EFFECTS OF THE COMBUSTION GASES ON THE RADIATION HEAT TRANSFER

Transcription

1 Intrnatonal Journal of Mchancal ngnrng and Tchnology (IJMT) Volum 9, Issu 9, Sptmr 018, pp. 30 7, Artcl ID: IJMT_09_09_08 Aalal onln at ISSN rnt: and ISSN Onln: IAM ulcaton Scopus Indxd FFCTS OF TH COMBUSTION GASS ON TH RADIATION HAT TRANSFR Dr. Aas Alw Sakhr Ad Unrsty of Al-adsyah, Collg of ngnrng, Iraq ABSTRACT A Mont Carlo program that smulats radat hat transfr n partcpatng mda for two-dmnsonal has n dlopd. Ths work has addd th alty to smulat th asorpton and msson of th comuston gass: CO,CO, H O, NO, SO, and CH. Th thory of gas radaton and th formulaton and mplmntaton of th thrmal modl ar prsntd. Sral aspcts of th smulaton ar rfd y numrcal solutons, comparson to pulshd rsults and xprmntal data. Th accuracy and utlty of th smulaton ar dmonstratd y comparson to othr ondmnsonal solutons for radat hat transfr n comuston gass and y comparson to xprmntal data. Ky words: radat, comuston gass, asorpton, msson. Ct ths Artcl: Dr. Aas Alw Sakhr Ad, ffcts of th comuston Gass on th Radaton Hat Transfr, Intrnatonal Journal of Mchancal ngnrng and Tchnology 9(8), 018, pp NOMNCATUR A : Ara, m. : Slf-roadnng to forgn-gas roadnng rato. C : Constant dfnd n q. (30). 1 d : Spacng twn th cntrs of spctral lns ( cm ). : nrgy of a photon packt (J), or nrgy stat (J). : Componnt of th msson ctor. F : xchang factor. f : Usr-dfnd cutoff fractons. k : Asorpton coffcnt (1/m). : ngth of a surfac (m), path lngth for asorpton (m), or dstanc twn two slas (m). C : ngth or whch asorpton s calculatd(m). n : Indx of rfracton, or mprcal factor dtor@am.com

2 Dr. Aas Alw Sakhr Ad : rssur (atm), or proalty. : ffct roadnng prssur paramtr. : nrgy pr unt tm (W ). q : Radat hat flux ( W / m ). R : Unformly dstrutd random numr twn 0 and 1. 1 S : n ntnsty ( cm. m / g ). s : ath lngth (m). T : Tmpratur (K). V: Volum( 3 m ) : Vratonal quantum numr. x : X coordnat (m). y : Y coordnat (m). : Gas asorptty. : Man ln wdth to spacng paramtr. 1 : Spctral half wdth of a ln( cm ) : Dsplacmnt twn nods (m), chang n a quantty, or ntral sz. nt : Stp sz for pcws ntgraton : Wanumr ntral : mssty. : n wdth to spacng paramtr : Dmnsonlss tmpratur. : Con angl(rad). : Mass asorpton coffcnt ( : Rflctty. : Gas transmssty. : Optcal dpth. : Dmnsonlss hat flux. m / 1 : xponntal dcay wdth ( cm ). SUBSCRITS 0 : Orgnal alu, or rfrnc stat. 1: Frst nod of a surfac. g ). : Scond nod of a surfac. D : Two-dmnsonal. as : Asord. : Black ody, gas asorptoon and, or ottom plat. : msson pont. : lsassr modl dtor@am.com

3 ffcts of th comuston Gass on th Radaton Hat Transfr l : lmnt. m : mttd. gp : Gauss ponts. I : Incdnt. : th of a st. nt : Intgraton, or ntrcpton. : th of a st. NR : Not rflctd. s : Spcular, or surfac. sm : Smoothd quantty. surf : Surfac. t : Top plat. u : Uppr lmt for a and. w : Wall. : Asorptanc. 1. INTRODUCTION Most arly work applyng Mont Carlo to gass dalt only wth gry gass [Howll and rlmuttr, 196a; rlmuttr and Howll, 196]. Howll and rlmuttr [196] wr th frst to consdr non gry mda wth Mont Carlo whn thy modld hydrogn gas twn nfnt lack plats. Th spctral asorpton coffcnt for hydrogn at hgh tmpraturs and prssurs was otand from xprmntal data. Stward and Cannon [1971] usd Mont Carlo and th wghd-sum-of-gry-gass mthod to modl a cylndrcal furnac. Sral good dscrptons of th us of th Mont Carlo mthod n radat hat transfr, ncludng partcpatng mda, xst [Howll, 1968; Ha-Shkh, 1988; Modst, 1993]. Mor rcntly, a rw of th Mont Carlo mthod was don y Waltrs and Buckus [199] and concntrats on th rrs Mont Carlo mthod and th modlng of scattrng. Mor rcntly thr has n much ntrst n th us of Mont Carlo n modlng th spctral haor of ral gass. Bsds th work don y Farmr and Howll mntond ao, Modst [199] has formulatd a Mont Carlo mthod for gass wth spctral ln structur. On-dmnsonal Mont Carlo smulatons ha n don for watr apor [u and Twar, 199] and for a mxtur of watr apor and caron doxd [Tanguch t al., 199]. Farmr and Howll [199] ha also nstgatd rducng th computatonal tm rqurd to modl spctral ffcts y usng th dffuson approxmaton [Modst, 1993; Sgl and Howll, 199] n optcally thck cass. Although much work has n don on modlng gas radaton y Mont Carlo, ry lttl work has n don usng ral gomtrs xcpt y Farmr and Howll. To fnd th ffcts of gas radaton for most furnacs, fforts must mad to modl th complx gomtrs and surfacs proprts as wll. Th prsnt work dscrs Mont Carlo tchnqus to modl ral gas proprts and surfac proprts whch ary wth ncdnt angl n complx, two-dmnsonal nclosurs. Th mplmntaton s asd on th programs MONTD and MONT3D [Malty t al, 199; Burns and ryor, 1987; Burns t al., 199; Burns t al., 1990] whch smulat radat hat transfr n gomtrs wth nonpartcpatng mda. Th currnt work rls on th sam photon tracng algorthm and gomtry and surfac proprty modls as MONTD. 3 dtor@am.com

4 Dr. Aas Alw Sakhr Ad. FORMAIZATION AND SOUTION Th gomtry s ddd nto N surf surfacs of unform tmpratur and mttanc and N grd grd clls contanng gas of unform composton, tmpratur, and prssur. Th sum of th N surf surfacs and th N grd grd clls gs N l lmnts whch mt and asor photons. Now, th total rat of nrgy,,, mttd from a surfac of ara A, s [Modst, 1993; Sgl and Howll, 199]: m surf, grd n T A (1) m Whl th total rat of nrgy, m, grd ncludng slf-asorpton s [Modst, 1993]:, mttd from a gas olum of olum V not n ( T V () m, grd ) whr () s th lanck man dnsty mass asorpton coffcnt product, n s ssntally on for most gas mxturs ncludng comuston gass. For ths rason, from ths pont on, n s assumd to on and s not xplctly shown n any of th quatons. Th amount of nrgy mttd y lmnt that s asord y lmnt,, s: as F m, a, (3) F s th xchang factor from lmnt to lmnt whch s dfnd as: F () whr s th total nrgy of photon packts rlasd y lmnt, and s th total nrgy of photon packts rlasd y lmnt whch ar asord y lmnt, no mattr what th path. Both and rsult drctly from th Mont Carlo photon tracng. nough photon undls must mttd n th Mont Carlo to rduc th statstcal scattr n an accptal ll. Th total nrgy msson rat for th lmnt, m N l 1, s:, Fm, (5) F to If hat transfr s only y radaton and th lmnt s at qulrum, s 0. If th lmnt s not at qulrum and/or thr s sgnfcant hat transfr y othr mans such as conducton, concton, or chmcal racton, s gnrally not 0. artcularly notworthy s that th summaton n q. (5) ncluds th lmnt. Ths s why q. () dos not nclud slf-asorpton. Slf-asorpton s accountd for n th F trm. Th xpandd rson of q. (5) for surfacs s: N surf 1 N l T A T A F [( ) ] T V F (6) 1,,3... N surf and for gass t s: N surf 1 N surf 1 [( ) ] T V T A F [( ) ] T V F (7) Nl N surf dtor@am.com

5 N surf 1, N l ffcts of th comuston Gass on th Radaton Hat Transfr For a two-dmnsonal cas, snc th gomtry s nfnt n th z drcton, f s not 0, thn t s nfnt. To crcumnt ths, qs. (6) and (7) ar ddd y a unt lngth n th z drcton and com: N surf 1 N l T T F [( ) ] T A F (8) 1,,3... N surf N surf 1 N surf 1 [( ) ] T A T F [( ) ] T A F (9) N surf 1, N l Nl N surf 1 whr s th two-dmnsonal lngth of a surfac n th x-y plan, and A s th ara of gas nsd ach grd n th x-y plan. Th prms sgnfy quantts pr unt lngth n th z drcton. Th gas asorptanc modl usd n ths work s th Goody narrow and modl wth paramtrs from th dwards xponntal-tald wd and modl [dwards, 1976; dwards and Balakrshnan, 1973]. By assumng that th spctrum dcays xponntally n th tals of th asorpton and, dwards has formulatd four paramtrs that ha n shown to g accurat rsults for th maor comuston gass whn usd wth th Goody narrow and modl. Th rasons for choosng ths modl ar th followng. n y ln ntgraton gs mor accurat rsults [Twar and Gupta, 1978], ut t rqurs hug dataass of nformaton on nddual spctral lns and nols lngthy calculatons. Sral othr wd and modls xst [dwards, 1976; Modst, 1993], such as nnr s ox modl. Th adantag of th dwards/goody modl or th othr modls, s that t has paramtrs dfnd for all maor comuston gass: CO, H O, CO, CH, NO, and SO and has n found to corrlat to wthn 15% of xprmntal data. Mor rcntly, t al. [1995] ha dlopd wd and paramtrs for CH and all of th ao gass xcpt SO whch whn usd wth th lsassr narrow and modl otan rsults as accurat as th dwards/goody modl. Th Goody modl thorzs that th ln spctral dstruton s a unform random dstruton of man ln spacng d. Th ln ntnsts of ach ln, S, follow an xponntal dstruton. Th proalty of a partcular alu of S, (S), s qual to: ( 1/ S)xp( S / S), whr S s th man ln ntnsty. For ths modl, () s not dfnd, xcpt statstcally, ut: ( S / d) X 1 g, sm [1 ( S / d) X / ) ] (10) g, sm xp 1 1/ Whr s th ln wdth to spacng paramtr, dfnd y: (11) d For a symmtrcal and wth a and cntr, c,.. for most ands: S / d ( / )xp( / ) (1) c For an asymmtrc and wth an uppr lmt, u, such as th.3m and of CO : S / d ( / )xp( / ) (13) u 3 dtor@am.com

6 Dr. Aas Alw Sakhr Ad For an asymmtrc and wth an lowr lmt, l, such as th rotatonal and of watr: S / d ( / )xp( u / ) (1) T o l o [dwards,1976] (15) T T o o (16) T o T T Th alus of 1 } o ( A B o,,,, and c u o o o l (17) for arous gass ar gn Tal 1. From qs. (11) and (17),, th ln wdth to spacng paramtr s dfnd as: d d T o o (18) T, th ffct roadnng prssur, s drd from q. (17). Usng th mol rato, x A /, s: ]} n {[ / o ][1 ( 1) x (19) Th xponnt, n, s an mprcal fudg factor, usd to otan ttr agrmnt wth xprmntal data [dwards, 1976]. n s proaly a corrcton for anharmonc ffcts and ars from 0.6 to 0.8. Valus for n and for arous gass ar gn n Tal 1. s a masur of th dgr to whch nddual spctral lns orlap. arg alus of rprsnt larg orlap. Th two paramtrs and rprsnt rspctly th ffcts of tmpratur and prssur on ln orlap. For watr apor, ths dson twn prssur and tmpratur dos not hold tru caus th slf-roadnng to forgn-gas roadnng rato,, us to calculat s found xprmntally to a functon of tmpratur. 100K H O T Ths may du to th fact that watr apor s much mor polar than th othr comuston gass. Th polar nds of th watr molculs ar attractd to ach othr, and ths wll promot collsons. For orlappng ands, th transmttanc s ssntally th products of th transmttanc n ach and [dwards and Balakrshnan, 1973]. For a mxtur of N gass, ach wth N () orlappng ands: N g N ( ) N g N ( ) 1 g,,, 1 (1 g,,, g,, mx A photon s mttd from a surfac n a drcton gn y th msson ctor,, from th msson pont, x. Th Cartsan quaton that dfns th msson ln s: x x x y y y z z z Th quaton that dscrd ach cartsan surfac, s: ) (1) () g 35 dtor@am.com

7 ffcts of th comuston Gass on th Radaton Hat Transfr x x1, y y1, z z1, x y z whr ( x1,, y1,, z1, ) ar th coordnats of th frst nod of th surfac, and, x y, and z ar th coordnat dsplacmnts to nod 1. Usng ths quatons, th surfac whch ntrcpts th photon n th shortst dstanc can found. Th partculars of th tracng algorthm for prsmatc gomtrs ar gn y Burns and ryor [1987]. To dtrmn whthr a st s an oct or an nclosur, th surfac ctor x for surfac s dfnd from nod 1 to as: x x, x1, Th cumulat proalty that an nt wll occur s dtrmnd y ntgratng th DF or th ndpndnt aral(s) and s trmd th cumulat dstruton functon, or CDF. For a gas [Modst, 1993]: CFD( ) 0 0 ( ) ( ) For a mxtur of N g d d N gass, ach wth N () ands: 1,, (3) (5) g N ( ) S ( ) (6) 1 d As th photon packt trals through th gomtry, th attnuaton of th photon packt n ach grd cll s calculatd, and a cumulat rcord of th nrgy asord y ach cll s kpt. Th photon packt s tracd untl ts nrgy drops low a spcfd fracton, f, of ts orgnal nrgy. f s gnrally st to Fgur 1 shows an xampl of th photon path. Whn th photon packt ntrs a nw grd cll, frst a chck s don to dtrmn f t s ntrcptd y any surfac. If t s ntrcptd y a surfac, thn s, th dstanc tralld through th grd cll, s computd to th pont of ntrcpton. If t s not ntrcptd, thn s s calculatd usng th pont on th grd whr th photon xts. Du to th ffcts of wanumr aragng, th calculaton of th amount of nrgy asord y a grd s complcatd whn th photon s rflctd from a surfac. For ths rason, th asorpton of nrgy y a surfac wll dscussd for th asorpton n a grd. For a surfac: ( ) (7) surf whr I nc sn 1 ( R), R Randomnumrstwn1and0 (8) Th amount of nrgy asord y th grd cll s: C 1)]} (9) grd { g, mx[ s( )] g, mx[ s( m whr m s th nrgy of th photon packt at msson, and s() s th dstanc tralld from th pont of msson to th pont th photon packt xts th grd. If th photon packt s rflctd, thn th calculatons ar don to fnd th nrgy asord for and aftr httng 36 dtor@am.com

8 Dr. Aas Alw Sakhr Ad th wall. C s a corrcton factor that ndcats what fracton of th nrgy of th nrgy s asord onc rflctons ar takn nto account. If thr ar no rflctons, th amount of nrgy aalal to asord y th gas past a crtan pont s th nrgy transmttd up to that pont,. Whn rflctons ar consdrd, ths amount must rducd y th tr, NR nrgy asord n all rflctons wth surfacs, as, surf. As a frst approxmaton, C mght consdrd ust ( ). Th rror n ths formulaton s that th dffrnc tr, NR as, surf / tr, NR twn th nrgy asord y th gas f thr wr no rflctons, asord f rflctons ar takn nto account, as NR,, and th nrgy as NR, s sutractd twc. It s consdrd part of nrgy asord f thr wr no rflctons and also as part of th nrgy asord y th surfacs. Countng ths nrgy only onc, th corrcton factor s: C tr, NR as, surf as, NR a, NR (30) tr, NR Th Mont Carlo rqur th calculaton of sral lngths, aras, and olums. For a two-dmnsonal surfac wth nods 1 and, snc th surfacs ar lnar, th lngth of th surfac s gn y: D, ( x, x1, ) ( y, y1, ) (31) To calculat th ara of st n a two-dmnsonal Cartsan gomtry, Grn s thorm n th plan s usd [Kryszg, 1993] A, D dxdy (31) A Agan, th N sg surfac sgmnts that mak up th st ar lnar. Snc th contour around th st mght trarsd n thr drcton, th asolut alu of th rsult s takn: A N y sg, N sg D, xdy 0.5( y, k y1, k )( x, k x1, k ) k 1 y k 1 1, k To stalsh spctral wndows, th followng procdur s usd. Frst, th gas msson 1 1 spctrum, 0 to 8,000 cm, s ddd nto ntrals of usually 5 cm. Nxt, an artrary charactrstc lngth,, s chosn. For th work n ths rsarch, s takn as 1 m. For C ach ntral, th total asorptty or th dstanc C s calculatd for th condtons n ach grd. Th maxmum of all calculatd asorptts s found. If, for all grd clls, th asorptty at th gnnng of th ntral s lss than a fractonal thrshold, f 1, of th maxmum asorptty, th ntral s markd as a spctral wndow. No asorpton calculatons ar don n a spctral wndow, and th gas walngth nrs CDF dos not nclud ths rgon. To furthr rduc calculatons, th asorptty of nddual asorpton ands also ar compard to th maxmum total asorptty. For ach ntral, only ands for whch th asorptty for at last on grd cll n th gomtry s ao a scond fractonal thrshold, f, ar usd n calculatons. Typcal alus for th cutoff fractons ar f and f 110. Th numrcal paramtrs,, f 1, and f must chosn as a comproms twn accuracy and ffcncy. Too small a alu of wasts computr mmory. Too larg a alu of causs th wrong asorpton ands to usd n calculatons. If f1 and C (3) 37 dtor@am.com

9 ffcts of th comuston Gass on th Radaton Hat Transfr f ar too small, too many asorpton ands wll usd n calculatons, and spctral wndows wll not usd to adantag. Choosng alus too larg for f 1 and f wll caus th long rang asorptty, whch occurs n th wngs whr th asorptty s low, to modld mproprly. 3. RSUTS AND DISCUSSION 3.1. Gas-Gas Transmttanc rfcaton Th concpt of a corrcton factor for gas to gas asorptanc s anothr da rqurng rfcaton. Th gas to gas asorptanc formulas ar not gn y Modst [199] ut ar drd from random numr rlatons h gs for a proalstc Mont Carlo formulaton. To rfy that th da s ng appld corrctly, th two Mont Carlo tst cass gn y Modst n hs papr [199] ar don. Both tsts us a on-dmnsonal gomtry of partcpatng mda twn two nfnt plats 1 mtr apart, as shown n Fg.. Btwn th nfnt plats s a Goody modl gas wth on spctral asorpton and. Ths and s 1 symmtrc wth th and cntr at 3,000 cm. For ths gas,, 1 do not ary wth tmpratur. 00 cm and and low. Th frst cas consdrd s an sothrmal sla of gas at 1,000 K twn 0 K nonmttng plats. For ths prolm, th quantty of ntrst s th nondmnsonal hat flux, whch s dfnd n ths cas as: q (33) T m Tm s th tmpratur of th mdum (1,000 K). from th cntr ln. In ths tst prolm, s masurd at th wall. Th scond cas s that of th gas n radat qulrum ound y two sothrmal lack plats. Th tmpratur of th top plat, T t, s 1,00 K whl th tmpratur of th lowr plat, T, s 800 K. Agan, th quantty of ntrst s whch s dfnd for ths cas as: q (3) ( T t T ) Snc th prolm s at radat qulrum, s constant. Th gomtry for whch GASD s usd to modl a sla of gas twn two nfnt plats s shown n Fg.. Th two plats ar modld y a rctangl wth totally spcularly rflctng sds. Th lft and rght oundars xht symmtry (.. thy ar prfct spcular rflctors). Th spac twn th slas s ddd nto a numr of gas rgons, ach of whch s sothrmal. For ths cass, th 7 5 alus of th paramtrs usd ar: f 110, f 110, f 110, C =1.0m, 1 5cm, N I, g 5, 001 N gp, s 1, and f nt, g 0. 01cm, gp, g 1 1 N, N I, s 1, 001, 1 nt, s 0. 5cm,. Varaton of ths paramtrs, ndcats that ths alus g rsults accurat to wthn %. Th rsults ar shown n Tal. Th rsults match narly xactly to two sgnfcant fgurs xcpt n th last st of alus for cas whr thr s a 1.3% dffrnc. Thrfor, th gas to gas asorptanc modl appars corrct. 3.. Gray gas n a rctangular nclosur Although an mphass of ths work s two-dmnsonal gomtrs, no smpl twodmnsonal cass wth whch to compar could found. To rfy that GASD conrgs to 38 dtor@am.com

10 Dr. Aas Alw Sakhr Ad th propr xchang fractons for two-dmnsonal surfacs, th cas of a two-dmnsonal rctangular nclosur flld wth gry gas s sold numrcally. From th asc dfnton of xchang fractons n Sgl and Howll [199], th xchang fracton twn surfacs 1 and wthn an ntrnng gry gas s formulatd as a thr-dmnsonal ntgral: F 1 1 A 1 A A 1 xp( ks)cos1 cos da da s 1 whr s s th path lngth twn da1 and da. A smlar xprsson can usd to otan th xchang fracton twn a olum of gas and a surfac. To tst th aldty of th cod for two-dmnsonal gomtrs, four cass ar sold for a squar wth sds of lngth on. Cas 1: all surfacs lack, no gas, k = 0 Cas : all surfacs lack, k=0.5 Cas 3: surfac prfctly spcularly rflctng, all othr surfacs lack, no gas Cas : surfac prfctly spcularly rflctng, all othr surfacs lack, k = 0.5 Fgur 3 shows th dtals of th gomtry. For ths trals, only sd 1 and th gas ar mttng, and th gas xsts ntrly n on grd. In th GASD runs, on mllon photons ach ar mttd from sd 1 and th gas. Gaussan quadratur s usd to prform th ntgratons wth 160 ponts pr ntgral. Th Gaussan quadratur ponts ar gnratd usng a suroutn from Numrcal Rcps n FORTRAN, nd dton [rss t al., 199]. Th soluton of th ntgrals n two dmnsons rqurs ntgratng from to n z drcton, ut Gaussan quadratur can only don or fnt ntrals. To orcom ths prolm, th susttutons ar mad, z tan(t) (36) (35) dz (1 tan( t) ) dt (37) For cass n whch thr s no gas, th xact soluton s otand usng Hottl s crossd strngs mthod [Modst, 1993]. For cass wth a spcularly rflctng sd, th soluton y numrcal ntgraton and th crossd strngs mthod ar otand y rplacng sd wth a mrror mag of th othr sds. Dtals ar shown n Fg.. For ths gomtry, th xchang fractons from mttng surfac or olum to sd ar rlatd to th xchang fractons to sd n th orgnal gomtry y: F F F (38) aa a Th rsults ar shown n Tals 3 and. In gnral, th numrcal soluton conrgs asymtotcally from low to th corrct answr as th numr of Gaussan ntgraton ponts s ncrasd. For th calculaton of F 1, th ntgraton conrgs to th soluton wth many fwr Gauss ponts. Th rsults ar xtrmly good. Th GASD solutons ar wthn 0.1% of th xact solutons and.1% of th numrcal solutons n all cass. Consdrng th rror of up to 1.% twn th numrcal and xact solutons, th GASD solutons ar ry clos to th numrcal solutons Gray gas twn nfnt paralll plats A common nchmark prolm n gas radaton s th on-dmnsonal non-sothrmal gry gas twn two paralll nfnt plats n radat qulrum. A wll known numrcal soluton for ths prolm s on drd y Haslt and Warmng [Haslt and Warmng, 1965; Modst, 1993]. For ths prolm th d, s dfnd n q. (3) dtor@am.com

11 ffcts of th comuston Gass on th Radaton Hat Transfr Snc th prolm s at radat qulrum, s constant. For lack slas, th nondmnsonal hat flux,, s a functon of optcal dpth,, only. k (39) Whr k s th asorpton coffcnt (a constant for a gry gas), and s th dstanc twn th two slas. Tal 5 lsts as a functon of optcal dpth. If th plats ar dffusly rflctng and mttng wth total hmsphrcal mssts, 1and, thn: Bsds actng as a gnral tst of GASD, ths prolm also tsts th accuracy of th nrgy alanc cod. Th frst tst dtrmns th numr of grd clls ndd to modl th prolm as. For ths tst, a gomtry of Fg. s usd. On mllon photons ar mttd smlar to that dscrd n gas-gas transmttanc tst s usd. On mllon photons ar mttd pr lmnt. Th rsults ar shown n Fg. 5. s orstmatd n ry cas xcpt for = 0.1. Th numr of grds rqurd to modl th prolm proprly ncrass as ncrass. rrors lss than 0.1% ar du to statstcal scattr n th Mont Carlo rsults. As sn n th graph, ths ll s rachd usng only on grd whn = 0.1, ut nn grds ar rqurd fo = 0.5, and n forty-f grds ar not suffcnt for = Watr Vapor twn Infnt lats Ths cas assumd th dstanc twn th plats s 1 m, th numr of grds s 0, and th numr of photons mttd pr grd s 50,000. From past xprnc wth MONTD, th numr of photons sms too low to yld accurat rsults. For ths rason, anothr cas whr on mllon photons ar mttd pr grd s don. Th rsults ar shown low n Fg. 6 and n Tal 6. As th graph shows, th GASD rsults for 50,000 photons sms as accurat as u s rsults for 50,000 photons, ut oth xht fluctuatons. Whn th numr of photons s ncrasd to on mllon, a smooth cur s otand. Tal 6 compars th calculatd alu of q, th hat flux at th wall, for all thr cass. For ach GASD cas, w thr runs ar don. Th rang shown for q w rsults from th statstcal scattr for ths thr runs. Th dffrnc n q w twn u s rsults and GASD s rangs from 8.6% to 10.5%. Snc th orgnal dwards gas modl only corrlats to aout 15% wth xprmntal data, th rsults ar as clos as can xpctd. Th ffct on q w of th largr numr of photons s not as dramatc as that for q / y. By ncrasng th numr of photons y factor of 0, 3 3 th statstcal scattr n q w droppd from 0.53 kw / m to 0.13 kw / m of photons has a roughly proportonal ffct on run tm. (0). Incrasng th numr 3.5. CO and H O twn nfnt paralll plats Anothr Mont Carlo prolm nolng molcular gass s on y Tanguch t al. [199]. Ths prolm nols a mxtur of CO : H O : N n a mol prcnt rato of 9.5:19.0:71.5 at on atmosphr prssur. Th gas mxtur s typcal of products of comuston of mthan. Hr, th gas mxtur s n radat qulrum twn two plats on mtr apart. Th ottom plat s at 1,500 K, and th top plat s at 1,000 K. To match th orgnal prolm, th gomtry s ddd nto f grds, and snc th numr of photons s not spcfd, on 0 dtor@am.com

12 Dr. Aas Alw Sakhr Ad mllon ar mttd from ach lmnt. To dtrmn th ffcts of ncrasng th numr of grds, ths cas s rpatd usng nn and twnty-sn grds. Snc th prolm s n radat qulrum, t must sold tratly. In addton, th surfac wanumr 1 paramtrs ar st to: N I, s 1, 001, nt, s 0. 5cm, N gp, s 1, and f 110. Th rsults ar shown n Fg. 7 and Tal 7. Th graph clarly shows that th GASD rsults for much hghr than Tanguch s, whl th tal shows that q w dffrs only y 6.1% to 6.5%. Whr, T T ( y) (1) T T t t Th rason th tmpratur profls dffr y so much may that Tanguch s gas modl s ncorrct. Th gas modl Tanguch uss s th lsassr modl wth dwards wd and paramtrs. dwards wd and paramtrs ha n formulatd to work wth th Goody modl and ha nr for n usd wth th lsassr modl. Whn prsntng ths modl, Tanguch pros that dwards wd and paramtrs can usd to calculat th mssty of gass; h dos not pro that hs formulaton of th lsassr modl s corrct. As dnt from th graph and th tal, ncrasng th numr of grds has ry lttl ffct; all th s ar aout th sam, and q w drops y 1.1 kw / m or 0.55% at most. F grds ar thrfor suffcnt. Whn th numr of grd clls s ncrasd, th numr of lmnts mttng photons s ncrasd, and th numr of grds clls a photon must trars also ncrass. For th on-dmnsonal gomtrs of th tst cass, thr ar only thr or four surfacs and many photons must tracd through all grd clls. In gnral, ncrasng th numr of grd clls n a on-dmnsonal gomtry y afactor of N ncrass th run tm y a factor of N. CONCUSIONS 1-Comparson of th calculatons of th dwards/goody modl to xprmntal rsults otand y Hns [196], pros that th modl can accuratly calculat th asorpton spctra of mxturs of caron doxd and watr. -Th comparson to u and Twar s [199] rsults dmonstrats th accuracy of th dwards/goody gas modl n GASD. Th rsults of Tanguch t al. [199] do not match as wll, ut ths s ld to causd y an rror n thr gas modl. 3-Th ral gas modl dos g much ttr answrs than usng th gry gas approxmaton or totally gnorng gas ffcts. Th rsults of usng th ral gas modl g answrs that ar two to thr tms mor accurat thn thos otand gnorng th ffcts of th gas compltly. Th gry gas approxmaton s found always to orstmat th wall hat flux y from % to 15%. -For th Goody modl, usng th Curts-Godson approxmaton, th spctrally smoothd surfac to gas asorptanc s ry dffrnt than th spctrally smoothd gas to gas asorptanc. Ths rsult suggsts that car must takn whn usng spctrally aragd quantts or larg rrors may occur. Snc most soluton mthods for radat hat transfr n partcpatng mda us som form of spctral aragng, ths rsult has far rangng mplcatons. 1 dtor@am.com

13 ffcts of th comuston Gass on th Radaton Hat Transfr Tal 1 Valus of,,,,, n c u o o o,, and l for Comuston Gass (dwards [1976]). aramtr c u o cm 1 l CO H O CH CO NO SO 1 cm cm cm o o / g. m cm K H O T n Tal Comparson to Modst s Rsults for C A S 1 C A S Vaus Modst GASD,000, ,000, ,000, ,000, ,000, ,000, Tal 3 Comparson of xchang Fractons from Sd 1 dtor@am.com

14 Dr. Aas Alw Sakhr Ad Tal Comparson of xchang Fractons from th Gas Tal 5 Nondmnsonal Radat Hat Transfr Tal 6 Comparson to u and 1, / Twar s Rsults for q w Cas ( kw / m ) q w u 50,000 photons -7.6 GASD 50,000 photons -5.3,-.7 GASD 1 mllon photons -5.07,-.9 Tal 7 Comparson to u and Tanguch t al. Rsults for Cas ( kw / m ) q w Tanguch 5 grds 13. GASD 5 grds 199.9,00.3 GASD 9 grds 199.,199.7 GASD 7 grds 199.,199.3 q w Fgur 1 hoton ath 3 dtor@am.com

15 ffcts of th comuston Gass on th Radaton Hat Transfr Fgur aralll Infnt Slas Fgur 3 Squar Gomtry Fgur Modl of a Spcularly Rflctng Squar Fgur 5 rcnt rror as a Functon of and Numr of Grd Clls dtor@am.com

16 Dr. Aas Alw Sakhr Ad Fgur 6 Comparson to u and Twar s Rsults for q / y RFRNCS Fgur 7 Comparson to Tanguch t al. Rsults for [1] Burns,. J. and ryor, D. V., Vctorzd Mont Carlo Radat Hat Transfr Smulaton of th asr Isotop Sparaton rocss, ISC Tchncal Rport 8700, Insttut for Scntfc Computng, Fort Collns, CO. [] Burns,. J., Malty, J. D., and Chrston, M. A., arg-scal Surfac to Surfac Transport for hotons and lctrons Va Mont Carlo, Computng Systms n ngnrng, Vol.1 No. 1, pp dtor@am.com

17 ffcts of th comuston Gass on th Radaton Hat Transfr [3] Burns,. J., ohrk, R. I., Dolaghan, J. S., and Malty, J. D., 199. hoton Tracng n Axsymmtrc nclosurs, Dlopmnts n Radat Hat Transfr, Amrcan Socty of Mchancal ngnrs, HTD-Vol. 03, pp [] dwards, D. K. and Balakrshnan, A., Thrmal Radaton y Comuston Gass, Intrnatonal Journal of Hat and Mass Transfr, Vol. 16, pp [5] dwards, D. K., Molcular Gas Band Radaton, n Vol. 1 of Adancs n Hat Transfr, Irn, T. F. and Hartntt, J.. (ds.), Acadmc rss, Nw York, pp [6] Farmr, J. T. and Howll, J. R Hyrd Mont Carlo/Dffuson Mthods for nhancd Soluton of Radat Transfr n Optcally Thck Nongray Mda, Radat Hat Transfr: Currnt Rsarch, Amrcan Socty of Mchancal ngnrs, HTD-Vol. 76, pp [7] Howll, J. R. and rlmuttr, M., 196a. Mont Carlo Soluton of Thrmal Transfr Through Radant Mda Btwn Gray Walls, ASM Journal of Hat Transfr, Vol. 86, pp [8] Haslt, M. A. and Warmng, R. F., Radat Transport and Wall Tmpratur Slp n an Asorng lanar Mdum, Intrnatonal Journal of Hat and Mass Transfr, Vol. 8, pp [9] Howll, J. R., Applcaton of Mont Carlo to Hat Transfr rolms, n Vol. 5 of Adancs n Hat Transfr, Irn, T. F. and Hartntt, J.. (ds.), Acadmc rss, Nw York, pp [10] Ha-Shkh, A., Mont Carlo Mthods, Chaptr 16 n Handook of Numrcal Mthods n Hat Transfr, Mnkowycz t al. (ds.), John Wly & Sons, Nw York, pp [11] Howll, J. R. and rlmuttr, M., 196. Mont Carlo Soluton of Radant Hat Transfr n a Nongray Nonsothrmal Gas wth Tmpratur Dpndnt roprts, Amrcan Insttut of Chmcal ngnrs Journal, Vol. 10, No., pp [1] Kryszg,., Adancd ngnrng Mathmatcs, 7th d., John Wly & Sons, Nw York. [13], W., Tong, T. W., Doranch, D., and Grtzo,. A., A Comnd Narrow- and Wd-Band Modl for Computng th Spctral Asorpton Coffcnt of CO, CO, H O, CH, CH and NO, Journal of uanttat Spctroscopy and Radat Transfr, Vol. 5, No. 6 pp [1] u, J. and Twar, S. N., 199. Instgaton of Radat Transfr n Nongray Gass Usng a Narrow Band Modl and Mont Carlo Smulaton, ASM Journal of Hat Transfr, Vol. 116, pp [15] Modst, M. F., Radat Hat Transfr, McGraw-Hll, St. ous. [16] Modst, M. F., 199. Th Mont Carlo Mthod Appld to Gass wth Spctral n Structur, Dlopmnts n Radat Hat Transfr, Amrcan Socty of Mchancal ngnrs, HTD-Vol. 03, pp dtor@am.com

18 Dr. Aas Alw Sakhr Ad [17] Malty, J. D., Z, C. N., Dolaghan, J., and Burns,. J., 199. Usr s Manual for MONTD - Vrson.6 and MONT3D - Vrson.3, Dpartmnt of Mchancal ngnrng, Colorado Stat Unrsty, Fort Collns, CO. [18] rlmuttr, M. and Howll, J. R Radat Transfr Through a Gray Gas Btwn Concntrc Cylndrs Usng Mont Carlo, ASM Journal of Hat Transfr, Vol. 86, pp [19] rss, W. H., Tukolsky, S. A., Vttrlng, W. T., and Flannry, B.., 199. Numrcal Rcps n FORTRAN: Th Art of Scntfc Computng, nd d., Camrdg Unrsty rss, Nw York. [0] Stward, F. R. and Cannon,., Th Calculaton of Radat Hat Flux n a Cylndrcal Furnac Usng th Mont Carlo Mthod, Intrnatonal Journal of Hat and Mass Transfr, Vol. 1, No., pp [1] Sgl, R. and Howll, J. R., 199. Thrmal Radaton Hat Transfr, 3rd d., Hmsphr ulshng, Washngton. [] Tanguch, H., Kudo, K., Otaka, M., Sumarsono, M., and Oata, M Dlopmnt of a Mont Carlo Mthod for Numrcal Analyss on Radat nrgy Transfr Through Non-Gry-Gas ayr, Intrnatonal Journal for Numrcal Mthods n ngnrng, Vol. 35, pp [3] Twar, S. N. and Gupta, S. K., Accurat Modllng for Infrard Radaton, ASM Journal of Hat Transfr, Vol. 100, pp [] Waltrs, D. V. and Buckus, R. O., 199. Mont Carlo Mthods for Radat Hat Transfr n Scattrng Mda, Chaptr 3 of Vol. 5 of Annual Rw of Hat Transfr, Tn, C.. (d.), CRC rss, Ann Aror, pp dtor@am.com