 J/kg K, ks ⳱ 0.225 J/m s K and k0/ks ⳱ 0.113 (taken from Refs. [7,15]), and using the results of Fig. 2 for Yo⬁ ⳱ 0.23 and SR ⳱ 0, the computed spread rate is 1.4 ⳯ 10ⳮ4 m/s in the infinite Damko¨hler number limit, which is justified for this cylinder radius. While part of the difference with the measured value can be due to uncertainties in the values used for the PMMA properties and in the experiments with rods of this fuel, the computed value is still too low. The discrepancy can be understood noticing that the boundary-layer approximation used here leaves out axial conduction in the gas and solid phases on the assumption that lc/dg ⳱ Gr1/8 and lc/ds ⳱ NGr1/8 are both large numbers, while their values for the present case are 3.21 and 0.69, respectively (using DTg/DTs ⳱ (Tf ⳮ 1)/(Tv ⳮ 1) to evaluate N, with Tf defined in the section on finite rate effects). It is clear that axial conduction in the solid should have been retained in equation 11 for this moderate Grashof number and a material as conductive as PMMA. Its effect is to heat up the solid ahead of the flame and thus to increase the spread rate. A criterion of applicability of the analysis

of this paper can be deduced from a comparison of equation 1, which predicts a spread rate increasing moderately with the radius of the cylinder, as a1/8, when finite rate effects are left out, and the estimate for NGr1/8 K 1 following equation 1, which predicts a spread rate decreasing as 1/a1/4. Fig. 5 of Sibulkin and Lee [2] shows a spread rate that decreases when the radius of the cylinder increases and is just leveling out for their largest cylinders.

Conclusions Flame spread along a horizontal solid cylinder in a quiescent atmosphere is due to the natural convection flow induced under the cylinder by the axial gradient of the hydrostatic pressure that gravity generates in the warm gas. The boundary layer approximation is applicable to describe this flow and determine the spread rate. Moreover, only the flow around the bottom of the cylinder needs to be computed. The spread rate has been determined as a function of the Damko¨hler number for a single-step Arrhenius gas-phase reaction, showing that extinction occurs at a value of the Damko¨hler number equal or close to that leading to extinction of the uniform flame left behind by the advancing front, and that the spread rate increases to an asymptotic value in the limit of infinitely fast kinetics. This limiting spread rate has been computed as a function of the oxygen concentration in the atmosphere surrounding the cylinder and of the strength of the heat losses from its surface. It is seen that a flame ceases to exist when the oxygen concentration falls below a certain limiting value that increases when the radiation losses increase. The additional aiding or opposing natural convection flow that appears when the cylinder is slightly tilted has been also described. Analysis of these flows bridges the gap between upward and downward flame spread. Acknowledgments This work was supported by DGES grant PB98-0142C04-04.

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COMMENTS Howard Baum, NIST, USA. Your analysis implies that the fuel rod is sufficiently large for the boundary layer to be much smaller than the radius of the rod. It also implies that the heat release is sufficiently small for the pressure gradients induced by the plume to be ignited in comparison with local effects. A worry exists that these two assumptions greatly limit the applicability of the analysis. Can you comment on this point? Author’s Reply. The analysis relies on the boundary layer approximation for the gas; it is an asymptotic analysis for large values of the Grashof number. An improved estimate of the boundary layer thickness that takes into account the thermal expansion of the gas is dg ⳱ (c1/2/Gr1/4)␣. For PMMA in air at normal gravity, it would be c1/2/Gr1/4 ⳱ 1 when ␣ ⬇ 1.2 mm, though this minimum radius increases in reduced gravity. The characteristic thickness and velocity of the plume at distances of order ␣ above the cylinder are dg and wc, as for the boundary layer around the cylinder. The entrainment of the plume and of the boundary layer induce velocities of order vc ⳱ g1/2␣1/2/Gr1/4 and pressure variations of order q0v2c ⳱ q0g␣/Gr1/2, which are small compared with the axial velocities of order uc ⳱ g1/2␣1/2/Gr1/8 induced by the pressure variations of order q0gdg ⳱ q0g␣/ Gr1/4 discussed in the paper. The ratio vc/uc ⳱ Grⳮ1/8 is taken to be small in the analysis. It may be only moderately small in some real cases, and then the effects of the plume, of the finite value of dg/lc, and some others, become more important. In principle, these effects can be taken into account in a rational perturbation scheme for which the present analysis gives the leading order approximation, but it is

clear that they complicate the theory and the interpretation of the experiments. ● Carlos Fernadez-Pello, University of California, Berkeley, USA. In the case where you assume a boundary layer in the gas and solid phases, what is the mechanism of heat transfer ahead of the flame? If it is convection, should the model be applicable to flow-assisted (concurrent) flame spread? Author’s Reply. The mechanisms of heat transfer ahead of the flame in the case no NGr1/8 k 1 analyzed in the paper is convection in the gas, similar to flow-assisted flame spread. The difference is in the cause of the flow along the axis of the cylinder. Here, it is due to the axial gradient of the buoyancy-induced pressure variation rather than a forced stream or the axial component of buoyancy. The boundary layer approximation is not applicable in the solid if NGr1/8 is not large. If NGr1/8 K 1, then heat transfer ahead of the flame is due mostly to conduction in the solid, because dg k lc, and the convection-conduction balance in the solid becomes qgcgUDTg/dg ⳱ jgDTg/d2g, whence U ⬃ Gr1/4Njg/qgcg␣. Keeping everything else constant, NGr1/8 increases with the radius of the cylinder, so that this regime (for which U ⬀ ␣ⳮ1/4) gives way to the regime analyzed in the paper (for which U ⬀ ␣1/8) when the radius of the cylinder increases. Many of the specimens used in the experiments of Ref. [2] in this paper are in the first of these regimes.