Mathematical modeling of the crown forest fires impact on buildings

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1 INTENATIONAL JOUNAL OF ATHEATIC AND COPUTE IN IULATION Volum 07 athmatal modlng of th rown forst frs mpat on buldngs Valry Prmno Abstrat Th mathmatal modlng of forst frs atons on buldngs and struturs has bn arrd out to study th ffts of fr ntnsty and wnd spd on possblty of gnton of buldngs. Th rown forst fr s ntrodud as a hat and mass sour dfnd by th mpral alus of arag rown fr tmpratur and rtal gas loty at th top rown surfa dpndnt on fr ntnsty. Th hydrodynam and thrmal ntratons btwn plum wnd flow and buldng ar analyzd. Th modlng approah s basd on th us of standard non-statonary thr-dmnsonal onsraton quatons that ar sold numrally undr th nput ondtons spf for larg rown forst frs. Kywords forst fr; rown fr; mathmatal modl; fnt olum mthod; gnton I. INTODUCTION Th protton of buldngs and struturs n a ommunty from dstruton by fr s a ry mportant onrn. Ths papr addrsss th dlopmnt of a mathmatal modl for mpat of rown frs wth buldngs. Th forst fr s a ry omplatd phnomnon. At prsnt fr srs an forast th dangr ratng of or th spf wathr lmnts rlatng to forst fr. Thr s nd to undrstand and prdt forst fr ntaton bhaour and mpat of fr on th buldngs and onstrutons. Ths papr s purposs ar th mpromnt of knowldg on th fundamntal physal mhansms that ontrol forst fr bhaor. A grat dal of work has bn don on th thortal problm of forst frs. Th frst aptd mthod for prdton of rown frs was gn by othrmal [] and Van Wagnr []. Th smmpral modls [-] allow to obtan a qut good data of th forst fr rat of sprad as a funton of ful bulk and mostur wnd loty and th trran slop. But ths modls us data for partular ass and do not g rsults for gnral fr ondtons. Also rown frs ntaton and hazard ha bn studd and modld n dtal Valry Prmno s wth th Dpartmnt of Eology and Bas afty Tomsk Polythn Unrsty Tomsk ussa -mal addrss: prmno@tpu.ru (g: Alxandr [] Van Wagnr [] Xanthopoulos [] Van Wagnr [] Cruz [6] Albn [7] ott J. H. and nhar E. D. [8]. Th dsusson of th problm of modlng forst frs s prodd by Grshn [9]. A mathmatal modl of forst frs was obtand by Grshn [9] basd on an analyss of known and orgnal xprmntal data and usng onpts and mthods from rat mda mhans. Th physal two-phas modls usd n [0] may b onsdrd as a dlopmnt and xtnson of th formulaton proposd by Grshn [9]. Howr th nstgaton of rown frs ntaton has bn lmtd manly to ass studd of forst frs propagaton wthout tak nto aount th mutual ntraton of rown forst frs wth dffrnt obstals (roads glads and t.) buldngs and onstrutons. In ths papr th mpats of rown forst frs on buldngs ar studd. Th dangrous dstans btwn forst and buldngs ar alulatd n ass whn th buldngs wll b gntd undr th nflun forst frs. II. PHYICAL AND ATHEATICAL ODEL It s assumd that th forst durng a forst fr an b modld as ) a mult-phas multstord spatally htrognous mdum; ) n th fr zon th forst s a porous-dsprsd two-tmpratur sngl-loty rat mdum; ) th forst anopy s supposd to b non dformd mdum (trunks larg branhs small twgs and ndls) whh affts only th magntud of th for of rsstan n th quaton of onsraton of momntum n th gas phas.. th mdum s assumd to b quas-sold (almost non-dformabl durng wnd gusts); ) lt thr b a so-alld ntlatd forst massf n whh th olum of fratons of ondnsd forst ful phass onsstng of dry organ mattr watr n lqud stat sold pyrolyss produts and ash an b ngltd ompard to th olum fraton of gas phas (omponnts of ar and gasous pyrolyss produts); ) th flow has a dlopd turbulnt natur and molular transfr s ngltd; 6) gasous phas dnsty dosn t dpnd on th prssur baus of th low lots of th flow n omparson wth th loty of th sound. Lt th pont x x x 0 s stuatd at th ntr of th surfa forst fr sour at th hght of th roughnss ll axs 0x drtd paralll to th Earth s surfa to th rght n th drton of th IN:

2 INTENATIONAL JOUNAL OF ATHEATIC AND COPUTE IN IULATION Volum 07 unprturbd wnd spd axs 0x drtd prpndular to 0x and axs 0x drtd upward (Fg. ). Th buldng s stuatd on th rght part of th ptur. Fg. Th shm of alulaton doman. Problm formulatd abo rdus to th soluton of systms of quatons ()-(7): + ( ) Q ; d p d dt P + ( ) s g Q ; ( T T ) + k d ( s T ) + q p g ( U σt ( ) + Q ; U ku + k T + k T k k k + k ; g T pϕ q q k ( U σt C C ); σ gσ 0 ) + ( T T ); 0; p T ( ) g (00 g). Th systm of quatons () (7) must b sold takng nto aount th ntal and boundary ondtons: () () () () () (6) (7) a. t 0 : T T T T ϕ ϕ ; s s x x V T T : 0 0 U + U / 0; k x : x T U 0 + U 0 ; k x x0: T U 0 + U 0 ; k x x : x T U 0 + U 0. k U x 0 : U 0 k + 0 T Tg x x 0 T T x > x > ; x x : x T U 0 + U 0. k (8) (9) (0) () () () () d Hr and abo s th symbol of th total (substantal) drat; s th offnt of phas xhang; - dnsty of gas dsprsd phas t s tm; - th loty omponnts; T T - tmpraturs of gas and sold phass U - dnsty of radaton nrgy k - offnt of radaton attnuaton P - prssur; p onstant prssur spf hat of th gas phas p ϕ spf hat dnsty and olum of fraton of ondnsd phas ( dry organ substan mostur ondnsd pyrolyss produts mnral part of forst ful) th mass rats of hmal ratons q thrmal ffts of hmal ratons; k g k - radaton absorpton offnts for gas and ondnsd phass; T - th ambnt tmpratur; - mass onntratons of - omponnt of gas - dsprsd mdum ndx... whr orrsponds to th dnsty of oxygn - to arbon monoxd CO - to arbon doxd and nrt omponnts of ar - to partls of blak - to partls of smok; unrsal gas onstant; C and molular mass of - omponnts of th gas phas arbon and ar mxtur; g s th graty alraton; d s an mpral offnt of th rsstan of th gtaton s s th spf surfa of th forst ful n th gn forst stratum. To dfn sour trms IN:

3 INTENATIONAL JOUNAL OF ATHEATIC AND COPUTE IN IULATION Volum 07 whh haratrz nflow (outflow of mass) n a olum unt of th gas-dsprsd phas th followng formula wr usd for th rat of formulaton of th gas-dsprsd mxtur Q outflow of oxygn hangng arbon monoxd gnraton of blak and smok partls. m ( ) νg( ) 0. + * E kϕ xp( T ) E kϕσ xp( ) T 0. k ϕ T 0. E xp( T. E k T xp( ). T Th ntal alus for olum of fratons of ondnsd phass ar dtrmnd usng th xprssons: d( ν z ) Wd ϕ ϕ ϕ ϕ whr d -bulk dnsty for surfa layr ν z offnt of ashs of forst ful W forst ful mostur ontnt. It s supposd that th optal proprts of a mdum ar ndpndnt of radaton walngth (th assumpton that th mdum s gry ) and th so-alld dffuson approxmaton for radaton flux dnsty wr usd for a mathmatal dsrpton of radaton transport durng forst frs. To los th systm () (7) th omponnts of th tnsor of turbulnt strsss and th turbulnt hat and mass fluxs ar dtrmnd usng th loal-qulbrum modl of turbuln (Grshn [9]). Th systm of quatons () (7) ontans trms assoatd wth turbulnt dffuson thrmal onduton and onton and nds to b losd. Th omponnts of th tnsor of turbulnt strsss as wll as th turbulnt fluxs of hat and mass T p ar wrttn n trms of th gradnts of th arag flow proprts usng th formulas µ t + Kδ T pt λ t Dt λ µ / Pr D µ / µ µ K / ε t t p t t t t t whr µ t λ t D t ar th offnts of turbulnt sosty thrmal ondutty and dffuson rsptly; Pr t t ar th turbulnt Pranl and hm numbrs whh wr assumd to b qual to. In dmnsonal form th offnt of dynam turbulnt sosty s dtrmnd usng loal ) qulbrum modl of turbuln [9]. Th lngth of th mxng path s dtrmnd usng th formula l xkt /( +.x d s / h) takng nto aount th fat that th offnt of rsstan d n th spa btwn th ground or and th forst anopy bas s qual to zro whl th onstants k t 0. and hh -h (h h hght of th tr rowns and th hght of th rown bas). It should b notd that ths systm of quatons dsrbs prosss of transfr wthn th ntr rgon of th forst massf whh nluds th spa btwn th undrlyng surfa and th bas of th forst anopy th forst anopy and th spa abo t whl th approprat omponnts of th data bas ar usd to alulat th spf proprts of th arous forst strata and th nar-ground layr of atmosphr. Ths approah substantally smplfs th thnology of solng problms of prdtng th stat of th mdum n th fr zon numrally. Th thrmodynam thrmophysal and strutural haratrsts orrspond to th forst fuls n th anopy of a dffrnt (for xampl pn [9]) typ of forst. III. NUEICAL OLUTION AND EULT Th boundary-alu problm () () s sold numrally. A dsrt analog was obtand by mans of th ontrol olum mthod usng th IPLE lk algorthm (Patankar []). Dffrn quatons that ars n th ours of samplng wr rsold by th mthod of IP []. In ordr to ffntly sol ths problm n a rat flow th mthod of splttng aordng to physal prosss was usd. Th bas da of ths mthod s basd on th nformaton that th physal tmsal of th prosss s grat than hmal. In th frst stag th hydrodynam pattrn of flow and dstrbuton of salar funtons was alulatd. Thn th systm of ordnary dffrntal quatons of hmal knts obtand as a rsult of splttng was thn ntgratd. Th tm stp for ntgratng ah funton has to b smallr than th haratrst tm of physal pross to nsur th onrgn of th numral mthod. Th tm stp was sltd automatally. Th auray of th program was hkd by th mthod of nsrtd analytal solutons. Analytal xprssons for th unknown funtons wr substtutd n () () and th losur of th quatons wr alulatd. Ths was thn tratd as th sour n ah quaton. Nxt wth th ad of th algorthm dsrbd abo th alus of th funtons usd wr nfrrd wth an auray of not lss than %. Th fft of th dmnsons of th ontrol olums on th soluton was studd by dmnshng thm. Flds of tmpratur loty omponnt mass fratons and olum fratons of phass wr obtand numrally. Th frst stag s rlatd to nrasng maxmum tmpratur n th pla of gnton wth th rsult that a rown fr sour appars. At ths pross stag or th fr sour a thrmal wnd s formd a zon of hatd forst fr pyrolyss produts whh ar mxd wth ar float up and pntrat nto th rowns of trs. As a rsult forst fuls n th tr rowns ar hatd mostur aporats and gasous and dsprsd pyrolyss produts ar gnratd. Ignton of IN:

4 INTENATIONAL JOUNAL OF ATHEATIC AND COPUTE IN IULATION Volum 07 gasous pyrolyss produts of th rown ours at th nxt stag and that of gasous pyrolyss produts n th forst anopy ours at th last stag. As a rsult of hatng of forst ful lmnts of rown mostur aporats and pyrolyss ours aompand by th rlas of gasous produts whh thn gnt and burn away n th forst anopy. At th momnt of gnton th gas ombustbl produts of pyrolyss burn away and th onntraton of oxygn s rapdly rdud. Th tmpraturs of both phass rah a maxmum alu at th pont of gnton. Th gnton prosss s of a gas - phas natur. At V 0 th wnd fld n th forst anopy ntrats wth th gas-t obstal that forms from th forst fr sour and from th gntd forst anopy and burn away n th forst anopy. Th sothrms of gas phas omponnts mod n th forst anopy by th aton of wnd. It s onludd that th forst fr bgns to sprad. Th rsults of th alulaton g an opportunty to onsdr forst fr sprad for dffrnt wnd loty anopy bulk dnsts and mostur forst ful. It s onsdrd th fft of forst fr front on th buldng whh s stuatd nar from th forst. Th nfluns of wnd loty and dstan btwn forst and buldng on gnton of buldng ar studd numrally. Th rsults of alulatons an b usd to aluat th thrmal ffts on th buldng loatd nar from th forst frs. Th tmpratur flds of rown forst fr at dfnt momnt wll b ntratd wth th obstal - buldng (Fgur a) and b)) and gntd t. Fg.. shos tmpratur fld at th ntal momnt of forst fr sprad for a wnd spd of m/s. buldng. In papr [] t s showd that th wood wll b gntd whn ts tmpratur xds 00C. Th rsults of alulatons prsntd on Fgurs - show that th surfa tmpratur rah ths alus at wnd lots mor than 6 m/s. Th hght of buldng n ths alulatons was H m. Fg. Th dstrbuton of tmpratur on th wall of th buldng for thr wnd spd alus. Th dstan btwn forst and buldng s m. Fg. Th dstrbuton of tmpratur on th wall of th buldng for thr wnd spd alus. Th dstan btwn forst and buldng s 6 m. Fg. Gas tmpratur fld at t s for a wnd spd of m/s;... ; T T/T T T /T T 00K. Durng ths pross th surfa of th wall of th buldng hats as a rsult of onton and radaton hat transfr. Th wood buldng wll b gntd at dfnt tmpratur. It dpnds on wnd lots dstans from th forst fr to buldng th hght of buldng and othrs paramtrs. Th Fgurs - rprsnt th prdtd dstrbutons of tmpratur on th surfas of th wall of th buldng as a funton of rtal oordnat for th thr sltd wnd spd alus and dffrnt dstans btwn forst and Fg. Th dstrbuton of tmpratur on th wall of th buldng for thr wnd spd alus. Th dstan btwn forst and buldng s m. IN:

5 INTENATIONAL JOUNAL OF ATHEATIC AND COPUTE IN IULATION Volum 07 As a rsult of ths alulatons t was dfnd maxmum safty dstans btwn forst and buldng whn th buldng would not ha bn gntd by forst fr (Fg. 6). Th wnd spd alus nras from 6 to m/s. Fg. 6 Th dpndn of safty dstans btwn forst and buldng as a funton of wnd spd alus; H m. Also t was studd th nflun of th hght of buldng on th alu of safty dstans. Whn th hght of buldng hangs from to 6 m th safty dstans l also nrass (Fgur 7) for dffrnt alus of wnd spd. Fg. 7 Th dpndn of safty dstans btwn forst and buldng as a funton of wnd spd alus. H6 m. IV. CONCLUION A multphas mathmatal modl of wnd-add rown forst frs propagatng through htrognous ful bds has bn prformd. It taks nto aount th hydrodynam aspts of th flow and uss Arrhnus knts to dsrb th bas physs and hmal prosss of thrmal domposton hatng dryng pyrolyss and ombuston. Turbuln and radaton ar onsdrd n ordr to mpro th physal nsght. It allows to nstgat th dynams of th mpat of forst frs on buldngs undr th nflun of arous xtrnal ondtons: a) mtorology ondtons (ar tmpratur wnd loty t.) b) typ (arous knds of forst ombustbl matrals) and thr stat (load mostur t.). Th alulatons lt to gt th maxmum dstan from th fr to th buldng n whh th obt possbl gnton. It has bn found that th fft of nrasng th wnd spd s to nras th safty dstans btwn forst and buldng. Th nrasng of buldng hght s obsrd also ld to nras th safty dstans btwn forst and buldng. pf xprmnts ar also ndd to obtan mor rlabl nformaton on aldaton of furthr soluton of ths problm. Aknowldgmnt Th papr was supportd from FB (prot od: р_а) and wthn th framwork of Tomsk Polythn Unrsty Compttnss Enhanmnt Program grant. frns [].C. othrmal Prdtng bhaour and sz of rown frs n th Northrn oky ountans //s.pap. INT-8. Ogdn UT: U Dpartmnt of Agrultur Forst r Intrmountan Forst and ang Exprmnt taton 97 6p. [] C.E. Van Wagnr Condtons for th start and sprad of rown fr Canadan Journal of Forst sarh pp. -. [].E. Alxandr Crown fr thrsholds n xot pn C.E. In '0th onfrn on fr and forst mtorology'. Ottawa Ontaro. (Eds D. C. air H. Auld and. Whtwood) 979 pp [] G. Xanthopoulos Dlopmnt of a wldland rown fr ntaton modl. PhD thss Unrsty of ontana ontana 990. [] C.E. Van Wagnr Prdton of rown fr bhaor n two stands of ak pn. Canadan Journal of Forst sarh. 999 pp. -9. [6].G. Cruz t al. Prdtng rown fr bhaor to support forst fr managmnt dson-makng In 'IV Intrnatonal onfrn on forst fr rsarh'. Luso-Combra Portugal. (Ed. D. X. Vgas) [CD- O]. (llprss). 00. [7] F.A. Albn t al odlng gnton and burnng rat of larg woody natural fuls Int. Journal of Wldland fr 99 pp [8] J.H. ott t al Assssng rown fr potntal by lnkng modls of surfa and rown fr bhaor. UDA Forst r oky ountan Forst and ang Exprmnt taton. Fort Collns: - P-9 (UA) 00. [9] A.. Grshn athmatal odlng Forst Fr and Nw thods Fghtng Thm. Tomsk: Publshng Hous of Tomsk Unrsty 997. [0] D. oran J.L. Dupuy odlng th propagaton of wldfr through a dtrranan shrub usng a multphas formulaton Combuston and Flam V pp [].V. Patankar Numral Hat Transfr and Flud Flow Hmsphr Publshng Corporaton Nw York 98. [] E.N. Valndk P.. ath.a.afromo Larg forst frs and fghtng wth thm. oskow n 979. (n ussan) IN: