Proceedings of the 7th IASME / WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09)

Transcription

1 athmatal modlng of forst fr sprad VALEIY PEINOV Dpartmnt of athmats and Natural Sn Blovo Branh of Kmrovo Stat Unvrsty Sovtskaya Strt Blovo Kmrovo rgon 6600 USSIA Abstrat: - Ths papr s dvotd a problm of dvlopmnt of a mathmatal modl for dsrpton of hat and mass transfr prosss at rown forst fr ntaton and sprad. athmatal modl of forst fr was basd on an analyss of xprmntal data and usng onpt and mthods from ratv mda mhans. Ths study gvs a two dmnsonal avragd mathmatal sttng and mthod of numral soluton of a problm of a forst fr sprad. Th boundary-valu problm s solvd numrally usng th mthod of splttng aordng to physal prosss. It was basd on numral soluton of two dmnsonal ynolds quatons for th dsrpton of turbulnt flow takng nto aount for dffuson quatons hmal omponnts and quatons of nrgy onsrvaton for gasous and ondnsd phass. In ths ontxt a study - mathmatal modlng - of th ondtons of forst fr spradng that would mak t possbl to obtan a dtald ptur of th hang n th vloty tmpratur and omponnt onntraton flds wth tm and dtrmn as wll as th lmtng ondtons of forst fr propagaton s of ntrst.. Ky-Words: athmatal modl forst fr gnton ombuston ontrol volum dsrt analogu. Introduton A grat dal of work has bn don on th thortal problm of forst fr ntaton. Crown frs ar ntatd by onvtv and radatv hat transfr from surfa frs. Howvr onvton s th man hat transfr mhansm. Th frst xplanaton of ths pross was gvn by Van Wagnr []. Th thory proposd thr dpnds on thr smpl rown proprts: rown bas hght bulk dnsty and mostur ontnt of forst ful. Also rown fr ntaton and hazard hav bn studd and modld n dtal (s for xampl Alxandr []; Van Wagnr []; Xanthopoulos []; othrml []; Cruz and othrs [6]; Albn and othrs [7]; Sott and nhardt [8]).Th mor omplt dsusson of th problm of modlng forst frs s provdd by a yl of works produd by a group of oworkrs at Tomsk Unvrsty (Grshn [9]; Grshn and Prmnov [0]; Prmnov []). In partular a mathmatal modl of forst frs was obtand by Grshn [9] basd on an analyss of known and orgnal xprmntal data (Grshn [9]; Konv [] and usng onpts and mthods from ratv mda mhans. Th physal two-phas modls usd by orvan and Dupuy [] may b onsdrd as a ontnuaton and xtnson of th formulaton proposd n [9]. Problm Formulaton Th bas assumptons adoptd durng th dduton of quatons and boundary and ntal ondtons: ) th forst rprsnts a mult-phas multstord spatally htrognous mdum; ) n th fr zon th forst s a porous-dsprsd two-tmpratur sngl-vloty ratv 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 onsrvaton 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 vntlatd forst massf n whh th volum 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 volum fraton of gas phas (omponnts of ar and gasous pyrolyss produts); ) th flow has a dvlopd turbulnt natur and molular transfr s ngltd; 6) gasous phas dnsty dosn t dpnd on th prssur baus of th low vlots of th flow n omparson wth th vloty of th sound. Lt th oordnat rfrn pont x x x = 0 b stuatd at th ntr of th surfa forst fr sour at th hght of th roughnss lvl axs 0x drtd paralll to th Earth s surfa to th rght n th drton of th unprturbd wnd spd axs 0x drtd prpndular to 0x and axs 0x drtd upward (Fg. ). Fgur ISSN: ISBN:

2 Baus of th horzontal szs of forst massf mor than hght of forst h systm of quatons of gnral mathmatal modl of forst fr [9] was ntgratd btwn th lmts from hght of th roughnss lvl - 0 to h. Bsds suppos that h 0 φ dx =φ h & φ & - avrag valu of φ. Th problm formulatd abov s rdud to a soluton of th followng systm of quatons: ρ + + ( ρv ) = Q ( m& m& ) / h = ; t x dv p r ρ = + ( ρv v ) ρsd v v ρg dt x x ρ p d ρ dt x = dt dt + Qv + ( τ τ ) / h = ; = ( ρ v T p ) + q v ( T Ts ) + x = x k ρ p ( q T + q ) / h; T + ( ρv ) + Q + ( J J ) / h = ; U k( U x TS ϕ = q t + ( T T V S ); ϕ ϕ ρ = ρ t t ϕ ρ = C t C σt S ) + ( q q q + k( U = ϕ ρ t = 0; + ) / h = 0; σt r r = p = ρ T v = ( v v v) g = (00 g). = = Th systm of quatons () (7) must b solvd takng nto aount th ntal and boundary ondtons t = 0 : v = 0 v = 0 v = 0 T = T = T = T ϕ = ϕ ; s v x = x v = V v = = T = T = : 0 0 x U + U /= 0; k x v v S ) = x : = 0 = 0 = 0 = 0 x x x x x T U = 0 + U = 0 ; x k x v (8) (9) v v v x = x0: = 0 = 0 = 0 = 0 x x x x (0) T U = 0 + U = 0 ; x k x () () () () () (6) (7) v v v x = x : = 0 = 0 = 0 = 0 x x x x () T U = 0 + U = 0. x k x U x = 0 : v = 0 v = 0 0 U 0 x = k x + = () v = v0 T = Tg x Δ x Δ v = 0 T = T x >Δ x >Δ; v v v x = x : = 0 = 0 = 0 = 0 x x x x () T U = 0 + U = 0. x k x Hr and abov d s th symbol of th t otal (subs tantal) dt drvatv; v s th offnt of phas xhang; ρ - dnsty of gas dsprsd phas t s tm; v - th vloty omponnts; T T S - 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 ρ ϕ spf hat dnsty and volum 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 S - 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; unvrsal gas onstant; C and molular mass of - omponnts of th gas phas arbon and ar mxtur; g s th gravty alraton; d s an mpral offnt of th rsstan of th vgtaton s s th spf surfa of th forst ful n th gvn forst stratum. To dfn sour trms whh haratrz nflow (outflow of mass) n a volum unt of th gas-dsprsd phas th followng formula wr usd for th rat of formulaton of th gasdsprsd mxtur m& outflow of oxygn hangng arbon monoxd. Q = ( ) + + = = ν ( ) = 0. g Hr ν g mass fraton of gas ombustbl produts of pyrolyss and mpral onstants. aton rats of ths varous ontrbutons (pyrolyss vaporaton ombuston of ok and volatl ombustbl produts of pyrolyss) ar approxmatd by Arrhnus laws whos paramtrs (pr-xponntal onstant k and atvaton ISSN: ISBN:

3 nrgy E ) ar valuatd usng data for mathmatal modls [9]. E 0. E = kρϕ xp = kρϕ T s xp Ts Ts E. E = kρϕsσ xp = k T xp. Ts T Th ntal valus for volum of fratons of ondnsd phass ar dtrmnd usng th xprssons: d( ν z ) Wd ϕ ρ ϕ = ϕ = ϕ = ρ ρ ρ 0. 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 wavlngth (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 onvton and nds to b losd. Th omponnts of th tnsor of turbulnt strsss ρ v as v wll as th turbulnt fluxs of hat and mass ρ vt p ρ v ar wrttn n trms of th gradnts of th avrag flow proprts usng th formulas ρv v v μ v = t + Kδ x x T ρvpt = λ t ρv = ρdt x x λ = μ / Pr ρ D = μ / S μ μ ρ K t t p t t t t t = / ε whr μ t λ t D t ar th offnts of turbulnt vsosty thrmal ondutvty and dffuson rsptvly; Pr t S t ar th turbulnt Prandtl and Shmdt numbrs whh wr assumd to b qual to. In dmnsonal form th offnt of dynam turbulnt vsosty s dtrmnd usng loal qulbrum modl of turbuln [9]. Th thrmodynam thrmophysal and strutural haratrsts orrspond to th forst fuls n th anopy of a dffrnt (for xampl pn [9]) typ of forst. Th systm of quatons () (7) must b solvd takng nto aount th ntal and boundary ondtons. Th thrmodynam thrmophysal and strutural haratrsts orrspond to th forst fuls n th anopy of a dffrnt typ of forst; for xampl pn forst (Grshn Prmnov []). Calulaton thod and sults Th boundary-valu problm () (7) s solvd numrally usng th mthod of splttng aordng to physal prosss (Prmnov []). In th frst stag th hydrodynam pattrn of flow and dstrbuton of salar funtons was alulatd. Th systm of ordnary dffrntal quatons of hmal knts obtand as a rsult of splttng was thn ntgratd. A dsrt analog was obtand by mans of th ontrol volum mthod usng th SIPLE lk algorthm (Patankar [6] ) Th auray of th program was hkd by th mthod of nsrtd analytal solutons. Analytal xprssons for th unknown funtons wr substtutd n () (7) 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 abov th valus of th funtons usd wr nfrrd wth an auray of not lss than %. Th fft of th dmnsons of th ontrol volums on th soluton was studd by dmnshng thm. Th tm stp was sltd automatally. Flds of tmpratur vloty omponnt mass fratons and volum fratons of phass wr obtand numrally. Th dstrbuton of bas funtons shows that th pross of rown forst fr ntaton gos through th nxt stags. Th frst stag s rlatd to nrasng maxmum tmpratur n th ground ovr wth th rsult that a surfa fr sour appars. At ths pross stag ovr 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 vaporats and gasous and dsprsd pyrolyss produts ar gnratd. Ignton of gasous pyrolyss produts of th ground ovr 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 vaporats 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 burns away and th onntraton of oxygn s rapdly rdud. Th tmpraturs of both phass rah a maxmum valu at th pont of gnton. Th gnton prosss s of a gas - phas natur. Not also that th transfr of nrgy from th fr sour taks pla du to radaton; th valu of radaton hat flux dnsty s small ompard to that of th onvtv hat flux. In th vnty of th sour of hat and mass rlas hatd ar masss and produts of pyrolyss and ombuston float up. At V 0 th wnd fld n th forst anopy ntrats wth th gas-t obstal that forms from th surfa forst fr sour and from th gntd forst anopy bas and burn away n th forst anopy. In th vnty of th sour of hat and mass rlas hatd ar masss and produts of pyrolyss and ombuston float up. At V 0 th wnd fld n th ISSN: ISBN:

4 forst anopy ntrats wth th gas-t obstal that forms from th surfa forst fr sour and from th gntd forst anopy bas. On th wndward sd th movmnt of th ar flowng past th gnton rgon alrats. Fgurs ab ab and ab prsnt th dstrbuton of tmpratur T ( T = T / T T = 00K) (-...) for gas phas oxygn ( ) volatl ombustbl produts of pyrolyss onntratons ( ) ( = с / с = 0. ) for wnd vloty V = 0 m/s: and a) t= s. b) t= s. W an not that th sothrms s movd n th forst anopy and dformd by th aton of wnd. Smlarly th flds of omponnt onntratons ar dformd. It s onludd that th forst fr bgns to sprad. Fgur. a. Fgur.a Fgur b. Fgur.b Fgur a. ISSN: ISBN:

5 Th rsults obtand agr wth th laws of physs and xprmntal data. frns Conluson Fgur b. Th rsults of alulaton gv an opportunty to valuat rtal ondton of th forst fr sprad whh allows applyng th gvn modl for prvntng frs. It ovrstmats th rown forst fr sprad that dpnds on rown proprts: bulk dnsty mostur ontnt of forst ful and t. Th modl proposd thr gv a dtald ptur of th hang n th vloty tmpratur and omponnt onntraton flds wth tm and dtrmn as wll as th nflun of dffrnt ondtons on th rown forst fr ntaton. Th rsults obtand agr wth th laws of physs and xprmntal data (Grshn [9]; Konv []). From an analyss of alulatons and xprmntal data t was found that for th ass n quston th mnmum total nndary hat puls s 600 kj/m (Grshn [9]).. Fgur 8 llustrats th tmpratur dpndn of tm gnton of rown for dffrnt bulk dnsts of surfa layr ( d=0 and - 0 kg/m - numral alulatons; - d=0 and x 0 kg/m xprmntal rsults (orshn [7]). Fgur 8. ostur ontnt of FF n rown s W=0% and d=0. kg/m. [] Van Wagnr C.E. Condtons for th start and sprad of rown fr. Canadan Journal of Forst sarh No pp. -. [] Alxandr.E. Crown fr thrsholds n xot pn plantatons of Australasa. PhD thss Dpartmnt of Forstry Australan Natonal Unvrsty 998. [] Van Wagnr C. E. Prdton of rown fr bhavor n onfr stands. In '0th onfrn on fr and forst mtorology'. Ottawa Ontaro. (Eds D. C. aivr H. Auld and. Whtwood) pp [] Xanthopoulos G. Dvlopmnt of a wldland rown fr ntaton modl. PhD thss Unvrsty of ontana 990. [] othrml.c. Crown fr analyss and ntrprtaton. In th Intrnatonal onfrn on fr and forst mtorology. ssoula ontana USA 99. [6] Cruz.G. t al Prdtng rown fr bhavor 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] Albn F.A. t al odlng gnton and burnng rat of larg woody natural fuls. Int. Journal of Wldland fr Vol. No. 99 pp [8] Sott J.H. t al Assssng rown fr potntal by lnkng modls of surfa and rown fr bhavor. USDA Forst Srv oky ountan Forst and ang Exprmnt Staton. Fort Collns: S-P-9 (Colorado USA) 00. [9] Grshn A.. athmatal odlng Forst Fr and Nw thods Fghtng Thm. Tomsk Publshng Hous of Tomsk Unvrsty ussa [0] Grshn A.. Prmnov V.A. athmatal modlng of th gnton of tr rowns. Combuston Exploson and Shok Wavs Vol. 998 pp [] Prmnov V.A. athmatal odlng of Crown and ass Forst Frs Intaton Wth th Allowan for th adatv - Convtv Hat and ass Transfr and Two Tmpraturs of dum Ph.D Thss Tomsk Stat Unvrsty ussa 99. [] Prmnov V.A. athmatal modlng of rown forst fr ntaton. In 'III Intrnatonal onfrn on forst fr rsarh and th onfrn on fr and forst mtorology'. Luso Portugal. (Ed. D.X.Vgas) 998 pp. 9-. [] Konv E.V. Th physal foundaton of vgtatv matrals ombuston.. Novosbrsk Nauka ussa [] orvan D. Dupuy J.L. odlng of fr sprad ISSN: ISBN:

6 through a forst ful bd usng a multphas formulaton. Combuston and Flam Vol pp [] orvan D. Dupuy J.L. odlng th propagaton of wldfr through a dtrranan shrub usng a multphas formulaton. Combuston and Flam Vol pp [6] Patankar S.V. Numral Hat Transfr and Flud Flow. Nw York: Hmsphr Publshng Corporaton 98. [7] orshn V.N. [986]. Calulaton mthod of th thn vgtabl partl gnton by onvtv flux LTA Lnngrad (ussa). ISSN: ISBN: