METHOD FOR DETERMINATION OF CONFIGURATION FACTORS IN RADIATION HEAT TRANSFER. Mirosław ŻUKOWSKI

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1 The Jourl of Polsh Socet for Geometr d Egeerg Grphcs Volume 19 (9), METHOD FOR DETERMINATION OF CONFIGURATION FACTORS IN RADIATION HEAT TRANSFER Mrosłw ŻUKOWSKI Blstok Techcl Uverst, Deprtmet of Het Egeerg 45E Wesk st Blstok, Pold eml: mukowsk@p.edu.pl Astrct: The curret pper presets method for estmtg the cofgurto fctor het trsfer rdto uldgs. The umercl procedure c e used to determe ectl ths qutt eclosure wth comple geometr d strog grdet of surfces temperture. A verfcto of the method s lso reported ths work. The umercl results from the preseted method, mplemeted s computer progrm, re compred wth the ltcl soluto. A ver good greemet hs ee foud etwee these two procedures for determto of the cofgurto fctor. The totl error of the method, for the recommeded dest of surfce dvso, s out 1%. Kewords: cofgurto fctor, rdto het trsfer, umercl tegrto. 1 Itroducto I egeerg prctce d reserch, we ofte del wth closed spces tht c e chrctered smmetr of the temperture dstruto o surfces (for emple: rooms wth floor hetg or coolg celg, ehto rooms wth lrge re of glg). The log-wve rdto het trsfer frequetl represets gret prt of the het lce these tpes of eclosures. It s well kow tht the rdtve flu emtted from rtrr surfce to surfce c e ppromted the followg equto: 4 4 q = σε F θ θ, (1) ( ),,, where: σ = W/m /K 4 the Stef-Boltm costt, ε emssvt of surfces, F, cofgurto fctor, θ solute temperture of emttg surfces. A correct determto of F, fctor s dffcult d tme cosumg prolem for the comple room geometr. The cofgurto fctors c e clculted etwee perso d the surroudg surfces or ol etwee surfces. The frst ssue ws clerl referred Ro d couthors [1], Oek et l. [], d Kuh et l. [3]. I ths pper ol the secod prolem s cosdered. The et chpter presets method for determto of the cofgurto fctor usg umercl tegrto. Descrpto of the method The fudmetl formul for cofgurto fctor etwee sotherml d lckod surfces d s gve : F, = A dada, () πr A A, where: A re of the rdtve surfce, R legth of the vector (see Fg. 1), β polr gle etwee vector R d ut orml to the surfce. ISSN / PLN PTGGI

2 88 M. Żukowsk: Method for Determto of Cofgurto Fctors Rdto Het Trsfer We c ppromte Eq. () for the umercl tegrto form Eq. (3) dvdg d surfces to fte m cells A. Z r. R r. β, Y r β, X Fg. 1: Geometrc reltos for determg of the cofgurto fctor. m m,, F =, A ν, A A, (3) = 1 = 1 πr, where: v, vslt fctor (v, = 1 f da s vsle to da d v, = otherwse). The frst step of the procedure cludes: detfg sotherml prts of the eclosure d the cuttg these ples to smll detcl su-surfces. Net, the cose of β gle for ech pr of cells s determed through utomtc geerto of the tegrto pots coordtes. The ceter pot of the Crtes coordte sstem c e plced corer of the tested spce. Fg. 1 shows the geometrcl reltos for evlutg ( ) d ( ) gles f the frst su-surfce s stuted o XOY ple d the secod o YOZ ple. T. 1 cots the lgerc reltos for clcultg of tht c e emploed Eq. 3. Posto O O O O O O O O O O Tle 1. The reltos for determg, d,,, The lst step for clcultg cofgurto fctor of surfce cludes summg of the prtculr cofgurto fctors. These procedures re mplemeted s computer code.

3 The Jourl of Polsh Socet for Geometr d Egeerg Grphcs Volume 19 (9), Verfcto of the method The ccurc of developed method s checked comprso wth clculto results from ltcl formuls, whch re preseted T. [4, pp ]. The other reltos for clcultg cofgurto fctors for comple rrgemet of surfces hve tke from [5, p. 199]. c Tle. The cofgurto fctor for sc rrgemet of sotherml surfces [4]. A Smols A A A h F, ( )( ) h 1 + h + h h h l rctg rctg ( ) + π + + h h h h 1 + h + h + rctg + rctg π + h + h 1 1 rc tg c rc tg c rctg l( d1d d3 ) π c + c 4π d 1 ( + )( + c ) ( + + c ) ( + + c ) c = d = ( + + c ) ( + )( + c ) d3 = ( + c )( + c ) It s cosdered tpcl room wth floor hetg sstem wth surfce temperture grdet, show Fg., wth = ples. The sotherml surfces re dvded o m m cells. c temperture grdet Fg. : The detled shpe of the tested room. The umer of su-surfces s rgg etwee 4 d 4 per oe ple. The results of clcultos re ccurte, f the followg relto (clled summto rule) s equl to 1: = 1 = 1 F 1. (4), = It s troduced f d f fctors, whch re descred elow equtos. The ccurc creses f these fctors (verge vlue clculted for the ll ples) ted to (Fg. 3). f = F,,,, f = F F. (5) = 1 = 1 = 1 = 1 ISSN / PLN PTGGI

4 9 M. Żukowsk: Method for Determto of Cofgurto Fctors Rdto Het Trsfer,7 f,6,5,4,3,,1 m m 1 3 4,8 f,7,6,5,4,3,,1 m m Fg. 3: The depedece of f d f fctors o the umer of cells. From Fg. 3, we c e cocluded, tht the optml umer of su-surfces s 15. The totl error of umercl procedure, for cosderg cse, osclltes out 1%. Icresg the umer of m m cells ove 15 s ot recommeded the uthor for the tpcl cses. 4 Cocludg remrks The clcultos, sed o trdtol equtos, gve ver good ccurc for determto of the cofgurto fctors. But ltcl method s complcted d tme cosumg for more comple geometr. The ove dsdvtge prctcll elmtes ths procedure for lg 3-D oects wth strog temperture grdets of surfces. The method, whch s preseted the curret pper, llows to utomtcll geerte cofgurto fctors for eclosure geometr. We c ot good ccurc f the umer of su-surfces s 15. The totl error of ths umercl estmto s equl out 1% for the optml dest of surfce sudvso. Ackowledgemets Ths vestgto ws crred out uder grts of Blstok Techcl Uverst. Refereces [1]. Ro G. et l.: Algorthms for the clculto of the vew fctors etwee hum od d rectgulr surfces prllelepped evromets. Eerg d Buldgs, Vol. 19 (1), pp. 51-6, 199. []. Kuh K. et l.: Predctg hum geometr-relted fctors for detled rdto lss door spces. I: Proceedgs of the 7 th Itertol IBPSA Coferece Edhove, Netherlds, August 11-14, 3. [3]. Oek Y. et l.: Numercl clculto of gle fctors etwee hum od d rectgulr ples. I: Proceedgs of the 7 th Itertol Coferece o Ar Dstruto Rooms ROOMVENT, Aw HB, ed. Elsever, Amsterdm,. [4]. Wśewsk S. et l.: Wm cepł. Wdwctw Nukowo-Techce, Wrsw. Wd. IV, [5]. Kostowsk E.: Prepłw cepł. Wdwctwo Poltechk Śląske. Glwce, Wd. I,.

5 The Jourl of Polsh Socet for Geometr d Egeerg Grphcs Volume 19 (9), METODA OBLICZANIA STOSUNKÓW KONFIGURACJI PŁASZ- CZYZN IZOTERMICZNYCH STOSOWANA W RADIACYJNEJ WY- MIANIE CIEPŁA W prktce żerske mm cęsto do ce pomescem chrkteruącm sę dużą smetrą tempertur powerch pregród (p. w prpdku stosow ogrew podłogowego lu chłodee suftowe, sloów wstwowch dużm preskleem). Chcąc w sposó dokłd dokoć l komfortu ceplego or uwględć promeowe długoflowe w cłkowtm lse cepł, musm olcć stosuk kofgurc płsc otermcch wch róweż współckm kofgurc. Moż to ucć podstwe worów lu omogrmów, gd roptruem proste ukłd geometrce. Ntomst stosowe metod ltce lu grfce do l wm cepł drode promeow długoflowego w pomescech o łożom kstłce est skomplkowe rdo prcochłoe. W referce propoowo metodę olc wrtośc współck kofgurc uącą umercm rowąu trdcego rów cłkowego. W celu określe dokłdośc preetowe metod porówo reultt smulc komputerowe wkm dokłdm otrmm podstwe olceń wkorstem worów ltcch. Pr stosowu optmle wrtośc gęstośc podłu płsc mksml łąd umercego oscow wos około 1%. ISSN / PLN PTGGI

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