HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD

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1 Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 45-5 HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs, Czesochowa Unversy of Technology Czesochowa, Poland sanslaw.kukla@m.pcz.pl, urszula.sedlecka@m.pcz.pl Absrac. In hs paper, a soluon of he hea conducon problem n a wo-layered hollow cylnder by usng he Green s funcon mehod s presened. The consderaons concern he hea conducon n he radal drecon whle akng no accoun me-dependen boundary condons. The connuy condon of he emperaure a he dvdng surface of he layers s wren n he form of a Volerra negral equaon. Keywords: hea conducon problem, wo-layered cylnder, Green s funcon mehod Inroducon The problems of he hea conducon n hollow and sold compose cylnders, n recen years, were he subec of many sudes (cf. refs [-3]). A soluon of he hea conducon problem n a compose mul-layered crcular cylnder can be deermned by usng analycal mehods. In he papers [-] he soluon of he problem was obaned by usng he Laplace ransform. The nverse ransform has been numercally deermned. However, he calculaons of he nverse ransform lead ofen o numercal nsables and ha way he search for new mehods o avod he possble nsably n numercal compuaon s purposeful. An analycal mehod for solvng he hea conducon problem n a compose crcular cylnder, whch s an exenson of an approach used n Caresan coordnaes, s proposed by Lu e al. n reference [3]. In he paper he emperaure soluon for n-layered fne cylnder by applcaon of he Laplace ransform has been obaned and he closed form soluon as he real par of a funcon s gven. The oher approach o solve he problems s he use of he properes of Green s funcons. The examples of applcaons of he mehod are gven by Beck e al. n he book [4]. The emperaure soluons n mul-layer bodes by usng he Green s funcon mehod are presened by Ha-Shekh and Beck n paper [5]. The sudy concerns he hree-dmensonal hea conducon wh examples of a wo-layer body n Caresan coordnaes. In he presen paper, he soluon of he problem of hea conducon n radal drecon n a wo-layered crcular cylnder s proposed. The soluon s obaned by he use of he Green s funcon mehod.

2 46 S. Kukla, U. Sedlecka. Formulaon of he problem We consder a hollow wo-layered cylnder n whch he hea ransfer n a radal drecon s held (Fg. ). The radal hea conducon n he layers are governed by he dfferenal equaons: T T T r k [ ] =, r r, r, =, () where T s he emperaure n -h layer, r s he radal coordnae, s me and k s he hermal dffusvy of -h layer. z L r r r r Fg.. The skech of he hollow cylnder We assume zero nal condon and boundary condons whch provde he free hea exchange a he nner ( r= r ) and ouer ( r= r ) surfaces of he cylnder (, ), [, ] T r = r r r, =, () T r = T r T (, ) ( (, ) ( )) T r = T T r (, ) ( ( ) (, )) (3) (4) where, are he surface hea ransfer coeffcens,, are hermal conducves, T are he me-dependen boundary emperaures. Moreover, a he conac surface of he layers, he connuy condons are n force

3 Hea conducon problem n a wo-layered hollow cylnder by usng he Green s funcon mehod 47 (, ) (, ) T r = T r (5) T T (, r) = (, r) (6). Dervaon of he emperaure feld We deermne he emperaure T n he layers of he cylnder by usng properes of Green s funcons whch correspond o he problem ()-(6). The Green s funcons G and G sasfy he auxlary equaon G G G r k r k ( r r ) ( ), r [ r, r] δ δ τ = and he homogeneous boundary condons and zero nal condon, =, (7) G r τ r G r τ r G r (, ;, ) = (, ;, ) G, ;, =, ;, = r ( r τ r ) ( r τ r ) (8) (9) G r τ r = G r τ r (, ;, ) (, ;, ) (, ; τ, ), [, ] () G r r = r r r, =, () In equaon (7) he δ() denoes a Drac dela funcon. The dervaon of he soluon of he problem (7)-() we begn wh use of he recprocy relaon [4] (, ; τ, ) ( τ, ;, ) G r r = G r r, =, () Takng no accoun he relaon () n equaon (7), one obans G G G δ ( r r ) δ( τ) = r k r k τ, =, (3) The emperaure equaons () we wre also n erms of r and τ as T T T = r k τ, =, (4)

4 48 S. Kukla, U. Sedlecka Mulplyng equaon (3) by T and equaon (4) by obaned equaons, one gves ( =, ) G and subracng he ( G T) T G r G r T δ( r r ) δ( τ) T = r r r r r k r k τ (5) Nex, we negrae equaon (5) wh respec o r from r o r ( =,) and negrae wh respec o τ from o. The resul s r r T G r G r T δ( r r ) δ( τ) T dr dτ r r r = r k r ( G T) r = r dr dτ,, k = τ (6) Usng now he properes of he Drac dela funcon, we oban r r T G τ= (, ) = τ ( ) τ= r (7) r T r k r G r T dr d r G T dr =, Inegrang by pars he negral on he rgh-hand sde of equaon (7), and nex usng he boundary condons (3)-(4) and (8)-(), we have (, ) = r = r T G T r k r G T, =, (8) r = r Applyng he boundary condons (3) and (4) n equaon (8), gves T T(, r) = k r G(, r; τ, r) r T ( τ) G(, r; τ, r) T T(, r) = k r T ( τ) G(, r; τ, r) r G(, r; τ, r) (9) The nex sep n he dervaon of he emperaure T s o deermne he T T unknown dervaves and a r = r. In order o do hs, he connuy

5 Hea conducon problem n a wo-layered hollow cylnder by usng he Green s funcon mehod 49 condons (5)-(6) wll be used. Because from (6) we have condon (5), we oban T T =, hen usng r T k r G(, r ; τ, r) r T ( τ) G(, r ; τ, r) T = k r T ( τ) G(, r ; τ, r) r G(, r ; τ, r) () The equaon () can be wren n he form: (, τ) ( τ) τ= ( ) K y d F () where y( τ) T = = and and K(, τ) k r G (, r ; τ, r) k r G (, r ; τ, r) F( ) = k r T ( τ) G(, r ; τ, r) k r T ( τ) G(, r ; τ, r) () Equaon (), as he Volerra negral equaon of he frs knd [6], wll be solved by usng a quadraure mehod. Fnally, he emperaures T n he layers of he cylnder are gven by equaons: T(, r) = k r y( τ) G(, r; τ, r) r T ( τ) G(, r; τ, r) T(, r) = k r T ( τ) G(, r; τ, r) r y( τ) G(, r; τ, r) (3) In order o compue he emperaures T we use a quadraure mehod o numercal calculaon of negrals occurrng n equaons () and (3). Afer usng he quadraure rule, he equaon () gves (, ) ( ) τ = ( ) n AK h y h d F (4) =

6 5 S. Kukla, U. Sedlecka where are he pons of negraon and rule. Subsung successvely: =,,..., h a sysem of equaons wh unknowns y y( h) can be wren n he marx form where A = AK(, ), =, [ y y y n ] n A are he weghs of he quadraure = n, n equaon (4) we oban =, =,..., n. The equaon sysem A Y = F (5) Y, ( ) ( ) ( )... T F = F F... F n. Nex, he soluon of he equaon (5) s used n dscrezed form of equaons (3) for he calculaon of emperaures T, T n he compose cylnder. T Conclusons The soluon of he hea conducon problem n a wo-layered hollow cylnder by usng he properes of he correspondng Green s funcons s deermned. The connuy condon on he nerface of he cylnder layers leads o he Volerra negral equaon of he frs knd whch can be numercally solved. Alhough he problem concerns he hea conducon n radal drecon, he formulaon and soluon of he problem can be easly expanded o he hea conducon n axs drecon of he cylnder. References [] Nezhad Y.R., Asem K., Akhlagh M., Transen soluon of emperaure feld n funconally graded hollow cylnder wh fne lengh usng mul-layered approach, Inernaonal Journal of Mechancs and Maerals n Desgn, 7, 7-8. [] Garba L., Krope J., Mehes S., Baral I., Transen hea conducon n compose sysems, Proceedngs of he 4h WSEAS Inernaonal Conference on Hea Transfer, Thermal Engneerng and Envronmen, Elounda, Greece, Augus -3, 6, [3] Lu X., Tervola P., Vlanen M., Transen analycal soluon o hea conducon n compose crcular cylnder, Inernaonal Journal of Hea and Mass Transfer 6, 49, [4] Beck J.V., Cole K.D., Ha-Shekh A., Lkouh B., Hea Conducon Usng Green s Funcons, Hemsphere, Washngon DC, 99. [5] Ha-Shekh A., Beck J.V., Temperaure soluon n mul-dmensonal mul-layer bodes, Inernaonal Journal of Hea and Mass Transfer, 45, [6] Grzymkowsk R., Hemanok E., Słoa D., Wykłady z modelowana maemaycznego, Wydawncwo Pracown Kompuerowe Jacka Skalmerskego, Glwce.