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A comparison of extinction limits and spreading rates in opposed and concurrent spreading flames over thin solids
Combustion and Flame 132 (2003) 667– 677
A comparison of extinction limits and spreading rates in opposed and concurrent spreading flames over thin solids Amit Kumara, Hsin Yi Shihb, James S. T’iena,* b
a Case Western Reserve University, Cleveland, Ohio, U.S.A. Industrial Technology Research Institute, Chutung Hsinchu, Taiwan R. O. C.
Received 26 February 2002; received in revised form 24 September 2002; accepted 25 September 2002
Abstract Flame-spread phenomena over thin solids are investigated for purely forced-opposing and concurrent flows. A two-dimensional, opposed-flow, flame-spread model, with flame radiation, has been formulated and solved numerically. In the first part of the paper, flammability limits and spread rates in opposed flow are presented, using oxygen percentage, free-stream velocity, and flow-entrance length as parameters. The comparison of the flammability boundaries and spread-rate curves for two different entrance lengths exhibits a cross-over phenomenon. Shorter entrance length results in higher spread rates and a lower oxygen-extinction limit in low free-stream velocities, but lower spread rates and a higher oxygen-extinction limit in high free-stream velocities. The entrance length affects the effective flow rate that the flame sees at the base region. This affects the radiation loss and gas residence-time in an opposing way to cause the cross-over. Radiation also affects the energy balance on the solid surface and is in part responsible for the solid-fuel non-burn-out phenomenon. In the second part of the paper, a comparison of flammability limits and flame-spreading rates between opposing and concurrent spreading flames are made; both models contain the same assumptions and properties. While the spread rate in concurrent spread increases linearly with free-stream velocity, the spread rate in opposed flow varies with free-stream velocity in a non-monotonic manner, with a peak rate at an intermediate free-stream velocity. At a given free-stream velocity, the limiting oxygen limits are lower for concurrent spread, except in the very low free-stream-velocity regime, where the spreading flame may be sustainable in opposed mode and not in concurrent mode. The cross-over disappears if the two spread modes are compared using relative flow velocities with respect to the flames rather than using free-stream velocities with respect to the laboratory. © 2003 The Combustion Institute. All rights reserved. Keywords: Flame spread model; Solid fuel; Extinction limit
1. Introduction The study of flame-spread over solid fuels in low-speed flows has gained impetus in recent years due to safety concerns for human endeavors in space.
* Corresponding author. Tel.: ⫹1-216-368-4581; fax: ⫹1-216-368-6445. E-mail address: [email protected] (J.S. T’ien).
Over the years, there has been significant progress in our understanding of flame-spread physics, which is inherently a complex phenomenon involving nonlinear interactions between flow, heat, and masstransfer, along with the chemistry in both solid and gas phases. Normally, the flame-spread phenomenon is classified as either the opposed-flow or concurrentflow mode, based on the direction of flame-spread relative to the direction of oxidizer flow. In opposedflow flame-spread, the flame spreads against the ox-
0010-2180/03/$ – see front matter © 2003 The Combustion Institute. All rights reserved. doi:10.1016/S0010-2180(02)00516-3
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idizer flow, whereas in concurrent-flow flame-spread, the direction of flame-spread is the same as that of the oxidizer. In the present paper, an attempt is made to compare the propagation and extinction processes of the two flame-spreading modes. Concurrent-flow flame-spread has been traditionally thought to be an accelerating process, since down-stream convection is the main solid-heat-up mechanism. However, in small-scale experiments using thin fuels [1–3] and modeling work with radiative heat loss [4 – 6], it has been shown that the flames can attain a constant flame-length and a steady spreadrate (i.e., both the flame tip and flame base move with a constant rate). Using a simple heat-balance argument, Honda and Ronney [2] have argued that radiative heat loss is needed for wide samples to achieve steady flame-length, and they deduced that the spread rate is proportional to the flow-approaching velocity (in forced flow) or to gravity (for buoyant, upward spread). These are consistent with existing modeling and experimental results. In the comparison to be presented, steady concurrent-flow flame-spread is to be compared with steady opposed-flow flame-spread in a forced-flow environment. Although concurrent flame-spread is generally considered more rapid than the opposed-flow mode, as the oxidizer flow direction favors flame propagation, there is a limiting flow velocity below which no concurrent-flow flame-spread can be sustained. On the contrary, the flame can self-propagate in a quiescent environment, which is a special case of opposedflow flame spreading. This suggests that, under certain conditions, opposed-flow flame-spread is possible where concurrent-flow flame-spread is forbidden. In a recent micro-gravity experiment and modeling study on the transition from ignition to flame-spread [7,8], ignition was initiated in the middle of a paper sample, and the flame may spread up-stream (opposed-flow mode) but not down-stream (concurrent-flow mode) if oxygen concentration and imposed free-stream velocity are sufficiently low. The authors attributed this to the “oxygen shadow” effect; i.e., oxygen is consumed by the flame-front up-stream, and there is not enough left to sustain a down-stream flame. Furthermore, by comparing the experimental data of Grayson et al. [9] (concurrentflow mode) and Olson [10] (opposed-flow mode), the concurrent-flow flame-spread rates are slightly lower than those of opposed flow in air and at 30% oxygen for velocities less than 5 cm/s. All the above observations are intriguing and puzzling and require further investigation on both concurrent-flow and opposed-flow flame-spread in a low free-streamvelocity regime (0 –10 cm/s) for a better understanding of the phenomenon. So far, research groups employing advanced com-
putational techniques have focused their work primarily on either opposed-flow flame-spread [11–13] or concurrent-flow flame-spread [3– 6]. They have used different mathematical models, fuel properties, and operating conditions, making it difficult to compare the results fairly between the two types of spreading flames. Although we are aware that in [8] a comparison of computed flame-spread trends versus the relative wind velocity at 30% oxygen for concurrent-flow and opposed-flow flames was made using the same model, a more complete picture of the flammability limits and the controlling mechanisms of these two types of flames, including the effect of flame radiation, is desirable. In this work, an opposed-flow model, which is consistent with a previously established model in concurrent-flow, has been formulated and numerically solved. The computed flammability map and spreading behaviors of opposed-flow flames in low-speed forced-flows are compared with the counterparts in concurrent-flow flames.
2. Flame-spread model 2.1. Model description A previous model of concurrent-flow flamespread over a thin solid [3,4,14] was adapted for the opposed-flow configuration. The mathematical formulations given in [14] are applied and modified accordingly, and all thermal and transport properties were kept the same. The gas-phase model consists of two-dimensional, steady, laminar, full Navier-Stokes equations, along with the conservation equations of mass, energy, and species. The energy equation contains a radiative divergence term that needs to be solved from the radiation-transfer equation. The species equations are for the fuel vapor, oxygen, carbon dioxide, and water vapor. A one-step, second-order, finite-rate Arrhenius reaction between fuel vapor and oxygen is assumed. The activation energy and preexponential factors are 1.13 ⫻ 105 J/g 䡠 mol and 1.58 ⫻ 1012 cm3 䡠 g⫺1 䡠 s⫺1, respectively. The heat of combustion is 1.675 ⫻ 104 J/g. The thin solid model (both thermally and aerodynamically thin) comprises continuity and energy equations whose solutions provide boundary conditions for the gas phase. The solid is assumed to be a thin, cellulosic material (Kimwipes paper having a half-thickness area density of 1 mg/cm2) with the chemical formula C6H10O5. The solid pyrolysis is modeled using one-step, zero-thorder Arrhenius kinetics, where the activation energy is 1.256 ⫻ 105 J/g mol and the pre-exponential factor is 3.8 ⫻ 107 cm/s. The solid is assumed to have a total emittance and a total absorptance of 0.92.
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Fig. 1. Schematic of domain of interest in flame fixed co-ordinates. The domain is open ( y 3 ⬁), U ⬁ is the uniform entrance flow-velocity with respect to the laboratory, and V f is the flame-spread rate. (A) Opposed-flow flame-spread configuration; x ⫽ 0 is located at the pyrolysis front (95% of fresh-fuel thickness). (B) Concurrent-flow flame-spread configuration; x ⫽ 0 is located at the solid-fuel burnout point.
A schematic of the domain of interest for the current opposed-flow flame configuration in an open domain is shown in Fig. 1A. The flame-fixed coordinate system is located on the pyrolysis front ( x ⫽ 0) of the solid, where the fuel thickness is 95% of the fresh-fuel thickness. The flame leading edge is shown stabilized over the fuel near the origin, with its tail extending downstream ( x ⬍ 0). The free-stream forced-flow with uniform velocity U ⬁ enters the domain at a distance x e (referred to as the flow entrance length in this paper) from the origin. The presence of fresh, solid fuel upstream ( x ⬎ 0) imposes no-slip conditions at the surface, which result in a nonuniform flow-profile approaching the flame. The flow approaches the flame (in the negative x direction) at a relative velocity of u r ( x, y) ⫽ u( x, y) ⫹ V f , where V f is the flame-spread rate and u( x, y) is the local flow-velocity relative to the laboratory. In this opendomain configuration, then, the flow-velocity profile always depends on the entrance length ( x e ), which is an additional parameter that needs to be considered in describing the opposed-flow flame-spread. Most of the opposed-flow computations in this work were performed with an entrance length of 6 cm. Additional computations were made with an entrance
length of 10 cm to study the effect of entrance length on flame-spread rate and flammability limits. Note that for a concurrent flame-spread configuration as shown in Fig. 1B, the flame-fixed coordinate system is fixed at the solid burnout point, and no solid is left after flame passage; therefore, the specification of entrance length is not needed. In both configurations, the domain is open to the atmosphere, and in the y direction, it extends far to the free-stream condition ( y 3 ⬁). An additional remark is made here on the relationship between the model and experiments on opposing spread over thin-solid fuel. In typical forced-flow experiments [15,16], a solid sample is placed in a wind tunnel and is ignited downstream; as the flame spreads upstream, the distance between the leading edge of the sample and the flame-front (entrance length, x e ) gradually decreases. If the rate of decrease is small compared to the characteristic time of the flame, then a quasi-steady entrance length can be defined; a steady entrance length can also be achieved in a specially designed experimental setup, like a plug flow-generating device moving upstream at the flame-spread rate. In this work, the entrance length is assumed to be constant for a given steady computation.
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2.2. Treatment of radiation Radiation (gaseous and/or surface) plays an important role in micro-gravity flames and is a key part of the model. It is responsible for the existence of the low-speed quenching-limit [17] and is also a heattransfer mechanism, besides conduction/convection in combustion systems. The importance of flame radiation in the spreading and extinction of flames over thin solid fuels in a microgravity environment has been substantiated in work by Jiang [4] for concurrent spread and by Bhattacharjee and Altenkirch [11] and Chen and Cheng [13] for opposed-flow spread. Because of the limitations of computing capability, gas-phase thermal radiation from carbon dioxide and water vapor is modeled using a mean absorption coefficient. We note that the spectral radiation computation in a one-dimensional flame has been carried out recently, e.g., [18], but the computational requirement is prohibitive for multi-dimensional flames at the present time. Soot radiation is neglected, which is based on experimental observations at low oxygen concentrations and in low-speed flows [9,10]. The solid radiation is assumed to be diffused. A mean absorption coefficient for the gas mixture is needed. The simplest way is to assign an absorption coefficient, which is treated as a parameter in the model [19]. A more sophisticated treatment of gas radiation has been proposed in [11]. While gas radiation is treated as a heat loss term in the energy equation with a constant absorption coefficient, the value of this coefficient is evaluated from a global radiation-energy balance. The radiation-heat feedback to the solid is then obtained in similar fashion. The overall radiative transfer is ensured in a selfconsistent manner, but only a single absorption coefficient is used for the entire flame, and it is believed that the absorption coefficient can vary significantly in the flame. The Planck mean absorption coefficient, K p , based on the local temperature and composition of the mixture, has been used in several modeling works [4,13]. The local Planck-mean absorption coefficient for the mixture can be given by K p ⫽ X CO 2 䡠 K p (CO 2 ) ⫹ X H2O 䡠 K p (H 2 O), where X i represents the molar fraction of species i. The values of K p for each species are from the data of Tien [20] as a function of temperature. However, by comparison with the more accurate results obtained from a narrow-band radiation model in one-dimensional flames, it was shown that the Planck mean results overpredict net emission from the flame [18,21]. A novel feature used here for both concurrent-flow and opposed-flow flames is the incorporation of a calibration procedure [21] for the mean absorption coefficient. Calibration of the absorption coefficient
against the narrow-band results through a quasi-onedimensional flame is done to account for the different optical lengths in different parts of the flame and the effect of spectral self-absorption of gaseous species. Therefore, the local absorption coefficient K used in this work is set to equal to CK p , where C is the correction factor. The distribution of C, determined iteratively in the solution procedure for the sample flame, will be shown in the results later. 2.3. Numerical scheme The SIMPLER algorithm [22] is used for the fluid-flow and combustion equations, and the radiation equation is solved using the S–N discrete ordinates method [23,24]. The two-dimensional S4 scheme with 12 ordinate directions is used in the current computations by considering the balance between numerical accuracy and computational expense [4]. Since the radiation equation and the rest of the combustion/fluid equations are coupled, they are solved iteratively. In addition, the gas-phase system is coupled to the solid-phase equations, which are solved by a finite-difference technique. The steady flame-spread rate (the eigenvalue of the whole system) is determined iteratively using a bisection method to force the burnout point in the concurrentflow mode and the pyrolysis front (95% of the freshfuel thickness) in opposed-flow mode to occur at x ⫽ 0. Computations are carried out on a non-uniform mesh. In the y direction, the grids are clustered near the solid and expand away from the surface. In the x direction, a fine grid structure is used in the flamestabilization region (the flame leading-edge) to capture the drastic variations in the flame; the grids then expand upstream and downstream. For the opposedflow configuration, a fine grid-structure near the solid burnout region (the trailing edge) is also needed, and since the pyrolysis length is not known a priori, adaptive gridding is used for this purpose. The grid structures are consistent near the flame-stabilization zone for all the calculations. The smallest grid size is 0.05 thermal lengths in the flame-stabilization zone. The domain extends to 200 thermal lengths in the y direction and 250 thermal lengths in the down-stream x direction measured from the pyrolysis front. The upstream extent of the domain is the prescribed entrance length, which is the dimensional constraint. The total number of grid nodes in the x direction varies with the free-stream velocity as dimensional entrance length is converted to a non-dimensional distance in the calculation. For a typical case of U ⬁ ⫽ 5 cm/s at 15% O2 there are 59 grid nodes in the y direction and 113 grid nodes in the x direction, whereas for U ⬁ ⫽ 10 cm/s, the number of grid nodes
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in the x direction increases to 132. The computational time, which varies from case to case, is typically about 1 to 2 hours on a Compaq XP1000 workstation.
3. Results 3.1. Effect of entrance length on opposing-flamespread There is fresh fuel up-stream of the flame in opposed-flow flame-spread; hence, a velocity boundary-layer develops over the fresh fuel surface (this is not the case for concurrent flame-spread if solid burnout occurs). Given a plug flow-velocity at the entrance, the velocity profile that the flame base sees varies with its distance from the plug flow-entrance ( x e ). Although this flow-profile influence has been recognized in the past [15,16], detailed investigation into this aspect has been lacking. We do, however, like to mention that in using the thermal diffusive model (i.e., prescribed velocity profiles without using momentum equations), Wichman [25] has chosen a linear velocity profile with velocity gradient as the flow parameter. The velocity-gradient prescription appears to be a reasonable representation when the velocity boundary layer thickness is much greater than the flame thickness, while the plug flow prescription by De Ris [26] is better when the boundary thickness is much smaller than the flame thickness. The most common cases, however, are the in-between situations. In such cases, at least two flow parameters are needed to specify the approaching flow. In this work, the flow entrance length ( x e ) and the plug flow-velocity at the entrance (U ⬁ ) are the specified parameters. We assume that for opposing spread, the fresh fuel extends all the way to the plug flow-entrance, from where the velocity boundary layer over the fuel surface starts to develop. Figure 2 shows the computed flame-spread rates versus the free-stream velocity at 15% oxygen for two different entrance-lengths (6 cm and 10 cm). The curves are non-monotonic and exhibit a maximum spread-rate at an intermediate free-stream velocity. The velocityextinction limits are the ends of the curves shown as vertical dotted lines. Near the extinction limit (dotted extension of the plotted curves), the computed flamespread rates were not constant and oscillated in a bounded fashion over a set of iterations. Whether this oscillation is physical [27] or numerical [28] is difficult to ascertain with the current steady-state model. The spread rates at a given free-stream velocity are different and with a cross-over at U ⬁ ⫽ 7 cm/s. For a free-stream velocity below 7 cm/s, the x e ⫽ 6 cm case is faster, and for the free-stream velocity above
Fig. 2. Effect of entrance-length ( x e ⫽ 6 cm and 10 cm) on opposed-flow flame-spread rate over thin solid fuel at 15% O2. The vertical dotted line indicates the extinction-velocity limits. The inset figure shows visible flame (fuel reactionrate contour ˙ F ⫽ 10 ⫺4 g/cm3 s) at U ⬁ ⫽ 5 cm/s for the two entrance lengths. Also shown are the corresponding velocity profiles (u r /U ⬁ ), 1 cm upstream of the pyrolysis front (95% fresh-fuel thickness).
7 cm/s, the x e ⫽ 10 cm case is faster. In addition, the high-speed blow-off and the low-speed quenching limits are also different. The reason for the shifting of limits and spread rates will be discussed later. In the inset of Fig. 2, the fuel-vapor reaction-rate contours (10⫺4 g 䡠 cm⫺3 䡠 s⫺1) and the velocity profiles 1 cm up-stream of the pyrolysis front are plotted for a representative case (U ⬁ ⫽ 5 cm/s). It is clear that the two velocity profiles are different, and the velocity that the flame encounters is smaller for the longerentrance-length case. So, although the two cases have the same entrance flow-velocity, the effective oxygen flow rates that the flames encounter are different. Note that the velocity plotted here is the relative velocity with respect to the flame, u r ⫽ u ⫹ V f , normalized against U ⬁ ⫽ 5 cm/s. On the fuel surface ( y ⫽ 0), u ⫽ 0, so u r ⫽ V f . 3.2. Surface heat-flux distribution and fuel-left-over phenomena in opposing spread It is observed that under certain conditions, not all fuel is pyrolyzed by the flame, and so there is leftover fuel downstream of the flame. It should be noted that the solid-fuel pyrolysis law model used here is of zeroth order [3], which allows for complete burn-out of the solid fuel. In previous models that use a firstorder pyrolysis law [11,12,16,29,30], the solid fuel will not burn out in a finite time (or in finite length). In the work by Frey and T’ien [30], for example, pyrolysis is assumed to cease when the amount of
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Fig. 3. Solid temperature profile and fuel thickness (nondimensionalized with fresh-fuel thickness) for U ⬁ ⫽ 4 cm/s, O2 ⫽ 15%, and entrance length ⫽ 6 cm.
fuel left over reaches 7%, a somewhat arbitrary criterion. This fuel leftover phenomenon has been observed in experiments [29,31], where a significant amount of potentially pyrolyzable solid was left unburned downstream of the flame. This phenomenon can be explained in terms of a vanishingly small heat-flux that reaches the solid in the region. The net amount of heat-flux going into the solid at a particular location is the sum of various types of radiation and convection contributions. In the following paragraph, a detailed examination of the heat-flux distribution will be made. A representative case with solid left-over is shown here (U ⬁ ⫽ 4 cm/s, O2 ⫽ 15%, and 6-cm entrance-length). Figure 3 gives the solid profiles. The fuel thickness (non-dimensionalized by the fresh-fuel thickness) decreases as the solid pyrolyzes. There is a considerable amount of solid left unburned for this case (about 16.6%). In the pre-heat zone ( x ⬎ 0), the solid temperature rises rapidly from ambient temperature to the pyrolysis temperature (the peak at 650 K). It becomes more flattened in the pyrolysis region and then continues to drop further downstream. Figure 4 presents the distribution of heatfluxes per unit-width of solid fuel (W/cm2) along the solid. The positive value indicates heat gain for the solid, and the negative value is a heat loss from the surface. q c is the conduction heat-flux from the gas. The gas-phase radiation feedback is represented by q r( gas) , and the radiation leaving the solid surface is q r(solid) , which is the sum of surface emission and reflection and transmission. The sum of q r( gas) and q r(solid) gives q r(net) , the net radiation flux. The sum total of q r(net) and q c is the net heat-flux on the solid surface q (net) . Near the flame leading-edge ( x ⫽ 0), q c is substantially greater than q r( gas) because the flame is very close to the solid. As we go downstream, the flame standoff distance increases, and q c drops in magnitude and q r( gas) is almost constant in the pyrolysis zone, then drops slowly further downstream. q r(solid) also peaks near the anchor point ( x ⫽ 0) and decreases slowly downstream, reflecting the strong emission from the high-temperature solid-
Fig. 4. Heat-flux distribution over solid surface for U ⬁ ⫽ 4 cm/s, O2 ⫽ 15%, and entrance length ⫽ 6 cm. Where q c is the conduction heat-flux, q r( gas) gas-phase radiation feedback, q r(solid) sum of emitted and reflected radiation from the solid, q r(net) is the net radiation heat flux (q r( gas) ⫹ q r(solid ), and q (net) is the net heat-flux (q r(net) ⫹ q c ).
surface. The net heat-flux, q (net) , distribution shows a maximum close to the flame anchor point, owing to dominance of q c , and drops in either direction; further, it is noted that for x ⬍ ⫺4.2 cm, q (net) becomes negative (shown enlarged in the inset A of Fig. 4). The pyrolysis rate along the solid fuel drops in the downstream direction to a very small value from a maximum at x ⫽ 0, which results in fuel leftover. This phenomenon can be understood in the following way. The distribution of fuel pyrolysis follows the solidsurface temperature distribution through a zeroth order Arrhenius-type dependence (activation energy ⫽ 1.256 ⫻ 105 J/g mol). The solid temperature in turn is decided by the heat balance on the surface. In the region 0 cm ⬎ x ⬎ ⫺4.2 cm, the solid-fuel temperature decreases (Fig. 3) owing to surface radiative loss and pyrolysis. This becomes evident by looking at the integrated heat-fluxes on the surface per unit width of fuel (in W/cm). The integrated incident heat-flux (q in ⫽ q r( gas) ⫹ q c ) is 3.12 W/cm, and the integrated solid-surface radiation loss [qr(solid))] is ⫺2.74 W/cm. So, the net available heating rate for the pyrolysis of fuel is qin ⫹ q r(solid) ⫽ 0.38 W/cm. The actual value of the energy consumption rate for the pyrolysis of fuel obtained by integration of local values over the fuel length is ⫺0.41 W/cm. The difference of 0.03 W/cm is supplied from the sensible heat of the solid fuel, which results in cooling of the solid fuel. It should be noted that the solid radiation loss term is a large fraction compared to the incoming heat-flux. In the region x ⬍ ⫺4.2 cm, q (net) ⬍ 0 is a consequence that the net radiative loss
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Fig. 5. Contours of mean absorption (emission) coefficient (K) distribution for the case with entrance length ( x e ) of 6 cm, 15% O2, and U ⬁ ⫽ 4 cm/s; also plotted is the fuel vapor reaction-rate contour of 10⫺4 g/cm3s (dotted curve). The inset figure shows spatial variation in the correction factor (C, K ⫽ CK p ) for the Planck mean-absorption coefficient (K p ) at U ⬁ ⫽ 4 cm/s and 10 cm/s.
Fig. 6. Flammability boundary and left-over-fuel boundary for opposed-flow flame-spread over a thin solid fuel with 6-cm entrance-length at 1 atmosphere.
from the solid exceeds the heat gain from the gasphase conduction and the gas-phase radiation feedback. In this portion, the surface temperature is substantially reduced, pyrolysis is minimal, and solid fuel is left unburnt, as demonstrated in Fig. 3. Whether there will be fuel left over and if so, how much, depends on the oxygen level and free-stream flow-velocity. This will be shown later. The inset B of Fig. 4 also shows enlarged plots in the pre-heat region. It is interesting to note that q r(net) has a positive value over a stretch of one preheat region, but q c is negative. This is because the gas-phase radiation feedback can reach farther upstream than flame conduction. In this region, net radiation from the gas phase provides energy to heat up the fuel, while gas-phase conduction plays the role of cooling. This heat-transfer mechanism is quite different from the corresponding case in concurrent flow (not shown here). With the same solid emittance (0.92), the net radiation-flux for the concurrent-flow flame is always a heat loss for the solid, although the radiation feedback could be greater than conduction in the preheat region. The spatial distribution of the mean-absorption coefficient K for the U ⬁ ⫽ 4 cm/s case is shown in Fig. 5. Recall that K ⫽ CK p , where K p is the Planck mean-absorption coefficient and C is the correction factor for the non-optically thin flame. The variation of C as a function of x is shown in the inset. In the downstream flame region of x ⬍ 0, C is determined by an optical traverse in the y direction (perpendicular to the solid) and using an empirical relation proposed in [21]. For the region of x ⬎ 0, where the flame is highly two-dimensional, two traverses are made from x ⫽ 0: one in the y direction and the other
in x direction toward upstream. The correction factor in this region is then the average of these two traverse-values. A constant C is assumed in this domain. If the value of C is close to unity, an optically thin approximation using the Planck mean-absorption coefficient is reasonable. From the inset in Fig. 5, one can see that for a free-stream velocity of 4 cm/s, the factor is close to 0.5 upstream but drops to around 0.3 downstream, indicating thickening of the radiating layer, consisting of combustion products (CO2 and H2O). The inset also shows the correction factor for the U ⬁ ⫽ 10 cm/s case. In this case, the values of C are higher than those for U ⬁ ⫽ 4 cm/s. At a higher free-stream velocity, the flame is thinner, so it is closer to the optically thin limit. Despite the semiempirical nature of the correction factor, the variation in local flame-emission power and its dependence on flow and environmental conditions has been included in an approximate manner in this model computation. The flammability boundary, along with the region of solid-fuel left-over as a function of free-stream velocity, is shown in Fig. 6. The flammability boundary consists of a blow-off branch and a radiative quenching branch. The flame goes out through the blow-off boundary due to an insufficient gas residence-time to complete the reaction, and the flame also goes out through the quenching boundary due to excessive radiative loss from the flame. Note that at U ⬁ ⫽ 0, the solid cannot propagate a flame if oxygen is below 19.85%. This is the computed limiting oxygen index in a quiescent atmosphere in zero gravity. This value is slightly lower than the experimental value of 21%, observed by Olson et al. [32], reflecting stronger kinetic parameters used in the model.
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Fig. 7. Left-over fuel (h leftover expressed as a percentage of fresh-fuel thickness) downstream of the flame as a function of free-stream velocity.
We have not attempted to calibrate these global kinetic constants in this study. Qualitatively, the Vshaped flammability-boundary in Fig. 6 resembles the experimental curve in Olson et al. [32]. It should be pointed out, however, that the flammability boundary in the above reference is a data composite from a combination of forced and buoyant experiments, while the present theoretical curve is entirely forced. Experimental data of a purely forced-flow flammability boundary that covers the whole range of free-stream velocity are not yet available. Further, Fig. 6 shows the complete solid-burnout boundary. In general, complete burnout lies on the high-velocity side, while fuel leftover lies on the low-velocity side. In a low-oxygen environment, those close to the fundamental oxygen limit, the burn-out boundary is more complex, and the amount of fuel left unburned is not necessarily monotonic with respect to the free-stream velocity or oxygen percentage. This is illustrated in Fig. 7. The trend of fuel leftover can be understood in terms of effect of free-stream velocity on the role of gas-phase heat feedback to the solid. The heat feed-back to the preheat region determines the flame spread-rate, and the heat feedback to the pyrolysis region in the time frame prescribed by the flame-spreading rate determines the amount of fuel leftover. As the free-stream velocity is lowered, both gas-phase conduction as well as gas-phase radiation feedback to the solidphase decrease in strength. Although the flame spread-rate decreases (i.e., increased available time for pyrolysis), a more drastic decrease in the heat feedback from a shorter flame over the pyrolysis region results in an increased amount of un-burned, leftover fuel. In the region very close to the quench curve at a low oxygen level (i.e., for 15% O2 in Fig. 7), gas-
Fig. 8. Comparison of extinction boundaries of opposedflow flame-spread with concurrent-flow flame-spread.
phase radiation dominates (about 70% of the incoming heat flux) heat feedback to the pre-heat region, and conduction heat-transfer is dominant (also about 70% of the incoming heat flux) in the pyrolysis region. The domination of radiation and conduction in preheat and pyrolysis regions, respectively, become more prominent as the oxygen level and/or flow velocity is lowered to the quenching limit. Since gas temperature drops fast near the quenching limit, the radiation-controlled flame-spread-rate drops very fast. This gives more time for pyrolysis in the conduction dominated region under the flame and thus, reverses the trend of leftover fuel. 3.3. Comparison of flammability limits and spreadrates in opposed and concurrent flames First, two flammability boundaries with entrance lengths of 6 cm and 10 cm in opposed-flow configuration are shown in Fig. 8 to show the effect of entrance length. With longer entrance length, the boundary moves to the right toward the higher freestream-velocity side, and the two boundaries cross over at U ⬁ ⫽ 6 cm/s. The self-propagation and the fundamental oxygen limits are not affected. This shift of the flammability boundary is consistent with the crossing-over behavior of the spread-rates shown in Fig. 2. For U ⬁ ⬎ 6 cm/s, the flame is in the gasresidence-time limited regime, and more flow will reduce the residence time, decrease the spread-rate, and push the flame closer to the blow-off limit. For shorter entrance length (i.e., 6 cm), the effective flow-rate that the flame sees is higher than the longer entrance length case (10 cm) at the same entrance velocity as shown previously in Fig. 3 (inset). Therefore, the blow-off velocity limit is lower. For U ⬁ ⬍
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6 cm/s, the flame is in the oxygen-limited regime, and a higher effective flow-rate enhances the flame. Therefore, the shorter-entrance case has a wider flammable domain at low velocity. The spread-rate behavior shown in Fig. 3 follows the same reasoning. Next, the extinction boundaries for opposed-flow flame-spread are compared to those of concurrentflow flame-spread in Fig. 8. The concurrent-flow flame-spread can occur at a lower oxygen level than opposed-flow flame for a free-stream velocity higher than 2 cm/s. The fundamental oxygen limit for concurrent flame is 11.6%, substantially lower than that of 13.76% for the opposed-flow configuration. However, only the opposed-flow flame exists at very low free-stream velocities (⬍ 2 cm/s and high oxygen concentration level). The crossover of the opposedflow and concurrent-flow flammability boundaries and the notion that a flame can be sustained in an opposing flow but not in concurrent flow seem perplexing at first. However, the true convective velocity that the flame sees is the relative flow velocity given by U r ⫽ U ⬁ ⫹ V f for opposed-flow flame-spread and U r ⫽ U ⬁ ⫺ V f for concurrent-flow flamespread, not U ⬁ alone. For a thin solid, the magnitude of the flame-spread rate can be substantial compared with that of the opposing velocity in this crossover regime. For example, at 21% O2 and U ⬁ ⫽ 0, V f ⫽ 2.17 cm/s ⫽ U r for opposed-flow flame-spread. The convective velocity relative to the flame in this case is entirely due to spreading of the flame into this quiescent ambient. For the concurrent-flow case at the same oxygen percentage, at U ⬁ ⫽ 1.4 cm/s (near the quench limit), V f ⫽ 0.144 cm/s, which gives U r ⫽ 1.356 cm/s. This is lower than the self-propagation opposed-flow case discussed above. If U ⬁ is decreased, then U r would be too low and oxygen supply will be insufficient to sustain the concurrent flame. To gain more physical insight, it then seems more reasonable to plot the flammability boundary in this low-speed regime using the relative flow velocity. This can be found in the inset of Fig. 8. In this new plot, the concurrent-flow flame is more flammable than the opposed-flow flame (in the sense of having a lower limiting relative velocity at the same oxygen percentage). The two limits approach each other as the oxygen content approaches 19.85%, the low-oxygen limit in a quiescent atmosphere. This analysis using the relative velocity can yield deeper insight and practical implications. For thermally thin solids, the spread-rate is expected to vary as the reciprocal of sample thickness. So, the crossover behavior will occur at a lower free-stream velocity as the fuel becomes thicker and the spread-rate becomes smaller. Flame-spread rates as a function of the free stream-velocities and several oxygen levels are pre-
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Fig. 9. Comparison of (A) spread-rates and (B) flamelengths in opposed (solid curves) and concurrent spreading flame (dotted curves), X, extinction.
sented in Fig. 9A. For opposed-flow situation, the flame-spread rate exhibits a non-monotonic behavior for all three oxygen concentrations (15%, 18%, 21%), which has been observed in many experiments [10,32]. The flame-spread rate increases sharply with increasing free-stream velocity near the quenching limit, reaches a maximum at some intermediate freestream velocity, and then falls gradually toward blow-off extinction. The computed flame-spread rates in concurrent flow at 15% O2 are also shown in Fig. 9A, where the concurrent-flow flame-spread-rate increases monotonically (approximately linear) with the free-stream velocity. Note that there is a crossover between these two flame-spreading modes at 15% O2. At low free-stream velocity (⬍7 cm/s), the opposed flame spreads faster than the concurrent flame, and the trend reverses at high free-stream velocity. The cross-over phenomenon in spread-rate values for concurrent-flow and opposed-flow spreading flames with flow velocity was also observed in low-speed microgravity simulations of [8] at 30% O2, where the fuel was ignited in the center to obtain both concurrent and opposed flames simultaneously. We note, however, that [8] does not include flame radiation. Flame-length variation with flow velocity is shown in Fig. 9B. The flame length (based on the 10⫺4 g 䡠 cm⫺3 䡠 s⫺1 fuel-reaction-rate contour) follows the same trend as the flame-spread rate for both concurrent-flow and opposed-flow flames. Again, a crossover is found in the low-speed flow regime. Note that although the flame sizes (and the spreadrates) of the concurrent-flow flames and opposedflow flames can be comparable at low free-stream velocities, they could be tremendously different at high free-stream velocities.
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4. Concluding remarks In order to quantitatively compare the extinction and flame-spreading characteristics between opposed-flow and concurrent-flow flame-spreads in purely forced flows, an opposed-flow flame-spread model has been formulated and solved. This model employs the same approximations and property constants as a previously established concurrent-flow flame-spread model. In particular, flame radiation, important in the low free-stream velocity regime, is treated in the same way. The radiation-transfer equation was solved using the discrete ordinate method with distributed absorption coefficients. A new procedure to calibrate these coefficients has been employed. It was found that for flame spreading in low-speed forced flow, gaseous flame radiation is an important contribution to the total heat feedback to the solid in addition to being a flame heat-loss. In the first part of the paper, certain new features in opposed-flow spread were presented. These include the effect of flow entrance length and the solid non-burnout phenomenon. Flow entrance length affects the spread-rate and the extinction limit differently, depending on the free-stream velocity. At low free-stream velocities, a short entrance length results in a higher spread-rate and a lower oxygen limit than those for the longer-entrance length case. At high free-stream velocities, the reverse is true. These trend reversals can be explained based on the velocity profiles the flames see in the flame base (flame front) region and the competing effects between radiation loss and gas residence time on oxygen supply. In an opposing-flame-spread mode, some solid fuel may be left over in the downstream region. Solid surface radiative heat-loss is a major contributor to this phenomenon. The surface radiation loss can result in negative net heat-flux to the solid, thus accelerating cooling of the solid downstream of the flame. In the free stream-velocity versus oxygen percentage map, the domain where fuel leftover occurs, leftover was presented along with a trend for leftover (in percentages) with free stream-velocity at various oxygen concentrations. The second part of the paper concentrated on the comparison between concurrent-flow and opposedflow flame-spreads. The trends of flame-spread rates between the two modes are qualitatively different. Spread-rate in concurrent flow increases linearly with free-stream velocity, but spread-rate in opposed-flow varies non-monotonically with free-stream velocity and peaks at an intermediate free-stream velocity. While these qualitative trends were previously known, the present quantitative comparison demonstrates that the two rate-curves have a crossover. Opposed-flow flame-spread can be faster than con-
current-flow flame-spread if the flow velocities are low enough, consistent with previous experimental data. A comparison of flammability boundaries in an oxygen (O2) percentage versus free-stream-velocity (U ⬁ ) map also shows the existence of a crossover between the two spreading configurations. Concurrent-flow flame in general has lower oxygen limits (including the fundamental limit), but the trend is reversed when the free-stream velocity is very small. This can be more easily understood when we consider the limiting case of zero flow-velocity. Flame spread in a quiescent atmosphere is a special case of opposed-flow flame-spread. When the oxygen percentage is sufficiently high, quiescent spread is possible, while there is no corresponding concurrent case. This limit reversal point (i.e., the free-stream velocity where the two boundaries crossover) depends on the flame-spread rates, hence, the solid thickness. It is reasoned that the domain where the opposed-flow flame is more flammable diminishes with solid thickness. If the flammability map is plotted using the relative flow-velocity with respect to the flame (rather than with respect to the laboratory coordinates), the limit reversal disappears. We note that the present work deals with pure opposed-flow and pure concurrent-flow flame-spreads only. Simultaneous opposing and concurrent spreads (i.e., ignition in the middle of the fuel sample) may have different conclusions [7,8]. In addition, the present models are two-dimensional. Although three-dimensional models have appeared recently [33,34], they have as yet to incorporate flame radiation, which is essential in this work.
Acknowledgment This research has been supported by NASA grants NCC3-669 and NCC3-633, Dr. Kurt Sacksteder, grant monitor.
References [1] L. Chu, C.H. Chen, J.S. T’ien, Upward spreading flames over paper samples, ASME, 81-WA/HF-42, 1981. [2] L.K. Honda, P.D. Ronney, Proceedings of the Combustion Institute, Vol. 28, The Combustion Institute, Pittsburgh, 2000, p. 2793. [3] P.V. Ferkul, J.S. T’ien, Combust. Sci. Technol. 99 (1994) 345–370. [4] C.B. Jiang, A model of flame spread over a thin solid in concurrent flow with flame radiation, Ph.D. Thesis, Case Western Reserve University, Cleveland, OH, 1995.
Kumar et al. / Combustion and Flame 132 (2003) 667– 677 [5] C.-B. Jiang, J.S. T’ien, H.Y. Shih, Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, p. 1353. [6] I.I. Feier, H.Y. Shih, K.R. Sacksteder, J.S. T’ien, Upward flame spread over thin solids in partial gravity, Proceedings of the Combustion Institute, Vol. 29, The Combustion Institute, Pittsburgh, 2002, in press. [7] T. Kashiwagi, K.B. McGrattan, S.L. Olson, O. Fujita, M. Kikuchi, K. Ito, Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, p. 1345. [8] K.B. McGrattan, T. Kashiwagi, H.R. Baum, S.L. Olson, Combust. Flame 106 (1996) 377–391. [9] G. Grayson, K.R. Sacksteder, P.V. Ferkul, J.S. T’ien, Micro-grav. Sci. Technol. 7 (1994) 187–193. [10] S.L. Olson, Combust. Sci. Technol. 76 (1991) 233– 249. [11] S. Bhattacharjee, R.A. Altenkirch, Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1990, p. 1627. [12] S. Bhattacharjee, R.A. Altenkirch, Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, p. 1669. [13] C.H. Chen, M.C. Cheng, Combust. Sci. Technol. 97 (1994) 63– 83. [14] J.S. T’ien, H.Y. Shih, C.B. Jiang, H.D. Ross, J. Miller, A.C. Fernandez-Pello, J.L. Torero, D. Walther, Microgravity combustion: fire in free fall, H. Ross (Ed.), New York, Academic Press, 2001. [15] S.R. Ray, Flame spread over solid fuels, Ph.D. Thesis, Princeton University, Princeton, NJ, 1981. [16] J. West, S. Bhattacharjee, R.A. Altenkirch, Combust. Sci. Technol. 83 (1992) 233–244. [17] J.S. T’ien, Combust. Flame 65 (1986) 31–34. [18] H. Bedir, J.S. T’ien, H.S. Lee, Combust. Theory Model. 1 (1997) 395– 404.
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[19] C. Di Blasi, Combust. Flame 100 (1995) 332–340. [20] C.L. Tien, Thermal radiation properties of gases, in Advances in heat transfer, Academic Press, New York 5 (1968) 234 –254. [21] J.L. Rhatigan, H. Bedir, J.S. T’ien, Combust. Flame 112 (1998) 231–241. [22] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere Publishing Co., New York, 1980. [23] W.A. Fiveland, J. Heat Transfer 106 (1984) 699 –706. [24] T.K. Kim, H.S. Lee, J. Quant. Spectrosc. Radiat. Transfer 42 (1989) 225–238. [25] I.S. Wichman, Prog. Energy Combust. Sci. 18 (1992) 553–593. [26] J.N. De Ris, Twelfth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1969, p. 241. [27] K. Sato, K. Miki, T. Hirano, J. Heat Transfer 106 (1984) 707–712. [28] J. Buckmaster, Y. Zhang, Combust. Theory Model. 3 (1999) 547–565. [29] P.A. Ramachandra, R.A. Altenkirch, S. Bhattacharjee, L. Tang, K. Sacksteder, M.K. Wolverton, Combust. Flame 100 (1995) 71– 84. [30] A.E. Frey, J.S. T’ien, Combust. Flame 36 (1979) 263– 289. [31] S. Bhattacharjee, R.A. Altenkirch, K. Sacksteder, Combust. Sci. Technol. 91 (1993) 225–242. [32] S.L. Olson, P.V. Ferkul, J.S. T’ien, Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, p. 1213. [33] W.E. Mell, T. Kashiwagi, Twenty-Seventh Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1998, p. 2635. [34] H.Y. Shih, J.S. T’ien, Proceedings of the Combustion Institute, Vol. 28, The Combustion Institute, Pittsburgh, 2000, p. 2777.
A comparison of extinction limits and spreading rates in opposed and concurrent spreading flames over thin solids Amit Kumara, Hsin Yi Shihb, James S. T’iena,* b
a Case Western Reserve University, Cleveland, Ohio, U.S.A. Industrial Technology Research Institute, Chutung Hsinchu, Taiwan R. O. C.
Received 26 February 2002; received in revised form 24 September 2002; accepted 25 September 2002
Abstract Flame-spread phenomena over thin solids are investigated for purely forced-opposing and concurrent flows. A two-dimensional, opposed-flow, flame-spread model, with flame radiation, has been formulated and solved numerically. In the first part of the paper, flammability limits and spread rates in opposed flow are presented, using oxygen percentage, free-stream velocity, and flow-entrance length as parameters. The comparison of the flammability boundaries and spread-rate curves for two different entrance lengths exhibits a cross-over phenomenon. Shorter entrance length results in higher spread rates and a lower oxygen-extinction limit in low free-stream velocities, but lower spread rates and a higher oxygen-extinction limit in high free-stream velocities. The entrance length affects the effective flow rate that the flame sees at the base region. This affects the radiation loss and gas residence-time in an opposing way to cause the cross-over. Radiation also affects the energy balance on the solid surface and is in part responsible for the solid-fuel non-burn-out phenomenon. In the second part of the paper, a comparison of flammability limits and flame-spreading rates between opposing and concurrent spreading flames are made; both models contain the same assumptions and properties. While the spread rate in concurrent spread increases linearly with free-stream velocity, the spread rate in opposed flow varies with free-stream velocity in a non-monotonic manner, with a peak rate at an intermediate free-stream velocity. At a given free-stream velocity, the limiting oxygen limits are lower for concurrent spread, except in the very low free-stream-velocity regime, where the spreading flame may be sustainable in opposed mode and not in concurrent mode. The cross-over disappears if the two spread modes are compared using relative flow velocities with respect to the flames rather than using free-stream velocities with respect to the laboratory. © 2003 The Combustion Institute. All rights reserved. Keywords: Flame spread model; Solid fuel; Extinction limit
1. Introduction The study of flame-spread over solid fuels in low-speed flows has gained impetus in recent years due to safety concerns for human endeavors in space.
* Corresponding author. Tel.: ⫹1-216-368-4581; fax: ⫹1-216-368-6445. E-mail address: [email protected] (J.S. T’ien).
Over the years, there has been significant progress in our understanding of flame-spread physics, which is inherently a complex phenomenon involving nonlinear interactions between flow, heat, and masstransfer, along with the chemistry in both solid and gas phases. Normally, the flame-spread phenomenon is classified as either the opposed-flow or concurrentflow mode, based on the direction of flame-spread relative to the direction of oxidizer flow. In opposedflow flame-spread, the flame spreads against the ox-
0010-2180/03/$ – see front matter © 2003 The Combustion Institute. All rights reserved. doi:10.1016/S0010-2180(02)00516-3
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idizer flow, whereas in concurrent-flow flame-spread, the direction of flame-spread is the same as that of the oxidizer. In the present paper, an attempt is made to compare the propagation and extinction processes of the two flame-spreading modes. Concurrent-flow flame-spread has been traditionally thought to be an accelerating process, since down-stream convection is the main solid-heat-up mechanism. However, in small-scale experiments using thin fuels [1–3] and modeling work with radiative heat loss [4 – 6], it has been shown that the flames can attain a constant flame-length and a steady spreadrate (i.e., both the flame tip and flame base move with a constant rate). Using a simple heat-balance argument, Honda and Ronney [2] have argued that radiative heat loss is needed for wide samples to achieve steady flame-length, and they deduced that the spread rate is proportional to the flow-approaching velocity (in forced flow) or to gravity (for buoyant, upward spread). These are consistent with existing modeling and experimental results. In the comparison to be presented, steady concurrent-flow flame-spread is to be compared with steady opposed-flow flame-spread in a forced-flow environment. Although concurrent flame-spread is generally considered more rapid than the opposed-flow mode, as the oxidizer flow direction favors flame propagation, there is a limiting flow velocity below which no concurrent-flow flame-spread can be sustained. On the contrary, the flame can self-propagate in a quiescent environment, which is a special case of opposedflow flame spreading. This suggests that, under certain conditions, opposed-flow flame-spread is possible where concurrent-flow flame-spread is forbidden. In a recent micro-gravity experiment and modeling study on the transition from ignition to flame-spread [7,8], ignition was initiated in the middle of a paper sample, and the flame may spread up-stream (opposed-flow mode) but not down-stream (concurrent-flow mode) if oxygen concentration and imposed free-stream velocity are sufficiently low. The authors attributed this to the “oxygen shadow” effect; i.e., oxygen is consumed by the flame-front up-stream, and there is not enough left to sustain a down-stream flame. Furthermore, by comparing the experimental data of Grayson et al. [9] (concurrentflow mode) and Olson [10] (opposed-flow mode), the concurrent-flow flame-spread rates are slightly lower than those of opposed flow in air and at 30% oxygen for velocities less than 5 cm/s. All the above observations are intriguing and puzzling and require further investigation on both concurrent-flow and opposed-flow flame-spread in a low free-streamvelocity regime (0 –10 cm/s) for a better understanding of the phenomenon. So far, research groups employing advanced com-
putational techniques have focused their work primarily on either opposed-flow flame-spread [11–13] or concurrent-flow flame-spread [3– 6]. They have used different mathematical models, fuel properties, and operating conditions, making it difficult to compare the results fairly between the two types of spreading flames. Although we are aware that in [8] a comparison of computed flame-spread trends versus the relative wind velocity at 30% oxygen for concurrent-flow and opposed-flow flames was made using the same model, a more complete picture of the flammability limits and the controlling mechanisms of these two types of flames, including the effect of flame radiation, is desirable. In this work, an opposed-flow model, which is consistent with a previously established model in concurrent-flow, has been formulated and numerically solved. The computed flammability map and spreading behaviors of opposed-flow flames in low-speed forced-flows are compared with the counterparts in concurrent-flow flames.
2. Flame-spread model 2.1. Model description A previous model of concurrent-flow flamespread over a thin solid [3,4,14] was adapted for the opposed-flow configuration. The mathematical formulations given in [14] are applied and modified accordingly, and all thermal and transport properties were kept the same. The gas-phase model consists of two-dimensional, steady, laminar, full Navier-Stokes equations, along with the conservation equations of mass, energy, and species. The energy equation contains a radiative divergence term that needs to be solved from the radiation-transfer equation. The species equations are for the fuel vapor, oxygen, carbon dioxide, and water vapor. A one-step, second-order, finite-rate Arrhenius reaction between fuel vapor and oxygen is assumed. The activation energy and preexponential factors are 1.13 ⫻ 105 J/g 䡠 mol and 1.58 ⫻ 1012 cm3 䡠 g⫺1 䡠 s⫺1, respectively. The heat of combustion is 1.675 ⫻ 104 J/g. The thin solid model (both thermally and aerodynamically thin) comprises continuity and energy equations whose solutions provide boundary conditions for the gas phase. The solid is assumed to be a thin, cellulosic material (Kimwipes paper having a half-thickness area density of 1 mg/cm2) with the chemical formula C6H10O5. The solid pyrolysis is modeled using one-step, zero-thorder Arrhenius kinetics, where the activation energy is 1.256 ⫻ 105 J/g mol and the pre-exponential factor is 3.8 ⫻ 107 cm/s. The solid is assumed to have a total emittance and a total absorptance of 0.92.
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Fig. 1. Schematic of domain of interest in flame fixed co-ordinates. The domain is open ( y 3 ⬁), U ⬁ is the uniform entrance flow-velocity with respect to the laboratory, and V f is the flame-spread rate. (A) Opposed-flow flame-spread configuration; x ⫽ 0 is located at the pyrolysis front (95% of fresh-fuel thickness). (B) Concurrent-flow flame-spread configuration; x ⫽ 0 is located at the solid-fuel burnout point.
A schematic of the domain of interest for the current opposed-flow flame configuration in an open domain is shown in Fig. 1A. The flame-fixed coordinate system is located on the pyrolysis front ( x ⫽ 0) of the solid, where the fuel thickness is 95% of the fresh-fuel thickness. The flame leading edge is shown stabilized over the fuel near the origin, with its tail extending downstream ( x ⬍ 0). The free-stream forced-flow with uniform velocity U ⬁ enters the domain at a distance x e (referred to as the flow entrance length in this paper) from the origin. The presence of fresh, solid fuel upstream ( x ⬎ 0) imposes no-slip conditions at the surface, which result in a nonuniform flow-profile approaching the flame. The flow approaches the flame (in the negative x direction) at a relative velocity of u r ( x, y) ⫽ u( x, y) ⫹ V f , where V f is the flame-spread rate and u( x, y) is the local flow-velocity relative to the laboratory. In this opendomain configuration, then, the flow-velocity profile always depends on the entrance length ( x e ), which is an additional parameter that needs to be considered in describing the opposed-flow flame-spread. Most of the opposed-flow computations in this work were performed with an entrance length of 6 cm. Additional computations were made with an entrance
length of 10 cm to study the effect of entrance length on flame-spread rate and flammability limits. Note that for a concurrent flame-spread configuration as shown in Fig. 1B, the flame-fixed coordinate system is fixed at the solid burnout point, and no solid is left after flame passage; therefore, the specification of entrance length is not needed. In both configurations, the domain is open to the atmosphere, and in the y direction, it extends far to the free-stream condition ( y 3 ⬁). An additional remark is made here on the relationship between the model and experiments on opposing spread over thin-solid fuel. In typical forced-flow experiments [15,16], a solid sample is placed in a wind tunnel and is ignited downstream; as the flame spreads upstream, the distance between the leading edge of the sample and the flame-front (entrance length, x e ) gradually decreases. If the rate of decrease is small compared to the characteristic time of the flame, then a quasi-steady entrance length can be defined; a steady entrance length can also be achieved in a specially designed experimental setup, like a plug flow-generating device moving upstream at the flame-spread rate. In this work, the entrance length is assumed to be constant for a given steady computation.
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2.2. Treatment of radiation Radiation (gaseous and/or surface) plays an important role in micro-gravity flames and is a key part of the model. It is responsible for the existence of the low-speed quenching-limit [17] and is also a heattransfer mechanism, besides conduction/convection in combustion systems. The importance of flame radiation in the spreading and extinction of flames over thin solid fuels in a microgravity environment has been substantiated in work by Jiang [4] for concurrent spread and by Bhattacharjee and Altenkirch [11] and Chen and Cheng [13] for opposed-flow spread. Because of the limitations of computing capability, gas-phase thermal radiation from carbon dioxide and water vapor is modeled using a mean absorption coefficient. We note that the spectral radiation computation in a one-dimensional flame has been carried out recently, e.g., [18], but the computational requirement is prohibitive for multi-dimensional flames at the present time. Soot radiation is neglected, which is based on experimental observations at low oxygen concentrations and in low-speed flows [9,10]. The solid radiation is assumed to be diffused. A mean absorption coefficient for the gas mixture is needed. The simplest way is to assign an absorption coefficient, which is treated as a parameter in the model [19]. A more sophisticated treatment of gas radiation has been proposed in [11]. While gas radiation is treated as a heat loss term in the energy equation with a constant absorption coefficient, the value of this coefficient is evaluated from a global radiation-energy balance. The radiation-heat feedback to the solid is then obtained in similar fashion. The overall radiative transfer is ensured in a selfconsistent manner, but only a single absorption coefficient is used for the entire flame, and it is believed that the absorption coefficient can vary significantly in the flame. The Planck mean absorption coefficient, K p , based on the local temperature and composition of the mixture, has been used in several modeling works [4,13]. The local Planck-mean absorption coefficient for the mixture can be given by K p ⫽ X CO 2 䡠 K p (CO 2 ) ⫹ X H2O 䡠 K p (H 2 O), where X i represents the molar fraction of species i. The values of K p for each species are from the data of Tien [20] as a function of temperature. However, by comparison with the more accurate results obtained from a narrow-band radiation model in one-dimensional flames, it was shown that the Planck mean results overpredict net emission from the flame [18,21]. A novel feature used here for both concurrent-flow and opposed-flow flames is the incorporation of a calibration procedure [21] for the mean absorption coefficient. Calibration of the absorption coefficient
against the narrow-band results through a quasi-onedimensional flame is done to account for the different optical lengths in different parts of the flame and the effect of spectral self-absorption of gaseous species. Therefore, the local absorption coefficient K used in this work is set to equal to CK p , where C is the correction factor. The distribution of C, determined iteratively in the solution procedure for the sample flame, will be shown in the results later. 2.3. Numerical scheme The SIMPLER algorithm [22] is used for the fluid-flow and combustion equations, and the radiation equation is solved using the S–N discrete ordinates method [23,24]. The two-dimensional S4 scheme with 12 ordinate directions is used in the current computations by considering the balance between numerical accuracy and computational expense [4]. Since the radiation equation and the rest of the combustion/fluid equations are coupled, they are solved iteratively. In addition, the gas-phase system is coupled to the solid-phase equations, which are solved by a finite-difference technique. The steady flame-spread rate (the eigenvalue of the whole system) is determined iteratively using a bisection method to force the burnout point in the concurrentflow mode and the pyrolysis front (95% of the freshfuel thickness) in opposed-flow mode to occur at x ⫽ 0. Computations are carried out on a non-uniform mesh. In the y direction, the grids are clustered near the solid and expand away from the surface. In the x direction, a fine grid structure is used in the flamestabilization region (the flame leading-edge) to capture the drastic variations in the flame; the grids then expand upstream and downstream. For the opposedflow configuration, a fine grid-structure near the solid burnout region (the trailing edge) is also needed, and since the pyrolysis length is not known a priori, adaptive gridding is used for this purpose. The grid structures are consistent near the flame-stabilization zone for all the calculations. The smallest grid size is 0.05 thermal lengths in the flame-stabilization zone. The domain extends to 200 thermal lengths in the y direction and 250 thermal lengths in the down-stream x direction measured from the pyrolysis front. The upstream extent of the domain is the prescribed entrance length, which is the dimensional constraint. The total number of grid nodes in the x direction varies with the free-stream velocity as dimensional entrance length is converted to a non-dimensional distance in the calculation. For a typical case of U ⬁ ⫽ 5 cm/s at 15% O2 there are 59 grid nodes in the y direction and 113 grid nodes in the x direction, whereas for U ⬁ ⫽ 10 cm/s, the number of grid nodes
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in the x direction increases to 132. The computational time, which varies from case to case, is typically about 1 to 2 hours on a Compaq XP1000 workstation.
3. Results 3.1. Effect of entrance length on opposing-flamespread There is fresh fuel up-stream of the flame in opposed-flow flame-spread; hence, a velocity boundary-layer develops over the fresh fuel surface (this is not the case for concurrent flame-spread if solid burnout occurs). Given a plug flow-velocity at the entrance, the velocity profile that the flame base sees varies with its distance from the plug flow-entrance ( x e ). Although this flow-profile influence has been recognized in the past [15,16], detailed investigation into this aspect has been lacking. We do, however, like to mention that in using the thermal diffusive model (i.e., prescribed velocity profiles without using momentum equations), Wichman [25] has chosen a linear velocity profile with velocity gradient as the flow parameter. The velocity-gradient prescription appears to be a reasonable representation when the velocity boundary layer thickness is much greater than the flame thickness, while the plug flow prescription by De Ris [26] is better when the boundary thickness is much smaller than the flame thickness. The most common cases, however, are the in-between situations. In such cases, at least two flow parameters are needed to specify the approaching flow. In this work, the flow entrance length ( x e ) and the plug flow-velocity at the entrance (U ⬁ ) are the specified parameters. We assume that for opposing spread, the fresh fuel extends all the way to the plug flow-entrance, from where the velocity boundary layer over the fuel surface starts to develop. Figure 2 shows the computed flame-spread rates versus the free-stream velocity at 15% oxygen for two different entrance-lengths (6 cm and 10 cm). The curves are non-monotonic and exhibit a maximum spread-rate at an intermediate free-stream velocity. The velocityextinction limits are the ends of the curves shown as vertical dotted lines. Near the extinction limit (dotted extension of the plotted curves), the computed flamespread rates were not constant and oscillated in a bounded fashion over a set of iterations. Whether this oscillation is physical [27] or numerical [28] is difficult to ascertain with the current steady-state model. The spread rates at a given free-stream velocity are different and with a cross-over at U ⬁ ⫽ 7 cm/s. For a free-stream velocity below 7 cm/s, the x e ⫽ 6 cm case is faster, and for the free-stream velocity above
Fig. 2. Effect of entrance-length ( x e ⫽ 6 cm and 10 cm) on opposed-flow flame-spread rate over thin solid fuel at 15% O2. The vertical dotted line indicates the extinction-velocity limits. The inset figure shows visible flame (fuel reactionrate contour ˙ F ⫽ 10 ⫺4 g/cm3 s) at U ⬁ ⫽ 5 cm/s for the two entrance lengths. Also shown are the corresponding velocity profiles (u r /U ⬁ ), 1 cm upstream of the pyrolysis front (95% fresh-fuel thickness).
7 cm/s, the x e ⫽ 10 cm case is faster. In addition, the high-speed blow-off and the low-speed quenching limits are also different. The reason for the shifting of limits and spread rates will be discussed later. In the inset of Fig. 2, the fuel-vapor reaction-rate contours (10⫺4 g 䡠 cm⫺3 䡠 s⫺1) and the velocity profiles 1 cm up-stream of the pyrolysis front are plotted for a representative case (U ⬁ ⫽ 5 cm/s). It is clear that the two velocity profiles are different, and the velocity that the flame encounters is smaller for the longerentrance-length case. So, although the two cases have the same entrance flow-velocity, the effective oxygen flow rates that the flames encounter are different. Note that the velocity plotted here is the relative velocity with respect to the flame, u r ⫽ u ⫹ V f , normalized against U ⬁ ⫽ 5 cm/s. On the fuel surface ( y ⫽ 0), u ⫽ 0, so u r ⫽ V f . 3.2. Surface heat-flux distribution and fuel-left-over phenomena in opposing spread It is observed that under certain conditions, not all fuel is pyrolyzed by the flame, and so there is leftover fuel downstream of the flame. It should be noted that the solid-fuel pyrolysis law model used here is of zeroth order [3], which allows for complete burn-out of the solid fuel. In previous models that use a firstorder pyrolysis law [11,12,16,29,30], the solid fuel will not burn out in a finite time (or in finite length). In the work by Frey and T’ien [30], for example, pyrolysis is assumed to cease when the amount of
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Fig. 3. Solid temperature profile and fuel thickness (nondimensionalized with fresh-fuel thickness) for U ⬁ ⫽ 4 cm/s, O2 ⫽ 15%, and entrance length ⫽ 6 cm.
fuel left over reaches 7%, a somewhat arbitrary criterion. This fuel leftover phenomenon has been observed in experiments [29,31], where a significant amount of potentially pyrolyzable solid was left unburned downstream of the flame. This phenomenon can be explained in terms of a vanishingly small heat-flux that reaches the solid in the region. The net amount of heat-flux going into the solid at a particular location is the sum of various types of radiation and convection contributions. In the following paragraph, a detailed examination of the heat-flux distribution will be made. A representative case with solid left-over is shown here (U ⬁ ⫽ 4 cm/s, O2 ⫽ 15%, and 6-cm entrance-length). Figure 3 gives the solid profiles. The fuel thickness (non-dimensionalized by the fresh-fuel thickness) decreases as the solid pyrolyzes. There is a considerable amount of solid left unburned for this case (about 16.6%). In the pre-heat zone ( x ⬎ 0), the solid temperature rises rapidly from ambient temperature to the pyrolysis temperature (the peak at 650 K). It becomes more flattened in the pyrolysis region and then continues to drop further downstream. Figure 4 presents the distribution of heatfluxes per unit-width of solid fuel (W/cm2) along the solid. The positive value indicates heat gain for the solid, and the negative value is a heat loss from the surface. q c is the conduction heat-flux from the gas. The gas-phase radiation feedback is represented by q r( gas) , and the radiation leaving the solid surface is q r(solid) , which is the sum of surface emission and reflection and transmission. The sum of q r( gas) and q r(solid) gives q r(net) , the net radiation flux. The sum total of q r(net) and q c is the net heat-flux on the solid surface q (net) . Near the flame leading-edge ( x ⫽ 0), q c is substantially greater than q r( gas) because the flame is very close to the solid. As we go downstream, the flame standoff distance increases, and q c drops in magnitude and q r( gas) is almost constant in the pyrolysis zone, then drops slowly further downstream. q r(solid) also peaks near the anchor point ( x ⫽ 0) and decreases slowly downstream, reflecting the strong emission from the high-temperature solid-
Fig. 4. Heat-flux distribution over solid surface for U ⬁ ⫽ 4 cm/s, O2 ⫽ 15%, and entrance length ⫽ 6 cm. Where q c is the conduction heat-flux, q r( gas) gas-phase radiation feedback, q r(solid) sum of emitted and reflected radiation from the solid, q r(net) is the net radiation heat flux (q r( gas) ⫹ q r(solid ), and q (net) is the net heat-flux (q r(net) ⫹ q c ).
surface. The net heat-flux, q (net) , distribution shows a maximum close to the flame anchor point, owing to dominance of q c , and drops in either direction; further, it is noted that for x ⬍ ⫺4.2 cm, q (net) becomes negative (shown enlarged in the inset A of Fig. 4). The pyrolysis rate along the solid fuel drops in the downstream direction to a very small value from a maximum at x ⫽ 0, which results in fuel leftover. This phenomenon can be understood in the following way. The distribution of fuel pyrolysis follows the solidsurface temperature distribution through a zeroth order Arrhenius-type dependence (activation energy ⫽ 1.256 ⫻ 105 J/g mol). The solid temperature in turn is decided by the heat balance on the surface. In the region 0 cm ⬎ x ⬎ ⫺4.2 cm, the solid-fuel temperature decreases (Fig. 3) owing to surface radiative loss and pyrolysis. This becomes evident by looking at the integrated heat-fluxes on the surface per unit width of fuel (in W/cm). The integrated incident heat-flux (q in ⫽ q r( gas) ⫹ q c ) is 3.12 W/cm, and the integrated solid-surface radiation loss [qr(solid))] is ⫺2.74 W/cm. So, the net available heating rate for the pyrolysis of fuel is qin ⫹ q r(solid) ⫽ 0.38 W/cm. The actual value of the energy consumption rate for the pyrolysis of fuel obtained by integration of local values over the fuel length is ⫺0.41 W/cm. The difference of 0.03 W/cm is supplied from the sensible heat of the solid fuel, which results in cooling of the solid fuel. It should be noted that the solid radiation loss term is a large fraction compared to the incoming heat-flux. In the region x ⬍ ⫺4.2 cm, q (net) ⬍ 0 is a consequence that the net radiative loss
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Fig. 5. Contours of mean absorption (emission) coefficient (K) distribution for the case with entrance length ( x e ) of 6 cm, 15% O2, and U ⬁ ⫽ 4 cm/s; also plotted is the fuel vapor reaction-rate contour of 10⫺4 g/cm3s (dotted curve). The inset figure shows spatial variation in the correction factor (C, K ⫽ CK p ) for the Planck mean-absorption coefficient (K p ) at U ⬁ ⫽ 4 cm/s and 10 cm/s.
Fig. 6. Flammability boundary and left-over-fuel boundary for opposed-flow flame-spread over a thin solid fuel with 6-cm entrance-length at 1 atmosphere.
from the solid exceeds the heat gain from the gasphase conduction and the gas-phase radiation feedback. In this portion, the surface temperature is substantially reduced, pyrolysis is minimal, and solid fuel is left unburnt, as demonstrated in Fig. 3. Whether there will be fuel left over and if so, how much, depends on the oxygen level and free-stream flow-velocity. This will be shown later. The inset B of Fig. 4 also shows enlarged plots in the pre-heat region. It is interesting to note that q r(net) has a positive value over a stretch of one preheat region, but q c is negative. This is because the gas-phase radiation feedback can reach farther upstream than flame conduction. In this region, net radiation from the gas phase provides energy to heat up the fuel, while gas-phase conduction plays the role of cooling. This heat-transfer mechanism is quite different from the corresponding case in concurrent flow (not shown here). With the same solid emittance (0.92), the net radiation-flux for the concurrent-flow flame is always a heat loss for the solid, although the radiation feedback could be greater than conduction in the preheat region. The spatial distribution of the mean-absorption coefficient K for the U ⬁ ⫽ 4 cm/s case is shown in Fig. 5. Recall that K ⫽ CK p , where K p is the Planck mean-absorption coefficient and C is the correction factor for the non-optically thin flame. The variation of C as a function of x is shown in the inset. In the downstream flame region of x ⬍ 0, C is determined by an optical traverse in the y direction (perpendicular to the solid) and using an empirical relation proposed in [21]. For the region of x ⬎ 0, where the flame is highly two-dimensional, two traverses are made from x ⫽ 0: one in the y direction and the other
in x direction toward upstream. The correction factor in this region is then the average of these two traverse-values. A constant C is assumed in this domain. If the value of C is close to unity, an optically thin approximation using the Planck mean-absorption coefficient is reasonable. From the inset in Fig. 5, one can see that for a free-stream velocity of 4 cm/s, the factor is close to 0.5 upstream but drops to around 0.3 downstream, indicating thickening of the radiating layer, consisting of combustion products (CO2 and H2O). The inset also shows the correction factor for the U ⬁ ⫽ 10 cm/s case. In this case, the values of C are higher than those for U ⬁ ⫽ 4 cm/s. At a higher free-stream velocity, the flame is thinner, so it is closer to the optically thin limit. Despite the semiempirical nature of the correction factor, the variation in local flame-emission power and its dependence on flow and environmental conditions has been included in an approximate manner in this model computation. The flammability boundary, along with the region of solid-fuel left-over as a function of free-stream velocity, is shown in Fig. 6. The flammability boundary consists of a blow-off branch and a radiative quenching branch. The flame goes out through the blow-off boundary due to an insufficient gas residence-time to complete the reaction, and the flame also goes out through the quenching boundary due to excessive radiative loss from the flame. Note that at U ⬁ ⫽ 0, the solid cannot propagate a flame if oxygen is below 19.85%. This is the computed limiting oxygen index in a quiescent atmosphere in zero gravity. This value is slightly lower than the experimental value of 21%, observed by Olson et al. [32], reflecting stronger kinetic parameters used in the model.
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Fig. 7. Left-over fuel (h leftover expressed as a percentage of fresh-fuel thickness) downstream of the flame as a function of free-stream velocity.
We have not attempted to calibrate these global kinetic constants in this study. Qualitatively, the Vshaped flammability-boundary in Fig. 6 resembles the experimental curve in Olson et al. [32]. It should be pointed out, however, that the flammability boundary in the above reference is a data composite from a combination of forced and buoyant experiments, while the present theoretical curve is entirely forced. Experimental data of a purely forced-flow flammability boundary that covers the whole range of free-stream velocity are not yet available. Further, Fig. 6 shows the complete solid-burnout boundary. In general, complete burnout lies on the high-velocity side, while fuel leftover lies on the low-velocity side. In a low-oxygen environment, those close to the fundamental oxygen limit, the burn-out boundary is more complex, and the amount of fuel left unburned is not necessarily monotonic with respect to the free-stream velocity or oxygen percentage. This is illustrated in Fig. 7. The trend of fuel leftover can be understood in terms of effect of free-stream velocity on the role of gas-phase heat feedback to the solid. The heat feed-back to the preheat region determines the flame spread-rate, and the heat feedback to the pyrolysis region in the time frame prescribed by the flame-spreading rate determines the amount of fuel leftover. As the free-stream velocity is lowered, both gas-phase conduction as well as gas-phase radiation feedback to the solidphase decrease in strength. Although the flame spread-rate decreases (i.e., increased available time for pyrolysis), a more drastic decrease in the heat feedback from a shorter flame over the pyrolysis region results in an increased amount of un-burned, leftover fuel. In the region very close to the quench curve at a low oxygen level (i.e., for 15% O2 in Fig. 7), gas-
Fig. 8. Comparison of extinction boundaries of opposedflow flame-spread with concurrent-flow flame-spread.
phase radiation dominates (about 70% of the incoming heat flux) heat feedback to the pre-heat region, and conduction heat-transfer is dominant (also about 70% of the incoming heat flux) in the pyrolysis region. The domination of radiation and conduction in preheat and pyrolysis regions, respectively, become more prominent as the oxygen level and/or flow velocity is lowered to the quenching limit. Since gas temperature drops fast near the quenching limit, the radiation-controlled flame-spread-rate drops very fast. This gives more time for pyrolysis in the conduction dominated region under the flame and thus, reverses the trend of leftover fuel. 3.3. Comparison of flammability limits and spreadrates in opposed and concurrent flames First, two flammability boundaries with entrance lengths of 6 cm and 10 cm in opposed-flow configuration are shown in Fig. 8 to show the effect of entrance length. With longer entrance length, the boundary moves to the right toward the higher freestream-velocity side, and the two boundaries cross over at U ⬁ ⫽ 6 cm/s. The self-propagation and the fundamental oxygen limits are not affected. This shift of the flammability boundary is consistent with the crossing-over behavior of the spread-rates shown in Fig. 2. For U ⬁ ⬎ 6 cm/s, the flame is in the gasresidence-time limited regime, and more flow will reduce the residence time, decrease the spread-rate, and push the flame closer to the blow-off limit. For shorter entrance length (i.e., 6 cm), the effective flow-rate that the flame sees is higher than the longer entrance length case (10 cm) at the same entrance velocity as shown previously in Fig. 3 (inset). Therefore, the blow-off velocity limit is lower. For U ⬁ ⬍
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6 cm/s, the flame is in the oxygen-limited regime, and a higher effective flow-rate enhances the flame. Therefore, the shorter-entrance case has a wider flammable domain at low velocity. The spread-rate behavior shown in Fig. 3 follows the same reasoning. Next, the extinction boundaries for opposed-flow flame-spread are compared to those of concurrentflow flame-spread in Fig. 8. The concurrent-flow flame-spread can occur at a lower oxygen level than opposed-flow flame for a free-stream velocity higher than 2 cm/s. The fundamental oxygen limit for concurrent flame is 11.6%, substantially lower than that of 13.76% for the opposed-flow configuration. However, only the opposed-flow flame exists at very low free-stream velocities (⬍ 2 cm/s and high oxygen concentration level). The crossover of the opposedflow and concurrent-flow flammability boundaries and the notion that a flame can be sustained in an opposing flow but not in concurrent flow seem perplexing at first. However, the true convective velocity that the flame sees is the relative flow velocity given by U r ⫽ U ⬁ ⫹ V f for opposed-flow flame-spread and U r ⫽ U ⬁ ⫺ V f for concurrent-flow flamespread, not U ⬁ alone. For a thin solid, the magnitude of the flame-spread rate can be substantial compared with that of the opposing velocity in this crossover regime. For example, at 21% O2 and U ⬁ ⫽ 0, V f ⫽ 2.17 cm/s ⫽ U r for opposed-flow flame-spread. The convective velocity relative to the flame in this case is entirely due to spreading of the flame into this quiescent ambient. For the concurrent-flow case at the same oxygen percentage, at U ⬁ ⫽ 1.4 cm/s (near the quench limit), V f ⫽ 0.144 cm/s, which gives U r ⫽ 1.356 cm/s. This is lower than the self-propagation opposed-flow case discussed above. If U ⬁ is decreased, then U r would be too low and oxygen supply will be insufficient to sustain the concurrent flame. To gain more physical insight, it then seems more reasonable to plot the flammability boundary in this low-speed regime using the relative flow velocity. This can be found in the inset of Fig. 8. In this new plot, the concurrent-flow flame is more flammable than the opposed-flow flame (in the sense of having a lower limiting relative velocity at the same oxygen percentage). The two limits approach each other as the oxygen content approaches 19.85%, the low-oxygen limit in a quiescent atmosphere. This analysis using the relative velocity can yield deeper insight and practical implications. For thermally thin solids, the spread-rate is expected to vary as the reciprocal of sample thickness. So, the crossover behavior will occur at a lower free-stream velocity as the fuel becomes thicker and the spread-rate becomes smaller. Flame-spread rates as a function of the free stream-velocities and several oxygen levels are pre-
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Fig. 9. Comparison of (A) spread-rates and (B) flamelengths in opposed (solid curves) and concurrent spreading flame (dotted curves), X, extinction.
sented in Fig. 9A. For opposed-flow situation, the flame-spread rate exhibits a non-monotonic behavior for all three oxygen concentrations (15%, 18%, 21%), which has been observed in many experiments [10,32]. The flame-spread rate increases sharply with increasing free-stream velocity near the quenching limit, reaches a maximum at some intermediate freestream velocity, and then falls gradually toward blow-off extinction. The computed flame-spread rates in concurrent flow at 15% O2 are also shown in Fig. 9A, where the concurrent-flow flame-spread-rate increases monotonically (approximately linear) with the free-stream velocity. Note that there is a crossover between these two flame-spreading modes at 15% O2. At low free-stream velocity (⬍7 cm/s), the opposed flame spreads faster than the concurrent flame, and the trend reverses at high free-stream velocity. The cross-over phenomenon in spread-rate values for concurrent-flow and opposed-flow spreading flames with flow velocity was also observed in low-speed microgravity simulations of [8] at 30% O2, where the fuel was ignited in the center to obtain both concurrent and opposed flames simultaneously. We note, however, that [8] does not include flame radiation. Flame-length variation with flow velocity is shown in Fig. 9B. The flame length (based on the 10⫺4 g 䡠 cm⫺3 䡠 s⫺1 fuel-reaction-rate contour) follows the same trend as the flame-spread rate for both concurrent-flow and opposed-flow flames. Again, a crossover is found in the low-speed flow regime. Note that although the flame sizes (and the spreadrates) of the concurrent-flow flames and opposedflow flames can be comparable at low free-stream velocities, they could be tremendously different at high free-stream velocities.
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4. Concluding remarks In order to quantitatively compare the extinction and flame-spreading characteristics between opposed-flow and concurrent-flow flame-spreads in purely forced flows, an opposed-flow flame-spread model has been formulated and solved. This model employs the same approximations and property constants as a previously established concurrent-flow flame-spread model. In particular, flame radiation, important in the low free-stream velocity regime, is treated in the same way. The radiation-transfer equation was solved using the discrete ordinate method with distributed absorption coefficients. A new procedure to calibrate these coefficients has been employed. It was found that for flame spreading in low-speed forced flow, gaseous flame radiation is an important contribution to the total heat feedback to the solid in addition to being a flame heat-loss. In the first part of the paper, certain new features in opposed-flow spread were presented. These include the effect of flow entrance length and the solid non-burnout phenomenon. Flow entrance length affects the spread-rate and the extinction limit differently, depending on the free-stream velocity. At low free-stream velocities, a short entrance length results in a higher spread-rate and a lower oxygen limit than those for the longer-entrance length case. At high free-stream velocities, the reverse is true. These trend reversals can be explained based on the velocity profiles the flames see in the flame base (flame front) region and the competing effects between radiation loss and gas residence time on oxygen supply. In an opposing-flame-spread mode, some solid fuel may be left over in the downstream region. Solid surface radiative heat-loss is a major contributor to this phenomenon. The surface radiation loss can result in negative net heat-flux to the solid, thus accelerating cooling of the solid downstream of the flame. In the free stream-velocity versus oxygen percentage map, the domain where fuel leftover occurs, leftover was presented along with a trend for leftover (in percentages) with free stream-velocity at various oxygen concentrations. The second part of the paper concentrated on the comparison between concurrent-flow and opposedflow flame-spreads. The trends of flame-spread rates between the two modes are qualitatively different. Spread-rate in concurrent flow increases linearly with free-stream velocity, but spread-rate in opposed-flow varies non-monotonically with free-stream velocity and peaks at an intermediate free-stream velocity. While these qualitative trends were previously known, the present quantitative comparison demonstrates that the two rate-curves have a crossover. Opposed-flow flame-spread can be faster than con-
current-flow flame-spread if the flow velocities are low enough, consistent with previous experimental data. A comparison of flammability boundaries in an oxygen (O2) percentage versus free-stream-velocity (U ⬁ ) map also shows the existence of a crossover between the two spreading configurations. Concurrent-flow flame in general has lower oxygen limits (including the fundamental limit), but the trend is reversed when the free-stream velocity is very small. This can be more easily understood when we consider the limiting case of zero flow-velocity. Flame spread in a quiescent atmosphere is a special case of opposed-flow flame-spread. When the oxygen percentage is sufficiently high, quiescent spread is possible, while there is no corresponding concurrent case. This limit reversal point (i.e., the free-stream velocity where the two boundaries crossover) depends on the flame-spread rates, hence, the solid thickness. It is reasoned that the domain where the opposed-flow flame is more flammable diminishes with solid thickness. If the flammability map is plotted using the relative flow-velocity with respect to the flame (rather than with respect to the laboratory coordinates), the limit reversal disappears. We note that the present work deals with pure opposed-flow and pure concurrent-flow flame-spreads only. Simultaneous opposing and concurrent spreads (i.e., ignition in the middle of the fuel sample) may have different conclusions [7,8]. In addition, the present models are two-dimensional. Although three-dimensional models have appeared recently [33,34], they have as yet to incorporate flame radiation, which is essential in this work.
Acknowledgment This research has been supported by NASA grants NCC3-669 and NCC3-633, Dr. Kurt Sacksteder, grant monitor.
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