Open Journal of Forestry
2012. Vol.2, No.1, 17-22
Published Online January 2012 in SciRes (http://www.SciRP.org/journal/ojf) http://dx.doi.org/10.4236/ojf.2012.21003
Copyright © 2012 SciRes. 17
Mathematical Modeling of Crown Forest Fire Spread
Department of Mathematics and Natural Sciences, Belovo Branch of Kemerovo State University, Belovo, Russia
Received August 20th, 2011; revised November 9th, 2011; accepted November 17th, 2011
Mathematical model of forest fire was based on an analysis of known experimental data and using con-
cept and methods from reactive media mechanics. In this paper the assignment and theoretical investiga-
tions of the problems of crown forest fire spread in windy condition were carried out. In this context, a
study—mathematical modeling—of the conditions of forest fire spreading that would make it possible to
obtain a detailed picture of the change in the temperature and component concentration fields with time,
and determine as well as the limiting condition of fire propagation in forest with fire break.
Keywords: Forest Fire; Mathematical Model; Turbulence; Ignition; Fire Spread; Control Volume;
A great deal of work has been done on the theoretical prob-
lem of crown forest fire initiation. Crown fires are initiated by
convective and radiative heat transfer from surface fires. How-
ever, convection is the main heat transfer mechanism (Van Wag-
ner, 1977). The theory proposed by Van Wagner (1977) de-
pends on three simple crown properties: crown base height,
bulk density of forest combustible materials and moisture con-
tent of forest fuel. Also, crown fire initiation and hazard have
be e n st udied and mo deled in deta ils later (Alexan der, 1998 ); Va n
Wagner (1989); Xanthopoulos (1990); Rothermel (1991); Cruz
and others (2002); Albini and others (1195); Scott and
Reinhardt (2001). The more complete discussion of the prob-
lem of crown forest fires is provided by coworkers at Tomsk
University (Grishin, 1997); Grishin et al. (1998); Perminov
(1995, 1998). In p art ic ul ar, a ma th ema ti ca l mo del of for est fi re s
was obtained by Grishin (1997) based on an analysis of known
and original experimental data; Konev (1977), and using con-
cepts and methods from reactive media mechanics. The physi-
cal two-phase models used by Morvan & Dupuy (2001, 2004)
may be considered as a continuation and extension of the for-
mulation proposed in (Grishin 1997). This study gives a two
dimensional averaged mathematical setting and method of nu-
merical solution of a problem of a forest fire spread. The bound-
ary-value problem is solved numerically using the method of
splitting according to physical processes. It was based on nu-
merical solution of two dimensional Reynolds equations for the
description of turbulent flow taking into account for diffusion
equations chemical components and equations of energy con-
servation for gaseous and condensed phases, volume of fraction
of condensed phase (dry organic substance, moisture, con-
densed pyrolysis products, mineral part of forest fuel).
It is assumed that the forest during a forest fire can be mod-
eled as 1) a multi-phase, multi storied, spatially heterogeneous me-
dium; 2) in the fire zone the forest is a porous-dispersed, two-tem-
perature, single-velocity, reactive medium; 3) the forest canopy
is supposed to be non- deformed medium (trunk s, large branches,
small twigs and needles), which affects only the magnitude of
the force of resistance in the equation of conservation of mo-
mentum in the gas phase, i.e., the medium is assumed to be
quasi-solid (almost non-deformable during wind gusts); 4) let
there be a so-called “ventilated” forest massif, in which the vo-
lume of fractions of condensed forest fuel phases, consisting of
dry organic matter, water in liquid state, solid pyrolysis prod-
ucts, and ash, can be neglected compared to the volume fraction
of gas pha se (component s of air and gaseou s pyroly sis pr oducts ) ;
5) the flow has a developed turbulent nature and molecular trans-
fer is neglected; 6) gaseous phase density doesn’t depend on the
pressure because of the low velocities of the flow in compare-
son with the velocity of the sound. Let the point x1, x2, x3 = 0 is
situated at the centre of the surface forest fire source at the
height of the roughness level, axis 0x1 directed parallel to the
Earth’s surface to the right in the direction of the unperturbed
wind speed, axis 0x2 directed perpendicular to 0x1 and axis 0x3
directed upward (Figure 1).
Because of the horizontal sizes of forest massif more than
height of forest—h, system of equations of general mathematic-
cal model of forest fire (Grishin, 1997) was integrated between
the limits from height of the roughness level—0 to h. Besides,
Basic scheme of forest fire initiat i o n and propagation.
—average value of
. The problem formulated above is
reduced to a solution of the following system of equation:
vp vvsc vg
The system of Equations (1)-(7) must be solved taking into
account the in itial and bound a ry conditions:
tv vv TTccTT
22 2 2 2
0, ,, ;
33 3 33
vv vc T
Here and above d
dt is the symbol of the total (substantial)
v is the coefficient of phase exchange;
of gas-dispersed phase, t is time; vi—the velocity components;
T, TS—temperatures of gas and solid phases, UR—density of
radiation energy, k—coefficient of radiation attenuation,
P—pressure; cp—constant pressure specific heat of the gas
i—specific heat, density and volume of fraction
of condensed phase (1—dry organic substance, 2—moisture, 3—
condensed pyrolysis products, 4—mineral part of forest fuel),
Ri—the mass rates of chemical reactions, qi—hermal effects of
chemical reactions; kg, kS—radiation absorption coefficients for
gas and condensed phases; Te—the ambient temperature; c
mass concentrations of
-component of gas-dispersed medium,
= 1, 2, 3, where 1 corresponds to the density of oxygen,
2—to carbon monoxide CO, 3—to carbon dioxide and inert
components of air, 4, 5—soot and ash; R—universal gas con-
, MC, and M molecular mass of
-components of the
gas phase, carbon and air mixture; g is the gravity acceleration;
cd is an empirical coefficient of the resistance of the vegetation,
s is the specific surface of the forest fuel in the given forest
stratum. In system of Equations (1)-(7) are introduced the next
Upper indexes “+” and “–” designate values of functions at
x3 = h and x3 = 0 correspondingly. It is assumed that heat and
mass exchange of fire front and boundary layer of atmosphere
are governed by Newton law and written using the formulas:
To define source terms which characterize inflow (outflow of
mass) in a volume unit of the gas-dispersed phase, the follow-
ing formulae were used for the rate of formulation of the gas-
dispersed mixture, outflow of oxygen m
, changing carbon
g—mass fraction of gas combustible products of py-
5—empirical constants. Reaction rates of these
various contributions (pyrolysis, evaporation, combustion of coke
and volatile combustible products of pyrolysis) are approxi-
mated by Arrhenius laws whose parameters (pre-exponential
constant ki and activation energy Ei) are evaluated using data
for mathematical models (Grishin, 1997; Perminov, 1995).
exp ,exp ,
8 Copyright © 2012 SciRes.
The initial values for volume of fractions of condensed phases
are determined using the expressions:
dv Wd a
where d—bulk density for surface layer,
ashes of forest fuel, W—forest fuel moisture content. It is sup-
proximation for radiation
flux density were used for a mathematical description of radia-
tion transport during forest fires.
sed that the optical properties of a medium are independent
of radiation wavelength (the assumption that the medium is
“grey”), and the so-called diffusion ap
To close the system (1)-(7), the components of the tensor of
turbulent stresses, and the turbulent heat and mass fluxes are
determined using the local-equilibrium model of turbulence
(Grishin, 1997). The system of Equations (1)-(7) contains terms
associated with turbulent diffusion, thermal conduction, and
convection, and needs to be closed. The components of the ten-
sor of turbulent stresses vw
, as well as the turbulent fluxes
of heat and mass
a are written in terms of the
gradients of the average flow properties using the formulas:
t, Dt are the coefficients of turbut viscosity, ther-
mal conductivity, and diffusion, respective; Prt, Sct are the
turbulent Prandtl and Schmidt numbers, which were assumed to
be equal to 1. In dimensional form, the coefficient of dynamic
turbulent viscosity is determined using local equilibrium model
of turbulence (Grishin, 1997). The system of Equations (1)-(7)
must be solved taking into account the initial and boundary
obtained as a result of splitting was then integrated. A discrete
analog wae method
using the SIMPLE likThe accu-
conditionsThe thermodynamic, thermophysical and structural
characteristics correspond to the forest fuels in the canopy of a
different type of forest; such as, pine forest (Grishin, 1997).
Numerical Methods and Results
The boundary-value problem (1)-(13) is solved numerically
using the method of splitting according to physical processes
(Perminov, 1995). In the first stage, the hydrodynamic pattern
of flow and distribution of scalar functions was calculated. The
system of ordinary differential equations of chemical kine
s obtained by means of the control volum
e algorithm (Patankar, 1981).
cy of the program was checked by the method of inserted
analy tical solutions. Analytical expressio ns for the unkno wn func -
tions were substituted in (1)-(7) and the closure of the equations
were calculated. This was then treated as the source in each
equation. Next, with the aid of the algorithm described above,
the values of the functions used were inferred with an accuracy
of not less than 1%. The effect of the dimensions of the control
volumes on the solution was studied by diminishing them. The
time step was selected automatically. Fields of temperature, ve-
lo city , component ma ss fraction s, and vol u me fractions of p hase s
were obtained numerically. The distribution of basic functions
shows that the process of crown forest fire initiation goes through
the next stages. The first stage is related to increasing maximum
temperature in the fire source. At this process stage the fire source
a thermal wind is formed a zone of heated forest fire pyrolysis
products which are mixed with air, float up and penetrate into
the crowns of trees. As a result, forest fuels in the tree crowns
ar e heated, moisture evaporates and gaseous and dispe rsed pyro-
lysis products are generated. Ignition of gaseous pyrolysis pro-
ducts of the ground cover occurs at the next stage, and that of
gaseous pyrolysis products in the forest canopy occurs at the
la st stage. As a result of heating of forest fuel elements of crown,
moisture evaporates, and pyrolysis occurs accompanied by the
release of gaseous products, which then ignite and burn away in
the forest canopy. At the moment of ignition the gas combusti-
ble products of pyrolysis burns away, and the concentration of
oxygen is rapidly reduced. The temperatures of both phases reach
a maximu m value at the point of ignition. The ign ition processes
is of a gas-phase nature. Note also that the transfer of energy
from the fire source takes place due to radiation; the value of
radiation heat flux density is small compared to that of the con-
vective heat flux. At Ve 0, the wind field in the forest canopy
interacts with the gas-jet obstacle that forms from the forest fire
source and from the ignited forest canopy and burn away in the
forest canopy. Figures 2-5 present the distribution of tempe-
TTTT T (1 - 2, 2 - 2.6, 3 - 3, 4 - 3.5, 5 -
4) for gas phase, oxygen 1
c (1 - 0.1, 2 - 0. 5 , 3 - 0. 6, 4 - 0. 7 , 5 -
0.8, 6 - 0.9), volatile combustible products of pyrolysis 2
concentrations (1 - 1, 2 - 0.1, 3 - 0.05, 4 - 0.01)
, temperature of condensed phase
TT TTT (1 - 2, 2 - 2.6, 3 - 3, 4 - 3.5, 5 - 4)
for wind velocity Ve = 10 m/s at h = 10 m: 1) t = 3 sec., 2) t =
10 sec, 3) t = 18 sec., 4) t = 24 sec.
Field of isotherms of the forest fire spre ad (gas phase).
The distribution of oxy g en 1
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20 Copyright © 2012 SciRes.
The distribution of 2
ield of isotherms of the forest fire spread ( solid phase).
al levels are
meters (Figure 8) the fire con-
We can note that the isotherms and lines of equ
oved in the forest canopy and deformed by the action of wind.
Similarly, the fields of component concentrations are deformed.
It is concluded that the forest fire begins to spread. Mathemati-
cal model and the result of the calculation give an opportunity
to consider forest fire spread for different wind velocity.
Figures 6(a)-(d) present the distribution of temperatu
s phase, concentration of oxygen and volatile combustible pro-
ducts of pyrolysis c2 concentrations and temperature of con-
densed phase for wind velocity Ve = 5 m/s at h = 10 m: 1) t = 3
sec., 2) t = 10 sec., 3) t = 18 sec., 4) t = 20 sec., 5) t = 31 sec.,
6) t = 40 sec. The results reported in Figure 6 show the de-
crease of the wind induces a decrease of the rate of fire spread.
One of the objectives of this paper could be to develop mod-
ing means to reduce forest fire hazard in forest or near towns.
In this paper it present s numerical results to study forest fire pro-
pagation through firebreak. This problem was considered by
Zverev (1985) in one dimensional mathematical model approach.
Figures 7 and 8 (Figure 8(b) is a continuation of Figure
a)) present the forest fire front movement using distributions
of temperature at different instants of time for two sizes of fire-
breaks (4.5 and 4 meters). The fire break is situated in the mid-
dle of domain (x1 =100 m). In the first case the fire could not
spread through this fire break.
If the fire break reduces to 4
ue to spread but the isotherm (isotherm 5) of forest fire is
decreased after overcoming of fire break. In the Figure 9. The
Fields of isotherm s of gas (a) and solid phase (d), isolines of oxygen (b) and gas products of pyrolysis (c).
Field of isotherms. Fire break equals 4.5 m.
Field of isotherms. Fire break equals 4 m.
The influence of wind velocity at the size of fire break.
dependence of critical fire break value for different wind velo-
cities is presented. Of course the size of safe distance depends
not only of wind velocity, but type and quality of forest com-
bustible materials, its moisture, height of trees and others con-
ditions. This model allows to study an influence all these main
The results of calculation give an opportunity to evaluate
critical condition of the forest fire spread, which allows apply-
ing the given model for preventing fires. It overestimates the
velocity of crown forest fire spread that depends on crown pro-
, moisture content of forest fuel and etc.
a detailed picture of the change
t concentration fields with time,
and determine as well as the influence of different conditions
n. The results obtained agree
with the laws of physics and experimental data (Konev, 1977;
Grishin, 1997). From an analysis of calculations and experi-
mental data it was found that for the cases in question the
minimum total incendiary heat pulse is 2600 kJ/m2 (Grishin,
1997). Calculations demonstrated that the value of the radiant
heat flux for both problems is considerably less than the con-
vective one, therefore radiation has a weak effect on local and
integral characteristics of the problem discoursed above.
Albini, F. A., et al. (1995). Modeling ignition and burning rate of large
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in the temperature and componen
on the crown forest fire initiatio
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