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Modeling of fire spread through a forest fuel bed using a multiphase formulation
Modeling of Fire Spread Through a Forest Fuel Bed Using a Multiphase Formulation D. MORVAN*
Universite´ de la Me´diterrane´e UNIMECA, 60 rue Joliot Curie Technopo ˆle de Cha ˆteau Gombert, 13453 Marseille Cedex 13 France
and J. L. DUPUY
Institut National de la Recherche Agronomique, Unite de Recherches Forestieres Mediterrane´ennes, Equipe de Pre´vention des Incendies de Foreˆt 20 avenue Antonio Vivaldi, 84000 Avignon, France We describe a multiphase formulation to study numerically the propagation of a line fire in a forest fuel bed. One of the objectives of these studies is the improvement of knowledge on the fundamental physical mechanisms that control the propagation of forest fires. In complement of the experimental approach, this simulation tool can also be used for the development of simplified operational models used for instance for the prediction of the rate of spread (ROS) of wildland fires. The decomposition of solid fuel constituting a forest fuel bed as well as the multiple interactions with the gas phase are represented by adopting a multiphase formulation. This approach consists in solving the conservation equations (mass, momentum, energy) averaged in a control volume at a scale sufficient to contain several solid particles in the surrounding gas mixture. After a presentation of the equations and closure sub-models used in this approach, some numerical results obtained for the propagation of a line fire in a pine needles litter are presented and compared with experimental data obtained in laboratory. These results show that the rate of spread of fire in the fuel bed is primarily controlled by the radiative heat transfer. By increasing the fuel load (with a constant packing ratio), the results show the existence of two modes of propagation. A first area where the ROS varies linearly with the fuel load followed of a second where the ROS becomes independent of the load. By introducing the optical thickness characterizing the fuel bed, this difference in mode of propagation was interpreted like the demonstration of two modes of radiative transfer (optically thin and thick, respectively). The analysis of the distributions of the mass fractions of fuel and oxidant present in the gas mixture integrated through the depth of the fuel bed shows that the propagation velocity could also be limited by the lack of oxygen or fuel available in the ignited zone to maintain the pilot flame. © 2001 by The Combustion Institute
NOMENCLATURE cp Cps F s,i fv gi h h s, ␣ , ⌬h char I J
specific heat of the gas mixture specific heat of solid particles drag force resulting from gas/particles interaction in the fuel bed soot volume fraction gravity acceleration in the i-direction enthalpy of the gas mixture reaction heat (drying, pyrolysis, charcoal oxydation) radiative intensity irradiance
* Corresponding author. E-mail: [email protected] COMBUSTION AND FLAME 127:1981–1994 (2001) © 2001 by The Combustion Institute Published by Elsevier Science Inc.
k kg ˙ M s, ␣
ms pg Qconv ROS t xi T Ts
turbulent kinetic energy conductivity of the gas mixture production term of species ␣ resulting from the decomposition of the solid particles (drying, pyrolysis, char oxydation) fuel bed load (kg/m 2 ) pressure of the gas mixture gas/particles heat transfer rate of spread time cartesian coordinate in the i-direction temperature of the gas mixture temperature of the solid particles 0010-2180/01/$–see front matter PII S0010-2180(01)00302-9
1982 ui s s Y is , Y H , Y char 2O
Y␣
D. MORVAN AND J. L. DUPUY velocity vector component in the i-direction mass fraction of dry wood, moisture content and char of solid particles mass fraction of species ␣
Greek symbols
␣ g, ␣ s ␦ ⑀ g t ˙␣ s s ˙ pyr , ˙H 2O
˙ char g, s a ij s
volume fraction of the gas and solid phase fuel bed depth turbulent kinetic energy dissipation rate burned rate of solid particles (mass fraction) viscosity of the gas mixture eddy viscosity production, destruction rate of species ␣ (combustion) pyrolysis and drying reaction rates charcoal oxydation rate density of the gas and solid phase Stephan–Boltzmann constant absorption coefficient of the gas/soot mixture stress tensor surface/volume ratio of solid particles
INTRODUCTION The understanding of the physical mechanisms that control the ignition and the spread of wildfires constitutes a major objective for the management and the preservation of forest areas. Various mathematical models based on statistical, semi-empirical or physical approach have been proposed to evaluate the spread rate of a surface forest fire [1]. The semi-empirical models are based on a relationship between the rate of spread (ROS) of a surface fire with the energy received by the combustible layer and the energy necessary to carry the fuel particle until a threshold enthalpy of ignition. The constants necessary to close the system, are then evaluated from experimental fires. This ap-
proach leads to the well-known Rothermel model [2], which was introduced inside the computer code BEHAVE, which is currently used for the risk management related to the forest fires. These simplified models allow to obtain a quite good approximation of the fire rate of spread as a function of the fuel load, the wind intensity and the terrain slope for particular conditions (close to experiments carried out for the calibration of the parameters used in such models). Unfortunately, the use of these models for more general fire conditions does not always give satisfactory results. The main reason is that the physical mechanisms which control the thermal degradation of solid fuel, the fire behavior and the solid-flame interactions, are uncompletelly described in such models. Four possible mechanisms for the heat transfer can be involved in fire spread process: (1) direct flame contact, (2) convective heating, (3) radiative heating, and (4) firebrand contact. In the absence of wind and on a flat ground, experimental investigations have shown that 60% of the thermal energy required for ignition and received by the fuel comes from burning embers by convective and radiative heat transfer [3]. This result seems to confirm the minor part plays by the flame to sustain a spreading fire for this configuration. Experimental observations show that the flame is slightly leaning back, it forms an angle (positive in the direction of fire spread) ␣ ⬇ ⫺10, ⫺30 degrees with the direction perpendicular to the fuel bed [4]. This configuration of the flame reduces the view factor and thus the radiation of the flame on the top of the fuel bed. The relationship between these various contributions of course depends on the physical characteristics of the fuel bed (packing ratio, depth, surface/volume ratio, moisture content . . . ). To improve the numerical predictions concerning the behavior and the trajectory of a fire front, a more complete approach based on physical considerations can be adopted. We can distinguish two kinds of physical models: ●
The radiative models based on the calculation of the heat transfer (mainly radiative) between the burning zone (of which the characteristics are evaluated empirically) and the
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part of the unburnt fuel bed ahead of the fire [4 –7], The complete models which consist in solving the various reactive problems (pyrolysis, combustion) from the conservation equations (mass, momentum, energy) in the gas mixture and in the fuel bed by using a multiphase formulation [8, 9].
The reasons why the radiation is retained as the single mode of heat transfer in the first approach are based on the following hypothesis: without wind, it is the only mechanism able to heat solid fuel ahead of the fire; with wind, the flame makes obstacle with the wind which cannot penetrate deeply in the fuel bed. The ROS predicted by the first approach too strongly depends on the characteristics of flame introduced into the model. Mathematical problems remain concerning the algorithm of determination of the ignition zone, iterative calculation seems to converge only for initial conditions close to the final solution. The extension of such model to complex situations (fire on an inclined ground, effects of the wind, fuel discontinuities) remains difficult. Associated with experiment performed in laboratory, these models made it possible to render by a physical approach some basic mechanisms (radiation of the flame and embers, convection of hot gases, cooling of the particles upstream of the face caused by fresh gases) associated with wildfires spread. In the multiphase approach, the multiple interactions between the gaseous phase and the solid fuel are described in details to obtain a complete representation of the physical mechanisms (gas flow, combustion, radiation, and convection heat transfer) contributing to the thermal decomposition (vaporization, pyrolysis, char oxidation) occurring during the development of a wildland fire. Adopting an averaging formulation to take into account the effects induced by the microstructures (subgrid scale particles) upon the behavior of the system at a resolved scale, the coupling between the solid and the gaseous phase is rendered through source terms in the mass (decomposition of solid fuel to combustible gas), momentum (drag force), and energy (heat transfer by convection and radiation) balance equations. The fuel bed is approximated as an uniformly distributed cylinders
Fig. 1. Lire fire spread through a forest fuel bed.
(solid particles), characterized by geometrical (surface-to-volume ratio, packing ratio, fuel depth) and physical (fuel density, reaction rates and composition of the pyrolysis products, moisture content) properties evaluated experimentally. The aim of the present work is to describe the physical behavior and the propagation of a small scale surface fire in a pine needles litter. The studied cases are limited to a two-dimensional configuration (see Fig. 1). This assumption can be justified considering that the observations show that widths of many spreading fires are in nature much greater that the fire depth [3]. Although the flames exhibit fully three dimensional behavior, the major contributions concerning the heat balance inside the fuel bed and above the litter can be assimilated as twodimensional over shorter portions of the fire front (see Fig. 1). MATHEMATICAL FORMULATION IN THE GAS PHASE We propose to study the propagation of a surface fire through a pine needles fuel bed on a horizontal plate (see Fig. 1). As we mentioned in the introduction, the model used in this study is based on a detailed physical approach, which consists in solving the conservation equations (mass, energy, momentum) of the system formed by the litter and the surrounding gas mixture. The fuel bed is represented as an initially homogeneous distribution of cylindrical particles whose dimensions are evaluated from experimental data. In the fuel bed, the conser-
1984
D. MORVAN AND J. L. DUPUY
vation equations (mass, momentum, energy, chemical species) are integrated on a control volume sufficient to be able to include several solid particles to define an average value. Considering the conditions of development of these fires, we can assume that the flow at the top of the litter is turbulent and that homogeneous combustion is primarily controlled by the turbulent mixture of the oxygen contained in the ambient air with the flammable part of gas coming from the decomposition of solid fuel by pyrolysis. To take into account the contribution of the fluctuations for the transport of mass, of energy and momentum in the gas phase, we adopted a statistical two-equations turbulence model (RNG-k⫺ ⑀ ) [10, 11] using mass weighted averages (Favre averages) [12]. This method allows to extend the domain of validity of this model of turbulence (initially adapted to the fully developed turbulent flows) to the zones where the level of turbulence remains low (close to the ground and in the litter). By preserving an unsteady formulation, this approach is able to correctly represent the formation and the time evolution of the large vortex structures observed during the expansion of hot gases above the burning zone. The diffusion flames characteristic of natural fires are dominated by gravity effects, which are revealed by a strongly nonstationary behavior (flames height, fresh air flow in the vicinity . . . ). This behavior is also observed for pool fires, it is characteristic of 2 flames at low Froude number (F r ⫽ U fuel /gd: U fuel fuel injection velocity, d dimension of the gaseous fuel production zone, g gravity acceleration) [12, 13]. According to these remarks and using density-weighted Favre averaging formulation, the conservation equations of mass, energy and momentum in the gas mixture can be written as follows: ⭸ ⭸ 共 兲 ⫹ 共 u ˜ j兲 ⫽ ⭸t ⭸ xj
冘
˙ s,␣ M
(1)
␣
⭸ ⭸ ⭸ 共 u ˜ i兲 ⫹ 共 u ˜ ju ˜ i兲 ⫽ 共 ␣ 兲 ⭸t ⭸ xj ⭸ x j g ij ⫺
⭸ u ⬙ju ⬙i ⫹ g i ⫺ F s,i ⭸ xj
(2)
冉
˜ ⭸ ⭸ ⭸T ⭸ ˜兲 ⫹ ˜兲 ⫽ 共 h 共 u ˜ jh ⭸t ⭸ xj ⭸ x j Pr ⭸ x j ⫺
冊
⭸u⬙j h⬙ ⭸ ␣ gp ⫺ Qconv ⫹ ⫹ ␣ g a共 J ⫺ 4 T 4兲 ⭸xj ⭸t
⫹
冘 M˙ ␣
s, ␣ hs, ␣
⫹ 共1 ⫺ ␣sg 兲⌬hchar ˙ char
(3)
冉 冊
˜ ⭸ ⭸ ˜ ˜兲 ⫽ ⭸ ⭸Y ␣ 共 Y ␣兲 ⫹ 共 u ˜ jY ␣ ⭸t ⭸ xj ⭸ x j Pr ⭸ x j ⫺
⭸ u ⬙jY ⬙␣ ˙ s,␣ ⫹ ˙ ␣ ⫹ M ⭸ xj
(4)
where the following notations have been introduced: ⫽ ␣ g g , ⫽ ␣ g g , ␣ g , and g design the volume fraction and the density of the gas phase. ˙ is the average production term of species M s, ␣
␣ resulting from the decomposition of the solid fuel (drying, pyrolysis, charcoal oxydation), ˙ ␣ the production/destruction of chemical species ␣ resulting from the homogeneous combustion, ˙ char the destruction of char resulting from the heterogeneous combustion. F s,i and Q conv design respectively the i-component of the drag force and the convective heat flux resulting from the interaction between the gas and the solid particles in the fuel bed. Assuming that the action of the solid phase can be approximated as the sum of each individual particle (the interaction between two particles is neglected), these terms are approximated as follows: F s,i ⫽
3 ␣ C 兩u兩u i␣ s s 8 g g d
Q conv ⫽ h conv␣ s s共T ⫺ T s兲
(5) (6)
where the drag (C d ) and the heat transfer coefficient (h conv) are approximated using empirical correlations obtained for a laminar or a turbulent flow around a cylinder [14]. The enthalpy-temperature dependence is treated using the CHEMKIN thermodynamic data base [15]. In addition to previous equations of conservation, two equations of state are needed to close the system:
FIRE SPREAD
冘 WY
␣
p g ⫽ gR 0T
␣
g ⫽
1985
␣
冘 冉 TT 冊 ␣
ref
␣
(7)
where P k and W k are, respectively, the shear and buoyancy turbulent production terms:
(8)
P k ⫽ ⫺ u ⬙iu ⬙j
Cref
Y␣
where R 0 is the universal gas constant, W ␣ , ␣ are the molecular weight and the viscosity of each chemical species, respectively T ref ⫽ 302 K, C ref ⫽ 0.75 are the coefficients of the Sutherland’s law which describes the variation of the viscosity with the temperature. Using the low Mach number flows approximation, the variations of pressure can be neglected in the energy conservation equation and in the equation of state of the gas mixture (acoustic filtering). With this approximation the stability of the scheme of resolution (CFL condition) depends only on the convection velocity, which then makes it possible to work with larger time step (⬇ 10⫺3–10⫺2 s). The double correlations representing the action of the fluctuations on the average transport equations are evaluated using the eddy viscosity concept and generalized gradient diffusion of the scalar quantities ():
冉
⫺ u ⬙ju ⬙i ⫽ t
冊 冉
冊
˜k ˜j 2 ⭸u ˜ i ⭸u ⭸u ⫹ ⫺ t ⫹ k ␦ ij ⭸ xj ⭸ xi 3 ⭸ xk (9)
2
t ⫽ C
k ⑀
⫺ u ⬙j ⬙ ⫽
(10)
˜ t ⭸ ⭸ xj
(11)
The turbulence field is completely described from two scalar variables the turbulent kinetic energy (k) and its dissipation rate (⑀) which are calculated from the two following transport equations: ⭸ ⭸ ⭸ 共 k兲 ⫹ 共 u ˜ jk兲 ⫽ ⭸t ⭸ xj ⭸ xj
冋冉
⫹
冊 册
t ⭸k k ⭸ xj
⫹ P k ⫹ W k ⫺ ⑀ ⭸ ⭸ ⭸ 共 ⑀ 兲 ⫹ 共 u ˜ j⑀ 兲 ⫽ ⭸t ⭸ xj ⭸ xj ⫹ 共C ⑀1 ⫺ R兲
冋冉
冊 册
t ⭸⑀ ⫹ ⑀ ⭸ xj
⑀ ⑀ ⑀2 P k ⫹ C ⑀3 W k ⫺ C ⑀2 k k k
共1 ⫺ / 0兲 1 ⫹  3
⫽
冑
Pk C ⑀
(13)
t ⭸ ⭸p 2 ⭸ x j ⭸ x j
Wk ⫽ ⫺
R⫽
(14)
0 ⫽ 4.38
 ⫽ 0.015
C ⫽ 0.0845 C⑀ 1 ⫽ 1.42
(15) C⑀ 2 ⫽ 1.68
k ⫽ 0.7179
C⑀ 3 ⫽ 1.5
⑀ ⫽ 1.3
(16)
We can note that the implementation of the RNG k ⫺ ⑀ turbulence model does not involve great modifications compared to the standard model. Only one additional term R is introduced into the transport equation for ⑀ which represents the dissipative action of the subgrid scales on the average motion. Using thermogravimetry analysis (TGA) it has been shown that for temperatures about 1000 K, the gas mixture produced by pyrolysis during the decomposition of forest fuel was primarily composed of CO, CO 2 , CH 4 and in less proportion of H 2 , C 2 H 6 [8]. To simplify the problem we have conserved only five chemical species (CO, CO 2 , H 2 O, O 2 , N 2 ). Assuming that the chemical reaction is quasi instantaneous, the combustion rate in the gaseous phase is primarily limited by the time necessary to fuel and oxidant to mix. Assuming that the mixing time depends on the turbulence scales, the production/destruction rates ˙ ␣ can be evaluated using the Eddy Dissipation Concept [12, 16]:
˙ CO ⫽ ⫺A (12)
⭸u ˜i ⭸ xj
冉
冊
˜ ⑀ ˜, Y O2 min Y CO g k O 2
g ˙ O2 ⫽ O ˙ CO 2
g ˙ CO2 ⫽ ⫺共1 ⫺ O 兲 ˙ CO 2
(17) (18)
where the constant A is equal to 2.5 [17], and the stoichiometric ratio of the chemical reg action CO ⫹ 1/2O2 3 CO2 is O ⫽ 4/7 (in 2 mass).
1986
D. MORVAN AND J. L. DUPUY
RADIATION HEAT TRANSFER MODELING The energy exchanged in radiative form is evaluated by integrating the radiative transfer equation (RTE) taking into account of the emission coming from the embers and the mixture formed by the gas (air ⫹ pyrolysis and combustion products) and soot particles formed in the flame [8, 9, 18] (see Eq. 20):
冉
冊
冉
d␣g I T4 s ␣s T4s ⫽ ␣g a ⫺I ⫹ ⫺I ds 4 J⫽
冕
4
I d⍀
冊
(19)
(20)
contribution because of the gradient of temperature (thermophoretic velocity u jth ) defined as following: u jth ⫽ ⫺0.54
0
For these two contributions (gas ⫹ soot and embers) we suppose that the emission is carried out as that of a black body, this assumption is relevant because the particles present are primarily composed of carbon. This calculation is carried out by a discrete ordinates method (DOM) [19] which is well adapted to solve the problem of the radiative transfer in the media presenting significant variations of the absorption coefficient. The radiation coming from the flame is primarily because of the presence of soot particles which are formed in the flame and which give it a very characteristic yellow color. Because of the lack of information on soot production in natural fire, we have fixed the production rate of soot from the production rate of gaseous pyrolysis products [8]. Assuming that the soot particles can be assimilated as carbon spheres of a diameter d soot ⫽ 1 m, the soot volume fraction field can be evaluated from the following transport equation [20, 21]: ⭸ ⭸ ⭸ th˜ ˜ ˜ 共 ˜f v兲 ⫹ 共 u ˜ j˜f v兲 ⫽ ⫺ 共 u ˙ fv j f v兲 ⫹ ⭸t ⭸ xj ⭸ xj (21) ˜ ˙ fv ⫽
Fig. 2. Fundamental physical mechanisms controlling the propagation of a fire in a fuel bed.
1
soot
˜ 关 soot ˙ pyr ⫺ W Ox ˜f v soot兴
(22)
where soot ⫽ 6/d soot is the surface/volume ratio of soot particles. The transport of the soot particles is ensured by the convective motion of the gas (velocity u j ) to which is added the
˜ g ⭸ ln T g ⭸ x j
(23)
The term W Ox corresponds to soot oxidation, it is evaluated from the rate for oxidation of pyrolytic graphite by O 2 (kg/m 2 .s) [22], W Ox ⫽ 120
冋
k AP O2 1 ⫹ k zP O2
⫹ P O2共1 ⫺ 兲
册
1
⫽ 1⫹
kT k BP O2
(24) (25)
where P O 2 is the partial pressure of oxygen. The various reaction rates k A , k B , k T , k z introduced into the expression of W ox depend on the temperature in the following way [22, 23]: ˜ ) g/cm 2.s.atm k A ⫽ 20 exp (⫺30000/RT
(26)
˜ ) g/cm 2.s.atm kB ⫽ 4.46 10⫺3 exp (⫺15200/RT (27) ˜ ) g/cm 2.s kT ⫽ 1.51 105 exp (⫺97000/RT
(28)
˜ 兲atm ⫺1 k z ⫽ 21.3 exp 共4100/RT
(29)
MATHEMATICAL FORMULATION IN THE SOLID PHASE Under the action of the intense heat flow released by the burning zone (flame ⫹ embers), the solid particles located ahead of the fire front will break up in the following way (see Fig. 2): ● ●
Dehydration, Pyrolysis (gaseous products and charcoal)
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1987
Reaction rates of these various contributions are approximated by Arrhenius laws whose parameters (frequency factors and activation energy) are evaluated from TGA [8]. During its degradation the solid fuel is described by the distribution (mass fraction) of four components s s (dry wood Y is , water content Y H , coal Y char , 2O ash) evaluated by integrating the system of equations formed by the mass conservation of the system gas ⫹ fuel particles. The decomposition of the dry wood by pyrolysis induces the production of gaseous products (CO, CO 2 ) and charcoal (char kg of char per 1 kg of dry wood). The stoichiometric ratio of CO 2 produced by pyrolysis is CO 2 (kg of CO 2 per 1 kg of gas produced by pyrolysis). By contact with the oxygen contained in the ambient air, the combustible part of the pyrolysis products ignites (homogeneous combustion), while the charcoal undergoes a surface oxidation (heterogeneous combustion). Taking into account all these assumptions, the evolution of the composition of the solid particles in the litter is controlled by the following equations:
˙ char ⫽
1 s k char␣ g gY O2 exp O 2
d s 共 ␣ Y s兲 ⫽ ⫺ ˙ pyr dt s s i
(35)
d 2 ˙H 共 ␣ Y s 兲 ⫽ ⫺ 2O dt s s H2O
(36)
d s 共 ␣ Y s 兲 ⫽ 共 char ⫺ soot兲 ˙ pyr dt s s char ⫺ d 共␣ 兲 ⫽ ⫺ dt s s
冘 M˙
˙ H2O ⫽
k H2O
冑T s
冋
⫺ E pyr RT s
s ␣ s sY H exp 2O
冋
册 RT s
册
冊
ash s ⫹1 ˙ char char
s,␣
s ⫽ 共 char ⫺ soot ⫺ 1兲 ˙ pyr
2 s ⫺ ˙H ⫺ ˙ char 2O
1 s d ˙ 共 ␣ s兲 ⫽ ⫺ dt s char
␣ s sC ps
(37)
(38) (39)
dT s ⫽ Q conv ⫺ dt
冘 M˙ ␣
s,␣h s,␣
s ⫹ ␣ sg⌬h char ˙ char
␣ s s 共 J ⫺ 4 T 4s 兲 4
(30)
⫹
(31)
s ˙ s,CO ⫽ 共1 ⫺ char ⫺ CO 兲 ˙ pyr M 2
(32)
⫺ E H2 O
冉
␣
⫹ 共1 ⫺ char兲关共1 ⫺ CO2兲CO ⫹ CO2CO兴
˙ pyr ⫽ k pyr␣ s sY is exp
册
⫺ E char ␣ s s RT s (34)
Solid Fuel 3 char共Char ⫹ Soot兲
s s O 2 3 共1 ⫹ O 兲CO 2 Char ⫹ O 2 2
冋
s ˙ s,H O ⫽ M ˙H 2 2O
(40)
(41)
s s ˙ s,O ⫽ ⫺ O M ˙ char 2 2
(33)
s s s ˙ s,CO ⫽ 共1 ⫹ O M 兲 ˙ char ⫹ CO2 ˙ pyr 2 2
(42)
Vaporization:
k H 2O ⫽ 6.05 10 5 K 1/ 2 s ⫺ 1
E H 2O /R ⫽ 5956 K
Pyrolysis:
kpyr ⫽ 3.64 10 3 s ⫺ 1
E pyr/R ⫽ 7250 K
Charoxidation:
kchar ⫽ 430ms ⫺ 1
Echar/R ⫽ 9000 K, ␣ sg ⫽ 0.5
Stoichiometriccoefficients:
char ⫽ 0.338, CO 2 ⫽ 0.2
ash ⫽ 0.033, soot ⫽ 0.05
(per unit mass)
s O ⫽8/3 2
For this study we also supposed that the particles in the litter remained at rest during the
major part of the reactions of pyrolysis and combustion, except at the end of the decompo-
1988
Fig. 3. Thermal decomposition of a solid fuel sample (pine needles): mass fraction of dry wood (Y is ), moisture content s s (Y H O ) and char (Y char ), and mass loss obtained by TGA in 2 inert atmosphere simulated with a heating rate dT/dt ⫽ 10 K/s (top) and 50 K/s (bottom).
sition when the ash state is reached. At the time when the particles were completely transformed into ash, the litter collapses abruptly and the gas/particles interactions can then be neglected. The oxidation of the charcoal is not limited to a heterogeneous reaction (C ⫹ O 2 3 CO 2 ), it is also made up of two reactions (heterogeneous and homogeneous) following: (C ⫹ 21 O 2 3 CO), (CO ⫹ 21 O 2 3 CO 2 ). The distribution of the heat of reaction between the two phases (solid and gas) is taken into account by introducing a sharing coefficient ␣ sg (⫽ 0.5 for the present calculation). The curves represented on the Fig. 3 show the evolution of the mass fractions (dry wood, moisture content, and char-
D. MORVAN AND J. L. DUPUY coal) as well as the mass loss obtained numerically by simulating a TGA in a furnace in inert atmosphere for two heating rate 10 K/s and 50 K/s (values representative of the conditions usually used for a TGA and met ahead of a spreading fire, respectively). The results obtained for a heating rate of 10 K/s (see Fig. 3 on top) representing the evolution of the mass fractions (dry wood, moisture content and charcoal) according to the temperature, show that the wood sample is completely dehydrated starting from 400 K, and degradation intensifies (by pyrolysis) with 460 K. With 600 K the degradation (charcoal transformation) of the sample in inert atmosphere is complete, the material then has lost 74% of its initial mass [curve in continuous line on the Fig. 3 (top)]. For a heating rate of 50 K/s the curves describe the same stages with a shift of ⬇ 60 K of the thresholds of temperature. The last test (50 K/s) was carried out once again with an oxygenated atmosphere, assuming that the mass fraction of oxygen remains constant and equal to 0.23. The results presented on the Fig. 4 have been obtained by supposing that the oxidation of the charcoal was limited to the surface of the particles (on top) and by taking of account porosities which allows a certain diffusion of oxygen in volume (on bottom). In this case the surface of reaction is calculated starting from the ratio of density of the charcoal with carbon, by assuming that the size of the pores is homogeneous and equal to 50 m. The results show that the oxidation of the charcoal continues until the sample has reached the state of ash, the material will then have lost 97% of its initial mass. RESULTS AND DISCUSSION The system of partial differential equations formed by the set of the conservation equations in the gas phase is solved numerically using a finite volume method. By using a high order upwind convective scheme (QUICK) associated with a flux limiting strategy (ULTRA-SHARP) [23], we obtain a stable numerical integration procedure, without numerical diffusion. The system describing the evolution of solid fuel is solved in a separated way using a Runge–Kutta
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1989
Fig. 5. Position (in meter) of the pyrolysis front (represented as the isotherm T s ⫽ 500 K) and mass loss in the fuel bed as a function of time.
Fig. 4. Thermal decomposition of a solid fuel sample (pine needles): mass fraction of dry wood (Y is ), moisture content s s (Y H ) and char (Y char ), and mass loss obtained by TGA in 2O air simulated with a heating rate 50 K/s [without (top) and with charcoal porosity effects (bottom)].
method. Without wind and on a flat ground, the spread of a surface fire is characterized by a flame leaning in the direction opposed to the direction of propagation. The heat flow received by the particles ahead of the fire is mainly radiative. One can then estimate that the distance to which the solid particles thermically will be affected by the fire is about l rad ⫽ (( s ␣ s )/4) ⫺1 , l rad is the mean penetration distance [19]. For the present study ( s ⫽ 4550 m ⫺1 , ␣ s ⫽ 0.029) this distance is l rad ⬇ 3 cm. To obtain a good representation of the thermal degradation zone in the fuel bed, it is thus necessary that the mesh size in this region is lower than 1 cm. On the other hand apart from the zone in flame, the conditions of temperature
and the gas mixture remain not very affected. To ensure a level of precision constant throughout calculation, we have used a adaptative grid technique, consisting in refining the grid in the vicinity of the front of decomposition of the fuel bed located from the gradient of the dry wood mass fraction (ⵜY is ). The intense zone of pyrolysis corresponds to the zone where the module 储ⵜY is 储 is maximum, in this area the minimal mesh size (⌬) is calculated so that the ratio l rad/⌬ ⬇ 5 (⌬ ⫽ 6 mm). We carried out a first series of calculations for a dead pine needles litter (Pinus pinaster) uniformly layered over a horizontal level and into absence of wind. The properties of the litter are identical to those retained for the experiments carried out in laboratory [24]. Fuel density ( s ): Packing ratio ( ␣ s ): Surface-to-Volume ratio ( s ): Fuel moisture content: Fuel depth (␦): Fuel load (m s ⫽ ␣ s s ␦ ):
680 kg/m 2 0.029 4550 m ⫺1 0.02 from 0.03 to 0.2 m from 0.6 to 4 kg/m 2
Lighting is simulated by introducing a volume heat source over the whole fuel bed depth and along a length of 5 cm. This heat supply will be maintained until the temperature of the particles reaches a value of 1200 K. The results on the Fig. 5 show the time evolution of the position of the isotherm T s ⫽ 500 K as well as the rate of mass loss in the litter obtained for a fuel bed depth of 5 cm (m s ⫽ 1 kg/m 2 ). The
1990 results show that the development of the fire in the litter follows an initial phase of growth (which must depend on the conditions of lighting) and 40 s after the beginning, the fire achieves a quasi steady-state. The trajectory of the pyrolysis front which reaches a recti-linear shape, allows to evaluate a constant rate of spread (ROS ⬇ 3.8 mm/s). The curve of mass loss is characterized by fluctuations (around an average value ⬇ 3 g/m.s) which are related to the movements which characterize this type of flame subjected to the effects of gravity (puffing instabilities) [26]. The flame is subjected to vertical oscillatory movements which modify the radiative heat flow received by the particles of the litter. The increase of the flame height causes an acceleration of the pyrolysis rate and an increase in quantity of fresh air aspired by the burning zone. These two phenomena induce, respectively, a heating and a cooling of the solid particles located ahead of the fire. It is the combination of these two contradictory effects which is at the origin of the oscillatory behavior observed in experiments and in the present calculations. These results are in agreement with the experimental measurements obtained for the spread of a surface fire in a pine needles litter (Pinus Pinaster), for which one finds a ROS equal to 3.9 mm/s and a mass loss rate of 3.69 g/m.s [25]. The velocity field represented with the fuel bed density are on Fig. 6 (top) (fire is propagated from left to right), is characterized by the formation of a hot gas column at the top of the ignited zone, inclined behind and forming an angle of 40 degrees with the vertical. Along the thermal plume the gas undergoes an acceleration, the velocity magnitude reaches 0.4 m/s near the flames and 2 m/s 35 cm above the litter. In the fuel bed the flow is strongly reduced (⬇ 0.1 m/s). We can also note that after the passage of the fire the fuel bed density decreases from 20 kg/m 3 (initial value) to 0.64 kg/m 3 . The decomposition of the solid particles in the litter is total and the state of ash is reached. The temperature field (gas phase) represented on Fig. 6 (bottom) is characterized by two hot zones, corresponding respectively to the heat released by the homogeneous and heterogeneous (charcoal oxidation) combustion. The temperature in the flame is about 1200 K, which is in conformity with the infrared
D. MORVAN AND J. L. DUPUY
Fig. 6. Instantaneous velocity vector field and bed density (top) and gas temperature field (bottom) (closer view) calculated for a fuel depth of 5 cm (fuel load m s ⫽ 1 kg/m 2 ).
measurements obtained on the experimental setup [27, 28]. To identify the principal mechanisms of heat transfer contributing to the fire spread, we have evaluated the average value of various variables per integration through the depth of the litter. Distributions of the average temperatures achieved in gas (T) and in the solid particles (T s ), and the density of the litter along the direction of propagation are represented on the Fig. 7. One distinguishes well the two hot zones described previously, marked by a significant reduction of the fuel bed density corresponding to the two mechanisms of mass loss: drying ⫹ pyrolysis and combustion of the charcoal. The decomposition area of the fuel bed, limited by the zone where the particles are still intact and ashes extends over a length ␦ f ⬇ 22 cm, which gives a residence time ␦ f /ROS ⬇ 58 s (in conformity with the experimental results). The two principal mechanisms involved in the pre-heating of the litter located ahead of the fire are the radiative (Q rad) and the convective transfer (Q conv) (the conductive transfer in the solid phase can be regarded as negligible), defined as follows:
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1991
Fig. 9. Average distribution of mass fraction of O 2 and CO and fuel moisture content in the fuel bed along the fire spread direction (fuel load m s ⫽ 1 kg/m 2 ).
Fig. 7. Average distribution of temperature (in the gas and solid particles) and fuel bed density along the fire spread direction (fuel load m s ⫽ 1 kg/m 2 ).
Q rad ⫽
␣ s s 共 J ⫺ 4 T 4s 兲 4
Q conv ⫽ h conv␣ s s共T ⫺ T s兲
(43) (44)
One can evaluate the importance of these two contributions by integrating these two quantities over the depth of the litter. At a given time, a distribution of the heat flows (by convection and radiation) received by the litter along the direction of propagation is then obtained (see Fig. 8). One can notice that for these conditions (without wind and on flat ground) the degradation of the particles starts due to radiative heat trans-
Fig. 8. Average distribution of convective and radiative heat transfer and fuel moisture content in the fuel bed along the fire spread direction (fuel load m s ⫽ 1 kg/m 2 ).
fer. The ratio between the two peaks of heat flow by radiation and convection in the zone located ahead of the fire (where the degradation of the fuel bed is initiated) is equal to 8.6 (see Fig. 8). By integrating these two curves one can evaluate the total quantity of energy received by the solid fuel bed, by radiation and convection which is, respectively, of 5.475 kW/m (radiation) and 3.575 kW/m (convection) (for a fuel bed depth of 5 cm). The ratio (radiation/ convection) between these two sources of energy is thus 1.53 (60% of the energy received by the solid fuel is due to radiative heat transfer). The propagation of fire in the litter is thus primarily controlled by the radiation, confirming the assumptions retained by certain simplified physical models [4, 7]. By analyzing the average distribution in the fuel bed of the mass fraction of oxygen (Y O2 ) and carbon monoxide (Y CO ) (see Fig. 9), one can note the existence of a completely under-oxygenated zone. The totality of the oxygen contained in the air aspired upstream is consumed by the zone on fire (flame ⫹ embers), it is thus a factor which can limit the progression of fire. The ROS (numerical and experimental values) obtained for various fuel loads ranging from 0.6 kg/m 2 to 5 kg/m 2 are shown on Fig. 10 and compared with the value obtained with a purely radiative fire spread model [6, 29] assuming that the fire can be assimilated as a vertical radiant heater with a temperature uniform and equal to 1100 K (the contribution of the flame above the fuel bed is neglected). These results show that the variation of the ROS with respect to the fuel load consists of three zones:
1992
D. MORVAN AND J. L. DUPUY
Fig. 11. Average distribution of temperature (in the gas and solid particles) and fuel bed density along the fire spread direction [fuel load m s ⫽ 2 (top) and 4 kg/m 2 (bottom)].
Fig. 10. ROS (top) and mass loss rate (bottom) as a function of the fuel load.
● ●
●
For weak fuel loads (m s ⬍ 1.2 kg/m 2 ), the ROS varies linearly with the load, For intermediate loads (m s ⬍ 4 kg/m 2 ), the ROS varies still linearly with the load (the slope is weaker), For high fuel loads (m s ⬎ 4 kg/m 2 ), the mode of propagation changes and the ROS tends to become independent of the load.
According to the importance of the role played by the radiative transfer, the limit between the first two modes of propagation can be expected to be related to the optical thickness characterizing the initial state of the litter: opt ⫽ ␣ s s ␦ /4 (using the depth of the litter as characteristic length scale). The change of mode for which the dependence of the ROS with the load starts to be weaker, occurs for an optical thickness higher than 2. That means that the medium becomes optically thick and that the propagation of the fire is no longer controlled by the radiation through the whole fuel bed depth, but
through a fraction of it (⬇l rad). The studies realized on the propagation of a fire on a solid plate of PMMA show that below a certain thickness of fuel, the propagation velocity was a function inversely proportional to the thickness (thermically thin mode), to tend thereafter towards a constant value (optically thick mode). The increase thickness of fuel contributes to an increase in the conductive losses in material, until reaching a limiting thickness beyond which the heating effects are too weak to take part in the propagation of the combustion front [26, 30, 31]. If the fire is propagated in a porous environment, the increase thickness of fuel first of all involves an increase in the size of the zone on fire [see Fig. 11 (top) representing the temperature distributions in the fuel bed] and consequently an increase in the flow of fresh air which feeds the reaction of combustion ahead of the fire front. This situation is similar to that encountered for downward flame spread in an opposed forced flow [32, 33]. In this case an increase in the upstream air flow produced first an increase in the ROS, then the ROS reaches a critical value and decreases until reaching the limit of extinction where the selfmaintained propagation of the flame front is no more possible. This limitation results from the increase of the convectif heat transfer, the flow of
FIRE SPREAD
1993
fresh air causes finally a cooling of the particles located ahead of the fire front [33] [see Fig. 11 (bottom)]. As indicated before, on flat ground and in absence of wind, the combustion front is propagated with a flame directed slightly behind. Ahead of the fire the products of pyrolysis combine with the ambient air entrained into the base of the developing fire and burn behind the fire front [26, 33]. This spread mechanism is described as ‘counter-current’ (or ‘opposed flow’) which means that the oxidizer flow and the direction of propagation are opposed. The highlighting of these two modes of propagation makes it possible to understand the apparent contradiction between various experimental results with regard to the effect of the fuel load on the ROS [2, 25, 34, 35]. Concerning the experimental fires performed in a wind tunnel, in the absence of wind, the results obtained could be influenced by the effect of confinement caused by the drain of the thermal plume along the walls. When the propagation of fire becomes independent of the fuel load, the calculated ROS is equal to 5.2 mm/s, this value is very close to the computed value starting from a recent empirical model obtained for experimental fires in a wind tunnel (ROS ⫽ 5.56 mm/s [35]). On the same Fig. 10 one can note that the ˙ ) is a linear function of the mass loss rate (M load (m s ): ˙ ⫽ K a共m s ⫺ m cs 兲 M
(45)
where K a has the same dimension that a rate of spread and m cs indicates the threshold of fuel load to maintain a sustained fire propagation. In the present calculations K a ⫽ 4.44 mm/s and m cs ⫽ 0.3 kg/m 2 which are comparable with the values evaluated from experimental results: K a ⫽ 4.54 mm/s, m cs ⫽ 0.18 kg/m 2 . In stationary regime the mass loss rate and the ROS moreover are connected by the following relation: ˙ ⫽ ⫻ ROS ⫻ m s M
(46)
where indicates the burned rate of solid particles in the fuel bed (by taking account of the residual ash content ⱕ 0.97). The following relation is then obtained:
冉
⫻ ROS ⫽ K a 1 ⫺
m cs ms
冊
(47)
Far from the extinction limit (m cs /m s 3 0) the product ⫻ ROS remains constant and equal to K a .
CONCLUSION The propagation of a surface fire in a pine needles fuel bed has been simulated numerically using a multiphase formulation. The influence of the fuel load on the ROS was studied. The numerical results show that the propagation of a surface fire on a flat ground and without wind, is mainly controlled by the radiative heat transfer. The results also show that the oxygen supply is ensured by a flow of fresh air aspired just upstream of the zone on fire. The influence of the depth of the fuel bed on the propagation velocity can be summarized in the following way, the increase of the fuel layer induces an increase in the heat losses and an increase in the flow of air entrained upstream of the fire front. For a low-size fire the increase in the flow of fresh air will cause an increase in the propagation velocity. For a more significant fire, the additional contribution of air on the contrary will cause a cooling of the particles upstream and will slow down the progression of the combustion front. By comparing the computational results (ROS and mass loss rate) with the experimental data, this study made it possible to validate the various sub-models composing the multiphase approach for the study of the propagation of a surface fire. The experimental conditions (flat ground and without wind) make it possible to obtain a line fire, the mode of heat transfer of the ignited zone towards the unburnt zone is then dominated by the radiation coming from the flame and of embers. By reducing the role of the flame on the propagation, these conditions constitute an extremely severe test to validate the thermal model of degradation (drying, pyrolysis, oxidation of the charcoal) of the solid particles composing the fuel bed. Following this phase of validation, it remains to generalize this study on a large scale (on inclined ground, with wind), for conditions which are more representative of the real conditions of propagation of a wildfire.
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Weber, R. O., Prog. Energy Combust. Sci. 17:67– 82 (1991). Rothermel, R., A mathematical model for predicting fire spread in wildland fuels. Technical report, USDA Forest Service Research paper INT-115, 1972. Pitts, W. M., Prog. Energy Combust. Sci. 17:83–134 (1991). Albini, F. A., Combust. Sci. Technol. 42:229 –258 (1985). Pagni, P. J., and Peterson, T. G., Flame spread through porous fuels. In Proceedings 14th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, pp. 1099 –1107. Albini, F. A., Combust. Sci. Technol. 45:101–113 (1986). De Mestre, N. J., Catchpole, E. A., Anderson, D. H., and Rothermel, R. C., Combust. Sci. Technol. 65:231– 244 (1989). Grishin, A. M., Mathematical Modeling of Forest Fires and New Methods of Fighting Them. Publishing house of the Tomsk University, Tomsk, Russia. Ed. by F. Albini, 1997. Larini, M., Giroud, F., Porterie, B., and Loraud, J. C., Int. J. Heat Mass Transfer 41:881– 897 (1998). Yakhot, V., and Orszag, S. A., J. Scientific Computing 1(1):3–51 (1986). Orszag, S. A., Introduction to Renormalization Group Modeling of Turbulence. Oxford University Press, 1996, pp. 155–183. Cox, G., Combustion Fundamentals of Fire. Academic Press, London, UK, 1995. Morvan, D., Porterie, B., Larini, M., and Loraud, J. C., Combust. Sci. Technol. 140:93–122 (1998). Incropera, F. P., and DeWitt, D. P., Fundamentals of Heat and Mass Transfer. John Wiley and Sons, Chichester, UK (4th Edition), 1996. Kee, R. J., Rupley, F. M., and Miller, J. A., The CHEMKIN thermodynamic data base. Technical report, Sandia National Laboratories, 1992. Magnussen, B. F., and Hjertager, H., On mathematical modeling of turbulent combustion with special emphasis on soot formation and combustion. In Proceedings 16th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, pp. 719 –729, 1976. Bai, X. S., and Fuchs, L., Turbulent reacting flow modeling in gas turbine combustors. In Proceedings Computational Fluid Dynamics ’94, 1994, pp. 807– 814. Morvan, D., Porterie, B., and Loraud, J. C., Numerical simulation of a buoyant methane/air diffusion flame. In Fire Safety Science Proceedings 6th Int. Symp., 1999, pp. 277–287. Siegel, R., and Howell, J. R. Thermal Radiation Heat
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Transfer. Hemisphere Publishing Corporation, Washington, DC, 3rd edition, 1992. Syed, K. J., Stewart, C. D., and Moss, J. B. Modelling soot formation and thermal radiation in buoyant turbulent diffusion flames. In 23rd Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1990, p. 1533. Moss, J. B., in Cox, G. (Ed.), Turbulent Diffusion Flames, Academic Press, London, UK, 1995, pp. 221– 272. Nagle, J., and Strickland-Constable, R. F., Oxidation of carbon between 1000 –2000°c. In Proceedings 5th Conference on Carbon, vol. 1, 1962, pp. 154 –164. Leonard, B. P., and Mokhtari, S., Int. J. Numerical Methods Engineering 30:729 –766 (1990). Dupuy, J. L., Int. J. Wildland Fire 5(3):153–164 (1995). Drysdale, D., An Introduction to Fire Dynamics. John Wiley and Sons, Chichester, UK, 1985. Valette, J. C., Efaistos: Experiments and simulations for improvement and validation of behaviour models of forest fires. Technical report, Commission of the European Communities, DG XII, 1997. Den Breejen, E., Roos, M., Schutte, K., De Vries, J. S., and Winkel, H., Infrared measurements of energy release and flame temperatures of forest fires. In Proceedings 3rd International Conf. on Forest Fire Research, 14th Conference on Fire and Forest Meteorology, vol. 1, 1998, pp. 517–532. Dupuy, J. L., Combust. Sci. Technol. 154:149 –180 (2000). Fernandez–Pello, A. C., and Williams, F. A., Laminar flame spread over PMMA surfaces. In 15th Symposium (International) on Combustion, The Combustion Institute, 1974, pp. 217–231. Fernandez–Pello, A. C., and Santoro, R. J., On the dominant mode of heat transfer in downward flame spread. In 17th Symposium (International) on Combustion, The Combustion Institute, 1980, pp. 1201–1209. Fernandez–Pello, A. C., Ray, S. R., and Glassman, I., Combust. Sci. Technol. 19:19 –30 (1978). Fernandez–Pello, A. C., in Cox, G. (Ed.), The Solid Phase, Academic Press, London, UK, 1995, pp. 31– 100. Gould, J. S., Validation of the rothermel fire spread model and related fue parameters in grassland fuels. In Proceedings of Conference on Bushfire Modeling and Fire Danger Rating Systems, 1991, p. 198. Catchpole, W. R., Catchpole, E. A., Butler, B. W., Rothermel, R. C., Morris, G. A., and Latham, D. J., Combust. Sci. Technol. 131:1–37 (1998).
Received 5 February 2001; revised 7 April 2001; accepted 17 July 2001
Universite´ de la Me´diterrane´e UNIMECA, 60 rue Joliot Curie Technopo ˆle de Cha ˆteau Gombert, 13453 Marseille Cedex 13 France
and J. L. DUPUY
Institut National de la Recherche Agronomique, Unite de Recherches Forestieres Mediterrane´ennes, Equipe de Pre´vention des Incendies de Foreˆt 20 avenue Antonio Vivaldi, 84000 Avignon, France We describe a multiphase formulation to study numerically the propagation of a line fire in a forest fuel bed. One of the objectives of these studies is the improvement of knowledge on the fundamental physical mechanisms that control the propagation of forest fires. In complement of the experimental approach, this simulation tool can also be used for the development of simplified operational models used for instance for the prediction of the rate of spread (ROS) of wildland fires. The decomposition of solid fuel constituting a forest fuel bed as well as the multiple interactions with the gas phase are represented by adopting a multiphase formulation. This approach consists in solving the conservation equations (mass, momentum, energy) averaged in a control volume at a scale sufficient to contain several solid particles in the surrounding gas mixture. After a presentation of the equations and closure sub-models used in this approach, some numerical results obtained for the propagation of a line fire in a pine needles litter are presented and compared with experimental data obtained in laboratory. These results show that the rate of spread of fire in the fuel bed is primarily controlled by the radiative heat transfer. By increasing the fuel load (with a constant packing ratio), the results show the existence of two modes of propagation. A first area where the ROS varies linearly with the fuel load followed of a second where the ROS becomes independent of the load. By introducing the optical thickness characterizing the fuel bed, this difference in mode of propagation was interpreted like the demonstration of two modes of radiative transfer (optically thin and thick, respectively). The analysis of the distributions of the mass fractions of fuel and oxidant present in the gas mixture integrated through the depth of the fuel bed shows that the propagation velocity could also be limited by the lack of oxygen or fuel available in the ignited zone to maintain the pilot flame. © 2001 by The Combustion Institute
NOMENCLATURE cp Cps F s,i fv gi h h s, ␣ , ⌬h char I J
specific heat of the gas mixture specific heat of solid particles drag force resulting from gas/particles interaction in the fuel bed soot volume fraction gravity acceleration in the i-direction enthalpy of the gas mixture reaction heat (drying, pyrolysis, charcoal oxydation) radiative intensity irradiance
* Corresponding author. E-mail: [email protected] COMBUSTION AND FLAME 127:1981–1994 (2001) © 2001 by The Combustion Institute Published by Elsevier Science Inc.
k kg ˙ M s, ␣
ms pg Qconv ROS t xi T Ts
turbulent kinetic energy conductivity of the gas mixture production term of species ␣ resulting from the decomposition of the solid particles (drying, pyrolysis, char oxydation) fuel bed load (kg/m 2 ) pressure of the gas mixture gas/particles heat transfer rate of spread time cartesian coordinate in the i-direction temperature of the gas mixture temperature of the solid particles 0010-2180/01/$–see front matter PII S0010-2180(01)00302-9
1982 ui s s Y is , Y H , Y char 2O
Y␣
D. MORVAN AND J. L. DUPUY velocity vector component in the i-direction mass fraction of dry wood, moisture content and char of solid particles mass fraction of species ␣
Greek symbols
␣ g, ␣ s ␦ ⑀ g t ˙␣ s s ˙ pyr , ˙H 2O
˙ char g, s a ij s
volume fraction of the gas and solid phase fuel bed depth turbulent kinetic energy dissipation rate burned rate of solid particles (mass fraction) viscosity of the gas mixture eddy viscosity production, destruction rate of species ␣ (combustion) pyrolysis and drying reaction rates charcoal oxydation rate density of the gas and solid phase Stephan–Boltzmann constant absorption coefficient of the gas/soot mixture stress tensor surface/volume ratio of solid particles
INTRODUCTION The understanding of the physical mechanisms that control the ignition and the spread of wildfires constitutes a major objective for the management and the preservation of forest areas. Various mathematical models based on statistical, semi-empirical or physical approach have been proposed to evaluate the spread rate of a surface forest fire [1]. The semi-empirical models are based on a relationship between the rate of spread (ROS) of a surface fire with the energy received by the combustible layer and the energy necessary to carry the fuel particle until a threshold enthalpy of ignition. The constants necessary to close the system, are then evaluated from experimental fires. This ap-
proach leads to the well-known Rothermel model [2], which was introduced inside the computer code BEHAVE, which is currently used for the risk management related to the forest fires. These simplified models allow to obtain a quite good approximation of the fire rate of spread as a function of the fuel load, the wind intensity and the terrain slope for particular conditions (close to experiments carried out for the calibration of the parameters used in such models). Unfortunately, the use of these models for more general fire conditions does not always give satisfactory results. The main reason is that the physical mechanisms which control the thermal degradation of solid fuel, the fire behavior and the solid-flame interactions, are uncompletelly described in such models. Four possible mechanisms for the heat transfer can be involved in fire spread process: (1) direct flame contact, (2) convective heating, (3) radiative heating, and (4) firebrand contact. In the absence of wind and on a flat ground, experimental investigations have shown that 60% of the thermal energy required for ignition and received by the fuel comes from burning embers by convective and radiative heat transfer [3]. This result seems to confirm the minor part plays by the flame to sustain a spreading fire for this configuration. Experimental observations show that the flame is slightly leaning back, it forms an angle (positive in the direction of fire spread) ␣ ⬇ ⫺10, ⫺30 degrees with the direction perpendicular to the fuel bed [4]. This configuration of the flame reduces the view factor and thus the radiation of the flame on the top of the fuel bed. The relationship between these various contributions of course depends on the physical characteristics of the fuel bed (packing ratio, depth, surface/volume ratio, moisture content . . . ). To improve the numerical predictions concerning the behavior and the trajectory of a fire front, a more complete approach based on physical considerations can be adopted. We can distinguish two kinds of physical models: ●
The radiative models based on the calculation of the heat transfer (mainly radiative) between the burning zone (of which the characteristics are evaluated empirically) and the
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1983
part of the unburnt fuel bed ahead of the fire [4 –7], The complete models which consist in solving the various reactive problems (pyrolysis, combustion) from the conservation equations (mass, momentum, energy) in the gas mixture and in the fuel bed by using a multiphase formulation [8, 9].
The reasons why the radiation is retained as the single mode of heat transfer in the first approach are based on the following hypothesis: without wind, it is the only mechanism able to heat solid fuel ahead of the fire; with wind, the flame makes obstacle with the wind which cannot penetrate deeply in the fuel bed. The ROS predicted by the first approach too strongly depends on the characteristics of flame introduced into the model. Mathematical problems remain concerning the algorithm of determination of the ignition zone, iterative calculation seems to converge only for initial conditions close to the final solution. The extension of such model to complex situations (fire on an inclined ground, effects of the wind, fuel discontinuities) remains difficult. Associated with experiment performed in laboratory, these models made it possible to render by a physical approach some basic mechanisms (radiation of the flame and embers, convection of hot gases, cooling of the particles upstream of the face caused by fresh gases) associated with wildfires spread. In the multiphase approach, the multiple interactions between the gaseous phase and the solid fuel are described in details to obtain a complete representation of the physical mechanisms (gas flow, combustion, radiation, and convection heat transfer) contributing to the thermal decomposition (vaporization, pyrolysis, char oxidation) occurring during the development of a wildland fire. Adopting an averaging formulation to take into account the effects induced by the microstructures (subgrid scale particles) upon the behavior of the system at a resolved scale, the coupling between the solid and the gaseous phase is rendered through source terms in the mass (decomposition of solid fuel to combustible gas), momentum (drag force), and energy (heat transfer by convection and radiation) balance equations. The fuel bed is approximated as an uniformly distributed cylinders
Fig. 1. Lire fire spread through a forest fuel bed.
(solid particles), characterized by geometrical (surface-to-volume ratio, packing ratio, fuel depth) and physical (fuel density, reaction rates and composition of the pyrolysis products, moisture content) properties evaluated experimentally. The aim of the present work is to describe the physical behavior and the propagation of a small scale surface fire in a pine needles litter. The studied cases are limited to a two-dimensional configuration (see Fig. 1). This assumption can be justified considering that the observations show that widths of many spreading fires are in nature much greater that the fire depth [3]. Although the flames exhibit fully three dimensional behavior, the major contributions concerning the heat balance inside the fuel bed and above the litter can be assimilated as twodimensional over shorter portions of the fire front (see Fig. 1). MATHEMATICAL FORMULATION IN THE GAS PHASE We propose to study the propagation of a surface fire through a pine needles fuel bed on a horizontal plate (see Fig. 1). As we mentioned in the introduction, the model used in this study is based on a detailed physical approach, which consists in solving the conservation equations (mass, energy, momentum) of the system formed by the litter and the surrounding gas mixture. The fuel bed is represented as an initially homogeneous distribution of cylindrical particles whose dimensions are evaluated from experimental data. In the fuel bed, the conser-
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D. MORVAN AND J. L. DUPUY
vation equations (mass, momentum, energy, chemical species) are integrated on a control volume sufficient to be able to include several solid particles to define an average value. Considering the conditions of development of these fires, we can assume that the flow at the top of the litter is turbulent and that homogeneous combustion is primarily controlled by the turbulent mixture of the oxygen contained in the ambient air with the flammable part of gas coming from the decomposition of solid fuel by pyrolysis. To take into account the contribution of the fluctuations for the transport of mass, of energy and momentum in the gas phase, we adopted a statistical two-equations turbulence model (RNG-k⫺ ⑀ ) [10, 11] using mass weighted averages (Favre averages) [12]. This method allows to extend the domain of validity of this model of turbulence (initially adapted to the fully developed turbulent flows) to the zones where the level of turbulence remains low (close to the ground and in the litter). By preserving an unsteady formulation, this approach is able to correctly represent the formation and the time evolution of the large vortex structures observed during the expansion of hot gases above the burning zone. The diffusion flames characteristic of natural fires are dominated by gravity effects, which are revealed by a strongly nonstationary behavior (flames height, fresh air flow in the vicinity . . . ). This behavior is also observed for pool fires, it is characteristic of 2 flames at low Froude number (F r ⫽ U fuel /gd: U fuel fuel injection velocity, d dimension of the gaseous fuel production zone, g gravity acceleration) [12, 13]. According to these remarks and using density-weighted Favre averaging formulation, the conservation equations of mass, energy and momentum in the gas mixture can be written as follows: ⭸ ⭸ 共 兲 ⫹ 共 u ˜ j兲 ⫽ ⭸t ⭸ xj
冘
˙ s,␣ M
(1)
␣
⭸ ⭸ ⭸ 共 u ˜ i兲 ⫹ 共 u ˜ ju ˜ i兲 ⫽ 共 ␣ 兲 ⭸t ⭸ xj ⭸ x j g ij ⫺
⭸ u ⬙ju ⬙i ⫹ g i ⫺ F s,i ⭸ xj
(2)
冉
˜ ⭸ ⭸ ⭸T ⭸ ˜兲 ⫹ ˜兲 ⫽ 共 h 共 u ˜ jh ⭸t ⭸ xj ⭸ x j Pr ⭸ x j ⫺
冊
⭸u⬙j h⬙ ⭸ ␣ gp ⫺ Qconv ⫹ ⫹ ␣ g a共 J ⫺ 4 T 4兲 ⭸xj ⭸t
⫹
冘 M˙ ␣
s, ␣ hs, ␣
⫹ 共1 ⫺ ␣sg 兲⌬hchar ˙ char
(3)
冉 冊
˜ ⭸ ⭸ ˜ ˜兲 ⫽ ⭸ ⭸Y ␣ 共 Y ␣兲 ⫹ 共 u ˜ jY ␣ ⭸t ⭸ xj ⭸ x j Pr ⭸ x j ⫺
⭸ u ⬙jY ⬙␣ ˙ s,␣ ⫹ ˙ ␣ ⫹ M ⭸ xj
(4)
where the following notations have been introduced: ⫽ ␣ g g , ⫽ ␣ g g , ␣ g , and g design the volume fraction and the density of the gas phase. ˙ is the average production term of species M s, ␣
␣ resulting from the decomposition of the solid fuel (drying, pyrolysis, charcoal oxydation), ˙ ␣ the production/destruction of chemical species ␣ resulting from the homogeneous combustion, ˙ char the destruction of char resulting from the heterogeneous combustion. F s,i and Q conv design respectively the i-component of the drag force and the convective heat flux resulting from the interaction between the gas and the solid particles in the fuel bed. Assuming that the action of the solid phase can be approximated as the sum of each individual particle (the interaction between two particles is neglected), these terms are approximated as follows: F s,i ⫽
3 ␣ C 兩u兩u i␣ s s 8 g g d
Q conv ⫽ h conv␣ s s共T ⫺ T s兲
(5) (6)
where the drag (C d ) and the heat transfer coefficient (h conv) are approximated using empirical correlations obtained for a laminar or a turbulent flow around a cylinder [14]. The enthalpy-temperature dependence is treated using the CHEMKIN thermodynamic data base [15]. In addition to previous equations of conservation, two equations of state are needed to close the system:
FIRE SPREAD
冘 WY
␣
p g ⫽ gR 0T
␣
g ⫽
1985
␣
冘 冉 TT 冊 ␣
ref
␣
(7)
where P k and W k are, respectively, the shear and buoyancy turbulent production terms:
(8)
P k ⫽ ⫺ u ⬙iu ⬙j
Cref
Y␣
where R 0 is the universal gas constant, W ␣ , ␣ are the molecular weight and the viscosity of each chemical species, respectively T ref ⫽ 302 K, C ref ⫽ 0.75 are the coefficients of the Sutherland’s law which describes the variation of the viscosity with the temperature. Using the low Mach number flows approximation, the variations of pressure can be neglected in the energy conservation equation and in the equation of state of the gas mixture (acoustic filtering). With this approximation the stability of the scheme of resolution (CFL condition) depends only on the convection velocity, which then makes it possible to work with larger time step (⬇ 10⫺3–10⫺2 s). The double correlations representing the action of the fluctuations on the average transport equations are evaluated using the eddy viscosity concept and generalized gradient diffusion of the scalar quantities ():
冉
⫺ u ⬙ju ⬙i ⫽ t
冊 冉
冊
˜k ˜j 2 ⭸u ˜ i ⭸u ⭸u ⫹ ⫺ t ⫹ k ␦ ij ⭸ xj ⭸ xi 3 ⭸ xk (9)
2
t ⫽ C
k ⑀
⫺ u ⬙j ⬙ ⫽
(10)
˜ t ⭸ ⭸ xj
(11)
The turbulence field is completely described from two scalar variables the turbulent kinetic energy (k) and its dissipation rate (⑀) which are calculated from the two following transport equations: ⭸ ⭸ ⭸ 共 k兲 ⫹ 共 u ˜ jk兲 ⫽ ⭸t ⭸ xj ⭸ xj
冋冉
⫹
冊 册
t ⭸k k ⭸ xj
⫹ P k ⫹ W k ⫺ ⑀ ⭸ ⭸ ⭸ 共 ⑀ 兲 ⫹ 共 u ˜ j⑀ 兲 ⫽ ⭸t ⭸ xj ⭸ xj ⫹ 共C ⑀1 ⫺ R兲
冋冉
冊 册
t ⭸⑀ ⫹ ⑀ ⭸ xj
⑀ ⑀ ⑀2 P k ⫹ C ⑀3 W k ⫺ C ⑀2 k k k
共1 ⫺ / 0兲 1 ⫹  3
⫽
冑
Pk C ⑀
(13)
t ⭸ ⭸p 2 ⭸ x j ⭸ x j
Wk ⫽ ⫺
R⫽
(14)
0 ⫽ 4.38
 ⫽ 0.015
C ⫽ 0.0845 C⑀ 1 ⫽ 1.42
(15) C⑀ 2 ⫽ 1.68
k ⫽ 0.7179
C⑀ 3 ⫽ 1.5
⑀ ⫽ 1.3
(16)
We can note that the implementation of the RNG k ⫺ ⑀ turbulence model does not involve great modifications compared to the standard model. Only one additional term R is introduced into the transport equation for ⑀ which represents the dissipative action of the subgrid scales on the average motion. Using thermogravimetry analysis (TGA) it has been shown that for temperatures about 1000 K, the gas mixture produced by pyrolysis during the decomposition of forest fuel was primarily composed of CO, CO 2 , CH 4 and in less proportion of H 2 , C 2 H 6 [8]. To simplify the problem we have conserved only five chemical species (CO, CO 2 , H 2 O, O 2 , N 2 ). Assuming that the chemical reaction is quasi instantaneous, the combustion rate in the gaseous phase is primarily limited by the time necessary to fuel and oxidant to mix. Assuming that the mixing time depends on the turbulence scales, the production/destruction rates ˙ ␣ can be evaluated using the Eddy Dissipation Concept [12, 16]:
˙ CO ⫽ ⫺A (12)
⭸u ˜i ⭸ xj
冉
冊
˜ ⑀ ˜, Y O2 min Y CO g k O 2
g ˙ O2 ⫽ O ˙ CO 2
g ˙ CO2 ⫽ ⫺共1 ⫺ O 兲 ˙ CO 2
(17) (18)
where the constant A is equal to 2.5 [17], and the stoichiometric ratio of the chemical reg action CO ⫹ 1/2O2 3 CO2 is O ⫽ 4/7 (in 2 mass).
1986
D. MORVAN AND J. L. DUPUY
RADIATION HEAT TRANSFER MODELING The energy exchanged in radiative form is evaluated by integrating the radiative transfer equation (RTE) taking into account of the emission coming from the embers and the mixture formed by the gas (air ⫹ pyrolysis and combustion products) and soot particles formed in the flame [8, 9, 18] (see Eq. 20):
冉
冊
冉
d␣g I T4 s ␣s T4s ⫽ ␣g a ⫺I ⫹ ⫺I ds 4 J⫽
冕
4
I d⍀
冊
(19)
(20)
contribution because of the gradient of temperature (thermophoretic velocity u jth ) defined as following: u jth ⫽ ⫺0.54
0
For these two contributions (gas ⫹ soot and embers) we suppose that the emission is carried out as that of a black body, this assumption is relevant because the particles present are primarily composed of carbon. This calculation is carried out by a discrete ordinates method (DOM) [19] which is well adapted to solve the problem of the radiative transfer in the media presenting significant variations of the absorption coefficient. The radiation coming from the flame is primarily because of the presence of soot particles which are formed in the flame and which give it a very characteristic yellow color. Because of the lack of information on soot production in natural fire, we have fixed the production rate of soot from the production rate of gaseous pyrolysis products [8]. Assuming that the soot particles can be assimilated as carbon spheres of a diameter d soot ⫽ 1 m, the soot volume fraction field can be evaluated from the following transport equation [20, 21]: ⭸ ⭸ ⭸ th˜ ˜ ˜ 共 ˜f v兲 ⫹ 共 u ˜ j˜f v兲 ⫽ ⫺ 共 u ˙ fv j f v兲 ⫹ ⭸t ⭸ xj ⭸ xj (21) ˜ ˙ fv ⫽
Fig. 2. Fundamental physical mechanisms controlling the propagation of a fire in a fuel bed.
1
soot
˜ 关 soot ˙ pyr ⫺ W Ox ˜f v soot兴
(22)
where soot ⫽ 6/d soot is the surface/volume ratio of soot particles. The transport of the soot particles is ensured by the convective motion of the gas (velocity u j ) to which is added the
˜ g ⭸ ln T g ⭸ x j
(23)
The term W Ox corresponds to soot oxidation, it is evaluated from the rate for oxidation of pyrolytic graphite by O 2 (kg/m 2 .s) [22], W Ox ⫽ 120
冋
k AP O2 1 ⫹ k zP O2
⫹ P O2共1 ⫺ 兲
册
1
⫽ 1⫹
kT k BP O2
(24) (25)
where P O 2 is the partial pressure of oxygen. The various reaction rates k A , k B , k T , k z introduced into the expression of W ox depend on the temperature in the following way [22, 23]: ˜ ) g/cm 2.s.atm k A ⫽ 20 exp (⫺30000/RT
(26)
˜ ) g/cm 2.s.atm kB ⫽ 4.46 10⫺3 exp (⫺15200/RT (27) ˜ ) g/cm 2.s kT ⫽ 1.51 105 exp (⫺97000/RT
(28)
˜ 兲atm ⫺1 k z ⫽ 21.3 exp 共4100/RT
(29)
MATHEMATICAL FORMULATION IN THE SOLID PHASE Under the action of the intense heat flow released by the burning zone (flame ⫹ embers), the solid particles located ahead of the fire front will break up in the following way (see Fig. 2): ● ●
Dehydration, Pyrolysis (gaseous products and charcoal)
FIRE SPREAD
1987
Reaction rates of these various contributions are approximated by Arrhenius laws whose parameters (frequency factors and activation energy) are evaluated from TGA [8]. During its degradation the solid fuel is described by the distribution (mass fraction) of four components s s (dry wood Y is , water content Y H , coal Y char , 2O ash) evaluated by integrating the system of equations formed by the mass conservation of the system gas ⫹ fuel particles. The decomposition of the dry wood by pyrolysis induces the production of gaseous products (CO, CO 2 ) and charcoal (char kg of char per 1 kg of dry wood). The stoichiometric ratio of CO 2 produced by pyrolysis is CO 2 (kg of CO 2 per 1 kg of gas produced by pyrolysis). By contact with the oxygen contained in the ambient air, the combustible part of the pyrolysis products ignites (homogeneous combustion), while the charcoal undergoes a surface oxidation (heterogeneous combustion). Taking into account all these assumptions, the evolution of the composition of the solid particles in the litter is controlled by the following equations:
˙ char ⫽
1 s k char␣ g gY O2 exp O 2
d s 共 ␣ Y s兲 ⫽ ⫺ ˙ pyr dt s s i
(35)
d 2 ˙H 共 ␣ Y s 兲 ⫽ ⫺ 2O dt s s H2O
(36)
d s 共 ␣ Y s 兲 ⫽ 共 char ⫺ soot兲 ˙ pyr dt s s char ⫺ d 共␣ 兲 ⫽ ⫺ dt s s
冘 M˙
˙ H2O ⫽
k H2O
冑T s
冋
⫺ E pyr RT s
s ␣ s sY H exp 2O
冋
册 RT s
册
冊
ash s ⫹1 ˙ char char
s,␣
s ⫽ 共 char ⫺ soot ⫺ 1兲 ˙ pyr
2 s ⫺ ˙H ⫺ ˙ char 2O
1 s d ˙ 共 ␣ s兲 ⫽ ⫺ dt s char
␣ s sC ps
(37)
(38) (39)
dT s ⫽ Q conv ⫺ dt
冘 M˙ ␣
s,␣h s,␣
s ⫹ ␣ sg⌬h char ˙ char
␣ s s 共 J ⫺ 4 T 4s 兲 4
(30)
⫹
(31)
s ˙ s,CO ⫽ 共1 ⫺ char ⫺ CO 兲 ˙ pyr M 2
(32)
⫺ E H2 O
冉
␣
⫹ 共1 ⫺ char兲关共1 ⫺ CO2兲CO ⫹ CO2CO兴
˙ pyr ⫽ k pyr␣ s sY is exp
册
⫺ E char ␣ s s RT s (34)
Solid Fuel 3 char共Char ⫹ Soot兲
s s O 2 3 共1 ⫹ O 兲CO 2 Char ⫹ O 2 2
冋
s ˙ s,H O ⫽ M ˙H 2 2O
(40)
(41)
s s ˙ s,O ⫽ ⫺ O M ˙ char 2 2
(33)
s s s ˙ s,CO ⫽ 共1 ⫹ O M 兲 ˙ char ⫹ CO2 ˙ pyr 2 2
(42)
Vaporization:
k H 2O ⫽ 6.05 10 5 K 1/ 2 s ⫺ 1
E H 2O /R ⫽ 5956 K
Pyrolysis:
kpyr ⫽ 3.64 10 3 s ⫺ 1
E pyr/R ⫽ 7250 K
Charoxidation:
kchar ⫽ 430ms ⫺ 1
Echar/R ⫽ 9000 K, ␣ sg ⫽ 0.5
Stoichiometriccoefficients:
char ⫽ 0.338, CO 2 ⫽ 0.2
ash ⫽ 0.033, soot ⫽ 0.05
(per unit mass)
s O ⫽8/3 2
For this study we also supposed that the particles in the litter remained at rest during the
major part of the reactions of pyrolysis and combustion, except at the end of the decompo-
1988
Fig. 3. Thermal decomposition of a solid fuel sample (pine needles): mass fraction of dry wood (Y is ), moisture content s s (Y H O ) and char (Y char ), and mass loss obtained by TGA in 2 inert atmosphere simulated with a heating rate dT/dt ⫽ 10 K/s (top) and 50 K/s (bottom).
sition when the ash state is reached. At the time when the particles were completely transformed into ash, the litter collapses abruptly and the gas/particles interactions can then be neglected. The oxidation of the charcoal is not limited to a heterogeneous reaction (C ⫹ O 2 3 CO 2 ), it is also made up of two reactions (heterogeneous and homogeneous) following: (C ⫹ 21 O 2 3 CO), (CO ⫹ 21 O 2 3 CO 2 ). The distribution of the heat of reaction between the two phases (solid and gas) is taken into account by introducing a sharing coefficient ␣ sg (⫽ 0.5 for the present calculation). The curves represented on the Fig. 3 show the evolution of the mass fractions (dry wood, moisture content, and char-
D. MORVAN AND J. L. DUPUY coal) as well as the mass loss obtained numerically by simulating a TGA in a furnace in inert atmosphere for two heating rate 10 K/s and 50 K/s (values representative of the conditions usually used for a TGA and met ahead of a spreading fire, respectively). The results obtained for a heating rate of 10 K/s (see Fig. 3 on top) representing the evolution of the mass fractions (dry wood, moisture content and charcoal) according to the temperature, show that the wood sample is completely dehydrated starting from 400 K, and degradation intensifies (by pyrolysis) with 460 K. With 600 K the degradation (charcoal transformation) of the sample in inert atmosphere is complete, the material then has lost 74% of its initial mass [curve in continuous line on the Fig. 3 (top)]. For a heating rate of 50 K/s the curves describe the same stages with a shift of ⬇ 60 K of the thresholds of temperature. The last test (50 K/s) was carried out once again with an oxygenated atmosphere, assuming that the mass fraction of oxygen remains constant and equal to 0.23. The results presented on the Fig. 4 have been obtained by supposing that the oxidation of the charcoal was limited to the surface of the particles (on top) and by taking of account porosities which allows a certain diffusion of oxygen in volume (on bottom). In this case the surface of reaction is calculated starting from the ratio of density of the charcoal with carbon, by assuming that the size of the pores is homogeneous and equal to 50 m. The results show that the oxidation of the charcoal continues until the sample has reached the state of ash, the material will then have lost 97% of its initial mass. RESULTS AND DISCUSSION The system of partial differential equations formed by the set of the conservation equations in the gas phase is solved numerically using a finite volume method. By using a high order upwind convective scheme (QUICK) associated with a flux limiting strategy (ULTRA-SHARP) [23], we obtain a stable numerical integration procedure, without numerical diffusion. The system describing the evolution of solid fuel is solved in a separated way using a Runge–Kutta
FIRE SPREAD
1989
Fig. 5. Position (in meter) of the pyrolysis front (represented as the isotherm T s ⫽ 500 K) and mass loss in the fuel bed as a function of time.
Fig. 4. Thermal decomposition of a solid fuel sample (pine needles): mass fraction of dry wood (Y is ), moisture content s s (Y H ) and char (Y char ), and mass loss obtained by TGA in 2O air simulated with a heating rate 50 K/s [without (top) and with charcoal porosity effects (bottom)].
method. Without wind and on a flat ground, the spread of a surface fire is characterized by a flame leaning in the direction opposed to the direction of propagation. The heat flow received by the particles ahead of the fire is mainly radiative. One can then estimate that the distance to which the solid particles thermically will be affected by the fire is about l rad ⫽ (( s ␣ s )/4) ⫺1 , l rad is the mean penetration distance [19]. For the present study ( s ⫽ 4550 m ⫺1 , ␣ s ⫽ 0.029) this distance is l rad ⬇ 3 cm. To obtain a good representation of the thermal degradation zone in the fuel bed, it is thus necessary that the mesh size in this region is lower than 1 cm. On the other hand apart from the zone in flame, the conditions of temperature
and the gas mixture remain not very affected. To ensure a level of precision constant throughout calculation, we have used a adaptative grid technique, consisting in refining the grid in the vicinity of the front of decomposition of the fuel bed located from the gradient of the dry wood mass fraction (ⵜY is ). The intense zone of pyrolysis corresponds to the zone where the module 储ⵜY is 储 is maximum, in this area the minimal mesh size (⌬) is calculated so that the ratio l rad/⌬ ⬇ 5 (⌬ ⫽ 6 mm). We carried out a first series of calculations for a dead pine needles litter (Pinus pinaster) uniformly layered over a horizontal level and into absence of wind. The properties of the litter are identical to those retained for the experiments carried out in laboratory [24]. Fuel density ( s ): Packing ratio ( ␣ s ): Surface-to-Volume ratio ( s ): Fuel moisture content: Fuel depth (␦): Fuel load (m s ⫽ ␣ s s ␦ ):
680 kg/m 2 0.029 4550 m ⫺1 0.02 from 0.03 to 0.2 m from 0.6 to 4 kg/m 2
Lighting is simulated by introducing a volume heat source over the whole fuel bed depth and along a length of 5 cm. This heat supply will be maintained until the temperature of the particles reaches a value of 1200 K. The results on the Fig. 5 show the time evolution of the position of the isotherm T s ⫽ 500 K as well as the rate of mass loss in the litter obtained for a fuel bed depth of 5 cm (m s ⫽ 1 kg/m 2 ). The
1990 results show that the development of the fire in the litter follows an initial phase of growth (which must depend on the conditions of lighting) and 40 s after the beginning, the fire achieves a quasi steady-state. The trajectory of the pyrolysis front which reaches a recti-linear shape, allows to evaluate a constant rate of spread (ROS ⬇ 3.8 mm/s). The curve of mass loss is characterized by fluctuations (around an average value ⬇ 3 g/m.s) which are related to the movements which characterize this type of flame subjected to the effects of gravity (puffing instabilities) [26]. The flame is subjected to vertical oscillatory movements which modify the radiative heat flow received by the particles of the litter. The increase of the flame height causes an acceleration of the pyrolysis rate and an increase in quantity of fresh air aspired by the burning zone. These two phenomena induce, respectively, a heating and a cooling of the solid particles located ahead of the fire. It is the combination of these two contradictory effects which is at the origin of the oscillatory behavior observed in experiments and in the present calculations. These results are in agreement with the experimental measurements obtained for the spread of a surface fire in a pine needles litter (Pinus Pinaster), for which one finds a ROS equal to 3.9 mm/s and a mass loss rate of 3.69 g/m.s [25]. The velocity field represented with the fuel bed density are on Fig. 6 (top) (fire is propagated from left to right), is characterized by the formation of a hot gas column at the top of the ignited zone, inclined behind and forming an angle of 40 degrees with the vertical. Along the thermal plume the gas undergoes an acceleration, the velocity magnitude reaches 0.4 m/s near the flames and 2 m/s 35 cm above the litter. In the fuel bed the flow is strongly reduced (⬇ 0.1 m/s). We can also note that after the passage of the fire the fuel bed density decreases from 20 kg/m 3 (initial value) to 0.64 kg/m 3 . The decomposition of the solid particles in the litter is total and the state of ash is reached. The temperature field (gas phase) represented on Fig. 6 (bottom) is characterized by two hot zones, corresponding respectively to the heat released by the homogeneous and heterogeneous (charcoal oxidation) combustion. The temperature in the flame is about 1200 K, which is in conformity with the infrared
D. MORVAN AND J. L. DUPUY
Fig. 6. Instantaneous velocity vector field and bed density (top) and gas temperature field (bottom) (closer view) calculated for a fuel depth of 5 cm (fuel load m s ⫽ 1 kg/m 2 ).
measurements obtained on the experimental setup [27, 28]. To identify the principal mechanisms of heat transfer contributing to the fire spread, we have evaluated the average value of various variables per integration through the depth of the litter. Distributions of the average temperatures achieved in gas (T) and in the solid particles (T s ), and the density of the litter along the direction of propagation are represented on the Fig. 7. One distinguishes well the two hot zones described previously, marked by a significant reduction of the fuel bed density corresponding to the two mechanisms of mass loss: drying ⫹ pyrolysis and combustion of the charcoal. The decomposition area of the fuel bed, limited by the zone where the particles are still intact and ashes extends over a length ␦ f ⬇ 22 cm, which gives a residence time ␦ f /ROS ⬇ 58 s (in conformity with the experimental results). The two principal mechanisms involved in the pre-heating of the litter located ahead of the fire are the radiative (Q rad) and the convective transfer (Q conv) (the conductive transfer in the solid phase can be regarded as negligible), defined as follows:
FIRE SPREAD
1991
Fig. 9. Average distribution of mass fraction of O 2 and CO and fuel moisture content in the fuel bed along the fire spread direction (fuel load m s ⫽ 1 kg/m 2 ).
Fig. 7. Average distribution of temperature (in the gas and solid particles) and fuel bed density along the fire spread direction (fuel load m s ⫽ 1 kg/m 2 ).
Q rad ⫽
␣ s s 共 J ⫺ 4 T 4s 兲 4
Q conv ⫽ h conv␣ s s共T ⫺ T s兲
(43) (44)
One can evaluate the importance of these two contributions by integrating these two quantities over the depth of the litter. At a given time, a distribution of the heat flows (by convection and radiation) received by the litter along the direction of propagation is then obtained (see Fig. 8). One can notice that for these conditions (without wind and on flat ground) the degradation of the particles starts due to radiative heat trans-
Fig. 8. Average distribution of convective and radiative heat transfer and fuel moisture content in the fuel bed along the fire spread direction (fuel load m s ⫽ 1 kg/m 2 ).
fer. The ratio between the two peaks of heat flow by radiation and convection in the zone located ahead of the fire (where the degradation of the fuel bed is initiated) is equal to 8.6 (see Fig. 8). By integrating these two curves one can evaluate the total quantity of energy received by the solid fuel bed, by radiation and convection which is, respectively, of 5.475 kW/m (radiation) and 3.575 kW/m (convection) (for a fuel bed depth of 5 cm). The ratio (radiation/ convection) between these two sources of energy is thus 1.53 (60% of the energy received by the solid fuel is due to radiative heat transfer). The propagation of fire in the litter is thus primarily controlled by the radiation, confirming the assumptions retained by certain simplified physical models [4, 7]. By analyzing the average distribution in the fuel bed of the mass fraction of oxygen (Y O2 ) and carbon monoxide (Y CO ) (see Fig. 9), one can note the existence of a completely under-oxygenated zone. The totality of the oxygen contained in the air aspired upstream is consumed by the zone on fire (flame ⫹ embers), it is thus a factor which can limit the progression of fire. The ROS (numerical and experimental values) obtained for various fuel loads ranging from 0.6 kg/m 2 to 5 kg/m 2 are shown on Fig. 10 and compared with the value obtained with a purely radiative fire spread model [6, 29] assuming that the fire can be assimilated as a vertical radiant heater with a temperature uniform and equal to 1100 K (the contribution of the flame above the fuel bed is neglected). These results show that the variation of the ROS with respect to the fuel load consists of three zones:
1992
D. MORVAN AND J. L. DUPUY
Fig. 11. Average distribution of temperature (in the gas and solid particles) and fuel bed density along the fire spread direction [fuel load m s ⫽ 2 (top) and 4 kg/m 2 (bottom)].
Fig. 10. ROS (top) and mass loss rate (bottom) as a function of the fuel load.
● ●
●
For weak fuel loads (m s ⬍ 1.2 kg/m 2 ), the ROS varies linearly with the load, For intermediate loads (m s ⬍ 4 kg/m 2 ), the ROS varies still linearly with the load (the slope is weaker), For high fuel loads (m s ⬎ 4 kg/m 2 ), the mode of propagation changes and the ROS tends to become independent of the load.
According to the importance of the role played by the radiative transfer, the limit between the first two modes of propagation can be expected to be related to the optical thickness characterizing the initial state of the litter: opt ⫽ ␣ s s ␦ /4 (using the depth of the litter as characteristic length scale). The change of mode for which the dependence of the ROS with the load starts to be weaker, occurs for an optical thickness higher than 2. That means that the medium becomes optically thick and that the propagation of the fire is no longer controlled by the radiation through the whole fuel bed depth, but
through a fraction of it (⬇l rad). The studies realized on the propagation of a fire on a solid plate of PMMA show that below a certain thickness of fuel, the propagation velocity was a function inversely proportional to the thickness (thermically thin mode), to tend thereafter towards a constant value (optically thick mode). The increase thickness of fuel contributes to an increase in the conductive losses in material, until reaching a limiting thickness beyond which the heating effects are too weak to take part in the propagation of the combustion front [26, 30, 31]. If the fire is propagated in a porous environment, the increase thickness of fuel first of all involves an increase in the size of the zone on fire [see Fig. 11 (top) representing the temperature distributions in the fuel bed] and consequently an increase in the flow of fresh air which feeds the reaction of combustion ahead of the fire front. This situation is similar to that encountered for downward flame spread in an opposed forced flow [32, 33]. In this case an increase in the upstream air flow produced first an increase in the ROS, then the ROS reaches a critical value and decreases until reaching the limit of extinction where the selfmaintained propagation of the flame front is no more possible. This limitation results from the increase of the convectif heat transfer, the flow of
FIRE SPREAD
1993
fresh air causes finally a cooling of the particles located ahead of the fire front [33] [see Fig. 11 (bottom)]. As indicated before, on flat ground and in absence of wind, the combustion front is propagated with a flame directed slightly behind. Ahead of the fire the products of pyrolysis combine with the ambient air entrained into the base of the developing fire and burn behind the fire front [26, 33]. This spread mechanism is described as ‘counter-current’ (or ‘opposed flow’) which means that the oxidizer flow and the direction of propagation are opposed. The highlighting of these two modes of propagation makes it possible to understand the apparent contradiction between various experimental results with regard to the effect of the fuel load on the ROS [2, 25, 34, 35]. Concerning the experimental fires performed in a wind tunnel, in the absence of wind, the results obtained could be influenced by the effect of confinement caused by the drain of the thermal plume along the walls. When the propagation of fire becomes independent of the fuel load, the calculated ROS is equal to 5.2 mm/s, this value is very close to the computed value starting from a recent empirical model obtained for experimental fires in a wind tunnel (ROS ⫽ 5.56 mm/s [35]). On the same Fig. 10 one can note that the ˙ ) is a linear function of the mass loss rate (M load (m s ): ˙ ⫽ K a共m s ⫺ m cs 兲 M
(45)
where K a has the same dimension that a rate of spread and m cs indicates the threshold of fuel load to maintain a sustained fire propagation. In the present calculations K a ⫽ 4.44 mm/s and m cs ⫽ 0.3 kg/m 2 which are comparable with the values evaluated from experimental results: K a ⫽ 4.54 mm/s, m cs ⫽ 0.18 kg/m 2 . In stationary regime the mass loss rate and the ROS moreover are connected by the following relation: ˙ ⫽ ⫻ ROS ⫻ m s M
(46)
where indicates the burned rate of solid particles in the fuel bed (by taking account of the residual ash content ⱕ 0.97). The following relation is then obtained:
冉
⫻ ROS ⫽ K a 1 ⫺
m cs ms
冊
(47)
Far from the extinction limit (m cs /m s 3 0) the product ⫻ ROS remains constant and equal to K a .
CONCLUSION The propagation of a surface fire in a pine needles fuel bed has been simulated numerically using a multiphase formulation. The influence of the fuel load on the ROS was studied. The numerical results show that the propagation of a surface fire on a flat ground and without wind, is mainly controlled by the radiative heat transfer. The results also show that the oxygen supply is ensured by a flow of fresh air aspired just upstream of the zone on fire. The influence of the depth of the fuel bed on the propagation velocity can be summarized in the following way, the increase of the fuel layer induces an increase in the heat losses and an increase in the flow of air entrained upstream of the fire front. For a low-size fire the increase in the flow of fresh air will cause an increase in the propagation velocity. For a more significant fire, the additional contribution of air on the contrary will cause a cooling of the particles upstream and will slow down the progression of the combustion front. By comparing the computational results (ROS and mass loss rate) with the experimental data, this study made it possible to validate the various sub-models composing the multiphase approach for the study of the propagation of a surface fire. The experimental conditions (flat ground and without wind) make it possible to obtain a line fire, the mode of heat transfer of the ignited zone towards the unburnt zone is then dominated by the radiation coming from the flame and of embers. By reducing the role of the flame on the propagation, these conditions constitute an extremely severe test to validate the thermal model of degradation (drying, pyrolysis, oxidation of the charcoal) of the solid particles composing the fuel bed. Following this phase of validation, it remains to generalize this study on a large scale (on inclined ground, with wind), for conditions which are more representative of the real conditions of propagation of a wildfire.
1994
D. MORVAN AND J. L. DUPUY
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Received 5 February 2001; revised 7 April 2001; accepted 17 July 2001