Analysis with boundary elements of heat conductivity in steady state regime

Transcription

1 Recen Advances n Fud Mechancs, Hea & Mass Transfer and Bgy Anayss wh bundary eeens f hea cnducvy n seady sae rege Prf. dr. eng. IOA SÂRBU Deparen f Budng Servces Pehnca Unversy f Tsara Paa Bserc, n. 4A, Tsara ROMAIA an.sarbu@c.up.r, Absrac: Svng he dfferena equan f hea cnducn he eperaure n each pn f he bdy can be deerned. Hwever, n he case f bdes wh bundary surface f sphscaed geery n anayca ehd can be used. In hs case he use f nuerca ehds beces necessary. In hs paper he basc deas f nuerca anayss wh bundary (cnsan) eeens f cnducve hera feds generaed r nduced n pane was n seady sae rege are deveped. The eperaure dsrbun s anayzed n w varans f a eac paque usng bundary eeen ehd, peened n sfware deveped by he auhrs and anayca ehd. Ths shws he gd perfrance f he prpsed ehd. Key wrds: Hea cnducn, Seady sae rege, Bundary eeens, Maheaca de, Cpuer prgra. Inrducn Mdern cpuana echnques facae svng prbes wh psed bundary cndns usng dfferen nuerca ehds [5], [6], [8], [ 5]. uerca anayss f hea ransfer [0], [] has been ndependeny hugh n excusvey, deveped n hree an sreas: he fne dfferences ehd [7], [9], he fne eeen ehd [], [6], [8] and he bundary eeen ehd [], [3], [4]. The fne dfferences ehd (FDM) s based n he dfferena equan f he hea cnducn, whch s ransfred n a nuerca ne. The eperaure vaues w be cacuaed n he ndes f he newrk. Usng hs ehd cnvergence and saby prbe can appear. The fne eeen ehd (FEM) s based n he negra equan f he hea cnducn. Ths s baned fr he dfferena equan usng varana cacuus. The bundary eeen ehd (BEM) have advan-age ver he FDM r he FEM ces fr he fac ha nsead f fu dan dscrezan, ny he bundary s dscrezed n eeens and nerna pn psn can be freey defned. In hs paper a nuerca de wh bundary eeens fr he anayss f cnducve hera feds generaed r nduced n pan was s deveped. The eperaure dsrbun s anayzed n a sd bdy, wh near varan f he prperes, usng a sfware reazed by he auhrs n bass f he BEM. Anayca de f hea cnducn The eperaure n a sd bdy s a funcn f he e and space crdnaes. The pn crrespndng he sae eperaure vaue beng an shera surface. Ths surface n w densna Caresan syse s ransfred n an shera curve. The hea fw rae Q represens he hea quany hrugh an shera surface S n he e un: Q q d s () S where he densy f hea fw rae q s gven by he Furer aw: q λ λ grad () n n whch λ s he hera cnducvy f he aera. The hera cnducvy f he budng aeras s he funcn f he eperaure and varan can accrdngy be expressed as: λ λ b (3) [ ( )] 0 + n whch: λ 0 s he hera cnducvy crrespndng he 0 eperaure; b aera cnsan. If here s hea cnducn whn an nhgeneus and ansrpy aera, cnsderng he hea cnducvy cnsan n e, he eperaure varan n space and e s gven by he Furer equan: ρ c λ x λ y + λ z + Q0 τ x x + y y z z (4) 0 ISB:

2 Recen Advances n Fud Mechancs, Hea & Mass Transfer and Bgy n whch: s he eperaure; τ e; ρ aera densy; c specfc hea f he aera; λ x, λ y, λ z hera cnducvy n he drecns x, y and z; Q 0 pwer f he nerna surces. T sve he dfferena equans s necessary have suppeenary equans. These equans cnan he geerca cndns f he anayss fed, he sarng cndns (a τ 0) and he bundary cndns. The bundary cndns (Fg. ) descrbe he neracn beween he anayzed fed and he surrundngs. In funcn f hese neracns dfferen cndns are pssbe: Fg. Bundary cndns he Drche (ype I) bundary cndns gve us he eperaure vaues n he bundary surface S f he anayzed fed ke a space funcn cnsan r varabe n e: f ( x, y, z, τ) (5) he euann (ype II) bundary cndns gves us he vaue f he densy f hea fw rae hrugh he S q bundary surface f he anayzed fed: q λ x nx + λ y n y + λ z nz (6) x y z n whch: n x, n y, n z are he csne drecrs crrespndng he nra drecn n he S q bundary surface. he Cauchy (ype III) bundary cndns gves us he exerna eperaure vaue and he cnvecve hea ransfer ceffcen vaue beween he S α bundary surface f he bdy and he surrundng fud: α ( e ) λ x nx + λ y n y + λ z nz (7) x y z n whch: α s yhe cnvecve hea ransfer ceffcen fr S α he fud (r nversey); e he fud eperaure. The anayca de descrbed by he equans (4) (7) can be cpeed wh he aera equans whch prvde us nfran abu varan f he aera prperes dependng n eperaure. In he case f aeras wh near physca prperes, hs equans (λ cns.) are n used n he de. Svng he dfferena equan f he hea cnducn (4) we can deerne he eperaure vaues n each pn f he bdy. Hwever, n he case f bdes wh bundary surface f sphscaed geery, he equan (4) cann be sved usng anayca ehds. In hs case nuerca ehds shud be apped. The ncreasng avaaby f cpuers has as ead n he drecn f re frequen use f hese ehds. 3 uerca de wh bundary eeens f hea cnducvy n seady sae rege Ahugh hera phenena ake pace n hru densna bdes, he hera feds ha ccur have prednan varans n ceran drecns. Ths s why he anayss f hera fed n pane r cyn-drca was s usuay perfre usng w den-sna cpuana des. In seady sae hea ransfer prcesses he eperaure s a cnsan f e (/ τ 0), and fr w densna prbes he eperaure des n vary n z drecn, hus / z 0. In he case f a fa wa, nsde he anayss fed, he hea cnducvy n seady sae rege s deed by he Lapace equan [3]: 0 (8) On prn f bundary f he anayss fed Drche bundary cndns are psed and efner prn q euann bundary cndns are psed. In rder deerne he eperaure n he bundary f he anayss fed ne uses he fwng negra equan [], [3], [5]: ζ ζ ζ ζ ( (9) c( ) ( ) + ( v (, d( u (, d( n where: ζ s he pn n whch ne wres he negra equan(surce pn); c(ζ) a ceffcen; X he curren negran pn; u ( ζ, n π r( ζ, fundaena sun; u v nra der- n vave f hs sun. X The dsance r(ζ, X ) beween he curren pn and he surce pn ζ s cacuaed wh he rean: ISB:

3 Recen Advances n Fud Mechancs, Hea & Mass Transfer and Bgy r ( ζ, x( x( ζ ) + y( y( ζ ) (0) Bundary s dscrezed n cnsan bundary eeens fr whch ne cnsders eperaures, respecvey he nra dervave (/ n) cnsan and equa he d pn (nde) vaue f he ee-en. Thus he negra equan s baned under he fwng dscrezed fr: c r + ζ () v (, d( u ( ζ, d( n c + ˆ B n () A d n whch ceffcens $ A and B have he expressns: Aˆ v ( ζ, d( ; B u ( ζ, d( (3) When hese bece: + ˆ A A ; B n (4) π Expcey, equan () generaes a nar and cpabe syse f equans wh unknwns [ ş (/ n) ] and afer peenng he bundary cndns, he nuber f unknwns s reduced. In he case f cnsan bundary eenens, cef-fcen c has he vaue /. Ceffcens A $ and B fr (3) s cpued usng a Gauss quadraure [7], [5]: A ˆ vk wk ; B uk wk (5) k k n whch s he engh f he bundary eeen. Inrducng nans: n x cs(n, x); n y cs(n, y) and usng, fr X, he paraerc equans: [,] x Aξ + B; y Cξ + D, ξ (6) where: x [x, x + ] and y [y, y + ], he fwng reans are baned: C A nx ; ny (7) A + C A + C n whch (x, y ) and (x +, y + ) are he exrees f he bundary eeen. The anayss fed s ransfred n a densness ne by repacng he densna varabes (x, y) wh densness nes (x, y ): x y x ; y (8) x x ax ax n whch x ax s he axu exensn f he anayss fed afer axs Ox. In rder deerne he eperaure nsde f he anayss fed s used he negra represenan: ( ( ζ ) u ( ζ, d( ( v (, d( ζ (9) n n whch: ζ Ω, where Ω represen he nsde f he anayss fed Ω ( Ω Ω U ). Afer he dscrezan f bundary n cnsan bundary eeens ne bans he negra equan under dscrezed fr: ( ζ ) ζ v (, d( n u ( ζ, d( whch can be wrhen as such: Ceffcens Gauss quadraure: A (0) B A () n A and B are evauaed usng a vk wk ; B k k u k w k () n whch: s he nuber f Gauss ype pns; w k wegh ceffcens. Teperaures fr pns ζ are easy deerned akng n accun ha vaues and (/ n ) are knwn n he anayss fed bundary, and ceffcens A and B are cpued wh equan (6). By knwng vaues and f he eperaure n he anayss fed bundary, he grup f crdnae pns (x, y ) fr whch cns. represens he shera curves. The nuerca de deveped abve, based n BEM, was peened n a cpuer prgra reazed n FORTRA prgrang anguage, fr IBM PC cpabe syses. 4 uerca appcans In fgures and 3 are cnsdered w varans f a eac paque, wh densns , fr whch ne deernes he eperaure fed usng BEM and anayca ehd (AM). In fgures 4 and 5 are presened he densness anayss dans geher wh xed bundary cndns fr hese bundares. ISB:

4 Recen Advances n Fud Mechancs, Hea & Mass Transfer and Bgy presened n Tabe, cparavey wh he nes baned wh AM []. Fg. Meac paque Fg. 3 Meac paque wh a secyndrca cu u Fg. 4 Bundary cndns fr eac paque Fg. 6 Dscrezan f he bundary and nerna pns f anayss fed The absue percenage vaue f he reave dfference ward he anayca sun, fr bh he eperaure ε and nra dervave ε nd s defned by: ε AM AM BEM 00; (3) n AM n BEM ε nd 00 n AM Takng n accun he resus fr Tabe when appyng equans (3), accepabe vaues have been baned fr ε and ε nd (ε <.3%, ε nd < 3.5%) even f he nuber f bundary eeens cnsdered s sa. Fr eac paque n Fgure 3 he bundary can be dscrezed n 56 cnsan bundary eeens (Fg. 7) and usng BEM was deerned shera curves presened n Fgure 8. Fg. 5 Bundary cndns fr eac paque wh secyndrca cu u Fr eac paque n fgure he bundary can be dscrezed n 6 bundary eeens, ne saes 9 nerna pns (Fg. 6) and ne appes he cpuana de based n BEM. The nuerca resus baned by eans f an IBM cpuer are Fg. 7 Bundary dscrezan fr he paque wh secyndrca cu u ISB:

5 Recen Advances n Fud Mechancs, Hea & Mass Transfer and Bgy Tabe. The vaues and n Pn Crdnaes BEM AM x y n n , Fg. 8 Teperaure dsrbun fr he paque wh secyndrca cu u 5 Cncusns In pracce here are any suans where s ndspensabe knw he eperaure dsrbun n a bdy (e.g. n dfferen echanc and eecrnc cpnens). In cv engneerng s pran anayse he eperaure dsrbun n hera brdges, n ppe was, n nsuan aeras [9]. The nuerca deng wh bundary eeens represens an effcen way ban eperaure dsrbun n seady sae cnducve hea ransfer prcesses. The nuerca cpuan f he eperaure fed, n he bass f he bundary eeen ehd, has ed cse vaues he nes deerned anaycay even f a sa nuber f bundary eeens and respecvey nerna pns f he anayss dan was used. Usng he presened ehd, dfferen suan prgras cud be reazed wha akes pssbe effecuae a f dfferen nuerca experens f pracca prbes. References: [] Asad, A.S. Raa, S. Fne eeen hea ransfer and srucura anayss, Prceedngs f he WSEAS/IASME In. Cnference n Hea ISB:

6 Recen Advances n Fud Mechancs, Hea & Mass Transfer and Bgy and Mass Transfer, Crfu, Greece, Augus 7-9, 004, pp. -0. [] Banergee, P.K. Buerfed, R. Bundary Eeen Mehds n Engneerng Scence, McGraw H, Lndn, ew Yrk, 98. [3] Brebba, C.A. Tees, J.C. Wrbe, I.C. Bundary Eeen Technques, Sprnger Verag, Bern, Hedeberg, ew Yrk, 984. [4] Chen, G. Zhu, J. Bundary Eeen Mehds, Acadec Press, ew Yrk, 99. [5] Gafţanu, M. Peraşu, V. Mhaache,. Fne and bundary eeens wh appcans cpuang f achne cpnens, Technca Pubshng Huse, Buchares, 987. [6] Gdunv, S.K. Reabenk, V.S. Cacuan schea wh fne dfferences, Technca Pubshng Huse, Buchares, 977. [7] Isf, A. uerca sun wh near bundary eeens f hera feds n seady sae rege, Prceedngs f he h In. Cnference n Budng Servces and Aben Cfr, Tsara, Apr 8-9, 00, pp [8] Irns, B.M. Ahad, S. Technques f fne eeens, Jhn Wey, ew Yrk, 980. [9] Jóhanessn, G. Lecures n budng physcs, Kung Teknska Högskan, Sckh, 999. [0] Kays, W.M. Crawfrd, M.E. Cnvecve hea and ass ransfer, McGraw H, ew Yrk, 993. [] Leca, A. Madn, C.E. San, M. Hea and ass ransfer, Technca Pubshng Huse, Buchares, 998. [] Parrdge, W.P. Brebba, A.C. Cpuer peenan f he BEM Dua recprcy ehd fr he sun f genera fed equans, Cuncans Apped uerca Me-hds, v. 6, n., 990, pp [3] Ra, S. The fne eeen ehd n engneerng, Pergan Press, ew Yrk, 98. [4] Reddy, J.. An nrducn he fne eeen ehd, McGraw H, ew Yrk, 993. [5] Sârbu, I. uerca dengs and pzans n budng servces, Pubshng Huse Pyechnca, Tsara, 00. [6] Sârbu, I. uerca anayss f w densna hea cnducvy n seady sae rege, Perdca Pyechnca Budapes, v 49, n., 005, pp [7] Wang, B.L. Tan, Z.H. Appcan f fne eeen fne dfference ehd he deernan f ransen eperaure fed n funcnay graded aeras, Fne Eeens n Anayss and Desgn, v 4, 005, pp [8] Wang, B.L. Ma, Y.W. Transen ne densna hea cnducn prbes sved by fne eeen, Inernana Jurna f Mechanca Scences, v. 47, 005, pp [9] Wu, Q. Sheng, A. A ne n fne dfference ehd anayss an psed prbe, Apped Maheacs and Cpuan, v 8, 006, pp ISB: