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Model calculation of steady upward flame spread over a thin solid in reduced gravity
Twenty-Sixth Symposium (International) on Combustion/The Combustion Institute, 1996/pp. 1353–1360
MODEL CALCULATION OF STEADY UPWARD FLAME SPREAD OVER A THIN SOLID IN REDUCED GRAVITY CHING-BIAU JIANG, JAMES S. T’IEN and HSIN-YI SHIH Department of Mechanical and Aerospace Engineering Case Western Reserve University Cleveland, Ohio 44106-7222, USA
Steady upward flame spread in two-dimensional laminar flow over a thin solid is solved numerically in reduced gravity based on a combustion model recently formulated for concurrent flows. This flame spread model is more sophisticated than most previous ones because it avoids the boundary layer approximation and uses full elliptic Navier–Stokes equations. In addition, two-dimensional flame radiation treatment is accomplished by using the S-N discrete ordinates method. The emitting gas media are carbon dioxide and water vapor, the combustion products. Computed flow and flame structure are presented. The details of the flame stabilization zone near the solid burnout is resolved. Downstream flame is found to deviate from the self-similar boundary layer scaling relation. The effect of gravity level is studied. Flame length and spread rate increase approximately linearly with gravity level. A low-gravity flame quenching limit is predicted. Gaseous flame radiation is found to be important for flame structure, flame dimension, and extinction limit. Flame radiative feedback is an essential part of the solid surface energy balance. However, predicted flame spread rates have similar magnitudes as those computed by the model neglecting flame radiation.
Introduction Upward flame spread over solid fuel surface in normal earth gravity has been a subject of intensive experimental and theoretical investigations [1–11] because of its importance to fire safety. For many thick solids, flames grow to large sizes and become turbulent and sooty. This complicates the theoretical description and very often one has to resort to semiempirical relations or experimental data to complete a solution. With the opportunity to perform experiments in space, one has the option of varying the gravity level [12]. With thin solids at sufficiently reduced gravity, the flame can be small and steady in size (i.e., flame advancing rate is equal to the fuel burnout rate) and can be analyzed using laminar flow equations. In addition, in sufficiently low oxygen atmosphere, the observed flame is blue, suggesting the absence of soot. In such a case, the radiation from the flame is mainly from CO2 and H2O, the gaseous combustion products. Since the concentration and distribution of these species in the flame can be predicted with reasonable accuracy, the effect of gaseous radiation from these species on flame propagation and extinction can be analyzed in a more fundamental manner. Background The upward spreading flame we will analyze is shown schematically in Fig. 1. A thin, noncharring
solid fuel is allowed to burn out. The coordinate system moves with the burnout point (x 4 y 4 0). In this coordinate system, the flame is steady with constant length and the spread rate, VF, is an eigenvalue of the system. In this problem, we are concerned not only with flame spread rate but also with flame structure and extinction phenomena. To study this, we need to include a finite-rate chemical kinetic representation in the gas phase along with an elliptic description in the flame base region (where the fuel vapor meets the oxygen first) [13,14]. This is a departure from conventional analyses that employ boundary layer approximation. A previous work on concurrent forced-flow flame spread that incorporates these elements is Ref. 15. In this work, the flame base region is solved by full elliptic Navier–Stokes equations and is matched by the parabolic boundary layer equations downstream. Although computational cost is less, the extension of this approach to buoyant upward spread has not been successful because of numerical difficulties. Instead, full Navier–Stokes equations are applied to all the regions in the present work. This latter procedure has been found successful for both forced and buoyant flows. An additional element that is added to Ref. 15 is the contribution of gaseous radiation from the combustion products CO2 and H2O. In Ref. 15, solid surface radiative loss is accounted for but flame radiation is neglected. In the present work, radiative
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steady, laminar and elliptic (Navier–Stokes) continuity, momentum, energy, and species equations. The energy equation contains a radiative flux 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. Transport properties are the same as those used in Ref. 18, which are based on suggestions in Ref. 19. A one-step finite-rate combustion reaction between fuel vapor and oxygen is assumed. Radiation Treatment
Fig. 1. Schematic of upward flame spread over a thin solid in gravity field g.
transfer equation is solved using S-N discrete ordinates method [16,17] to account for the multidimensional nature of flame radiation. The formulation of the combustion and radiation models and the results for forced concurrent flame spread case with flame radiation are reported in Ref. 18. The results of buoyant upward spreading case are reported in this article. Outline of Theoretical Model The physical-chemical model is the same as that in Ref. [18], which is modified from Ref. 15. We will outline only the essential elements of the model here and refer the readers to these two references. The solid is both thermally thin (in the sense that the temperature gradient perpendicular to the solid surface is negligible) and aerodynamically thin (in the sense that the flame standoff distance is much greater than the thickness of the solid). The solid is allowed to burn out and a steady solution is sought with the coordinate system moving with the burnout point. The solid pyrolysis is modeled using a onestep Arrhenius kinetics of zero order [15]. In the computation performed, the solid is assumed to be a cellulosic material (Kimwipes) with chemical formula C6H10O5. Its area density is 1013 g/cm2 based on half the thickness of the solid sample. The gas-phase model consists of two-dimensional,
Since the thickness of the layer containing CO2 and H2O is small, no more than a couple of centimeters (to be shown later), the flame is assumed to be optically thin. Thus, the use of Planck-mean absorption coefficient for the mixture is adopted and is computed locally in the flame according to Ap 4 XCO2 • Ap (CO2) ` XH2O • Ap(H2O), where Xi are molar fractions for species i, and the value of Ap for CO2 and H2O, as a function of temperature, are obtained from Ref. 20. Because of a lack of detailed information, emission and absorption by fuel vapor have been neglected. The radiative transfer equation is solved by the two-dimensional S4 discrete ordinates scheme, which uses 12 ordinates. The solid fuel surface is assumed to be gray and diffused with emissivity and absorptivity equal to 0.92. Further details on the radiation model can be found in Ref. 18. Numerical Scheme The SIMPLER scheme [21] is used for the fluid flow and combustion equations, and the finite-difference positive matching scheme [22] is used in the discrete ordinates equation for radiative transfer. Both of these schemes need to be solved iteratively by themselves (iteration is needed for the radiative transfer equation because the solid reflects radiation) and between each other since they are also coupled. In addition, the gas-phase system is coupled to the solid heat transfer equation, which is solved using the trapezoidal rule. The spread rate, VF, is determined iteratively using the condition that burnout is located at x 4 0 [15]. Because of the disparity of the two length scales (stabilization versus pyrolysis zone), variable grid distribution is used in the numerical program. The small grid is 0.1 of the thermal-diffusional length to capture the details in the flame stabilization zone near the solid burnout point. The grids expand in both upstream and downstream directions to ensure that the computational domain is large enough (with a finite number of grids) to be compatible to the boundary conditions at infinity and to ensure that all the downstream pyrolysis and preheat zones are included. Both grid and domain independence have been checked.
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In the solution of the radiative transfer equation, higher ordinates S6 (24 ordinates) and S8 (40 ordinates) schemes have also been investigated [18]. S4 has been found to give adequate resolution (within 6%) on the radiative flux distribution to the solid. A similar conclusion was reached previously [23]. To achieve convergence, a case typically requires 15 h CPU time on a DEC 3000. Computed Results Flame Structure at 5 2 1012 ge, 15% O2, and 1 atm Pressure
Fig. 2. Profiles for (from top to bottom) temperature (nondimensionalized by T` 4 300 K), fuel vapor consumption rate, mass fraction of fuel vapor and oxygen, fuel/ oxygen local equivalence ratio, mass fraction of water vapor, mass fraction of carbon dioxide, Planck absorption coefficient, and nondimensional radiation flux gradient (1 unit 4 0.157 w/cm3). 15% O2, 5 2 1012 ge and 1 atm pressure. Gravity is toward the left.
Fig. 3. Contours of (top) relative pressure (nondimensionalized by 1/2 q` U2R). Bottom velocity vectors and stream lines. 1 unit of velocity is UR 4 7.26 cm/s.
Flame and solid profiles are presented for a gravity level of 5 2 1012 of that on earth and in a 15% O2, 85% N2 mixture at 1 atm pressure. Figure 2 shows the profiles for gas temperature, fuel vapor consumption rate, mass fractions of fuel and oxidizer, local fuel/oxygen equivalence ratio, mass fraction of water vapor and mass fraction of carbon dioxide, local Planck-mean absorption coefficient, and the divergence of radiative flux (only one-half of the flame profiles is plotted because of symmetry). The fuel vapor consumption rate gives the best indication of a visible flame in combustion models using one-step kinetics. Following Ref. 15, we selected a consumption rate of 1014 g/cm3/s as the visible flame contour for later discussion and comparison. The temperature and fuel consumption rate contours show that the flame standoff distance from the solid increases very fast near the flame base region but stays nearly constant in the downstream (7 to 8 mm from the surface). This rather large downstream flame standoff distance implies a weak conductive feedback from the flame, which will be discussed in later sections. Fig. 2 also shows the local Planck-mean absorption coefficient for the mixture Ap, which is used in the radiation calculation. The radiation flux divergence term ¹ • qr, which appears in the energy equation, is shown at the bottom of Fig. 2. The radiation flux divergence profile resembles that of the gas temperature, reflecting the strong temperature dependence of radiative emission and the assumption of optically thin flame. Figure 3 shows relative pressure distribution, stream lines, and velocity vectors around the flame. The pressure distribution is distinctly different from that in purely forced flow [15,18]. In a buoyant environment, flow is driven by density gradient. As can be seen in Fig. 3, the velocity vectors show a strong flow acceleration inside the flame stabilization zone but gradually approach boundary layer type profiles. The stream line is first deflected away from the centerline (y 4 0) in the upstream because the pressure increase created by the stagnation point and the fuel vapor blowing from the solid surface. It then bends toward the centerline because of flow acceleration
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Fig. 4. Radiation flux vectors. One unit is rT4`, T` 4 300 K.
Fig. 5. Solid profiles. Surface temperature Ts (nondimensionalized by T` 4 300 K), solid thickness h (nondimensionalized by ho 4 1.9 2 1013 cm) and blowing velocity Vw (nondimensionalized by UR 4 7.26 cm/s).
Fig. 6. Heat fluxes along the solid surface: qc, heat conduction, (qr)in incident radiation, (qr)out, outgoing radiation, (qr)net, net radiative flux to the surface, and qnet, total net heat flux to the surface.
in the flame zone. This is qualitatively different from that of the purely forced flow case in which the stream line is always directed away from the solid. Consequently, in the downstream region, the flame stays close to the solid for the buoyant flow case, as seen in Fig. 2. This increases the importance of convective heat transfer compared to the corresponding case in forced flow. Figure 3 also shows that much of the flow of oxygen into the flame zone comes from the upstream
instead of the side, as implied by the classical boundary layer theory. Furthermore, examination of the variation in flame standoff distance (distance from the solid to the location of maximum flame temperature or the maximum fuel consumption rate contour) with x indicates it does not follow the x1/4 dependence, as implied by the self-similar solution of boundary layer. Similarly, the blowing rate of the solid Vw does not follow the x11/4 dependence either. This could be due to a combination of the following reasons: (1) the flame length is not long enough (compared to standoff distance) to escape the upstream influence by the stabilization zone, (2) finiterate chemical kinetics and the variations of flame and surface temperature with x preclude the self-similar solution [15], and (3) radiative feedback to the solid does not follow the boundary layer scaling rule. For short flames at reduced gravity, the use of elliptic Navier–Stokes equation is suggested not only from the consideration of flame stability but also from the accuracy of fluid mechanics. Note that even when nonsimilar boundary layer is applicable downstream, it requires upstream boundary conditions that can only be provided accurately by the Navier–Stokes equations. Figure 4 shows the net radiative heat flux vector. The two-dimensional character of radiative heat transfer in the flame base region can be seen clearly. In the downstream, the radiative flux becomes approximately one dimensional. Note also that all the net radiative fluxes are directed away from the solid fuel surface. This is because the surface emits more than it absorbs from the flame. Figure 5 gives the solid fuel profiles. The fuel thickness h (nondimensionalized by the unburned solid thickness) decreases as the solid pyrolyzes. The fuel vapor blowing rate, Vw (nondimensionalized by the reference velocity UR 4 7.26 cm/s at g 4 5 2 1012 ge), decreases away from the solid burnout point. Solid temperature increases in the preheat zone and becomes more flattened in the pyrolysis region. For later usage, pyrolysis front is defined at the location Vw 4 1012 (Vw)max. This is different from Ref. 15, where the pyrolysis front is defined based on the incident heat flux. Figure 6 presents the heat fluxes along the solid. Heat flux by conduction is given by qc. Radiation influx (including that from the ambient) is given by (qr)in. Note that near the flame burnout, qc is substantially greater than (qr)in because of the closeness of flame to the solid. Downstream the flame standoff distance increases, qc drops in magnitude. Between x 4 2 and x 4 13 cm, (qr)in is actually slightly greater than qc. An integrated heat flux in the preheat zone as a function of gravity levels will be given later. Radiation leaving the surface is given by (qr)out, the sum of emission plus reflection. The magnitude of (qr)out is greater than (qr)in, which results in a negative net
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Fig. 7. Comparison of nondimensionalized temperature profiles: with flame radiation (top), without flame radiation (bottom). Ambient oxygen 15%, g 4 1012 ge, p 4 1 atm.
radiative flux on the surface, as illustrated by (qr)net. The total net heat flux to the solid is given by qnet.
Fig. 9. Flame spread rates versus gravity; 15% O2, 1 atm pressure.
Comparison with Model Calculation without Gas Radiation
Effect of Gravity Level
To evaluate the effect of flame radiation, comparison is made with model calculation neglecting gasphase radiation (but including surface radiative loss). Figure 7 shows the temperature contours for the corresponding cases at g 4 1012ge. With gas radiation, the flame is shorter because of the additional heat loss from the flame. This flame shortening is observed at all the gravity levels studied. In upward spread, flame length influences spread rate. However, part of the radiation emitted from the flame is absorbed by the solid, which also affects spread rate [24]. This will be examined later.
Figure 8 shows the influence of gravity level on fuel consumption contours. We note that the flame length increases approximately linearly with gravity level. When steady state is reached, the flame is rather long for two-dimensional flow, even at 0.1 ge. Figure 9 shows the influence of gravity level on flame spread rate with and without flame radiation. First, we notice that quenching occurs at low gravity. With gas radiation, the limit is at a higher gravity level. Gas radiation induces additional heat loss from the system and a narrower range of flammable gravity levels. This is similar to the finding in downward
Fig. 8. Fuel consumption rate contours at various gravity levels. Inner contour is 1014 g/cm3/s and a factor of 10 separates each contour; ambient oxygen 15%.
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Fig. 10. Pyrolysis length (lp) and integrated heat fluxes over the preheat zone as a function of gravity level in 15% O2.
flame spread [25,26]. Flame approaches the quench limit by shortening its length and decreasing the maximum temperature qualitatively similar to the flame behavior in low-speed forced flow when the free stream velocity is reduced [15,27]. Because of the shorter flame length, spread rate is lower when gas radiation is included. However, flame radiative feedback makes up part of the spread rate reduction. As gravity level is increased, the percentage of difference between the two cases is diminished. In thermal theory of flame spread, the rate of solid heat up in the preheat zone is often used to derive the expression for spread rate. In Fig. 10, integrated heat fluxes over the preheat zone per unit width, denoted by [], are shown as a function of the gravity level. Also shown is the pyrolysis length, which increases approximately linearly with gravity level. The conductive rate, [qc], is about twice as large as the rate of incident flame radiation to the surface, [(qr)in 1 rT4`]. The surface radiative emission, not shown in the figure, is larger than the absorbed radiation, which results in a net radiative loss from the surface shown by 1[(qr)net]. The net rate of heat input into the solid preheat zone, [qnet], is the difference of the convective rate and the net radiative loss rate. It is rather small, as shown in Fig. 10, since it is the difference of two large quantities.
Conclusion Numerical solutions have been obtained for twodimensional laminar, steady, upward flame spread over a thin solid in reduced gravity (1011 earth gravity and below) using a combustion model with flame radiation. A one-step finite-rate gas-phase combustion reaction and a zero-order solid pyrolysis reac-
tion are assumed. The fluid mechanical treatment in the model is free of the conventional boundary layer approximation; full elliptic Navier–Stokes momentum, energy, and species equations are used. This enables studies of the extinction phenomena and provides a more precise description of the flow field. The flow pattern affects the flame structures, especially for shorter flames in reduced gravity. The flame radiation treatment uses the S-N discrete ordinates method, which is capable of treating multidimensional radiative transfer and can be used to study the role of flame radiation on the preheating and the pyrolization of the solid fuel. Flame structures for one case (15% O2, 5 2 1012 ge 1 atm) are presented in detail. In the flame zone downstream of the flame base, the buoyancy draws the stream lines and the flame toward the fuel surface, which increases the rate of conductive heat transfer (as compared with that in the purely forced flow case). The computed solid vaporization rate and flame standoff distance suggest that the scaling according to the self-similar boundary layer solution is not followed. Up to 0.1 ge, flame spread rate and pyrolysis length increase approximately linearly with gravity level. The computation is terminated at 0.1 ge since the flame becomes too long above this gravity level and the laminar flow assumption can be violated. A low-gravity quenching extinction limit is found to occur. This is higher than the low-gravity limit for the same fuel when flame radiation is neglected (but surface radiative loss is included). The shift of limit shows that the radiation provides additional heat loss from the flame. Flame radiation can be a substantial portion of the total heat feedback to the solid in both the pyrolysis and the preheat zones. Flame radiation shortens the flame length and reduces the flame temperature. Despite this, the computed spread rates with and without flame radiation are close (except in the nearlimit region). Flame spread rate should not be used as the sole indicator for the soundness of a flame model. Although the present model for upward flame spread is already quite complicated, a number of improvements can be made in the future. Other than the simplified description of gas-phase kinetics and solid pyrolysis reactions, radiation treatment in the transfer equation can be improved. For optically thin situations, Planck-mean absorption coefficient provides the correct spectrally integrated emission characteristics. However, CO2, H2O, and fuel vapor (which we ignore in the radiation treatment) can absorb more radiation than we predicted in this work. With more computational effort and resources, more advanced radiation models (e.g., wide-band) can be incorporated into the upward flame spreading formulation. The present work provides the framework upon which future improvements can be made. A
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comparison with experiment has not been made because of the lack of data. Acknowledgments We would like to thank Dr. Paul Ferkul, whose model and numerical algorithm in Ref. 15 provided the basis upon which the present work is built. Thanks are also due to Mr. Kurt Sacksteder and Mr. Hasan Bedir for their consultation. This work has been supported by NASA Microgravity Science and Applications Division through Grant NAG31046. REFERENCES 1. Markstein, G. H. and de Ris, J., Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, pp. 1085–1097. 2. Orloff, L., de Ris, J., and Markstein, G. H., Fifteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1974, pp. 183–192. 3. Fernandez-Pello, A. C., Combust. Flame 35:135 (1978). 4. Chu, L., Chen, C. H., and T’ien, J. S., “Upward Spreading Flames over Paper Samples,” ASME, 81WA/HF-42, 1981. 5. Hasemi, Y., Proc. First Int. Symp. on Fire Safety Science, Hemisphere Pub. Co., Washington, D.C., 1985, pp. 87–96. 6. Saito, K., Quintiere, J. G., and Williams, F. A., Proc. First Int. Symp. on Fire Safety Science, Hemisphere Pub. Co., Washington, D.C., 1986, pp. 75–86. 7. Delichatsios, M. M., Mathew, M. K., and Delichatsios, M. A., Proc. Third Int. Symp. on Fire Safety, Elsevier, New York, 1991, pp. 207–216. 8. Mitler, H., Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1990, pp. 1715–1721. 9. Kulkarni, A. K. and Fischer, S. J., “Heat Transfer in Combustion Systems,” ASME HTD 122:53–61 (1989).
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10. Thomas, P. H., Fire Safety J. 22:89–99 (1994). 11. Honda, L. and Ronney, P. D., “Mechanism of Concurrent-Flow Flame Spread over Thin Solid Fuels,” Paper presented at Fall Meeting/Combustion Institute/Western States, Stanford, CA, Oct. 30–31, 1995. 12. Sacksteder, K. R. and T’ien, J. S., Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994. pp. 1685–1692. 13. Chen, C. H. and T’ien, J. S., Combust. Sci. Technol. 50:283–306 (1986). 14. DiBlasi, C., Progress in Astronautics and Aeronautics, Vol. 135, Chapter 21, AIAA. 15. Ferkul, P. V. and T’ien, J. S., Combust. Sci. Technol. 99:345–370 (1994). 16. Fiveland, W. A., J. Heat Transfer, 106:699–706 (1984). 17. Kim, T. K. and Lee, H. S., J. Quant. Spectros. Radiat. Transfer 42:225–238 (1989). 18. Jiang, C. B., “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. 19. Smooke, M. D. and Giovangigli, V., Lecture Notes in Physics, Springer Verlag, New York, 1991, pp. 1–28. 20. Abu-Romia, M. M. and Tien, C. L., (1967), J. Heat Transfer 89(4):321–327 (1967). 21. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere Pub. Co., New York, 1980. 22. Lathrop, K. D., J. Comput. Phys. 4:475–498 (1969). 23. Kim, J. S., Baek, S. W., and Kaplan, C. R., Combust. Sci. Technol. 88:133–150 (1993). 24. Bhattacharjee, S. and Altenkirch, R. A., Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1991, pp. 1627–1633. 25. West, J., Bhattacharjee, S., and Altenkirch, R. A., Combust. Sci. Technol. 83:233–244 (1992). 26. Chen, C. H. and Cheng, M. C., Combust. Sci. Technol. 97:63–83 (1994). 27. Grayson, G., Sacksteder, K. R., Ferkul, P. V., and T’ien, J. S., Microgravity Sci. Technol. 7(2):187–193 (1994).
COMMENTS Howard D. Ross, NASA Lewis Research Center, USA. Are any experiments available for comparison with the model? If so, do any of them show steady spread? Author’s Reply. At the present time, there isn’t any good experimental data on upward spread at one atmosphere in the low-gravity range (0.1 ge. and below) investigated in this work. An attempt has been made using aircraft flying parabolic trajectories to simulate this low-g environment, but the g-jitter influence was found to be too great.[1] On the other hand, steady upward spreads have been observed in a normal gravity, low-pressure environment for paper samples in a wide range of sample widths in our lab (see Ref. 4) and elsewhere (see Ref. 11).
REFERENCE 1. Sacksteder, K., private communication. ● Subrata Bhattacharjee, San Diego State University, USA. One of the most significant conclusions of your paper is the discovery of a steady-state flame in a concurrent configuration. However, your model assumes away the time terms in the N-S equations. I would suggest that the unsteady terms be retained. If the solution is steady, an unsteady model will definitely converge on the correct solution, which is not guaranteed by the steady-state model.
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Author’s Reply. Clearly, a steady combusting solution can be obtained from an unsteady model when the boundary conditions are steady and proper ‘hot’ initial conditions are prescribed. Although we did not use an unsteady model, the governing partial differential equations are elliptic in
our work and we have to iterate to obtain the steady solution. The iteration procedure has time-like characteristics. We are confident that when a steady spreading flame is obtained by this procedure, the mathematical system will allow the existence of such a solution.
MODEL CALCULATION OF STEADY UPWARD FLAME SPREAD OVER A THIN SOLID IN REDUCED GRAVITY CHING-BIAU JIANG, JAMES S. T’IEN and HSIN-YI SHIH Department of Mechanical and Aerospace Engineering Case Western Reserve University Cleveland, Ohio 44106-7222, USA
Steady upward flame spread in two-dimensional laminar flow over a thin solid is solved numerically in reduced gravity based on a combustion model recently formulated for concurrent flows. This flame spread model is more sophisticated than most previous ones because it avoids the boundary layer approximation and uses full elliptic Navier–Stokes equations. In addition, two-dimensional flame radiation treatment is accomplished by using the S-N discrete ordinates method. The emitting gas media are carbon dioxide and water vapor, the combustion products. Computed flow and flame structure are presented. The details of the flame stabilization zone near the solid burnout is resolved. Downstream flame is found to deviate from the self-similar boundary layer scaling relation. The effect of gravity level is studied. Flame length and spread rate increase approximately linearly with gravity level. A low-gravity flame quenching limit is predicted. Gaseous flame radiation is found to be important for flame structure, flame dimension, and extinction limit. Flame radiative feedback is an essential part of the solid surface energy balance. However, predicted flame spread rates have similar magnitudes as those computed by the model neglecting flame radiation.
Introduction Upward flame spread over solid fuel surface in normal earth gravity has been a subject of intensive experimental and theoretical investigations [1–11] because of its importance to fire safety. For many thick solids, flames grow to large sizes and become turbulent and sooty. This complicates the theoretical description and very often one has to resort to semiempirical relations or experimental data to complete a solution. With the opportunity to perform experiments in space, one has the option of varying the gravity level [12]. With thin solids at sufficiently reduced gravity, the flame can be small and steady in size (i.e., flame advancing rate is equal to the fuel burnout rate) and can be analyzed using laminar flow equations. In addition, in sufficiently low oxygen atmosphere, the observed flame is blue, suggesting the absence of soot. In such a case, the radiation from the flame is mainly from CO2 and H2O, the gaseous combustion products. Since the concentration and distribution of these species in the flame can be predicted with reasonable accuracy, the effect of gaseous radiation from these species on flame propagation and extinction can be analyzed in a more fundamental manner. Background The upward spreading flame we will analyze is shown schematically in Fig. 1. A thin, noncharring
solid fuel is allowed to burn out. The coordinate system moves with the burnout point (x 4 y 4 0). In this coordinate system, the flame is steady with constant length and the spread rate, VF, is an eigenvalue of the system. In this problem, we are concerned not only with flame spread rate but also with flame structure and extinction phenomena. To study this, we need to include a finite-rate chemical kinetic representation in the gas phase along with an elliptic description in the flame base region (where the fuel vapor meets the oxygen first) [13,14]. This is a departure from conventional analyses that employ boundary layer approximation. A previous work on concurrent forced-flow flame spread that incorporates these elements is Ref. 15. In this work, the flame base region is solved by full elliptic Navier–Stokes equations and is matched by the parabolic boundary layer equations downstream. Although computational cost is less, the extension of this approach to buoyant upward spread has not been successful because of numerical difficulties. Instead, full Navier–Stokes equations are applied to all the regions in the present work. This latter procedure has been found successful for both forced and buoyant flows. An additional element that is added to Ref. 15 is the contribution of gaseous radiation from the combustion products CO2 and H2O. In Ref. 15, solid surface radiative loss is accounted for but flame radiation is neglected. In the present work, radiative
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steady, laminar and elliptic (Navier–Stokes) continuity, momentum, energy, and species equations. The energy equation contains a radiative flux 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. Transport properties are the same as those used in Ref. 18, which are based on suggestions in Ref. 19. A one-step finite-rate combustion reaction between fuel vapor and oxygen is assumed. Radiation Treatment
Fig. 1. Schematic of upward flame spread over a thin solid in gravity field g.
transfer equation is solved using S-N discrete ordinates method [16,17] to account for the multidimensional nature of flame radiation. The formulation of the combustion and radiation models and the results for forced concurrent flame spread case with flame radiation are reported in Ref. 18. The results of buoyant upward spreading case are reported in this article. Outline of Theoretical Model The physical-chemical model is the same as that in Ref. [18], which is modified from Ref. 15. We will outline only the essential elements of the model here and refer the readers to these two references. The solid is both thermally thin (in the sense that the temperature gradient perpendicular to the solid surface is negligible) and aerodynamically thin (in the sense that the flame standoff distance is much greater than the thickness of the solid). The solid is allowed to burn out and a steady solution is sought with the coordinate system moving with the burnout point. The solid pyrolysis is modeled using a onestep Arrhenius kinetics of zero order [15]. In the computation performed, the solid is assumed to be a cellulosic material (Kimwipes) with chemical formula C6H10O5. Its area density is 1013 g/cm2 based on half the thickness of the solid sample. The gas-phase model consists of two-dimensional,
Since the thickness of the layer containing CO2 and H2O is small, no more than a couple of centimeters (to be shown later), the flame is assumed to be optically thin. Thus, the use of Planck-mean absorption coefficient for the mixture is adopted and is computed locally in the flame according to Ap 4 XCO2 • Ap (CO2) ` XH2O • Ap(H2O), where Xi are molar fractions for species i, and the value of Ap for CO2 and H2O, as a function of temperature, are obtained from Ref. 20. Because of a lack of detailed information, emission and absorption by fuel vapor have been neglected. The radiative transfer equation is solved by the two-dimensional S4 discrete ordinates scheme, which uses 12 ordinates. The solid fuel surface is assumed to be gray and diffused with emissivity and absorptivity equal to 0.92. Further details on the radiation model can be found in Ref. 18. Numerical Scheme The SIMPLER scheme [21] is used for the fluid flow and combustion equations, and the finite-difference positive matching scheme [22] is used in the discrete ordinates equation for radiative transfer. Both of these schemes need to be solved iteratively by themselves (iteration is needed for the radiative transfer equation because the solid reflects radiation) and between each other since they are also coupled. In addition, the gas-phase system is coupled to the solid heat transfer equation, which is solved using the trapezoidal rule. The spread rate, VF, is determined iteratively using the condition that burnout is located at x 4 0 [15]. Because of the disparity of the two length scales (stabilization versus pyrolysis zone), variable grid distribution is used in the numerical program. The small grid is 0.1 of the thermal-diffusional length to capture the details in the flame stabilization zone near the solid burnout point. The grids expand in both upstream and downstream directions to ensure that the computational domain is large enough (with a finite number of grids) to be compatible to the boundary conditions at infinity and to ensure that all the downstream pyrolysis and preheat zones are included. Both grid and domain independence have been checked.
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In the solution of the radiative transfer equation, higher ordinates S6 (24 ordinates) and S8 (40 ordinates) schemes have also been investigated [18]. S4 has been found to give adequate resolution (within 6%) on the radiative flux distribution to the solid. A similar conclusion was reached previously [23]. To achieve convergence, a case typically requires 15 h CPU time on a DEC 3000. Computed Results Flame Structure at 5 2 1012 ge, 15% O2, and 1 atm Pressure
Fig. 2. Profiles for (from top to bottom) temperature (nondimensionalized by T` 4 300 K), fuel vapor consumption rate, mass fraction of fuel vapor and oxygen, fuel/ oxygen local equivalence ratio, mass fraction of water vapor, mass fraction of carbon dioxide, Planck absorption coefficient, and nondimensional radiation flux gradient (1 unit 4 0.157 w/cm3). 15% O2, 5 2 1012 ge and 1 atm pressure. Gravity is toward the left.
Fig. 3. Contours of (top) relative pressure (nondimensionalized by 1/2 q` U2R). Bottom velocity vectors and stream lines. 1 unit of velocity is UR 4 7.26 cm/s.
Flame and solid profiles are presented for a gravity level of 5 2 1012 of that on earth and in a 15% O2, 85% N2 mixture at 1 atm pressure. Figure 2 shows the profiles for gas temperature, fuel vapor consumption rate, mass fractions of fuel and oxidizer, local fuel/oxygen equivalence ratio, mass fraction of water vapor and mass fraction of carbon dioxide, local Planck-mean absorption coefficient, and the divergence of radiative flux (only one-half of the flame profiles is plotted because of symmetry). The fuel vapor consumption rate gives the best indication of a visible flame in combustion models using one-step kinetics. Following Ref. 15, we selected a consumption rate of 1014 g/cm3/s as the visible flame contour for later discussion and comparison. The temperature and fuel consumption rate contours show that the flame standoff distance from the solid increases very fast near the flame base region but stays nearly constant in the downstream (7 to 8 mm from the surface). This rather large downstream flame standoff distance implies a weak conductive feedback from the flame, which will be discussed in later sections. Fig. 2 also shows the local Planck-mean absorption coefficient for the mixture Ap, which is used in the radiation calculation. The radiation flux divergence term ¹ • qr, which appears in the energy equation, is shown at the bottom of Fig. 2. The radiation flux divergence profile resembles that of the gas temperature, reflecting the strong temperature dependence of radiative emission and the assumption of optically thin flame. Figure 3 shows relative pressure distribution, stream lines, and velocity vectors around the flame. The pressure distribution is distinctly different from that in purely forced flow [15,18]. In a buoyant environment, flow is driven by density gradient. As can be seen in Fig. 3, the velocity vectors show a strong flow acceleration inside the flame stabilization zone but gradually approach boundary layer type profiles. The stream line is first deflected away from the centerline (y 4 0) in the upstream because the pressure increase created by the stagnation point and the fuel vapor blowing from the solid surface. It then bends toward the centerline because of flow acceleration
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Fig. 4. Radiation flux vectors. One unit is rT4`, T` 4 300 K.
Fig. 5. Solid profiles. Surface temperature Ts (nondimensionalized by T` 4 300 K), solid thickness h (nondimensionalized by ho 4 1.9 2 1013 cm) and blowing velocity Vw (nondimensionalized by UR 4 7.26 cm/s).
Fig. 6. Heat fluxes along the solid surface: qc, heat conduction, (qr)in incident radiation, (qr)out, outgoing radiation, (qr)net, net radiative flux to the surface, and qnet, total net heat flux to the surface.
in the flame zone. This is qualitatively different from that of the purely forced flow case in which the stream line is always directed away from the solid. Consequently, in the downstream region, the flame stays close to the solid for the buoyant flow case, as seen in Fig. 2. This increases the importance of convective heat transfer compared to the corresponding case in forced flow. Figure 3 also shows that much of the flow of oxygen into the flame zone comes from the upstream
instead of the side, as implied by the classical boundary layer theory. Furthermore, examination of the variation in flame standoff distance (distance from the solid to the location of maximum flame temperature or the maximum fuel consumption rate contour) with x indicates it does not follow the x1/4 dependence, as implied by the self-similar solution of boundary layer. Similarly, the blowing rate of the solid Vw does not follow the x11/4 dependence either. This could be due to a combination of the following reasons: (1) the flame length is not long enough (compared to standoff distance) to escape the upstream influence by the stabilization zone, (2) finiterate chemical kinetics and the variations of flame and surface temperature with x preclude the self-similar solution [15], and (3) radiative feedback to the solid does not follow the boundary layer scaling rule. For short flames at reduced gravity, the use of elliptic Navier–Stokes equation is suggested not only from the consideration of flame stability but also from the accuracy of fluid mechanics. Note that even when nonsimilar boundary layer is applicable downstream, it requires upstream boundary conditions that can only be provided accurately by the Navier–Stokes equations. Figure 4 shows the net radiative heat flux vector. The two-dimensional character of radiative heat transfer in the flame base region can be seen clearly. In the downstream, the radiative flux becomes approximately one dimensional. Note also that all the net radiative fluxes are directed away from the solid fuel surface. This is because the surface emits more than it absorbs from the flame. Figure 5 gives the solid fuel profiles. The fuel thickness h (nondimensionalized by the unburned solid thickness) decreases as the solid pyrolyzes. The fuel vapor blowing rate, Vw (nondimensionalized by the reference velocity UR 4 7.26 cm/s at g 4 5 2 1012 ge), decreases away from the solid burnout point. Solid temperature increases in the preheat zone and becomes more flattened in the pyrolysis region. For later usage, pyrolysis front is defined at the location Vw 4 1012 (Vw)max. This is different from Ref. 15, where the pyrolysis front is defined based on the incident heat flux. Figure 6 presents the heat fluxes along the solid. Heat flux by conduction is given by qc. Radiation influx (including that from the ambient) is given by (qr)in. Note that near the flame burnout, qc is substantially greater than (qr)in because of the closeness of flame to the solid. Downstream the flame standoff distance increases, qc drops in magnitude. Between x 4 2 and x 4 13 cm, (qr)in is actually slightly greater than qc. An integrated heat flux in the preheat zone as a function of gravity levels will be given later. Radiation leaving the surface is given by (qr)out, the sum of emission plus reflection. The magnitude of (qr)out is greater than (qr)in, which results in a negative net
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Fig. 7. Comparison of nondimensionalized temperature profiles: with flame radiation (top), without flame radiation (bottom). Ambient oxygen 15%, g 4 1012 ge, p 4 1 atm.
radiative flux on the surface, as illustrated by (qr)net. The total net heat flux to the solid is given by qnet.
Fig. 9. Flame spread rates versus gravity; 15% O2, 1 atm pressure.
Comparison with Model Calculation without Gas Radiation
Effect of Gravity Level
To evaluate the effect of flame radiation, comparison is made with model calculation neglecting gasphase radiation (but including surface radiative loss). Figure 7 shows the temperature contours for the corresponding cases at g 4 1012ge. With gas radiation, the flame is shorter because of the additional heat loss from the flame. This flame shortening is observed at all the gravity levels studied. In upward spread, flame length influences spread rate. However, part of the radiation emitted from the flame is absorbed by the solid, which also affects spread rate [24]. This will be examined later.
Figure 8 shows the influence of gravity level on fuel consumption contours. We note that the flame length increases approximately linearly with gravity level. When steady state is reached, the flame is rather long for two-dimensional flow, even at 0.1 ge. Figure 9 shows the influence of gravity level on flame spread rate with and without flame radiation. First, we notice that quenching occurs at low gravity. With gas radiation, the limit is at a higher gravity level. Gas radiation induces additional heat loss from the system and a narrower range of flammable gravity levels. This is similar to the finding in downward
Fig. 8. Fuel consumption rate contours at various gravity levels. Inner contour is 1014 g/cm3/s and a factor of 10 separates each contour; ambient oxygen 15%.
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Fig. 10. Pyrolysis length (lp) and integrated heat fluxes over the preheat zone as a function of gravity level in 15% O2.
flame spread [25,26]. Flame approaches the quench limit by shortening its length and decreasing the maximum temperature qualitatively similar to the flame behavior in low-speed forced flow when the free stream velocity is reduced [15,27]. Because of the shorter flame length, spread rate is lower when gas radiation is included. However, flame radiative feedback makes up part of the spread rate reduction. As gravity level is increased, the percentage of difference between the two cases is diminished. In thermal theory of flame spread, the rate of solid heat up in the preheat zone is often used to derive the expression for spread rate. In Fig. 10, integrated heat fluxes over the preheat zone per unit width, denoted by [], are shown as a function of the gravity level. Also shown is the pyrolysis length, which increases approximately linearly with gravity level. The conductive rate, [qc], is about twice as large as the rate of incident flame radiation to the surface, [(qr)in 1 rT4`]. The surface radiative emission, not shown in the figure, is larger than the absorbed radiation, which results in a net radiative loss from the surface shown by 1[(qr)net]. The net rate of heat input into the solid preheat zone, [qnet], is the difference of the convective rate and the net radiative loss rate. It is rather small, as shown in Fig. 10, since it is the difference of two large quantities.
Conclusion Numerical solutions have been obtained for twodimensional laminar, steady, upward flame spread over a thin solid in reduced gravity (1011 earth gravity and below) using a combustion model with flame radiation. A one-step finite-rate gas-phase combustion reaction and a zero-order solid pyrolysis reac-
tion are assumed. The fluid mechanical treatment in the model is free of the conventional boundary layer approximation; full elliptic Navier–Stokes momentum, energy, and species equations are used. This enables studies of the extinction phenomena and provides a more precise description of the flow field. The flow pattern affects the flame structures, especially for shorter flames in reduced gravity. The flame radiation treatment uses the S-N discrete ordinates method, which is capable of treating multidimensional radiative transfer and can be used to study the role of flame radiation on the preheating and the pyrolization of the solid fuel. Flame structures for one case (15% O2, 5 2 1012 ge 1 atm) are presented in detail. In the flame zone downstream of the flame base, the buoyancy draws the stream lines and the flame toward the fuel surface, which increases the rate of conductive heat transfer (as compared with that in the purely forced flow case). The computed solid vaporization rate and flame standoff distance suggest that the scaling according to the self-similar boundary layer solution is not followed. Up to 0.1 ge, flame spread rate and pyrolysis length increase approximately linearly with gravity level. The computation is terminated at 0.1 ge since the flame becomes too long above this gravity level and the laminar flow assumption can be violated. A low-gravity quenching extinction limit is found to occur. This is higher than the low-gravity limit for the same fuel when flame radiation is neglected (but surface radiative loss is included). The shift of limit shows that the radiation provides additional heat loss from the flame. Flame radiation can be a substantial portion of the total heat feedback to the solid in both the pyrolysis and the preheat zones. Flame radiation shortens the flame length and reduces the flame temperature. Despite this, the computed spread rates with and without flame radiation are close (except in the nearlimit region). Flame spread rate should not be used as the sole indicator for the soundness of a flame model. Although the present model for upward flame spread is already quite complicated, a number of improvements can be made in the future. Other than the simplified description of gas-phase kinetics and solid pyrolysis reactions, radiation treatment in the transfer equation can be improved. For optically thin situations, Planck-mean absorption coefficient provides the correct spectrally integrated emission characteristics. However, CO2, H2O, and fuel vapor (which we ignore in the radiation treatment) can absorb more radiation than we predicted in this work. With more computational effort and resources, more advanced radiation models (e.g., wide-band) can be incorporated into the upward flame spreading formulation. The present work provides the framework upon which future improvements can be made. A
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comparison with experiment has not been made because of the lack of data. Acknowledgments We would like to thank Dr. Paul Ferkul, whose model and numerical algorithm in Ref. 15 provided the basis upon which the present work is built. Thanks are also due to Mr. Kurt Sacksteder and Mr. Hasan Bedir for their consultation. This work has been supported by NASA Microgravity Science and Applications Division through Grant NAG31046. REFERENCES 1. Markstein, G. H. and de Ris, J., Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, pp. 1085–1097. 2. Orloff, L., de Ris, J., and Markstein, G. H., Fifteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1974, pp. 183–192. 3. Fernandez-Pello, A. C., Combust. Flame 35:135 (1978). 4. Chu, L., Chen, C. H., and T’ien, J. S., “Upward Spreading Flames over Paper Samples,” ASME, 81WA/HF-42, 1981. 5. Hasemi, Y., Proc. First Int. Symp. on Fire Safety Science, Hemisphere Pub. Co., Washington, D.C., 1985, pp. 87–96. 6. Saito, K., Quintiere, J. G., and Williams, F. A., Proc. First Int. Symp. on Fire Safety Science, Hemisphere Pub. Co., Washington, D.C., 1986, pp. 75–86. 7. Delichatsios, M. M., Mathew, M. K., and Delichatsios, M. A., Proc. Third Int. Symp. on Fire Safety, Elsevier, New York, 1991, pp. 207–216. 8. Mitler, H., Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1990, pp. 1715–1721. 9. Kulkarni, A. K. and Fischer, S. J., “Heat Transfer in Combustion Systems,” ASME HTD 122:53–61 (1989).
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10. Thomas, P. H., Fire Safety J. 22:89–99 (1994). 11. Honda, L. and Ronney, P. D., “Mechanism of Concurrent-Flow Flame Spread over Thin Solid Fuels,” Paper presented at Fall Meeting/Combustion Institute/Western States, Stanford, CA, Oct. 30–31, 1995. 12. Sacksteder, K. R. and T’ien, J. S., Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994. pp. 1685–1692. 13. Chen, C. H. and T’ien, J. S., Combust. Sci. Technol. 50:283–306 (1986). 14. DiBlasi, C., Progress in Astronautics and Aeronautics, Vol. 135, Chapter 21, AIAA. 15. Ferkul, P. V. and T’ien, J. S., Combust. Sci. Technol. 99:345–370 (1994). 16. Fiveland, W. A., J. Heat Transfer, 106:699–706 (1984). 17. Kim, T. K. and Lee, H. S., J. Quant. Spectros. Radiat. Transfer 42:225–238 (1989). 18. Jiang, C. B., “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. 19. Smooke, M. D. and Giovangigli, V., Lecture Notes in Physics, Springer Verlag, New York, 1991, pp. 1–28. 20. Abu-Romia, M. M. and Tien, C. L., (1967), J. Heat Transfer 89(4):321–327 (1967). 21. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere Pub. Co., New York, 1980. 22. Lathrop, K. D., J. Comput. Phys. 4:475–498 (1969). 23. Kim, J. S., Baek, S. W., and Kaplan, C. R., Combust. Sci. Technol. 88:133–150 (1993). 24. Bhattacharjee, S. and Altenkirch, R. A., Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1991, pp. 1627–1633. 25. West, J., Bhattacharjee, S., and Altenkirch, R. A., Combust. Sci. Technol. 83:233–244 (1992). 26. Chen, C. H. and Cheng, M. C., Combust. Sci. Technol. 97:63–83 (1994). 27. Grayson, G., Sacksteder, K. R., Ferkul, P. V., and T’ien, J. S., Microgravity Sci. Technol. 7(2):187–193 (1994).
COMMENTS Howard D. Ross, NASA Lewis Research Center, USA. Are any experiments available for comparison with the model? If so, do any of them show steady spread? Author’s Reply. At the present time, there isn’t any good experimental data on upward spread at one atmosphere in the low-gravity range (0.1 ge. and below) investigated in this work. An attempt has been made using aircraft flying parabolic trajectories to simulate this low-g environment, but the g-jitter influence was found to be too great.[1] On the other hand, steady upward spreads have been observed in a normal gravity, low-pressure environment for paper samples in a wide range of sample widths in our lab (see Ref. 4) and elsewhere (see Ref. 11).
REFERENCE 1. Sacksteder, K., private communication. ● Subrata Bhattacharjee, San Diego State University, USA. One of the most significant conclusions of your paper is the discovery of a steady-state flame in a concurrent configuration. However, your model assumes away the time terms in the N-S equations. I would suggest that the unsteady terms be retained. If the solution is steady, an unsteady model will definitely converge on the correct solution, which is not guaranteed by the steady-state model.
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Author’s Reply. Clearly, a steady combusting solution can be obtained from an unsteady model when the boundary conditions are steady and proper ‘hot’ initial conditions are prescribed. Although we did not use an unsteady model, the governing partial differential equations are elliptic in
our work and we have to iterate to obtain the steady solution. The iteration procedure has time-like characteristics. We are confident that when a steady spreading flame is obtained by this procedure, the mathematical system will allow the existence of such a solution.