Mathematical Modeling of Crown Forest Fire Spread

Transcription

1 Open Journal of Forestry 0. Vol., No., 7- Published Online January 0 in Sies ( Mathematial Modeling of Crown Forest Fire Spread Valeriy Perminov Department of Mathematis and Natural Sienes, Belovo Branh of Kemerovo State University, Belovo, ussia p_valer@mail.ru eeived August 0 th, 0; revised November 9 th, 0; aepted November 7 th, 0 Mathematial model of forest fire was based on an analysis of known experimental data and using onept and methods from reative media mehanis. In this paper the assignment and theoretial investigations of the problems of rown forest fire spread in windy ondition were arried out. In this ontext, a study mathematial modeling of the onditions of forest fire spreading that would make it possible to obtain a detailed piture of the hange in the temperature and omponent onentration fields with time, and determine as well as the limiting ondition of fire propagation in forest with fire break. Keywords: Forest Fire; Mathematial Model; Turbulene; Ignition; Fire Spread; Control Volume; Disrete Analogue Introdution A great deal of work has been done on the theoretial problem of rown forest fire initiation. Crown fires are initiated by onvetive and radiative heat transfer from surfae fires. However, onvetion is the main heat transfer mehanism (Van Wagner, 977). The theory proposed by Van Wagner (977) depends on three simple rown properties: rown base height, bulk density of forest ombustible materials and moisture ontent of forest fuel. Also, rown fire initiation and hazard have been studied and modeled in details later (Alexander, 998); Van Wagner (989); Xanthopoulos (990); othermel (99); Cruz and others (00); Albini and others (95); Sott and einhardt (00). The more omplete disussion of the problem of rown forest fires is provided by oworkers at Tomsk University (Grishin, 997); Grishin et al. (998); Perminov (995, 998). In partiular, a mathematial model of forest fires was obtained by Grishin (997) based on an analysis of known and original experimental data; Konev (977), and using onepts and methods from reative media mehanis. The physial two-phase models used by Morvan & Dupuy (00, 004) may be onsidered as a ontinuation and extension of the formulation proposed in (Grishin 997). This study gives a two dimensional averaged mathematial setting and method of numerial solution of a problem of a forest fire spread. The boundary-value problem is solved numerially using the method of splitting aording to physial proesses. It was based on numerial solution of two dimensional eynolds equations for the desription of turbulent flow taking into aount for diffusion equations hemial omponents and equations of energy onservation for gaseous and ondensed phases, volume of fration of ondensed phase (dry organi substane, moisture, ondensed pyrolysis produts, mineral part of forest fuel). Mathematial Model It is assumed that the forest during a forest fire an be modeled as ) a multi-phase, multistoried, spatially heterogeneous medium; ) in the fire zone the forest is a porous-dispersed, two-temperature, single-veloity, reative medium; ) the forest anopy is supposed to be non-deformed medium (trunks, large branhes, small twigs and needles), whih affets only the magnitude of the fore of resistane in the equation of onservation of momentum 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-alled ventilated forest massif, in whih the volume of frations of ondensed forest fuel phases, onsisting of dry organi matter, water in liquid state, solid pyrolysis produts, and ash, an be negleted ompared to the volume fration of gas phase (omponents of air and gaseous pyrolysis produts); 5) the flow has a developed turbulent nature and moleular transfer is negleted; 6) gaseous phase density doesn t depend on the pressure beause of the low veloities of the flow in ompareson with the veloity of the sound. Let the point x, x, x = 0 is situated at the entre of the surfae forest fire soure at the height of the roughness level, axis 0x direted parallel to the Earth s surfae to the right in the diretion of the unperturbed wind speed, axis 0x direted perpendiular to 0x and axis 0x direted upward (Figure ). Beause of the horizontal sizes of forest massif more than height of forest h, system of equations of general mathematial model of forest fire (Grishin, 997) was integrated between the limits from height of the roughness level 0 to h. Besides, suppose that Figure. Basi sheme of forest fire initiation and propagation. Copyright 0 Sies. 7

2 V. PEMINOV h dx 0 h average value of. The problem formulated above is redued to a solution of the following system of equation: v Qm m h,,,; () t x dvi p vv i sdvi v gi dt x x Qv h, i,,; i i i 5 5 q dt p pvt q v T Ts qt T dt x () h ; () d v 5 Q ( J J)/ h,, 5; (4) dt x U 4 ku 4 TS q q h0; (5) x k x T S ipii qq k U TS V T S ; i t,, t t M a, C 4 C 4 t M t M a a, pe T, v v, v, v, g 0, 0, g a a a T (6) The system of Equations ()-(7) must be solved taking into aount the initial and boundary onditions: t : v 0, v 0, v 0, T T,, T T, ; e a ae s e ie v x x v V v T T e : e, 0, 0, e, a ae, x U U 0; k x v v v T x x 0, e : 0, 0, 0, 0, x x x x x U U 0; k x x x v v v T 0, 0: 0, 0, 0, 0, x x x x x U U 0; k x v v v T xe : 0, 0, 0, 0, 0 x x x x x x U U 0; k x U x 0 : v 0, v 0, 0, U 0, x k x v v0, T Tg, x, x v 0, T T, x, x ; e, (7) (8) (9) (0) () () v v v T x xe : 0, 0, 0, 0, 0, x x x x x () U U 0. k x Here and above d is the symbol of the total (substantial) dt derivative; v is the oeffiient of phase exhange; density of gas-dispersed phase, t is time; v i the veloity omponents; T, T S temperatures of gas and solid phases, U density of radiation energy, k oeffiient of radiation attenuation, P pressure; p onstant pressure speifi heat of the gas phase, pi, i, i speifi heat, density and volume of fration of ondensed phase ( dry organi substane, moisture, ondensed pyrolysis produts, 4 mineral part of forest fuel), i the mass rates of hemial reations, q i hermal effets of hemial reations; k g, k S radiation absorption oeffiients for gas and ondensed phases; T e the ambient temperature; mass onentrations of -omponent of gas-dispersed medium, index =,,, where orresponds to the density of oxygen, to arbon monoxide CO, to arbon dioxide and inert omponents of air, 4, 5 soot and ash; universal gas onstant; M, M C, and M moleular mass of -omponents of the gas phase, arbon and air mixture; g is the gravity aeleration; d is an empirial oeffiient of the resistane of the vegetation, s is the speifi surfae of the forest fuel in the given forest stratum. In system of Equations ()-(7) are introdued the next designations: m v, i v iv, J v, JT vt Upper indexes + and designate values of funtions at x = h and x = 0 orrespondingly. It is assumed that heat and mass exhange of fire front and boundary layer of atmosphere are governed by Newton law and written using the formulas: T T e e q q h T T h, J J h h. To define soure terms whih haraterize inflow (outflow of mass) in a volume unit of the gas-dispersed phase, the following formulae were used for the rate of formulation of the gasdispersed mixture m, outflow of oxygen 5, hanging arbon monoxide 5. M M Q, 5 5, M M, 0. 5 g 5 5 Here g mass fration of gas ombustible produts of pyrolysis, 4 and 5 empirial onstants. eation rates of these various ontributions (pyrolysis, evaporation, ombustion of oke and volatile ombustible produts of pyrolysis) are approximated by Arrhenius laws whose parameters (pre-exponential onstant k i and ativation energy E i ) are evaluated using data for mathematial models (Grishin, 997; Perminov, 995). E 0.5 E k exp, k Ts exp, Ts Ts E k s exp, Ts 0.5 M M.5 E5 5 5 M M T k M T exp. p 8 Copyright 0 Sies.

3 V. PEMINOV The initial values for volume of frations of ondensed phases are determined using the expressions: z e d v Wd a,, e e e where d bulk density for surfae layer, z oeffiient of ashes of forest fuel, W forest fuel moisture ontent. It is supposed that the optial properties of a medium are independent of radiation wavelength (the assumption that the medium is grey ), and the so-alled diffusion approximation for radiation flux density were used for a mathematial desription of radiation transport during forest fires. To lose the system ()-(7), the omponents of the tensor of turbulent stresses, and the turbulent heat and mass fluxes are determined using the loal-equilibrium model of turbulene (Grishin, 997). The system of Equations ()-(7) ontains terms assoiated with turbulent diffusion, thermal ondution, and onvetion, and needs to be losed. The omponents of the tensor of turbulent stresses vw, as well as the turbulent fluxes of heat and mass vt p, v a are written in terms of the gradients of the average flow properties using the formulas: v v i vv i t Ki, x x i T vt v D a p t, a t x x t tp Pr t, Dt t St, t K, where t, t, D t are the oeffiients of turbulent visosity, thermal ondutivity, and diffusion, respetive ly; Pr t, S t are the turbulent Prandtl and Shmidt numbers, whih were assumed to be equal to. In dimensional form, the oeffiient of dynami turbulent visosity is determined using loal equilibrium model of turbulene (Grishin, 997). The system of Equations ()-(7) must be solved taking into aount the initial and boundary onditions. The thermodynami, thermophysial and strutural harateristis orrespond to the forest fuels in the anopy of a different type of forest; suh as, pine forest (Grishin, 997). Numerial Methods and esults The boundary-value problem ()-() is solved numerially using the method of splitting aording to physial proesses (Perminov, 995). In the first stage, the hydrodynami pattern of flow and distribution of salar funtions was alulated. The system of ordinary differential equations of hemial kinetis obtained as a result of splitting was then integrated. A disrete analog was obtained by means of the ontrol volume method using the SIMPLE like algorithm (Patankar, 98). The auray of the program was heked by the method of inserted analytial solutions. Analytial expressions for the unknown funtions were substituted in ()-(7) and the losure of the equations were alulated. This was then treated as the soure in eah equation. Next, with the aid of the algorithm desribed above, the values of the funtions used were inferred with an auray of not less than %. The effet of the dimensions of the ontrol volumes on the solution was studied by diminishing them. The time step was seleted automatially. Fields of temperature, veloity, omponent mass frations, and volume frations of phases were obtained numerially. The distribution of basi funtions shows that the proess of rown forest fire initiation goes through the next stages. The first stage is related to inreasing maximum temperature in the fire soure. At this proess stage the fire soure a thermal wind is formed a zone of heated forest fire pyrolysis produts whih are mixed with air, float up and penetrate into the rowns of trees. As a result, forest fuels in the tree rowns are heated, moisture evaporates and gaseous and dispersed pyrolysis produts are generated. Ignition of gaseous pyrolysis produts of the ground over ours at the next stage, and that of gaseous pyrolysis produts in the forest anopy ours at the last stage. As a result of heating of forest fuel elements of rown, moisture evaporates, and pyrolysis ours aompanied by the release of gaseous produts, whih then ignite and burn away in the forest anopy. At the moment of ignition the gas ombustible produts of pyrolysis burns away, and the onentration of oxygen is rapidly redued. The temperatures of both phases reah a maximum value at the point of ignition. The ignition proesses is of a gas-phase nature. Note also that the transfer of energy from the fire soure takes plae due to radiation; the value of radiation heat flux density is small ompared to that of the onvetive heat flux. At V e 0, the wind field in the forest anopy interats with the gas-et obstale that forms from the forest fire soure and from the ignited forest anopy and burn away in the forest anopy. Figures -5 present the distribution of temperature T T T Te, Te 00 K ( -, -.6, -, 4 -.5, 5-4) for gas phase, oxygen ( - 0., - 0.5, - 0.6, 4-0.7, 5-0.8, 6-0.9), volatile ombustible produts of pyrolysis onentrations ( -, - 0., , 4-0.0) с, с 0., temperature of ondensed phase e e TS TS TS Te, Te 00 K ( -, -.6, -, 4 -.5, 5-4) for wind veloity Ve = 0 m/s at h = 0 m: ) t = se., ) t = 0 se, ) t = 8 se., 4) t = 4 se. Figure. Field of isotherms of the forest fire spread (gas phase). Figure. The distribution of oxygen. Copyright 0 Sies. 9

4 V. PEMINOV Figure 4. The distribution of. Figure 5. Field of isotherms of the forest fire spread (solid phase). We an note that the isotherms and lines of equal levels are moved in the forest anopy and deformed by the ation of wind. Similarly, the fields of omponent onentrations are deformed. It is onluded that the forest fire begins to spread. Mathematial model and the result of the alulation give an opportunity to onsider forest fire spread for different wind veloity. Figures 6(a)-(d) present the distribution of temperature for gas phase, onentration of oxygen and volatile ombustible produts of pyrolysis onentrations and temperature of ondensed phase for wind veloity Ve = 5 m/s at h = 0 m: ) t = se., ) t = 0 se., ) t = 8 se., 4) t = 0 se., 5) t = se., 6) t = 40 se. The results reported in Figure 6 show the derease of the wind indues a derease of the rate of fire spread. One of the obetives of this paper ould be to develop mod- means to redue forest fire hazard in forest or near towns. eling In this paper it presents numerial results to study forest fire propagation through firebreak. This problem was onsidered by Zverev (985) in one dimensional mathematial model approah. Figures 7 and 8 (Figure 8(b) is a ontinuation of Figure 8( a)) present the forest fire front movement using distributions of temperature at different instants of time for two sizes of firebreaks (4.5 and 4 meters). The fire break is situated in the middle of domain (x =00 m). In the first ase the fire ould not spread through this fire break. If the fire break redues to 4 meters (Figure 8) the fire ontinue to spread but the isotherm (isotherm 5) of forest fire is dereased after overoming of fire break. In the Figure 9. The (a) (b) () Figure 6. Fields of isotherms of gas (a) and solid phase (d), isolines of oxygen (b) and gas produts of pyrolysis (). (d) 0 Copyright 0 Sies.

5 V. PEMINOV dependene of ritial fire break value for different wind veloities is presented. Of ourse the size of safe distane depends not only of wind veloity, but type and quality of forest ombustible materials, its moisture, height of trees and others onditions. This model allows to study an influene all these main fators. Figure 7. Field of isotherms. Fire break equals 4.5 m. (a) Conlusion The results of alulation give an opportunity to evaluate ritial ondition of the forest fire spread, whih allows applying the given model for preventing fires. It overestimates the veloity of rown forest fire spread that depends on rown properties: bulk density, moisture ontent of forest fuel and et. The model proposed there gives a detailed piture of the hange in the temperature and omponent onentration fields with time, and determine as well as the influene of different onditions on the rown forest fire initiation. The results obtained agree with the laws of physis and experimental data (Konev, 977; Grishin, 997). From an analysis of alulations and experimental data it was found that for the ases in question the minimum total inendiary heat pulse is 600 kj/m (Grishin, 997). Calulations demonstrated that the value of the radiant heat flux for both problems is onsiderably less than the onvetive one, therefore radiation has a weak effet on loal and integral harateristis of the problem disoursed above. (b) Figure 8. Field of isotherms. Fire break equals 4 m. Figure 9. The influene of wind veloity at the size of fire break. EFEENCES Albini, F. A., et al. (995). Modeling ignition and burning rate of large woody natural fuels. International Journal of Wildland Fire, 5, 8-9. doi:0.07/wf Alexander, V. E. (998). Crown fire thresholds in exoti pine plantations of Australasia. Ph.D. Thesis, Aton: Australian National University. Cruz, M. G., Alexander, M. E., & Wakimoto,. H. (00). Prediting rown fire behaviour to support forest fire management dei- sion-making. In D. X. Viegas (Ed.), Proeedings of 4th international onferene on Forest fire researh/00 wildland fire safety summit. otterdam: Millpress Siene Publishers. Grishin, A. M. (997). Mathematial modeling forest fire and new methods fighting them. Tomsk: Publishing House of Tomsk University. Grishin, A. M., & Perminov, V. A. (998). Mathematial modeling of the ignition of tree rowns. Combustion, Explosion, and Shok Waves, 4, doi:0.007/bf Konev, E. V. (977). The physial foundation of vegetative materials ombustion. Novosibirsk: Nauka. Morvan, D., & Dupuy, J. L. (00). Modeling of fire spread through a forest fuel bed using a multiphase formulation. Combustion and Flame, 7, doi:0.06/s000-80(0)000-9 Morvan, D., & Dupuy, J. L. (004). Modeling the propagation of wildfire through a Mediterranean shrub using a multiphase formulation. Combustion and Flame, 8, doi:0.06/.ombustflame Patankar, S. V. (98). Numerial heat transfer and fluid flow. New York: Hemisphere Publishing Corporation. Perminov, V. A. (995). Mathematial modeling of rown and mass forest fires initiation with the allowane for the radiative Convetive heat and mass transfer and two temperatures of medium. Ph.D Thesis, Tomsk: Tomsk State University. Perminov, V. A. (998). Mathematial modeling of rown forest fire initiation. Proeedings of III International onferene on forest fire researh and 4th onferene on fire and forest meteorology. Luso. othermel,. C. (99). Crown fire analysis and interpretation. Proeedings of th international onferene on Fire and forest meteor- Copyright 0 Sies.

6 V. PEMINOV ology. Bethesda, MA: Soiety of Amerian Foresters. Sott, J. H., et al. (00). Assessing rown fire potential by linking models of surfae and rown fire behavior. USDA Forest Servie, oky Mountain Forest and ange Experiment Station. Colorado: Fort Collins. Van Wagner, C. E. (977). Conditions for the start and spread of rown fire. Canadian Journal of Forest esearh, 7, -4. doi:0.9/x Van Wagner, C. E. (989). Predition of rown fire behavior in onifer stands. Proeedings of 0th onferene on fire and forest meteorology. Ottawa, Ontario: Forestry Canada. Xanthopoulos, G. (990). Development of a wildland rown fire initiation model. Ph.D. Thesis, Missoula: University of Montana. Zverev, V. G. (985). Mathematial modeling of aerodynamis and heat and mass transfer at rown forest fire spread. Ph.D. Thesis, Tomsk: Tomsk State University. Copyright 0 Sies.