Kemerovo State University(Russia)

Mathematical Modeling of Large Forest Fires

Valeriy A. Perminovpva@belovo.kemsu.ru, p_valer@mail.ru

2. Determination of the reaction and thermophysical properties of the medium, the transfer coefficients and structural parameters of the medium, and deduction of the basic system of equations with corresponding additional (boundary and initial) conditions.

Mathematical modeling using the deterministic approach contains the following relatively independent stages:

1. Physical analysis of the phenomenon of forest fire spread; definition of the medium type (biogeocenosis), and creation of a physical model of the phenomenon.

3. Selection of a method of numerical solution of the problem, and derivation of differentialequations approximating the basic system of equations.

5. Testing to see how well the derived results comply with the real phenomenon, their physical interpretation; development of new technical suggestions for ways of fighting forest fires.

4. Programming, test check of program; evaluation of the accuracy of the difference scheme; numerical solution of the system of differential equations.

3) models for predicting the characteristics of flow, heat and mass transfer at the forest fires due to

the natural and technogeneous catastrophes .

2) models for predicting the rate of spread of fire and the contours of forest fires;

1) models for initiation of forest fire;

Mathematical models of forest fires

The basic assumptions adopted during the deduction of equations, and boundary and initial conditions:

1) the forest represents a multi-phase, multistoried, spatially heterogeneous medium;

2) in the fire zone the forest is a porous-dispersed, multi-phase, two-temperature, single-velocity, reactive medium;

3) the forest canopy is supposed to be non - deformed medium, affects only the magnitude of the force of resistance in the equation of conservation of momentum in the gas phase;

4) the energetics and physico-chemical processes at the forest fire front are took into account;

5) the length of the free path of a photon during a forest fire is much smaller than the characteristic dimensions of the forest biogeocenosis l* and , where l* is the horizontal dimension of the forest massif, and is the average height of the forest fuel layer;

hh

6) let there be a so-called “ventilated” forest massif, in which the volume of fractions of condensed forest fuel phases, consisting of dry organic matter, water in liquid state, solid pyrolysis products, and ash, can be neglected compared to the volume fraction of gas phase (components of air and gaseous pyrolysis products);

7) the flow has a developed turbulent nature and molecular transfer is neglected;

8) gaseous phase density doesn’t depend on the pressure because of the low velocities of the flow in comparison with the velocity of the sound;

9) diffusion approximation is applied to describe energy transfer by radiation.

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v m j it x

( ) | | ;ii j d i i i

i j

dv Pv v sc v v g mv

dt x x

45 5( ) ( ) ( 4 );p p j v S g R

j

dTc c v T q R T T k cU T

dt x

5( ) , 1,5;jj

dcv c R mc

dt x

Mathematical model of forest fires

4 44 4 0, ;3

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j j

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x k x

4

43 3 2 2

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i

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t

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exp , exp ,

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x

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, ,j p t j tj j

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Mathematical modeling of forest fire initiation

(axisymmetrical formulation)

1( ) ( ) ;n

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t r r z

2 2

1 1( ) ( ) ( ) ( ) ( )

;

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d

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t r r z r r r z

sc v v w

2

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( ) ;

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t r r z z r r

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1 1( ) ( ) ( ) ( )

( ) ( 4 ) ( );

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p g R v s

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c w T k cU T q R T Tz

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, 1, 2;

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3 3

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r r k r z k z

44

3 3 2 21

( 4 ) ( );si pi i s R s V s

i

Tc q R q R k cU T T T

t

31 2 41 1 2 2 3 1 3 4

1

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t t t M t

3 3

1 1

1, e

cc P RT

M

The initial and boundary conditions

1 2 10 : 0, 0, , , , ;e e s e iet v v T T c c T T

0: 0, 0, 0, 0, 0;Rc Uw Tr v

r r r r

: 0, 0, 0, 0, 0;3 2

Re R

c Uv w T c cr r U

r r r r k r

40 0

1: 0, , 0, 0, (4 )

3 2R

s R

c UT cz z v w h m T cU

z z k z

2

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2

0 02

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;

( ) exp ,

e e

e e

t rT T T t t

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: 0, 0, 0, 0, 0.3 2

Re R

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z z z z k z

3 2 1 22 2 2 222

1

2 12 ( ) .

3 PrTT

c w v v w v w gl r v

c z r r r z z r r T z

The local – equilibrium model of turbulence.

/(1 2.5 / ).T dl zk z c s h

a

M *

b c

Fig.1 a,b,c. Relationships of dimensionless temperatures , concentrations and volume fractions in the lower boundary of the forest canopy at r=0 :

)/3,/2,/1(

,)/,2,1(,)300,/2,/1(

11333222111

121

ecce

eeesse

c

cccccbkTTTTTTTa

a b

Fig.2 a,b. Relationships between and on the axis of symmetry and dimensionless height .

T Ts, c c c e / 1

z z h / 1

The distribution of temperatures and velocities for axisymmetrical case.

Figure 4. Isotherms of gas (I) and condensed (II) phases at different instants of time t=2.0 s (a), t=2.2 s (b), t=2.4 s (c). Arabic numerals 1,2,3 corresponded to the isotherms 2., 3. and 5 respectively.

, ST T

I III II

 

I II

 I II

Figure 5. Lines of equal relative concentrations of oxygen and carbon monoxide at the same moments as in Figure 2. Arabic numerals correspond to -I: , II: .

eccc 1/ 1 11 1, 2 0.5,c c 13 0.4c ;1.04 1 c

4.03,5.02,11 222 ccc

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j

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dt x x

45 5( ) ( ) ( 4 );p p j v S g R

j

dTc c v T q R T T k cU T

dt x

5( ) , 1,5;jj

dcv c R mc

dt x

4 44 4 0, ;3

RR S S g g S

j j

UckcU k T k T k k k

x k x

44

3 3 2 21

( 4 ) ( );Si pi i S R S V S

i

Tc q R q R k cU T T T

t

31 21 1 2 2 3 1 3

1

44 3 55

1

, , ,

,

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C

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t t t M

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t

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( ) exp [( ) / ] ,

( ) exp [( ) / ] , ,

e e f

e e f

tT T T x x t t

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: 0, 0, 0, 0, 0.3 2

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x x x x k x

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j j

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x x

1 22 2 2 2

2 1 2 1 2 1 2

1 2 2 1 1 2 2

22 ,

3 Prtt

v v v v v v gl

x x x x x x T x

Vectorial fields of velocity and distributions of temperatures

1 2,2 2.5, 3 3;

/ , 300 , 1.3.e e

T T T

T T T T K t

1 1.1, 2 1.5, 3 2, 4 2.5, 5 3;

/ , 1.1S

S S e

T

T T T t

Mass concentrations of oxygen and combustible products of pyrolysis

Mathematical Modeling of Forest Fire Spread Initiation

Because of the horizontal sizes of forest massif more than height of forest – h

hdxh

3

0

- average value of .

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j

( ) | | ( ) / , (0, ) ;ii j d i i i i i

j j

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dt x x

5 5( ) ( ) ( ) / ;p p j v s T Tj

dTc c v T q R T T q q h

dt x

5( ) ( ) / , 1,3;jj

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dt x

4( 4 ) ( ) / 0;3

RR S R R

j j

Uck cU T q q h

x k x

44

3 3 2 21

( 4 ) ( );Si pi i R S V S

i

Tc q R q R k cU T T T

t

31 2 41 1 2 2 3 1 3 4

1

, , , 0;CC

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t t t M t

;),(,,1 2

4

11

3

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11 2 3 51 3 5 52 1 5 53

1 2

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M M

;,,0,0:0 21 ee ccTTvvt

;,,0,: 21101 eee ccTTvVvxx

;0,0,0,0:111

2

1

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x

T

x

c

x

v

x

vxx e

;0,0,0,0:222

2

2

1022

x

T

x

c

x

v

x

vxx

.0,0,0,0:222

2

2

122

x

T

x

c

x

v

x

vxx e

Figure1. Isotherms of the forest fire for t=5 s and 10 s:

.54,5.33,.6.22,5.11 TTTT

Figure 2. Isotherms of solid phase at t=5 s and 10 s: .54,5.33,6.22,5.11 sTsTsTsT

1c

.9.015

,85.014,75.013,6.012,5.011

c

cccc

2c

.4.024,12.023,03.022,01.021 cccc

Figure The distribution of at t=5 and 10 s:Figure The distribution of oxygen at t=5 and 10 s:

Mathematical Modeling of Large Forest Fire Initiation

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1 – x, y = 0; 2 – x = -10 km, y=0; 3 – x= - 15 km, y = 0.

/ ; / , 300 .e s s e eT T T T T T T K 1 – x, y = 0; 2 – x = -10 km, y=0; 3 – x= - 15 km, y = 0;

. 23.01,1/ eсecсc

1 – x, y = 0; 2 – x = -10 km, y=0; 3 – x= - 15 km, y = 0.

ce /,/_____ 222111

1 – t=7.0 sec, 2 – t=5 sec, 3 t=4.3 sec. 1 – t=7.0 sec, 2 – t=5 sec, 3 t=4.3 sec.

1 – t=7.0 sec, 2 – t=5 sec, 3 t=4.3 sec.

t v w T T c c T Te e s e i i e 0 0 0: , , , , , ;

r vw

r

T

r

c

r

U

rR 0 0 0 0 0 0: , , , , ;

r rv

r

w

r

c

r

T

r

c

k

U

r

cUe

RR : , , , , ;

0 0 0 03 2

0

40 : 0, 0, 0, 0, (4 );

3 2(2 )R

s R

c Uv w T cz z T cU

z z z z k z

z zv

z

w

z

T

z

c

z

z hc

k

U

z

cU q r z

e

RR R

: , , , ,

: ( , ) .

0 0 0 0

3 222

The distribution of basic functions in the region of forest ignition .