MATHEMATICAL MODELING OF FOREST FIRES INITIATION, SPREAD AND IMPACT ON ENVIRONMENT

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1 Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp.9-99 Got., Const. at. & Env., IN:86-990, Japan, DOI: Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp.9-99 ATHEATICAL ODELING OF FOET FIE INITIATION, PEAD AND IPACT ON ENVIONENT *Valriy Prminov and Alxandr Goudov Dpartmnt of Eology Basi afty, Tomsk Polythni Univrsity, ussia; Institut of Fundamntal sins, Kmrovo tat Univrsity, ussia *Corrsponding Author, ivd: 08 July 06, visd: 0 D, 06, Aptd: D. 06 ABTACT: A mathmatial modl of surfa and rown forst firs sprad and impat is onsidrd. A thr-dimnsional multiphas, physi basd modl is usd. Th boundary-valu problm is solvd numrially by finit volum mthod. This modl has bn applid to dsrib th pross of initiation and sprad of surfas forst firs and thir transfr into rown of forst firs. Th rsults of numrial solutions prsnt th distribution of th main funtions of th pross (th vloity fild, th tmpratur of gas and solid phas, th onntration of th oxygn, gas produt of pyrolysis and inrt omponnts, t.) ovr tim. narios modld within this study rprsnt a possibl approah to th prliminary assssmnt of risk and should b vrifid by mor dtaild omputational fluid dynami (CFD) modling Kywords: Crown Fir, urfa Fir, athmatial odling, Finit Volum thod. INTODUCTION On of th obtivs of ths studis is th improvmnt of knowldg on th fundamntal physial mhanisms that ontrol forst firs initiation and sprad. A grat dal of work has bn don on th thortial problm of how forst fir initiation. In forst, thr ar two stps for rown forst fir initiation: sprad of fir from row to rown and rown firs ar initiatd by onvtiv and radiativ hat transfr from surfa firs. Howvr, onvtion is th main hat transfr mhanism. Firstly, rown forst fir initiation has bn studid and modld by Van Wagnr []. Thr ar thr simpl rown proprtis: rown bas hight, bulk dnsity and moistur ontnt of forst ful in this thory. Also, rown fir initiation has bn studid and modld in dtail (g: Alxandr [], Van Wagnr [], Xanthopoulos, [], othrml [5,6], Van Wagnr, [7], Cruz [8], Albini [9], ott, J. H. and inhardt, E. D. [0]. Th disussion of th problm of modling forst firs is providd by a group of o-workrs at Tomsk Univrsity (Grishin [], Grishin and Prminov []. Th gnral mathmatial modl of forst firs was obtaind by Grishin [] basd on an analysis of known and original xprimntal data [,], and using onpts and mthods from rativ mdia mhanis. Th physial two-phas modls usd in [-5] may b onsidrd as a ontinuation and xtnsion of th formulation proposd by Grishin and Prminov []. Howvr, th invstigation of rown firs has bn limitd mainly to ass studid of forst firs initiation in two dimnsional sttings and did not tak into aount spa proprtis of ths phnomna. It is assumd that th forst during a forst fir an b modld as ) a multi-phas, multistorid, spatially htrognous mdium; ) in th fir zon th forst is a porous-disprsd, two-tmpratur, singl-vloity, rativ mdium; ) th forst anopy is supposd to b nondformd mdium (trunks, larg branhs, small twigs and ndls), affts only th magnitud of th for of rsistan in th quation of onsrvation of momntum in th gas phas, i.., th mdium is assumd to b quasi-solid (almost non-dformabl during wind gusts); ) lt thr b a so-alld vntilatd forst massif, in whih th volum of frations of ondnsd forst ful phass, onsisting of dry organi mattr, watr in liquid stat, solid pyrolysis produts, and ash, an b ngltd ompard to th volum fration of gas phas (omponnts of air and gasous pyrolysis produts); 5) th flow has a dvlopd turbulnt natur and molular transfr is ngltd; 6) gasous phas dnsity dosn t dpnd on th prssur baus of th low vloitis of th flow in omparison with th vloity of th sound. Lt th oordinat rfrn point x, x, x 0 (Figur ) b situatd. PHYICAL AND ATHEATICAL ODEL 9

2 Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp.9-99 Fig.. hmati of a forst fir domain at th ntr of th domain of surfa forst fir sour at th hight of th roughnss lvl, axis 0x dirtd paralll to th Earth s surfa to th right in th dirtion of th unprturbd wind spd, axis 0x dirtd prpndiular to 0x and axis 0x dirtd upward. Using th rsults of [- ] and known xprimntal data [] w hav th following suffiintly gnral quations, whih dfin th stat of th mdium in th forst fir zon, writtn using tnsor notation. ρ + ( ρv ) m,,,, i,,; () t x dvi P ρ + ( ρv i v ) ρsd vi v dt xi x () ρgi mv i; dt ρ p ( ρ v T p ) + q55 α v ( T Ts ) + dt x () + k g ( U σt ) dα ρ ( ρv α ) + 5α dt x m α, α,; U ku + k σt + x k x + k gσt 0, k k g + k, T ρi pi ϕi k s ( U σt ) + i t + q q + αv ( T T ); ϕ ϕ ρ, ρ, t t ϕ C ϕ ρ α C, ρ 0; t t α α α, p ρt, v ( v, v, v α α Th systm of quations () (7) must b solvd taking into aount th initial and boundary onditions t 0 : v 0, v 0, v 0, T T, α Ts Ts, ϕ i ϕ i ; x x : v V, v 0, v 0, T T, α α, U с + U 0, k x ). α () (5) (6) (7), (8) (9) x v v v α x : 0, 0, 0, 0, x x x x T U 0, + U 0 ; x k x v v v α x x0: 0, 0, 0, 0, x x x x T U 0, + U 0 ; x k x v v v α x x : 0, 0, 0, 0, x x x x T U 0, + U 0. x k x U x 0 : v 0, v α 0, 0, U 0, x k x + v v0, T Tg, x, x v 0, T T, x >, x > ; v v v α x x : 0, 0, 0, 0, x x x x T U 0, + U 0. x k x (0) () () () () d Hr and abov is th symbol of th total (substantial) dt drivativ: d + v + v + v, α v is th offiint of dt t x x x phas xhang; ρ - dnsity of gas disprsd phas, t is tim; v i - th vloity omponnts; T, T, - tmpraturs of gas and solid phass, U - dnsity of radiation nrgy, k - offiint of radiation attnuation, P - prssur; p onstant prssur spifi hat of th gas phas, pi, ρ i, ϕ i spifi hat, dnsity and volum of fration of ondnsd phas ( dry organi substan, moistur, ondnsd pyrolysis produts, minral part of forst ful), i th mass rats of hmial rations, q i thrmal ffts of hmial rations; k g, k - radiation absorption offiints for gas and ondnsd phass; T - th ambint tmpratur; α - mass onntrations of α - omponnt of gas - disprsd mdium, indx α,,, whr orrsponds to th dnsity of oxygn, - to arbon monoxid CO, - to arbon dioxid and inrt omponnts of air; univrsal gas onstant; α, C, and molular mass of α -omponnts of th gas phas, arbon and air mixtur; g is th gravity alration; d is an mpirial offiint of th rsistan of th vgtation, s is th spifi surfa of th forst ful in th givn forst stratum. To dfin sour trms whih haratriz inflow (outflow of mass) in a volum unit of th gas-disprsd phas, th following formula wr usd for th rat of formulation of th gas-disprsd mixtur m, outflow of oxygn 5 and 9

3 Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp.9-99 hanging arbon monoxid 5. m ( α ) + +, 5 ' 5 ν g ( α с ) 5, 5 5 E kρϕ xp( T ), E kρϕσ xp( ), T 0.5 k ρ ϕ T 0.5 0, E xp( T.5 E5 5 k5 T xp( ). T Th initial valus for volum of frations of ondnsd phass ar dtrmind using th xprssions: d( ν z ) Wd αϕ.,, ρ ϕ ϕ ϕ ρ ρ ρ whr d -bulk dnsity of forst ombustibl matrials, ν z offiint of ashs of forst ful, W forst ful moistur ontnt. It is supposd that th optial proprtis of a mdium ar indpndnt of radiation wavlngth (th assumption that th mdium is gry ), and th so-alld diffusion approximation for radiation flux dnsity wr usd for a mathmatial dsription of radiation transport during forst firs. To los th systm () (7), th omponnts of th tnsor of turbulnt strsss, and th turbulnt hat and mass fluxs ar dtrmind using th loal-quilibrium modl of turbuln (Grishin, []. Prosss of transfr within th ntir rgion of th forst massif, whih inluds th spa btwn th undrlying surfa and th bas of th forst anopy, th forst anopy and th spa abov it, whil th appropriat omponnts of th data bas ar usd to alulat th spifi proprtis of th various forst strata and th nar-ground layr of atmosphr. This approah substantially simplifis th thnology of solving problms of prditing th stat of th mdium in th fir zon numrially. Th thrmodynami, thrmophysial and strutural haratristis orrspond to th forst fuls in th anopy of a diffrnt (for xampl pin [0,]) typ of forst. NUEICAL OLUTION AND EZULT Th boundary-valu problm () (7) w solv numrially using th mthod of splitting aording to physial prosss []. In th first stag, th hydrodynami pattrn of flow and distribution of salar funtions was alulatd. Th systm of ordinary diffrntial quations of hmial kintis obtaind as a rsult of splitting [] was thn intgratd. A disrt analog was obtaind by mans of th ontrol volum mthod using th IPLE lik algorithm [,6,7]. Th auray of th program was hkd by th mthod of insrtd analytial solutions. Th tim stp was sltd automatially. Filds of tmpratur, ), vloity, omponnt mass frations, and volum frations of phass wr obtaind numrially. Figur illustrat th tim dpndn of dimnsionlss tmpraturs of gas () and ondnsd phass (), Figur. mass onntrations of gas omponnts (- oxygn, - gas produts of pyrolysis), and Fig. - rlativ volum frations of solid phass (), moistur () and ok () at rown bas of th forst (V 5m/s). At th momnt of ignition, th gas ombustibl produts of pyrolysis burn away, and th onntration of oxygn is rapidly rdud. Th tmpraturs of both phass rah a maximum valu at th point of ignition. Th ignition prosss ar of a gas - phas natur, i.. initially hating of solid and gasous phass ours, moistur is vaporatd. Thn domposition pross into ondnsd and volatil pyrolysis produts starts, th lattr bing ignitd in th forst anopy. Fig. Tmpratur of gas ( - T ) and solid phas ( - T ); T T/T, T T /T, T 00K Fig. Conntration of oxygn (- C ) and gas produts of pyrolysis ( - C );, с 0.. α α / Fig. Volum frations of solid phas: ϕ ϕ / ϕ, ϕ ϕ / ϕ, ϕ ϕ / ϕ. 95

4 Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp.9-99 Th vtor fild of vloitis and isothrms of gas phas (Fig. 5): -6 orrspond to th isothrms T.,.5,.7,, and. In th viinity of th sour of hat and mass rlas, hatd air masss and produts of pyrolysis and ombustion float up. Th wind fild in th forst anopy intrats with th gas-t obstal that forms from th surfa forst fir sour and from th ignitd forst anopy bas. irulating flow forms byond th zon of hat and mass rlas, whil on th windward sid th movmnt of th air flowing past th ignition rgion alrats. Undr th influn of th wind th tilt angl of th flam is inrasd. As a rsult, this part of th forst anopy, whih is shiftd in th dirtion of th wind from th ntr of th surfa forst fir sour, is subtd to a mor intnsiv warming up. Th isothrms of th gas and ondnsd phass ar dformd in th dirtion of th wind. At V 0 th forst fir sour ignits th forst rown and burn away in th forst anopy. Forst fir bgins to sprad in th forst anopy. Th distribution of tmpratur, onntrations of gas produts of pyrolysis and oxygn in th Fig.7. Isolins of gas produts of pyrolysis (t s.) forst fir front ar prsntd in th Figur 8. It is sn that th ombustion wav looks lik as a soliton. Th oxygn onntration drops to nar zro in front of a fir. It is onsumd in th ombustion of th pyrolysis produts, th onntration of whih is rahd thir maximum bfor th maximum of tmpratur. Figurs 9-0 prsnt th distribution of tmpratur T ( T T / T, T 00K) (-.5, -.,.6,, 5.5, 6.) for gas phas, onntrations of oxygn ( 0., 0.5, 0.6, 0.7, 5 0.8, 6 0.9) and volatil ombustibl produts of pyrolysis (., - 0., 0.05, 0.0) ( α сα /, с 0. ) at diffrnt instants of tim for wind vloity V 5 m/s (Fig.9) and V 0 m/s (Fig.0). Fig.5. Th vtor fild of vloitis and isothrms of gas phas; (t s.) Figur 6 prsnts isolins of oxygn for diffrnt instants of tim (t,, s.): -5 orrspond to th isolins 0.9, 0.8, 0.7, 0.5 and 0.. Figur 7 prsnts isolins of gas produts of pyrolysis: - orrspond to th isolins 0.0, 0.05, 0. and 0.5. Fig.8 Th distribution of gas tmpratur, onntrations of gas produts of pyrolysis and oxygn Fig.6 Isolins of oxygn (t s.) Th distribution of isothrms of ombustion tmpratur shows th moving of forst fir front with tim. Figur 0 shows that with th inras of wind spd up to 0 m/s 96

5 Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp.9-99 inrass th rat of fir sprad to 5 m/s. Ignition of forst du to spotting is on of th most diffiult aspts to undrstand th bhavior of firs. Th phnomnon of spotting firs ompriss thr squntial mhanisms: gnration, transport and ignition of ripint ful. Th prsnt mathmatial modl and rsults of alulation ar usd to illustrat pitur of th formation of larg firs by ombining small ombustion sours arising from th transfr of firbrands. In ordr to undrstand ths mhanisms, many alulation xprimnts hav bn prformd. In th Figurs - th pross of formation larg forst fir front as a rsult of intgration of various sours of ombustion is showd. Th distributions of tmpratur for gas phas(a), onntrations of oxygn (b) and volatil ombustibl produts of pyrolysis () at diffrnt tims (I - t s., II - t6 s, III - t s.) for V 5 m/s ar prsntd. Thr ar prsnt th sam valus of isothrms and isolins of onntration for and as wll as in th Figurs -. If th sours of ignition from th burning partils ar arrangd in a triangl, th prosss of formation of th forst fir front ar shown in Figurs -. In th sond as (Figur ), th right ignition sour in th x dirtion was tims lss than in th first as (Figur ). Fig. 9 Th distribution of a) tmpratur for gas phas, b) onntration of oxygn and ) volatil ombustibl produts of pyrolysis; V 5 m/s, at diffrnt instants of tim: I - t s., II - t6 s, III - t s., IV - t 0 s. Fig.0 Th distribution of a) tmpratur for gas phas, b) onntrations of oxygn and ) volatil ombustibl produts of pyrolysis; V 0 m/s, at diffrnt instants of tim: I - t s., II - t6 s, III - t s., IV - t 0 s. Fig.. Th distribution of tmpratur, onntrations of oxygn and volatil ombustibl produts of pyrolysis. 97

6 Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp.9-99 Fig.. Th distribution of tmpratur, onntrations of oxygn and volatil ombustibl produts of pyrolysis. 98

7 Intrnational Journal of GEOATE, July, 07, Vol., Issu 5, pp CONCLUION sults of alulation giv an opportunity to dsrib th diffrnt onditions of th forst fir sprad taking aount diffrnt wathr onditions and stat of forst ombustibl matrials, whih allows applying th givn modl for prdition and prvnting firs. It ovrstimats th rat of rown forst fir sprad that dpnds on rown proprtis: bulk dnsity, moistur ontnt of forst ful, wind vloity and t. Th modl proposd hr givs a dtaild pitur of th hang in th tmpratur and omponnt onntration filds with tim, and dtrmin as wll as th influn of diffrnt onditions on th rown forst fir sprading for th diffrnt ass of inhomognous of distribution of sours of forst firs initiation. 5. ACKNOWLEDGEENT This rsarh was finanially supportd by th ussian Foundation for Basi sarh in 06 (th prot р_а) and on th stat task 0/6, th stat prot intifi rsarhs organization. 6. EFEENCE. Van Wagnr CE, Conditions for th start and sprad of rown fir, Canadian Journal of Forst sarh, Vol. 7, 977, pp. -.. Alxandr E, Crown fir thrsholds in xoti pin plantations of Australasia, PhD thsis, Dpartmnt of Forstry, Australian National Univrsity, Australia, Van Wagnr CE, Prdition of rown fir bhavior in onifr stands, Pro. '0th onfrn on fir and forst mtorology. Ottawa, Ontario. (Eds D. C. aivr, H. Auld and. Whitwood), 989, pp Xanthopoulos G., Dvlopmnt of a wildland rown fir initiation modl, PhD thsis, Univrsity of ontana, UA, othrml C. Crown fir analysis and intrprtation, Pro. Intrnational onfrn on fir and forst mtorology. issoula, ontana, UA, othrml C, Prditing bhavior of th 988 Yllowston Firs: protions vrsus rality, Int. Journal of Wildland Fir, Vol., 99, pp Van Wagnr CE, Prdition of rown fir bhavior in two stands of ak pin, Canadian Journal of Forst sarh, Vol., 999, pp Cruz G, Prditing rown fir bhavior to support forst fir managmnt disionmaking, Pro. IV Intrnational onfrn on forst fir rsarh, Luso-Coimbra, Portugal. (Ed. D. X. Vigas), [CD-O]. (illprss), Albini FA, odling ignition and burning rat of larg woody natural fuls, Int. Journal of Wildland fir, vol. 5, 995, pp ott JH, Assssing rown fir potntial by linking modls of surfa and rown fir bhavior, UDA Forst rvi, oky ountain Forst and ang Exprimnt tation. Fort Collins: -P-9, (Colorado, UA), 00.. Grishin A, athmatial odling Forst Fir and Nw thods Fighting Thm, Publishing Hous of Tomsk Univrsity, Tomsk, ussia, Grishin A, Prminov VA, athmatial modling of th ignition of tr rowns, Combustion, Explosion, and hok Wavs, Vol. No, 998, pp Konv EV, Th physial foundation of vgtativ matrials ombustion, Nauka, Novosibirsk, ussia, orvan D, Dupuy JL, odling of fir sprad through a forst ful bd using a multiphas formulation, Combustion and Flam, Vol. 7, 00, pp orvan D, Dupuy JL, odling th propagation of wildfir through a ditrranan shrub using a multiphas formulation, Combustion and Flam, Vol. 8, 00, pp Patankar V, Numrial Hat Transfr and Fluid Flow, Hmisphr Publishing Corporation, Nw York, Goudov A, Prminov VA, athmatial imulation of Contaminant Flow in th quar Closd srvoir, Intrnational Journal of Gomat, Vol., No 6, 06, pp Copyright Int. J. of GEOATE. All rights rsrvd, inluding th making of opis unlss prmission is obtaind from th opyright propritors. 99