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Generalized Byram's formula for arbitrary fire front geometries
International Journal of Thermal Sciences 110 (2016) 222e228
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Generalized Byram's formula for arbitrary fire front geometries Nouredine Zekri a, *, Omar Harrouz a, Ahmed Kaiss b, Jean-Pierre Clerc b, Xavier Domingos Viegas c Universit e des Sciences et Technologie d’Oran, LEPM, BP 1505 El Mnaouer, Oran, Algeria Aix Marseille Universit e, CNRS, IUSTI UMR 7343, 13453, Marseille, France c ADAI/CEIF, University of Coimbra, Rua Luis Reis dos Santos, Coimbra, 3030-788, Portugal a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 November 2015 Received in revised form 19 June 2016 Accepted 13 July 2016 Available online 21 July 2016
The expression proposed a half century ago by Byram allowing the estimation of wild land fire front heat flux was extensively used in fire research and by firefighters. However, this formula does not account for wild land fuel heterogeneity and local weather changes. Its validity is limited to stationary spreads, and fails in describing the behavior of curved fronts and non-spreading fires. A new formula is derived to generalize heat flux estimation to any fire front with arbitrary geometry. Heat flux is found to depend on the front local radius as well as its convexity/concavity. The front curvature induces also a quadratic dependence on the rate of spread. The derived equation is validated by numerical simulations using the small world network model for line and curved fronts. Extensions of generalized Byram formula including fractal dimension of the front and wind speed effect are also proposed. © 2016 Elsevier Masson SAS. All rights reserved.
Keywords: Byram law Heat transfer Fire front
1. Introduction Millions hectares of vegetation are burned each year because of wildland fires [1]. Control methods to be implemented against forest fires depend on their intensities which are divided into several classes [2]. In the late 50s, Byram proposed an empirical equation to estimate the intensity of a fire front [3]. This law is now widely used both by scientists for laboratory or field experiments, and by operational fire managers. It indicates that the heat flux f (kW/m) delivered per meter of fire front is related to the consumed fuel load m (kg/m2), the combustion heat or enthalpy H (kJ/kg) and the rate of spread v (m/s) by
f¼mHv
(1)
The heat of combustion (enthalpy) H used here is a proportionality constant assuming a linear dependence of the heat released rate on the fuel load. In case of a nonlinear dependence, the mass load m appears with a nonlinear exponent in (1). This relationship, valid for stationary propagation, has two major disadvantages: firstly it is not predictive, as the rate of spread cannot be estimated beforehand, and secondly it does not account for the
* Corresponding author. E-mail address: [email protected] (N. Zekri). http://dx.doi.org/10.1016/j.ijthermalsci.2016.07.006 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.
fuel heterogeneity (wild land-urban interfaces, highways, fuel breaks etc.), non-stationary spreading fronts, as well as nonspreading fires (v ¼ 0). The rate of spread strongly fluctuates from a zone to another one in heterogeneous fuel beds, and depends on the combustion region. This strongly decreases the accuracy of the fire intensity estimation, which moves from a risk class to another. In 1976, Tangren [4] noticed that (1) was introduced for a line front and that the rate of spread is in fact the ratio of the front width to fire residence time (v ¼ w/tc, w being the front thickness). He raised the problem of the spread direction (the main or the lateral direction). A curved geometry of the front could influence significantly fire intensity, and may be felt differently by operational fire managers. A reformulation of (1) is thus necessary to make it predictive and to account for other important parameters involved in the spread such as a curved and non-stationary spreading front. In this paper, a new formula of fire intensity is derived for various geometrical aspects of the front. The estimated intensity from this new formula is compared to fire spread simulations using the small world network model [5]. Such simulations were realized under various ignition outbreaks and wind conditions (circular or elliptical fronts for point like outbreaks and a line front for line outbreaks). Its behaviour is also compared to experimental data. Further improvements of the derived expression introduce the fractal dimensions of the front perimeter and area as well as the
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wind speed instead of the rate of spread. 2. Derivation of Byram formula The intensity per unit length expressed by Byram's formula (1) is derived from the heat released from the unit length perimeter of the front. This energy depends on the front width, its perimeter, the fuel load and the heat of combustion. The idea in this paper is to estimate the heat released from the front by the use of satellite images whose resolution analysis (size effect) allows the dependence of this energy on the fractal dimension of burning area of the fire front or a portion of it in which important changes of curvature or fuel load may occur. Time resolution of images is here assumed to be sufficient to determine the front rate of spread, necessary for the heat flux estimation. Satellite images currently available do not have sufficient time resolution (for example image rates of less than one per minute and a space resolution of 20 m are necessary for a front spread rate of 0.3 m/s). The price of satellite images being still high, the use of aerial images (aircrafts and UAVs) would solve this problem. Although the formula we derive here can be used in the general case, it will be validated in stationary propagation situations, which overcomes this problem of time resolution. Let us consider an L-sized square heterogeneous fuel system, with a fire front spreading in y direction (Fig. 1). The system heterogeneity may be geometrical (i.e. randomly distributed fuel through the system) and/or a fuel heterogeneity (fuel of different nature). Assuming that the fuel is of the same kind, the enthalpy (H) is constant. The energy released per unit length in the lateral (x) direction is given by:
EðtÞ H ¼ ∬ mðx; y; tÞ Pðx; y; tÞdxdy L L
As char cannot contribute to fire spread, the consumed mass load m(x,y,t) used in (1) is the effective fuel mass load per unit surface (i.e., the mass load without char and other unburned fuel). The function p(x,y,t) appearing in (2) defines the probability distribution to find a surface element dx dy of fuel burning at time t centered at position (x,y). Using the fire front surface element dP dxdy) of width dw(x,t) and perimeter dP(y,t) (dS ¼ dwdP ¼ dw dx dy (Fig. 1), Equation (2) can be rewritten as
E H vwðx; tÞ vPðy; tÞ ¼ ∬ mðx; y; tÞ dxdy L L vx vy
Fig. 1. Fire spread representation at a given time with unburned fuel (green) and fire front (red). Burned area is represented in black. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(3)
The integral Equation (3) includes a product of 3 functions of time and space. Deriving (3) with respect to time yields the heat flux per unit length released by the system, f ¼ 1L dEðtÞ (the system dt size L being fixed) reads
1 dE L dt H dmðx; y; tÞ vwðx; tÞ vPðy; tÞ ∬ dxdy ¼ L dt vx vy d vwðx; tÞ vPðy; tÞ þ mðx; y; tÞ dxdy dt vx vy
fðL; tÞ ¼
(4)
Time derivatives in (4) involve partial derivatives ! d ¼ v þ vx v þ vy v ¼ v þ ! v $ V . Thus (4) becomes vt vy vt vt vx vt dt
fðL; tÞ ¼
H L
! _ ∬ mðx; y; tÞ þ ! v $ V mðx; y; tÞ dS þ ∬ mðx; y; tÞdS_
þ ∬ mðx; y; tÞ (2)
223
vx vPðy; tÞ vwðx; tÞ vy þ vx rx vy ry
!
! dxdy (5)
Here m_ ¼ vm=vt and S_ ¼ vS=vt, the surface element being vP dS ¼ dP dw ¼ vw vx vy dxdy. The third term of (5) corresponds to the ! ! part v $ V of the time partial derivative. The gradient part of the time partial derivative yields the second derivatives of the front width and perimeter, defining the radius of front curvature in x and y directions ðv2 P=vy2 ¼ 1=ry Þ and ðv2 w=vx2 ¼ 1=rx Þ). Equation (5) is much more complicated than the initial Byram's law (1). In the first term of this equation appears the rate of mass loss m_ (describing the local fuel combustion), and its spatial variation. The second terms of this equation reveals the dependence of the heat release rate on the non-stationary behavior of the burning area (where the burning surface is time varying). The geometrical aspects of the front (the lateral (rx) and longitudinal (ry) curvature radius appear in the third term. Note here that the radius appearing in the third term has an algebraic value. It is negative in case of a concave front leading to the decrease of the heat release, and positive for a convex one increasing of the heat release rate. The heat flux dependence on the rate of spread is here much more complicated than (1), since both its lateral and longitudinal parts are weighted by the mass gradient (first term) and the front curvature (last term). Firefighters are usually looking at the head of the front mainly in the spreading direction (y), neglecting the lateral part. Furthermore, it is difficult to estimate the mass load gradient and the nonstationarity of the front spread from satellite or aircraft images. In the following, the fuel is assumed to be uniformly distributed with a load density depending only on time (m(t)). The surface coverage rate is the doping parameter (p). This parameter has values smaller than unity in heterogeneous mediums and is p ¼ 1 for homogeneous ones. Obviously, during fire spread there is a mass load difference between the forward and backward positions of the front,
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yielding a gradient load within the burning area, but inside this area the rate of spread (defined only at the interface burning/non ! burning zone) vanishes. The term with the gradient ! v $ V thus disappears in this case. Equation (3) becomes
t H p mðtÞ E ¼ L L
Z
Z dwðx; tÞ
vPðy; tÞ dy vy
(6)
Because of uniform fuel distribution, integrating dw yields the average front width (w). The heat flux per unit length is obtained by deriving (6) using the same process as before:
8 H p
(7)
Here r and v are the average radius and rate of spread in the main (y) direction. This formula is much simpler than (5). It is valid only for the heading part of the front in the main spreading direction. It allows the estimation of the heat release rate in the stationary (first and last terms) as well as the non-stationary wildfire spread (second and third terms). The last term of (7) depends on the mean front curvature radius. It is positive and enhances the heat release rate in the case of convex front, and is negative for a concave one. Following Mandelbrot [6], the perimeter and surface of an inhomogeneous image are power-law increasing with its size, the power exponents define their fractal dimensions (Dp and Ds). Assuming this power-law dependence for fire pattern, size analysis of (7) reveals the heat release rate dependence on fractal dimensions of the front perimeter ðPðL; tÞfLDP Þ and burning area ðSb ðL; tÞfLDS Þ, with (Sb ¼ w P). Caldarelli et al. estimated the fractal dimensions of some Mediterranean wildfire satellite images D p¼ 1.3 and D S¼ 1.8 [7]. Other analysis of several wildfires front perimeter in north America [8] led to a fractal dimension of around Dp z 1.15. These power-law size-dependences suggest that satellite images are self-similar (do not depend on the image resolution) [6]. However, from recent wildfire simulations analysis of Baara et al. [9], the size effects of perimeter and burning surface were found not to follow the above mentioned power-law behaviors. The burning zone showed a multi-fractal size behavior, indicating actually resolution dependence of wildfire image satellites [10]. This method may allow a more accurate estimation of the heat release rate from wildfire satellite or airplane images using (7).
3. The small world network applied to wildfire spread Equation (7) is validated here using Small World Network model (SWN) wildfire simulations [5]. The model was validated for real scale fires. This model is built on an L-sized square or triangular network, whose elementary cell has a size a ¼ 1m. In heterogeneous systems, fuel occupying elementary cells are randomly uniformly distributed through the system with an occupation probability p [11]. Due to heating and combustion process, occupied cells can be in one of the three following states: healthy (or heated), burning or burned (see Fig. 2). Interaction process can take place only between a burning and a healthy or heated cell. Two kinds of long-range interaction (beyond the nearest neighbors) are considered: probabilistic (corresponding to firebrand emission and spotting process) and deterministic (corresponding to heating by flame radiation and convection inside the interaction domain) [12]. As the present study concerns only one front (the primary one),
Fig. 2. The small world network applied to wildfire spread. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
only deterministic interaction is considered. Because of field effects (e.g., wind speed or terrain slope), the interaction domain is defined as two joined half-ellipses, one in the main propagation direction, y, and the other one in the opposite direction (see Fig. 2). All occupied cells inside the interaction domain of a burning cell are heated. They can be ignited if the residence time tc of the burning cell is sufficiently large. The axes ly and lx of the ellipse in the main spread direction (l'y and lx in the opposite direction) are the ignition characteristic lengths of the flame [12]. The interaction domain is 0 circular in the absence of wind and for a flat terrain ð lx ¼ ly ¼ ly Þ. The outbreak of fire can be linear or point-like. Each combustible cell i located inside the interaction domain of a burning cell j is submitted to a radiative heat flux which depends on the distances Dxij and Dyij in each spread direction [11]:
Pij ¼
Dxij lx
2
P0 2 b r 2 Dy þ l ij
(8)
y
P0is the minimal radiation heat flux below which an exposed cell cannot be ignited [12]. The cut-off Pij ¼ P0 defines thus the interaction domain edges of a burning cell. The power-law decrease of the heat flux with a power exponent br is a generalization of the observed behavior from a flame of cylindrical shape [12,13] (as shown in Fig. 2b of [12]). Close to the flame, the power-law exponent is small so that radiation heat flux remains large, and the convection contribution is significant in the presence of wind or a terrain slope. The model includes also a weighting process based on the flame residence time tc and the time at which the accumulated energy received by the fuel reaches its ignition energy (or in other words its ignition temperature). The weighting parameter R is thus defined as the ratio of the ignition energy Eign to that acquired by the fuel located at the edge of the flame interaction domain during its residence time [11].
R¼
Eign P0 tc
(9)
Obviously, if R 1 the whole interaction domain is ignited after a time t tc. In the case R > 1, only a part 1/R of this domain is ignited after tc, the remaining cells are only heated. As the ratio R increases, the ignited area decreases. Because of the discrete nature of the network, there is a critical weight Rc for which only the nearest neighbor cells in the main spread direction (Dxi j¼ 0 and Dyi j¼ a) are ignited during tc. No ignition occurs above this critical
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weight. From (8) and (9), the critical weight is Rc ¼ ðly =aÞbr for individual contributions of burning cells. It is slightly higher for collective contributions of burning cells. 4. Results and validation As described above, the main novelty of (5) and (7) compared to Byram's formula (1) is the heat flux dependence on front curvature in addition to the non-stationarity effect. Although extensive studies were carried out investigating the rate of spread and the heat release rate, Equations (5)e(7) can be experimentally validated only qualitatively from correlations between the spread and front curvature. Actually, an application for linear flame fronts that are generated in a fire tunnel using Quercus coccifera shrubs as vegetal fuel [14] shows clearly that convex burned zones of the front (black zones) become concave in the burning zones (red zones) due to significant heat flux enhancement induced by positive curvature in these zones. While the burnt concave zones become convex due to the heat flux decrease (see Fig. 11 of [14]). Another laboratory scale experimental setup using straw as fuel shows the time evolution of the fire front with different curvatures induced by fuel heterogeneity (see Fig. 3 of [15]). The behavior of instantaneous rate of spread is also correlated to the front curvature (last term of Equation (7)). We replotted in Fig. 3 the time dependent furthermost front position (see table 3 and Fig. 5 of [15]). This figure shows 4 different spread phases of the front. In the first one (until 188s) slight curvatures induced by the fuel heterogeneity are shifted by the same process as that observed in Ref. [14], but the front remains globally linear inducing a rate of spread of about 0.57 m/s. The front at 188s shows a larger convex curvature in its center. This convexity seems to induce an enhancement of the rate of spread up to 0.89 m/s for the two next fronts (second phase). The front shape becomes thus concave at 231s, which decreases the heat release rate, in agreement with (7), and then the rate of spread is reduced to 0.38 m/s (third phase). The front at 253s shows a large convex curvature at its left side, which appears to induce a significant increase of the rate of spread to 1.09 m/s (fourth phase). Another curvature induced strong increase of the rate of spread was observed in two junction line fires (corresponding to strongly convex fronts) observed in Canberra fire in 2003, and realized in laboratory scale experiments by Viegas et al. [16]. There is evidence that the strong increase of the fire spread rate observed in the symmetry axis is
225
induced by the positive curvature effect of (7). Therefore, the measurement errors in the above described experiments do not allow quantitative validation of (5) and (7). This validation is done by numerical simulations, where fire spread is simulated using SWN model (described in Section 3). A square system of 4ha is considered with basic cells of size a ¼ 1m. Two kinds of fire outbreaks are considered: A line fire (a line at the edge is ignited) and point-like outbreak (the center of an edge is ignited). As mentioned above, the quantity m(t) used in (1) is the effective mass load, i.e. the mass without the char. The residence time of the flame is here tc ¼ 30s. In fact, during the combustion process, the mass load was found to decrease as et=t [17,18], with t the characteristic chemical time scale of the global mass loss process (flaming, embers etc.). Assuming the characteristic time comparable to the flame residence time (t z tc), the rate of mass loss ðm_ ¼ m=tc Þ is assumed in the present simulations to be constant during tc at the first order approximation. For a basic cell (1m 1m), a constant heat release rate of ðH m_ ¼ 100kW=m2 Þ is considered with flames of height h ¼ 10m. The total heat release rate is pro_ For inhoportional to the number N of burning cells ðN H mÞ. mogeneous systems (with a fuel doping p < 1), a statistical averaging on 100 configurations is considered. Without loss of generality, only stationary spread is considered (second a third terms of (7) vanish). 4.1. Linear ignition In the case of linear fronts the curvature radius is expected to diverge (r ¼ ∞). The fourth term of (7) should thus vanish. Furthermore, the stationary spread reduces (7) to its first term. As the front perimeter P ¼ L and its width w ¼ v tc are constant, the heat flux is conserved. Equation (7) is becomes:
_ tc p v fðL; tÞ ¼ H m
(10)
The heat flux conservation is clearly observed in Fig. 4a, in the stationary regime (t > tc), both in the presence or absence of wind speed. This is due to the constant amount of fuel available for consumption by the fire front (L ly). The quantity f=ðH m_ tc Þ is plotted in Fig. 4b as a function of the rate of spread for different doping values p (simulation data is obtained by varying the wind speed from zero to 60 km/h). Extreme wildfire spread conditions of fuel load, doping and flame length (10 m) are considered in simulation. This explains the high values of the rate of spread obtained from simulations compared to typical ones. Similar conditions were observed in savanna fires with rates of spread close to 2 m/s [19]. An excellent linear fit of data appears with R2 > 0.999, confirming the linear dependence of the flux on the rate of spread in (10). The line slope appears slightly lower than the expected doping values from (10). This is due to the weak concave curvature of the front caused by the edge effects and the fuel heterogeneity as shown in Fig. 4c. Slightly concave parts of the front introduce a negative contribution of the fourth term of (7) to the heat flux. Byram's law (1) remains thus applicable for line and even slightly curved fronts, but the proportionality constant is lower. In the case of non-spreading fire, the Equation (7) is also reduced to its first term, and the heat release rate depends on the rate of mass loss f ¼ H m_ w p (w being the width of the nonspreading fire). 4.2. Point ignition
Fig. 3. Front position (inverse method) vs. time at different propagation phases (from Fig. 3 and Fig. 5 of [15]).
The slight difference observed below with respect to the predicted values of the heat release rate is a signature of curvature effects even with linear ignition outbreak. In order to examine the
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By using the above mentioned constant mass loss rate m_ ¼ m=tc , Equation (11) becomes
_ Hm L _ tc p Hm L q v jrj þ v2 tc q ¼ L jr j
fðL; tÞ ¼ N
(12)
Fig. 4. Linear outbreak: a) f=ðH m_ tc Þ vs. time for p ¼ 075, b) f=ðH m_ tc Þ vs. rate of spread for p ¼ 0.65, 0.70, 0.75, c) fire pattern after 38s of spread for a wind speed of 14 m/s and a doping p ¼ 0.75.
front curvature effect on thermal heat release rate, a point-like fire ignition is considered, where ignition is initiated from the middle of one side of the system. In order to avoid statistical errors and to reduce complexity, homogeneous systems (p ¼ 1) are considered. In a first step a front spread is considered on a flat terrain without wind. The front is thus a half circle (its concave part). In a second step a wind speed of 2 m/s is introduced in simulations so that the front becomes elliptical. In both cases, only a small concave part of the front is examined i.e., within an angle q in the main spreading direction. In order to avoid the effect of lateral spread in (5), the angle q is as small as 0.1 radian. The curvature radius is thus the distance of the foremost burning cell to the outbreak point position, and is related to the perimeter by P ¼ jrj q. During fire spread the curvature radius increases with time. The rate of increase defines thus the rate of spread. In case of stationary spread, the rate of spread is constant and (7) becomes:
fðL; tÞ ¼
Hp L
_ tc v jrj þ m v2 tc q qm
L
The negative sign (), in the second term is due to the front concavity (the curvature v2 P=vy2 is negative). For convex front the sign is positive leading thus a larger heat release. The front curvature induces clearly in (12) a non-linear dependence on the rate of spread as well as a heat flux dependence on the inverse radius (not shown in Byram's law). As the spread is stationary, the heat release rate is influenced only by the curvature radius r. Obviously from (12) the first term (linear term) is dominant for asymptotically large radius, whereas the terms with 1=jrj becomes dominant for small radius. The simulated heat release rate increases linearly with time (see Fig. 5) both in the presence or absence of wind. This nonconservation of the heat flux is easily explained by the increase of the fuel available for ignition by fire front. The decreasing region of the heat flux in Fig. 5 corresponds to fronts reaching the other side of the system, where the burning area decreases. The heat release rate dependence on the front radius is shown in Fig. 6a, as a double logarithmic scale, for no wind no slope case of spread (where the front is circular). For large values of radius jrj, the heat release rate increases nearly linearly with jrj (the power-law fit varies as jrj1:1 ) where the first term of (12) is dominant. The second term of (12) is significant for small values of the radius. The heat flux difference Df with the fit for large values of radius is plotted in Fig. 6b as a function of jrj. Four regions appear in this figure. The first one, at the beginning of the spread corresponds to the non-stationary regime where the rate of spread increases before reaching a constant value where collective contribution of different burning cells occurs (see the inset of Fig. 6a). In the second region the stationary spread is reached but the front radius remains small (this region corresponds to the contribution of the second term of (12)). In the third region the front radius becomes large, and the heat flux increases linearly with radius (as shown in Fig. 6a). In the last region the front reaches the edge of the system. We are then concerned with the second region, where the flux difference Df is expected to decrease with the radius as
jrj
(11)
Fig. 5. The quantity f=ðH m_ tc Þ vs. time for point-like outbreak with, and without wind speed.
N. Zekri et al. / International Journal of Thermal Sciences 110 (2016) 222e228
Fig. 6. a) Heat release rate vs. r for a point-like outbreak with no wind and no slope. The line is the power-law fit for large values of r. The inset is the same plot for small radius in the linear scale. b) Df vs. the radius. The line is the power-law fit for small radius and stationary spread.
227
Fig. 7. Same figure as 4, with a wind speed of 14 m/s.
_ Hm L _ tc p Hm L q Ud jrj þ U2d tc q ¼ L jrj
fðL; tÞ ¼ N .
_ p v2 t2c jrj DfðL; tÞ ¼ H m
(13)
In the stationary regime of the spread, and for small values of jrj, the heat flux difference Df decreases as a power-law (see the solid line of Fig. 6b) with a power exponent (0.84 ± 0.11) not far from the expected power exponent 1, confirming (13) within the statistical errors. Now let us consider fire spread with a wind speed of 2 m/s. The front becomes elliptical in this case. The elliptical shape of the front can be obtained also by the terrain slope. As observed for the spread with circular fronts (no wind and no slope), the heat flux behaves as a power-law with an exponent close to unity (1.10 ± 0.01) asymptotically for large values of the radius (see Fig. 7a), as predicted by the first term of (12). The difference Df (defined above) in the stationary region for small radius as a power-law (see the solid line of Fig. 7b) with an exponent close to unity within statistical errors (0.83 ± 0.09) confirming (13). Therefore, Equation (12) is validated and the role of curvature on the heat release rate clearly highlighted. The concave shape of the front induces a negative contribution to the heat release rate, which is smaller than the fit of large values of r. It is expected that a spreading fronts of a convex shape induce a larger heat flux, and thus enhances the rate of fire spread as observed for junction fires [16]. The behavior of Figs. 6 and 7) occurs also for heterogeneously distributed fuel (p < 1), but the estimation of the radius induces additional statistical errors. In order to make (12) predictive, a recent result of Khelloufi et al. [20] connecting the rate of spread v to the wind speed U is used:
vfUd
(15)
5. Conclusion A new form of Byram's law was formulated analytically to reflect the geometric aspects of the fire front (concavity, convexity) both for stationary and non-stationary fire spread. It includes dependence on both curvature radius and a quadratic dependence of the heat released flux on the rate of spread in the main and lateral directions. This formulation was validated qualitatively by some fire spread experiments. It was also quantitatively validated by fire spread simulations using the small world network model. Two particular ignition configurations of the fire were considered: the linear outbreak (straight front) and point-like outbreak (curved front). The heat release rate is decreased for concave fronts and increased for convex ones. This allows firefighters to pay attention to some hazardous fire zones. This formulation makes it also possible to study the size effects of the heat release rate from image analysis by determining the fractal dimension of the perimeter and the surface fire. This allows accurate estimation of the front heat flux only from its satellite or aircraft images. This formulation becomes more predictive if the heat released is expressed in terms of the wind speed instead of the rate of spread. Quantitative experimental validation of this formula is under study. Further simulations are in course in order to reveal particular front geometries involving extreme fire behavior.
(14)
The exponent d depends on radiation view factor dependence on the distance to the flame (the exponent br in Equation (5) of [12]). It is inversely proportional to br if the front is discontinuous (individual burning trees), and linearly increasing with br and tends to 1 for continuous fronts (collective contribution of flames to radiation heat transfer). Equation (12) becomes thus:
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Generalized Byram's formula for arbitrary fire front geometries Nouredine Zekri a, *, Omar Harrouz a, Ahmed Kaiss b, Jean-Pierre Clerc b, Xavier Domingos Viegas c Universit e des Sciences et Technologie d’Oran, LEPM, BP 1505 El Mnaouer, Oran, Algeria Aix Marseille Universit e, CNRS, IUSTI UMR 7343, 13453, Marseille, France c ADAI/CEIF, University of Coimbra, Rua Luis Reis dos Santos, Coimbra, 3030-788, Portugal a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 November 2015 Received in revised form 19 June 2016 Accepted 13 July 2016 Available online 21 July 2016
The expression proposed a half century ago by Byram allowing the estimation of wild land fire front heat flux was extensively used in fire research and by firefighters. However, this formula does not account for wild land fuel heterogeneity and local weather changes. Its validity is limited to stationary spreads, and fails in describing the behavior of curved fronts and non-spreading fires. A new formula is derived to generalize heat flux estimation to any fire front with arbitrary geometry. Heat flux is found to depend on the front local radius as well as its convexity/concavity. The front curvature induces also a quadratic dependence on the rate of spread. The derived equation is validated by numerical simulations using the small world network model for line and curved fronts. Extensions of generalized Byram formula including fractal dimension of the front and wind speed effect are also proposed. © 2016 Elsevier Masson SAS. All rights reserved.
Keywords: Byram law Heat transfer Fire front
1. Introduction Millions hectares of vegetation are burned each year because of wildland fires [1]. Control methods to be implemented against forest fires depend on their intensities which are divided into several classes [2]. In the late 50s, Byram proposed an empirical equation to estimate the intensity of a fire front [3]. This law is now widely used both by scientists for laboratory or field experiments, and by operational fire managers. It indicates that the heat flux f (kW/m) delivered per meter of fire front is related to the consumed fuel load m (kg/m2), the combustion heat or enthalpy H (kJ/kg) and the rate of spread v (m/s) by
f¼mHv
(1)
The heat of combustion (enthalpy) H used here is a proportionality constant assuming a linear dependence of the heat released rate on the fuel load. In case of a nonlinear dependence, the mass load m appears with a nonlinear exponent in (1). This relationship, valid for stationary propagation, has two major disadvantages: firstly it is not predictive, as the rate of spread cannot be estimated beforehand, and secondly it does not account for the
* Corresponding author. E-mail address: [email protected] (N. Zekri). http://dx.doi.org/10.1016/j.ijthermalsci.2016.07.006 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.
fuel heterogeneity (wild land-urban interfaces, highways, fuel breaks etc.), non-stationary spreading fronts, as well as nonspreading fires (v ¼ 0). The rate of spread strongly fluctuates from a zone to another one in heterogeneous fuel beds, and depends on the combustion region. This strongly decreases the accuracy of the fire intensity estimation, which moves from a risk class to another. In 1976, Tangren [4] noticed that (1) was introduced for a line front and that the rate of spread is in fact the ratio of the front width to fire residence time (v ¼ w/tc, w being the front thickness). He raised the problem of the spread direction (the main or the lateral direction). A curved geometry of the front could influence significantly fire intensity, and may be felt differently by operational fire managers. A reformulation of (1) is thus necessary to make it predictive and to account for other important parameters involved in the spread such as a curved and non-stationary spreading front. In this paper, a new formula of fire intensity is derived for various geometrical aspects of the front. The estimated intensity from this new formula is compared to fire spread simulations using the small world network model [5]. Such simulations were realized under various ignition outbreaks and wind conditions (circular or elliptical fronts for point like outbreaks and a line front for line outbreaks). Its behaviour is also compared to experimental data. Further improvements of the derived expression introduce the fractal dimensions of the front perimeter and area as well as the
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wind speed instead of the rate of spread. 2. Derivation of Byram formula The intensity per unit length expressed by Byram's formula (1) is derived from the heat released from the unit length perimeter of the front. This energy depends on the front width, its perimeter, the fuel load and the heat of combustion. The idea in this paper is to estimate the heat released from the front by the use of satellite images whose resolution analysis (size effect) allows the dependence of this energy on the fractal dimension of burning area of the fire front or a portion of it in which important changes of curvature or fuel load may occur. Time resolution of images is here assumed to be sufficient to determine the front rate of spread, necessary for the heat flux estimation. Satellite images currently available do not have sufficient time resolution (for example image rates of less than one per minute and a space resolution of 20 m are necessary for a front spread rate of 0.3 m/s). The price of satellite images being still high, the use of aerial images (aircrafts and UAVs) would solve this problem. Although the formula we derive here can be used in the general case, it will be validated in stationary propagation situations, which overcomes this problem of time resolution. Let us consider an L-sized square heterogeneous fuel system, with a fire front spreading in y direction (Fig. 1). The system heterogeneity may be geometrical (i.e. randomly distributed fuel through the system) and/or a fuel heterogeneity (fuel of different nature). Assuming that the fuel is of the same kind, the enthalpy (H) is constant. The energy released per unit length in the lateral (x) direction is given by:
EðtÞ H ¼ ∬ mðx; y; tÞ Pðx; y; tÞdxdy L L
As char cannot contribute to fire spread, the consumed mass load m(x,y,t) used in (1) is the effective fuel mass load per unit surface (i.e., the mass load without char and other unburned fuel). The function p(x,y,t) appearing in (2) defines the probability distribution to find a surface element dx dy of fuel burning at time t centered at position (x,y). Using the fire front surface element dP dxdy) of width dw(x,t) and perimeter dP(y,t) (dS ¼ dwdP ¼ dw dx dy (Fig. 1), Equation (2) can be rewritten as
E H vwðx; tÞ vPðy; tÞ ¼ ∬ mðx; y; tÞ dxdy L L vx vy
Fig. 1. Fire spread representation at a given time with unburned fuel (green) and fire front (red). Burned area is represented in black. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(3)
The integral Equation (3) includes a product of 3 functions of time and space. Deriving (3) with respect to time yields the heat flux per unit length released by the system, f ¼ 1L dEðtÞ (the system dt size L being fixed) reads
1 dE L dt H dmðx; y; tÞ vwðx; tÞ vPðy; tÞ ∬ dxdy ¼ L dt vx vy d vwðx; tÞ vPðy; tÞ þ mðx; y; tÞ dxdy dt vx vy
fðL; tÞ ¼
(4)
Time derivatives in (4) involve partial derivatives ! d ¼ v þ vx v þ vy v ¼ v þ ! v $ V . Thus (4) becomes vt vy vt vt vx vt dt
fðL; tÞ ¼
H L
! _ ∬ mðx; y; tÞ þ ! v $ V mðx; y; tÞ dS þ ∬ mðx; y; tÞdS_
þ ∬ mðx; y; tÞ (2)
223
vx vPðy; tÞ vwðx; tÞ vy þ vx rx vy ry
!
! dxdy (5)
Here m_ ¼ vm=vt and S_ ¼ vS=vt, the surface element being vP dS ¼ dP dw ¼ vw vx vy dxdy. The third term of (5) corresponds to the ! ! part v $ V of the time partial derivative. The gradient part of the time partial derivative yields the second derivatives of the front width and perimeter, defining the radius of front curvature in x and y directions ðv2 P=vy2 ¼ 1=ry Þ and ðv2 w=vx2 ¼ 1=rx Þ). Equation (5) is much more complicated than the initial Byram's law (1). In the first term of this equation appears the rate of mass loss m_ (describing the local fuel combustion), and its spatial variation. The second terms of this equation reveals the dependence of the heat release rate on the non-stationary behavior of the burning area (where the burning surface is time varying). The geometrical aspects of the front (the lateral (rx) and longitudinal (ry) curvature radius appear in the third term. Note here that the radius appearing in the third term has an algebraic value. It is negative in case of a concave front leading to the decrease of the heat release, and positive for a convex one increasing of the heat release rate. The heat flux dependence on the rate of spread is here much more complicated than (1), since both its lateral and longitudinal parts are weighted by the mass gradient (first term) and the front curvature (last term). Firefighters are usually looking at the head of the front mainly in the spreading direction (y), neglecting the lateral part. Furthermore, it is difficult to estimate the mass load gradient and the nonstationarity of the front spread from satellite or aircraft images. In the following, the fuel is assumed to be uniformly distributed with a load density depending only on time (m(t)). The surface coverage rate is the doping parameter (p). This parameter has values smaller than unity in heterogeneous mediums and is p ¼ 1 for homogeneous ones. Obviously, during fire spread there is a mass load difference between the forward and backward positions of the front,
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yielding a gradient load within the burning area, but inside this area the rate of spread (defined only at the interface burning/non ! burning zone) vanishes. The term with the gradient ! v $ V thus disappears in this case. Equation (3) becomes
t H p mðtÞ E ¼ L L
Z
Z dwðx; tÞ
vPðy; tÞ dy vy
(6)
Because of uniform fuel distribution, integrating dw yields the average front width (w). The heat flux per unit length is obtained by deriving (6) using the same process as before:
8 H p
(7)
Here r and v are the average radius and rate of spread in the main (y) direction. This formula is much simpler than (5). It is valid only for the heading part of the front in the main spreading direction. It allows the estimation of the heat release rate in the stationary (first and last terms) as well as the non-stationary wildfire spread (second and third terms). The last term of (7) depends on the mean front curvature radius. It is positive and enhances the heat release rate in the case of convex front, and is negative for a concave one. Following Mandelbrot [6], the perimeter and surface of an inhomogeneous image are power-law increasing with its size, the power exponents define their fractal dimensions (Dp and Ds). Assuming this power-law dependence for fire pattern, size analysis of (7) reveals the heat release rate dependence on fractal dimensions of the front perimeter ðPðL; tÞfLDP Þ and burning area ðSb ðL; tÞfLDS Þ, with (Sb ¼ w P). Caldarelli et al. estimated the fractal dimensions of some Mediterranean wildfire satellite images D p¼ 1.3 and D S¼ 1.8 [7]. Other analysis of several wildfires front perimeter in north America [8] led to a fractal dimension of around Dp z 1.15. These power-law size-dependences suggest that satellite images are self-similar (do not depend on the image resolution) [6]. However, from recent wildfire simulations analysis of Baara et al. [9], the size effects of perimeter and burning surface were found not to follow the above mentioned power-law behaviors. The burning zone showed a multi-fractal size behavior, indicating actually resolution dependence of wildfire image satellites [10]. This method may allow a more accurate estimation of the heat release rate from wildfire satellite or airplane images using (7).
3. The small world network applied to wildfire spread Equation (7) is validated here using Small World Network model (SWN) wildfire simulations [5]. The model was validated for real scale fires. This model is built on an L-sized square or triangular network, whose elementary cell has a size a ¼ 1m. In heterogeneous systems, fuel occupying elementary cells are randomly uniformly distributed through the system with an occupation probability p [11]. Due to heating and combustion process, occupied cells can be in one of the three following states: healthy (or heated), burning or burned (see Fig. 2). Interaction process can take place only between a burning and a healthy or heated cell. Two kinds of long-range interaction (beyond the nearest neighbors) are considered: probabilistic (corresponding to firebrand emission and spotting process) and deterministic (corresponding to heating by flame radiation and convection inside the interaction domain) [12]. As the present study concerns only one front (the primary one),
Fig. 2. The small world network applied to wildfire spread. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
only deterministic interaction is considered. Because of field effects (e.g., wind speed or terrain slope), the interaction domain is defined as two joined half-ellipses, one in the main propagation direction, y, and the other one in the opposite direction (see Fig. 2). All occupied cells inside the interaction domain of a burning cell are heated. They can be ignited if the residence time tc of the burning cell is sufficiently large. The axes ly and lx of the ellipse in the main spread direction (l'y and lx in the opposite direction) are the ignition characteristic lengths of the flame [12]. The interaction domain is 0 circular in the absence of wind and for a flat terrain ð lx ¼ ly ¼ ly Þ. The outbreak of fire can be linear or point-like. Each combustible cell i located inside the interaction domain of a burning cell j is submitted to a radiative heat flux which depends on the distances Dxij and Dyij in each spread direction [11]:
Pij ¼
Dxij lx
2
P0 2 b r 2 Dy þ l ij
(8)
y
P0is the minimal radiation heat flux below which an exposed cell cannot be ignited [12]. The cut-off Pij ¼ P0 defines thus the interaction domain edges of a burning cell. The power-law decrease of the heat flux with a power exponent br is a generalization of the observed behavior from a flame of cylindrical shape [12,13] (as shown in Fig. 2b of [12]). Close to the flame, the power-law exponent is small so that radiation heat flux remains large, and the convection contribution is significant in the presence of wind or a terrain slope. The model includes also a weighting process based on the flame residence time tc and the time at which the accumulated energy received by the fuel reaches its ignition energy (or in other words its ignition temperature). The weighting parameter R is thus defined as the ratio of the ignition energy Eign to that acquired by the fuel located at the edge of the flame interaction domain during its residence time [11].
R¼
Eign P0 tc
(9)
Obviously, if R 1 the whole interaction domain is ignited after a time t tc. In the case R > 1, only a part 1/R of this domain is ignited after tc, the remaining cells are only heated. As the ratio R increases, the ignited area decreases. Because of the discrete nature of the network, there is a critical weight Rc for which only the nearest neighbor cells in the main spread direction (Dxi j¼ 0 and Dyi j¼ a) are ignited during tc. No ignition occurs above this critical
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weight. From (8) and (9), the critical weight is Rc ¼ ðly =aÞbr for individual contributions of burning cells. It is slightly higher for collective contributions of burning cells. 4. Results and validation As described above, the main novelty of (5) and (7) compared to Byram's formula (1) is the heat flux dependence on front curvature in addition to the non-stationarity effect. Although extensive studies were carried out investigating the rate of spread and the heat release rate, Equations (5)e(7) can be experimentally validated only qualitatively from correlations between the spread and front curvature. Actually, an application for linear flame fronts that are generated in a fire tunnel using Quercus coccifera shrubs as vegetal fuel [14] shows clearly that convex burned zones of the front (black zones) become concave in the burning zones (red zones) due to significant heat flux enhancement induced by positive curvature in these zones. While the burnt concave zones become convex due to the heat flux decrease (see Fig. 11 of [14]). Another laboratory scale experimental setup using straw as fuel shows the time evolution of the fire front with different curvatures induced by fuel heterogeneity (see Fig. 3 of [15]). The behavior of instantaneous rate of spread is also correlated to the front curvature (last term of Equation (7)). We replotted in Fig. 3 the time dependent furthermost front position (see table 3 and Fig. 5 of [15]). This figure shows 4 different spread phases of the front. In the first one (until 188s) slight curvatures induced by the fuel heterogeneity are shifted by the same process as that observed in Ref. [14], but the front remains globally linear inducing a rate of spread of about 0.57 m/s. The front at 188s shows a larger convex curvature in its center. This convexity seems to induce an enhancement of the rate of spread up to 0.89 m/s for the two next fronts (second phase). The front shape becomes thus concave at 231s, which decreases the heat release rate, in agreement with (7), and then the rate of spread is reduced to 0.38 m/s (third phase). The front at 253s shows a large convex curvature at its left side, which appears to induce a significant increase of the rate of spread to 1.09 m/s (fourth phase). Another curvature induced strong increase of the rate of spread was observed in two junction line fires (corresponding to strongly convex fronts) observed in Canberra fire in 2003, and realized in laboratory scale experiments by Viegas et al. [16]. There is evidence that the strong increase of the fire spread rate observed in the symmetry axis is
225
induced by the positive curvature effect of (7). Therefore, the measurement errors in the above described experiments do not allow quantitative validation of (5) and (7). This validation is done by numerical simulations, where fire spread is simulated using SWN model (described in Section 3). A square system of 4ha is considered with basic cells of size a ¼ 1m. Two kinds of fire outbreaks are considered: A line fire (a line at the edge is ignited) and point-like outbreak (the center of an edge is ignited). As mentioned above, the quantity m(t) used in (1) is the effective mass load, i.e. the mass without the char. The residence time of the flame is here tc ¼ 30s. In fact, during the combustion process, the mass load was found to decrease as et=t [17,18], with t the characteristic chemical time scale of the global mass loss process (flaming, embers etc.). Assuming the characteristic time comparable to the flame residence time (t z tc), the rate of mass loss ðm_ ¼ m=tc Þ is assumed in the present simulations to be constant during tc at the first order approximation. For a basic cell (1m 1m), a constant heat release rate of ðH m_ ¼ 100kW=m2 Þ is considered with flames of height h ¼ 10m. The total heat release rate is pro_ For inhoportional to the number N of burning cells ðN H mÞ. mogeneous systems (with a fuel doping p < 1), a statistical averaging on 100 configurations is considered. Without loss of generality, only stationary spread is considered (second a third terms of (7) vanish). 4.1. Linear ignition In the case of linear fronts the curvature radius is expected to diverge (r ¼ ∞). The fourth term of (7) should thus vanish. Furthermore, the stationary spread reduces (7) to its first term. As the front perimeter P ¼ L and its width w ¼ v tc are constant, the heat flux is conserved. Equation (7) is becomes:
_ tc p v fðL; tÞ ¼ H m
(10)
The heat flux conservation is clearly observed in Fig. 4a, in the stationary regime (t > tc), both in the presence or absence of wind speed. This is due to the constant amount of fuel available for consumption by the fire front (L ly). The quantity f=ðH m_ tc Þ is plotted in Fig. 4b as a function of the rate of spread for different doping values p (simulation data is obtained by varying the wind speed from zero to 60 km/h). Extreme wildfire spread conditions of fuel load, doping and flame length (10 m) are considered in simulation. This explains the high values of the rate of spread obtained from simulations compared to typical ones. Similar conditions were observed in savanna fires with rates of spread close to 2 m/s [19]. An excellent linear fit of data appears with R2 > 0.999, confirming the linear dependence of the flux on the rate of spread in (10). The line slope appears slightly lower than the expected doping values from (10). This is due to the weak concave curvature of the front caused by the edge effects and the fuel heterogeneity as shown in Fig. 4c. Slightly concave parts of the front introduce a negative contribution of the fourth term of (7) to the heat flux. Byram's law (1) remains thus applicable for line and even slightly curved fronts, but the proportionality constant is lower. In the case of non-spreading fire, the Equation (7) is also reduced to its first term, and the heat release rate depends on the rate of mass loss f ¼ H m_ w p (w being the width of the nonspreading fire). 4.2. Point ignition
Fig. 3. Front position (inverse method) vs. time at different propagation phases (from Fig. 3 and Fig. 5 of [15]).
The slight difference observed below with respect to the predicted values of the heat release rate is a signature of curvature effects even with linear ignition outbreak. In order to examine the
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By using the above mentioned constant mass loss rate m_ ¼ m=tc , Equation (11) becomes
_ Hm L _ tc p Hm L q v jrj þ v2 tc q ¼ L jr j
fðL; tÞ ¼ N
(12)
Fig. 4. Linear outbreak: a) f=ðH m_ tc Þ vs. time for p ¼ 075, b) f=ðH m_ tc Þ vs. rate of spread for p ¼ 0.65, 0.70, 0.75, c) fire pattern after 38s of spread for a wind speed of 14 m/s and a doping p ¼ 0.75.
front curvature effect on thermal heat release rate, a point-like fire ignition is considered, where ignition is initiated from the middle of one side of the system. In order to avoid statistical errors and to reduce complexity, homogeneous systems (p ¼ 1) are considered. In a first step a front spread is considered on a flat terrain without wind. The front is thus a half circle (its concave part). In a second step a wind speed of 2 m/s is introduced in simulations so that the front becomes elliptical. In both cases, only a small concave part of the front is examined i.e., within an angle q in the main spreading direction. In order to avoid the effect of lateral spread in (5), the angle q is as small as 0.1 radian. The curvature radius is thus the distance of the foremost burning cell to the outbreak point position, and is related to the perimeter by P ¼ jrj q. During fire spread the curvature radius increases with time. The rate of increase defines thus the rate of spread. In case of stationary spread, the rate of spread is constant and (7) becomes:
fðL; tÞ ¼
Hp L
_ tc v jrj þ m v2 tc q qm
L
The negative sign (), in the second term is due to the front concavity (the curvature v2 P=vy2 is negative). For convex front the sign is positive leading thus a larger heat release. The front curvature induces clearly in (12) a non-linear dependence on the rate of spread as well as a heat flux dependence on the inverse radius (not shown in Byram's law). As the spread is stationary, the heat release rate is influenced only by the curvature radius r. Obviously from (12) the first term (linear term) is dominant for asymptotically large radius, whereas the terms with 1=jrj becomes dominant for small radius. The simulated heat release rate increases linearly with time (see Fig. 5) both in the presence or absence of wind. This nonconservation of the heat flux is easily explained by the increase of the fuel available for ignition by fire front. The decreasing region of the heat flux in Fig. 5 corresponds to fronts reaching the other side of the system, where the burning area decreases. The heat release rate dependence on the front radius is shown in Fig. 6a, as a double logarithmic scale, for no wind no slope case of spread (where the front is circular). For large values of radius jrj, the heat release rate increases nearly linearly with jrj (the power-law fit varies as jrj1:1 ) where the first term of (12) is dominant. The second term of (12) is significant for small values of the radius. The heat flux difference Df with the fit for large values of radius is plotted in Fig. 6b as a function of jrj. Four regions appear in this figure. The first one, at the beginning of the spread corresponds to the non-stationary regime where the rate of spread increases before reaching a constant value where collective contribution of different burning cells occurs (see the inset of Fig. 6a). In the second region the stationary spread is reached but the front radius remains small (this region corresponds to the contribution of the second term of (12)). In the third region the front radius becomes large, and the heat flux increases linearly with radius (as shown in Fig. 6a). In the last region the front reaches the edge of the system. We are then concerned with the second region, where the flux difference Df is expected to decrease with the radius as
jrj
(11)
Fig. 5. The quantity f=ðH m_ tc Þ vs. time for point-like outbreak with, and without wind speed.
N. Zekri et al. / International Journal of Thermal Sciences 110 (2016) 222e228
Fig. 6. a) Heat release rate vs. r for a point-like outbreak with no wind and no slope. The line is the power-law fit for large values of r. The inset is the same plot for small radius in the linear scale. b) Df vs. the radius. The line is the power-law fit for small radius and stationary spread.
227
Fig. 7. Same figure as 4, with a wind speed of 14 m/s.
_ Hm L _ tc p Hm L q Ud jrj þ U2d tc q ¼ L jrj
fðL; tÞ ¼ N .
_ p v2 t2c jrj DfðL; tÞ ¼ H m
(13)
In the stationary regime of the spread, and for small values of jrj, the heat flux difference Df decreases as a power-law (see the solid line of Fig. 6b) with a power exponent (0.84 ± 0.11) not far from the expected power exponent 1, confirming (13) within the statistical errors. Now let us consider fire spread with a wind speed of 2 m/s. The front becomes elliptical in this case. The elliptical shape of the front can be obtained also by the terrain slope. As observed for the spread with circular fronts (no wind and no slope), the heat flux behaves as a power-law with an exponent close to unity (1.10 ± 0.01) asymptotically for large values of the radius (see Fig. 7a), as predicted by the first term of (12). The difference Df (defined above) in the stationary region for small radius as a power-law (see the solid line of Fig. 7b) with an exponent close to unity within statistical errors (0.83 ± 0.09) confirming (13). Therefore, Equation (12) is validated and the role of curvature on the heat release rate clearly highlighted. The concave shape of the front induces a negative contribution to the heat release rate, which is smaller than the fit of large values of r. It is expected that a spreading fronts of a convex shape induce a larger heat flux, and thus enhances the rate of fire spread as observed for junction fires [16]. The behavior of Figs. 6 and 7) occurs also for heterogeneously distributed fuel (p < 1), but the estimation of the radius induces additional statistical errors. In order to make (12) predictive, a recent result of Khelloufi et al. [20] connecting the rate of spread v to the wind speed U is used:
vfUd
(15)
5. Conclusion A new form of Byram's law was formulated analytically to reflect the geometric aspects of the fire front (concavity, convexity) both for stationary and non-stationary fire spread. It includes dependence on both curvature radius and a quadratic dependence of the heat released flux on the rate of spread in the main and lateral directions. This formulation was validated qualitatively by some fire spread experiments. It was also quantitatively validated by fire spread simulations using the small world network model. Two particular ignition configurations of the fire were considered: the linear outbreak (straight front) and point-like outbreak (curved front). The heat release rate is decreased for concave fronts and increased for convex ones. This allows firefighters to pay attention to some hazardous fire zones. This formulation makes it also possible to study the size effects of the heat release rate from image analysis by determining the fractal dimension of the perimeter and the surface fire. This allows accurate estimation of the front heat flux only from its satellite or aircraft images. This formulation becomes more predictive if the heat released is expressed in terms of the wind speed instead of the rate of spread. Quantitative experimental validation of this formula is under study. Further simulations are in course in order to reveal particular front geometries involving extreme fire behavior.
(14)
The exponent d depends on radiation view factor dependence on the distance to the flame (the exponent br in Equation (5) of [12]). It is inversely proportional to br if the front is discontinuous (individual burning trees), and linearly increasing with br and tends to 1 for continuous fronts (collective contribution of flames to radiation heat transfer). Equation (12) becomes thus:
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