MATHEMATICAL MODELING OF LARGE FOREST FIRE INITIATION

Transcription

1 . INTODUCTION MATHEMATICAL MODELING OF LAGE FOEST FIE INITIATION Valeri A.Perinov * Belovo Branch of Keerovo State University Belovo Keerovo region ussia A large technogeneous or space catastrophe as a rule is known to accopany by the initiation of ass forest fires (Glastone (96) and Goldin (99)). In connection with the estiate of ecological and cliatic ipacts of severe fires the prediction of the process influence on forest phytosenoses and ground layer state of the atosphere is of interest. Considering that natural investigations of these probles are erely ipossible ethods of atheatical odeling are urgent. The ost coplete discussion of the proble of forest fire odeling is provided by a group of co-workers at Tosk State University. A fairly coplete bibliography of these works is given by Grishin (997). In particular general atheatical odel of forest fire based on an analysis of known experiental data and using concept and ethods fro reactive edia echanics. Within the fraework of this odel the forest and cobustion products during a fire represent a non deforable porous dispersed ediu. Based on this odel of forest fires the probles of forest fire initiation and spread are studied with due consideration for the effect of a turbulent atosphere and the actual structure of the forest biogeocenosis. In papers of Grishin (997) Grishin and Perinov (99) and Perinov (995) attention is given to questions of description of the initial stage in the developent of a assive forest fires initiated by heat radiation (for exaple fro a Tunguska celestial body fall).. THEOETICAL ANALYSIS AND PHYSICAL MATHEMATICAL MODEL It is known that in the case of entering of body in atosphere with supersonic the powerful ballistic shock wave is arose at the around stagnation point and the gas teperature has high value (Goldin (99)). As a result of this the subliation of celestial body atter is took place and teperature tension is arose. Therefore the celestial body is destroyed in Earth atosphere or its reains are falling with foration of crater. During the celestial body flying a fraction of its kinetic energy transfored into radiation and the heating of Earth surface and forest phytocenoces are took place. As a rule the sizes of celestial bodies are sall as copare with radius of Earth and thickness of overterrestrial layer it ay be considered to be a point source of radiation (Goldin (99)). It is supposed that the celestial body is destroyed as a result of explosion in Earth atosphere. Let the radiant energy source be at a height of H fro the Earth surface at the initial oent (Fig.). * Corresponding author address: Valeri A.Perinov Belovo Branch of Keerovo University Belovo Keerovo reg. 656 ussia; e-ail: pva@belovo.kesu.ru Where - the distance fro source of radiation to vegetation cover h - the height of forest assif O - epicenter of explosion - is the center of decart coordinate syste. An upper boundary zh of the forest assif is acted upon by an intensive radiant flux q (rt) which defined at the flight stage fro (Goldin(99) Grishin and Perinov(99)). q Figure. t p ISinL ( r t) I.5C H ρv S π where I V S the brightness velocity and idship section square of Tunguska fireball C H - the fraction of kinetic energy transfored into radiation; L - angle between radiative heat flux and vegetation cover ρ - density of atosphere at a height H. After explosion of celestial body (at oent tt ) the light flux is defined according to the data Glastone (96) t pp sin L ( t t) / t t < t q( r t) π exp( k(( t t)/ t )) t t.5.5 t t +.W P.W. - the distance fro the source of radiation to forest; t p - atospheric transissivity coefficient; kt/sec P - axiu value of heat radiative ipulse at oent tt ; W - weapon yield kt/sec; k - epirical coefficient. When the radiant energy reaches vegetation cover it causes heating forest fuels evaporation of oisture and subsequent theral decoposition of solid aterial with evaporating pyrolysis products liberation. The last aterial is burning in the atosphere and interacting with the oxygen of air. Since the intensities of radiant and convective fluxes to the forest canopy in x and y directions are low as copared to the z direction it enabled the proble to be treated in the quasy- one- diensional setting up. The forest canopy is considered as a hoogeneous two-teperatures reacting non-defored ediu. Teperatures of condensed (solid) T s and gaseous T

2 phases are separated out. The first includes a dry organic substance oisture condensed pyrolysis products and ineral part of forest fuels. In the gaseous phase we separate out only the coponents C necessary to describe reactions of cobustion ( - oxygen - pyrolysis cobustion products of forest fuels (CO and etc.) - the rest of coponents). The solid phase constituting forest fuels has no intrinsic velocity and its voluetric fraction as copared to the gaseous phase can be neglected in appropriate equations. adiation is the governing echanis of the energy transfer in this case. The solid phase ainly absorbs reflects and reradiates. Diffusion approxiation is used to describe the transfer in this specific continuous ediu. The syste of equations for the celestial body is dv d C x ρvst Sin g V C x ρv ST 6 Sin d CY ρ ST g + ctg Sin () z V dt S VSin d! Sin T T πρt Q π σ C x where T ρ T- - ass radius and density of celestial body C x C y - coefficients of drag and lifting t - tie - angle of trajectory inclination! - the distance along of trajectory g - constant acceleration azz - Earth radius σ - ablation coefficient Λ - heat transfer coefficient Q - celestial body specific energy of ablation. Setting the initial point of considering trajectory H6 k the height of explosion k.the initial values of velocity and ass of celestial body are set up according to the data Goldin (99). To describe convective transfer controlled by the wind and gravity in forest canopy we use eynolds equations for the description of turbulent flow taking into account diffusion equations for cheical coponents and equations of energy conservation for gaseous and condensed phases. For the objective of the present studies wind (velocity) speed was considered to be relatively not high and the energy was considered ainly to be transferred due to radiation. Since the intensities of radiant and convective fluxes to the forest canopy in horizontal directions (x and y) are low as copared to the z direction it enabled the proble to be treated in the quasy- one- diensional setting up and we supposed all paraeters are depended on t and z vertical coordinate ρ + ( ρw) " ; t () p ( ρw) + ( ρw ) + t () + ( ρw ) ρscd w ρ g; ( ρct p ) + ( ρwct p ) t ( ρcp wt ) + k( cu σt ) + () + q v( Ts T); ( ρc) + ( ρwc) t ( ρwc ) 5 ; (5) c U ( ) k( cu σ Ts ) ; k (6) Ts ρicpiϕi q q + i t (7) + kcu ( σts ) + V( T Ts); ϕ ϕ ρ ρ t t (8) ϕ ρ M ϕ ρ ; c c t M t c pe ρt ρϕ E k exp TS c M.5 E kρϕ TS exp TS E kρϕsσ cexp TS.5 cm cm.5 E5 5 Mk5 T exp. M M T The syste of equations () (9) ust be solved taking into account the following initial and boundary conditions: t : w T Te c ce Ts Te (9) ϕi ϕie; w T c z z : () c U ε ( σts cu); k ( ε)

3 w T c z h : c U c + U q ( r z ). k () Here and above z read fro the ground cover w are the velocity coponents; t is tie; U - density of radiation energy k - coefficient of forest fuel adsorption p - pressure; c p constant pressure specific heat of the gas phase c pi ρ i ϕ i specific heat density and volue of fraction of condensed phase ( dry organic substance oisture condensed pyrolysis products ineral part of forest fuel) V coefficient of heat exchange q i theral effects of cheical reactions. To define source ters which characterize inflow (outflow of ass) in a volue unit of the gas-dispersed phase the following forulae were used for the rate of forulation of the gas-dispersed ixture " outflow of oxygen 5 changing carbon onoxide 5 i the rates of cheical reactions. Thus the solution of the syste of equations ()- (8) with initial and boundary conditions (9)-() ay result in defining the of velocity teperature coponent concentrations and radiation density. The last conditions in () were obtained in P - approxiation of the spherical haronics ethod. It should be noted that this syste of equations describes processes of transfer within the entire region of the forest assif which includes the space between the underlying surface and the base of the forest canopy (for this region the coefficients " v ) the forest canopy h < z < h for which " v and the space above it for which " v. The therodynaic therophysical and structural characteristics correspond to forest fuels in the canopy of a pine forest and are given nuerically (following Grishin(997)) during the solution of the first proble. The syste of equations () (8) contains ters associated with turbulent diffusion theral conduction and convection and needs to be closed. The coponent of the tensor of turbulent stresses ρ w as well as the turbulent fluxes of heat and ass wt wc are written in ters of the gradients of the average flow properties.. NUMEICAL METHOD AND VEIFICATION OF THE MODEL The boundary-value proble () () was solved nuerically using the ethod of splitting according to physical processes. In the first stage the hydrodynaic pattern of flow and distribution of scalar functions was calculated. The syste of ordinary differential equations of cheical kinetics obtained as a result of splitting was then integrated. A discrete analog was obtained by eans of the control volue ethod using the SIMPLE algorith (Patankar (98)). The accuracy of the progra was checked by the ethod of inserted analytical solutions. Analytical expressions for the unknown functions were substituted in () (8) and the closure of the equations was calculated. This was then treated as the source in each equation. Next with the aid of the algorith described above the values of the functions used were inferred with an accuracy of not less than %. The effect of the diensions of the control volues on the solution was studied by diinishing the. The tie interval was selected autoatically.. ESULTS AND DISCUSSION The distribution of teperature of gas and condensed phases velocity coponent ass fractions and volue fractions of phases were obtained nuerically at different distances fro the source of radiation to forest and different instants of tie. The full energy of celestial body (E) is 6 J which consists of kinetic energy (K ) and energy of explosion E. A fraction of the celestial body energy transfored into radiation equals to.. Figure. x y ; x - k y; x - 5 k y ; T T / T e ; Ts Ts / Te Te K. Figure. x y ; x - k y; x - 5 k y ; c с / ce с e..

4 ellipse extended in the flight trajectory projection direction of Tunguska celestial body (Figures 5 7). As distinct fro collision catastrophes ignition contours take the for of a circuference as illustrated as the result of nuerical experients for the ignition of a hoogeneous vegetation layer by radiation fro the air nuclear explosion. Figures 5 7 present the dynaic of the developent of forest fire contours for different types of forest (pine larch and birch). Figure. x y ; x - k y; x -5k y. ϕ ϕ/ ϕ e ϕ ρ / ϕ ρc. Figures illustrate the tie dependence of diensionless teperatures of gas and condensed phases concentrations of coponents and relative volue fractions of solid phases at upper boundary zh of the forest for various distances fro the epicenter (solid curves teperature of gas phase; dash curves teperature of solid phase). Fig. (solid curves concentration of oxygen; broken curves concentration of cobustible products of pyrolysis (CO)) illustrates the distribution of concentrations of coponents of the gas phase. At the oent of ignition the CO burns away and the concentration of oxygen is rapidly reduced. The teperatures of both phases reach a axiu value at the point of ignition. The ignition processes is of a gas - phase nature i.e. initially heating of solid and gaseous phases occurs oisture is evaporated. Then decoposition process into condensed and volatile pyrolysis products starts the later being ignited at the upper boundary of the forest canopy. At the ignition zone boundary gaseous fuel products are also generated but they are not ignited because of not high enough radiant flux power. On the basis of data calculated for this proble as the ignition condition the condition was Figure 5. t7. sec t5 sec t. sec Figure 6. t7. sec t5 sec t. sec. T t t t x x* y y* z h Ќ where h - is an upper boundary height of the forest canopy and t t is the ignition tie which Ќ corresponds to the value of tie at which there is a second bending point in the teperature curve T x x ( ) * y y* z h T t. Fro the calculation results of forest canopy ignitions (Fig. - ) it is seen that three conditions are realized: the first is factual cobustion the second is so - called noral state of ignition and third is non - ignition (non - flaability). Within the fraework of the proble entioned above the sizes of the ignition zones were defined. Contours derived for collision catastrophes look like a circle arc in the neighborhood of epicenter of the explosion and take the for of the Figure 7. t7. sec t5 sec t. sec. 5. CONCLUSION A atheatical odel has been developed for the siulation of the proble on the vegetation ignition as eteorites fall down in the Earth's atosphere. The results obtained agree with the laws of physics and experiental data of Goldin (99). Thus the odel can be potentially utilized for the odeling of forest ignition by radiant energy and for the prediction of forest fire contours.

5 6. EFEENCES Glastone S. (Ed.) 96: The effects of nuclear weapons U.S. Gov t. Printing Office Washington. Goldin V.D. 99: On interpreting soe geophysical phenoena accopanying the fall of Tunguska eteorite. In Space substance and Earth. Novosibirsk: Nauka (in ussian). Grishin A.M. 997: Matheatical Modeling Forest Fire and New Methods Fighting The F.Albini (ed.) Publishing House of Tosk University Tosk (ussia). Grishin A.M. Perinov V.A. 99: On ignition of forest assifs by the action of Tunguska Meteorite Explosion Physics of cobustion and explosion 9 8-.(in ussian). Patankar S.V. 98: Nuerical Heat Transfer and Fluid Flow Heisphere Publishing Corporation New York 98. Perinov V.A. 995: Matheatical Modeling of Crown and Mass Forest Fires Initiation With the Allowance for the adiative - Convective Heat and Mass Transfer and Two Teperatures of Mediu Ph.D Thesis Tosk State University Tosk (ussia).