Numerical modelling of two dimensional heat transfer in steady state regime

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1 Issu 3, u 5, 435 ura g f w sa a rasfr sa sa rg Ia aru sra vg ffra ua f a u praur a p f a r. Hwvr, as f s w uar surfa f spsa gr aaa a us. I s as us f ura s s ssar. f s as gra ua f a u. s s a fr ffra ua usg varaa auus. praur vaus w aua f s., as s para sus, su fr r vu w r. Usg s w a v s as fs w a rr. s, ura g w uar s s us fr aass f a u. I s papr ar vp as as f ura aass w f s a uar (sa) s f uv ra fs gra r u s sa sa rg. praur sru s s s a pp sua s aa usg aaa a f a uar s, p w pur prgras vp aur. s sws g prfra f prps ura s. Kwrs Ha rasfr, a sa rg, s, Buar s, araa auus, ura s, Cpur prgras. M I. IROUCIO OER puaa us faa svg prs w ps uar s usg ffr ura s [6], [7], [9], [6 9]. ura aass f a rasfr [], [3] as p ug usv, vp r a sras: f ffrs [], [4], f [], [], [3] a uar [3], [4], [5]. f ffrs (M) s as ffra ua f a u, w s rasfr a ura. praur vaus w aua s f wr. Usg s vrg a sa pr a appar. f (EM) a uar (BEM) s as gra ua f a u. s s a fr ffra ua usg varaa auus. I frs as praur Ia aru s urr a Prfssr a par Ha a Bug rvs par, Pa Uvrs f sara, 333 ROMI, -a arss: a.saru@.up.r vaus w aua f s., as s para sus, su fr r vu w ra. Usg s w a v s as fs w urgua rr. I BEM as uar s sr s a ra p ps a fr f. I s papr praur sru s aa s s, w ar vara f prprs, usg EM a BEM. II. YIC MOE O HE COUCIO praur a s s a fu f a spa ras. ps rrspg sa praur vau g a sra surfa. s surfa a w sa Carsa ss s rasfr a sra urv. a fw ra rprss a ua rug a sra surfa u: s () wr s f a fw ra s gv urr aw: gra () w s ra uv f ara. ra uv f ug aras s fu f praur a vara a arg prss as: (3) [ ( )] w: s ra uv rrspg praur ara sa. If r s a u w a gus a asrp ara, srg a uv sa, praur vara spa a s gv urr ua: ρ τ w: s praur τ ρ ara s spf a f ara,, ra uv rs, a pwr f ra surs. sv ffra uas s ssar av suppar uas. s uas a g- (4) IERIO OUR f EERGY a EIROME

2 Issu 3, u 5, 436 ra s f aass f, sarg s (a τ ) a uar s. uar s (g. ) sr ra w aa f a surrugs. I fu f s ras ffr s ar pss: g. Buar s r (p I) uar s gv us praur vaus uar surfa f aa f a spa fu sa r vara : (,,, τ) f (5) ua (p II) uar s gvs us vau f s f a fw ra rug uar surfa f aa f: (6) w:,, ar s rrs rrspg ra r uar surfa. Cau (p III) uar s gvs us ra praur vau a vv a rasfr ff vau w uar surfa f a surrug fu: ( ) (7) w: s vv a rasfr ff fr fu (r vrs) fu praur. aaa sr uas (4) (7) a p w ara uas w prv us fra au vara f ara prprs pg praur. I as f aras w ar psa prprs, s uas ( s.) ar us. vg ffra ua f a u (4) w a r praur vaus a p f. Hwvr, as f s w uar surfa f spsa gr, ua (4) a sv usg aaa s. I s as ura s su app. rasg avaa f purs as as a r f r fru us f s s. III. ORMUIO O UMERIC MOE WIH IIE EEME us EM, rasfra f uas (4) (7) gra s ssar. ra s rasfra w a us vara auus. praur (,,, τ ) w rprs a su fr ffra a u ua (4) a fr s (5), (6), (7), as rprss a su fr sa sa ua f f: δ (8) w s uva, fr aaa p f vw, w uas (4) (7) a wr s fua f a u. ρ (9) τ w: s psv w ra surs pru a a gav w s surs asr a s ps-v w rvs a a gav w s a surrug fu s psv surfas wr a rasfr apps fr fu a s gav vrs. a f fua s rrspg a f. su fr r f s a g para sus. ug a u s arr u w r sa s, praur sru vara s sgfa ra rs. us, aass f praur sru ars, pa r ra was s usg a w sa. I sa sa a rasfr prsss praur s p, us ua (9) / τ. I a, a w sa prs, praur s var r, us /.. Gra Euas f EM I ur as ua (9) a prss as: () ag au a praur fu s uus r f, ua () a gra f s. O r f fua a wr as a su f fuas, wr s ur f f s: () IERIO OUR f EERGY a EIROME

3 () wr p rfrs a f. r a gv f praur a aua as praur vaus s: [ ]{ }... (3) wr: s ur f f s [] fr ar f f {} vr f praur vaus s. I prss () appar para rvas f praur, rfr ua (3) su rv: { } [ ]{ } B (4) If ra uvs ar wr ar fr: [ ] (5) ua () a arg prss as: [ ]{ } ( ) [ ][ ]{ } [ ]{ } [ ]{ } [ ]{ } ( ) [ ]{ } (6) Baus [ ]{ } ( ) { } [ ] [ ]{ } ( ) [ ]{ } ( ) [ ]{ } ( ) { } [ ] [ ]{ } (7) ua (6) a prss as: { } [ ] [ ][ ]{ } [ ]{ } [ ]{ } { } [ ] [ ]{ } [ ]{ } (8) If w rv ar ua (8) furr ua s a: { } [ ] [ ][ ] [ ] [ ] { } [ ] [ ] [ ] (9) Baus a, wr s ss f f, ara f f a g f f s, rsu: { } [ ] [ ][ ] [ ] [ ] { } [ ] [ ] [ ] () f ss s sr sa a ua w. ua (), a wr as prss fr: { } [ ]{ } { } p () wr: [ ] [ ] [ ][ ] [ ] [ ] () { } [ ] [ ] [ ] p (3) w: [] s ar f a u rrspg a f, frs r s ra u a s r v s f uar surfa {p} vr f a surs ag ra surs, s f a fw ra uar surfa a v uar surfa. a f fua suppss ua w r f frs rva a p f su f. ag au f () rsus: { } { } { } (4) Irug ua () (4) w a ua ss rrspg r f: [ ]{ } { } P K (5) wr: [ ] [ ] { } { } p P K (6) w: [K] s ar f a u f r f {P} vr f a surs rrspg sus f {} vr f uw praurs. ua (5) rprss fr w f f ffra ua f a u, w as a ur f uas ua ur f s w uw praur vaus. B. Mar f Ha Cu If w us f s w rag fr a ra p f f, usg ra (3) praur (g. ), a wr as: [ ] [ ]{ } (7) w:,, ar praurs,, s (s f rag f ) [] fr ar f f [7], [9]. Issu 3, u 5, 437 IERIO OUR f EERGY a EIROME

4 g.. w rag fr u ar f a f s: [ ] [ ] [ ] (8) wr: [ ] [ ] [ ][ ] [ ] [ ] [ ] (9) [] ar, usg ra (4) a prss as: { } [ ]{ } B (3) If w rv s f fr ar: [ ] (3) wr s ara f f, a rspv a wr as [9]: (3) Csu, [] ar s sa. Baus a ra uvs var fr a f, [] ar s as sa, us: [ ] [ ] [ ][ ] [ ] [ ][ ] (33) Irug prss f [] ar fr (3) a prss f [] ar fr (5) (33) rsus: [ ] 4 (34) ar [ ] fr ua (8) a wr as: [ ] (35) Usg aura ras a srg a vv a rasfr ss s f f, w a: [ ] (36) sv ua (36), fwg ra su us: ( )! )!(! (37) Csu fr prus w sa s r s a: ( ) 3!!! (38) a fr prus w ffr s a s a: ( ) 6!!! (39) usug ua (36) rsus: [ ] 6 (4) If vv a rasfr ss r ss f f ar: [ ] [ ] 6 6 (4) ar [ ] ss as w a as, s f f a rasfr s ra v. C. r f Ha urs s vr s as ua (3) fr r rs, w a aua usg aura ras. uppsg a s sa fr a f, usg fwg ra: ( )!!!! γ γ γ (4) w a a: { } [ ] p 3 (43) s r, fr a ra s f a fw ra, rrsps a rasfr uar surfa f Issu 3, u 5, 438 IERIO OUR f EERGY a EIROME

5 Issu 3, u 5, 439 su f. uppsg a rvs a fw rug s f f, usg ra (37) w a: { } [ ] p (44) r r, fr ua (3), rrsps vv a rasfr ( ) s f f. Usg ra (37) w a: { p } [ ] (45) I u srv a r vr (44) a (45) a up a ps, rrspg s f f w a rasfr. Bas ua sss a fr f s, a ra ua ss fr r su f. s ss a sv usg aaa r rav s. I prs r ar ffr prgras sfwar ar w pr ura aass f praur sru (.g. WEBRU) u s prgras ar psv a ur par a u. I s aa praur sru a s ur sa sa a rasfr rg usg ura prs av EM sfwar as vp aur f s ar. ua ss s sv usg Gauss. I. EEOPME O UMERIC MOE WIH BOURY EEME I as f a pa wa, s aass f, a uv sa sa rg s apa ua [4]: (46) O Γ pr f uar Γ f aass f r uar s ar ps a fr pr Γ ua uar s ar ps. I rr r praur uar f aass f uss fwg gra ua [3], [4], [6]: ( ζ ) ( ζ ) ( ) v ( ζ, )Γ( ) u ( ζ, )Γ( ) ( ) (47) Γ Γ wr: ζ s p w wrs gra ua (sur p) (ζ) a ff urr gra p u ( ζ, ) ( / π) / r( ζ, ) fuaa su v u / ra rvav f s su. sa r(ζ, ) w urr p sur p ζ s aua w ra: a r ( ζ, ) ( ) ( ζ ) ( ) ( ζ ) (48) Buar Γ s sr sa uar s fr w srs praurs, rspv ra rvav (/) sa a ua p () vau f. us gra ua s a ur fwg sr fr: r v ( ζ, )Γ( ) u ( ζ, )Γ( ) (49) Γ Γ ˆ B (5) w ffs $ a B av prsss: ˆ Γ v ( ζ, )Γ( ) B u ( ζ, )Γ( ) (5) Γ W s : ˆ B (5) π Ep, ua (5) gras a ar a pa ss f uas w uws [ a (/) ] a afr pg uar s, ur f uws s ru. I as f sa uar s, ff as vau /. Cffs $ a B fr (5) s pu usg a Gauss uaraur [8], [9]: ˆ v w B u w (53) w s g f uar. Irug as: s(, ) s(, ) a usg, fr Γ, parar uas: [,] ξ B Cξ, ξ (54) wr: [, ] a [, ], fwg ras ar a: C (55) C C w (, ) a (, ) ar rs f uar. aass f s rasfr a sss rpag sa varas (, ) w sss s (, ): a a (56) w a s au s f aass f afr as O. IERIO OUR f EERGY a EIROME

6 Issu 3, u 5, 44 I rr r praur s f aass f s us gra rprsa: ( ( ) ζ ) u ( ζ, )Γ( ) ( ) v (, )Γ( ) ζ (57) Γ Γ w: ζ Ω, wr Ω rprs s f aass f Ω ( Ω Ω U Γ). fr sra f uar Γ sa uar s as gra ua ur sr fr: ( ζ ) ζ v (, )Γ( ) Γ Γ u ( ζ, )Γ( ) w a wr as su: Cffs a B ar vaua usg a Gauss uaraur: B (58) (59) v w B u w (6) w: s ur f Gauss p ps w wg ffs. praurs fr ps ζ ar as r ag au a vaus a (/ ) ar w aass f uar, a ffs a B ar pu w ua (54). B wg vaus a f praur aass f uar, grup f ra ps (, ) fr w s. rprss sra urvs. ura as BEM as p pur prgra BEM, ra rra prgrag aguag, fr IBM PC pa sss.. PPICIO. praur sru Orrp B praur sru s aa a s 5 4 sa ss (g. 3). rvs a fw w ss: 3 W, 98 W. O r w ss ras a v 3. W/( K). ara f as rrp prprs w fwg vaus f ra uvs:.6 W/( K), 5.8 W/( K). sus f s v 4 f s w 3 s. Rug EM prgra vaus f praurs s av a a prs a I. praur sru s prs gur 4. W s rrp ara w s us v grg, a s prpr su a au a a ss ra f ugs (.g. a fw r prpuar r para fr). g. 3 ass f a I. praur vaus s Cras Cras [ C] [ C] g. 4 praur sru su IERIO OUR f EERGY a EIROME

7 Issu 3, u 5, 44 B. praur sru Pp Isua praur sru pp sua was aa (g. 5) usg EM prgra. auus was a fr a pp w 8 a ar a war praur was 5 C. a praur was sr C. aass as gr w uar s fr s uars. g. 8 Ma pau g. 5 ruur f sua pp wa a, sua ars 3 pr a a rsus w sr ra sua as a as pss vv a rasfr ff ra sua surfa was sr vara w vaus w a 5.6 W/( K). I gurs 6 a 7 aa f a praur sru ar prs pp s. I a srv a u vara uar s sua surfa sr urvs ar ruar urvs w ar a w assa auus s us. g. 9 Ma pau w a sra u u g. 6. ass f g. Buar s fr a pau g. 7. praur sru pp sua C. praur sru Ma Pau I fgurs 8 a 9 ar sr w varas f a a pau, w ss 4 4 7, fr w rs praur f usg BEM a aaa (M). I fgurs a ar prs sss g. Buar s fr a pau w sra u u IERIO OUR f EERGY a EIROME

8 Issu 3, u 5, 44 r a pau fgur uar a sr 6 uar s, sas 9 ra ps (g. ) a apps puaa as BEM. ura rsus a as f BEM prgra ar prs a II, parav w s a w M [3]. asu prag vau f rav ffr war aaa su, fr praur ε a ra rvav ε s f : ε ε M M BEM M M BEM (6) g. sra f uar a ra ps f aass f a II. vaus a P Cras BEM M , ag au rsus fr a II w appg uas (6), apa vaus av a fr ε a ε (ε <.3%, ε < 3.5%) v f ur f uar s sr s sa. r a pau fgur 3 uar a sr 56 sa uar s (g. 3) a usg BEM was r sra urvs prs gur 4. I. COCUIO I pra r ar a suas wr s spsa w praur sru a (.g. ffr a a r ps). I v grg s pra aas praur sru ra rgs, pp was, sua aras []. IERIO OUR f EERGY a EIROME

9 Issu 3, u 5, 443 g. 3 Buar sra fr pau w sra u u g. 4 praur sru fr pau w sra u u ura g w f a uar s rprss a ff wa a praur sru sa sa uv a rasfr prsss. ura pua f praur f, ass f uar, as s vaus s r aaa v f a sa ur f uar s a rspv ra ps f aass a was us. Usg prs s, ffr sua prgras u ra wa as pss ffua a f ffr ura prs f praa prs. REERECE [].. sa a. Raa, a rasfr a sruura aass, Prgs f WE/IME I. Cfr Ha a Mass rasfr, Crfu, Gr, ugus 7-9, 4, pp. -. [].. sa, Ha rasfr asr saga fw a f ruar r, Prgs f 5 WE I. Cfr Ha a Mass rasfr, apu, M, auar 5-7, 8, pp [3] P.K. Barg a R. Burf, Buar s grg s, M.Graw H,, w Yr, 98. [4] C.. Bra,.C. s a I.C. Wr, Buar us, prgr rag, Br, Hrg, w Yr, 984. [5] G. C a. Zu, Buar s, a Prss, w Yr, 99. [6] M. Gafţau,. Praşu a. Maa, a uar s w appas pua f a ps, a Pusg Hus, Buars, 987. [7].K. Guv a.. Ra, Caua sa w f ffrs, a Pusg Hus, Buars, 977. [8]. Isf, ura su w ar uar s f ra fs sa sa rg, Prgs f I. Cfr Bug rvs a Cfr, sara, pr 8-9,, pp. 4-. [9] B.M. Irs a. a, us f f s, W, w Yr, 98. [] G. óass, urs ug pss, Kug sa Högsa,, 999. [].H. Ka, Buar aass grg uu as, Pr Ha, w rs, 994. [] W.M. Kas a M.E. Crawfr, Cvv a a ass rasfr, MGraw H, w Yr, 993. [3]. a, C.E. Ma a M. a, Ha a ass rasfr, a Pusg Hus, Buars, 998. [4].. va a.c. Bra, up rpr : w appra fr rasfrg BEM a gras uar, Egrg ass, v. 6,. 3, 989, pp [5]. Pars a. Cañas, Buar : fuaas a appas, Ofr Uvrs Prss, Ofr, 997. [6] W.P. Parrg a.c. Bra, Cpur pa f BEM ua rpr fr su f gra f uas, Cuas pp ura Ms, v. 6,., 99, pp [7]. Ra, f grg, Prga Prss, w Yr, 98. [8].. R, ru f, MGraw H, w Yr, 993. [9] I. âru, ura gs a pas ug srvs ( Raa), Pusg Hus Pa, sara,. [] I. âru, ura aass f w sa a uv sa sa rg, Pra Pa Buaps, v 49,., 5, pp [] I. âru a. Isf, ura aass f v a praur f r auar u fr aar fr a v, Prgs f 8 IME/WE I. Cfr Ha rasfr, ra Egrg a Evr, ap, awa, ugus -,. [] B.. Wag a Z.H. a, ppa f f f ffr ra f ras praur f fua gra aras, Es ass a sg, v 4, 5, pp [3] B.. Wag a Y.W. Ma, ras sa a u prs sv f, Iraa ura f Maa s, v. 47, 5, pp [4]. Wu a. g, f ffr aass a ps pr, pp Maas a Cpua, v 8, 6, pp IERIO OUR f EERGY a EIROME