ϕ direction (others needed boundary conditions And we have following boundary conditions in

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1 Exact Analytical 3-D Steady-State Solution fo Cylindical Fin ANITA PILIKSERE, ANDRIS BUIKIS Institute of Mathematics and Compute Science Univesity of Latvia Raina ulv 9, Riga, LV459 LATVIA lzalv/scientists/uikishtm Astact: - In this pape we constuct two exact analytical thee-dimensional solutions fo cylindical wall and fin We assume that the heat tansfe pocess in the wall and the fin is stationay Key-Wods: - steady-state, thee-dimensional, heat exchange, cylindical fin, analytical solution Intoduction Usually mathematical modeling of systems with extended sufaces is ealized y one dimensional steady-state assumptions []-[4] In ou pevious papes we have constucted two and thee dimensional analytical appoximate [5]-[8] solutions In pape [9] was constucted exact -D solution fo ectangula fin Hee we constuct the solution in diffeent fom [9] way This way gives moe suitale fom of the solution in the fom of Fedholm integal equation We educe exact 3-D polem to two dimensional and otain exact analytical two-dimensional solution y the Geen function method Mathematical Fomulation of 3-D Polem and Exact its Reduction to Non-homogeneous -D Polem We will stat with accuate thee-dimensional fomulation of steady-state polem fo system of cylindical wall and fin The one element of the wall (ase) is placed in the domain { % [ R, R], z% [, H], ϕ [, ] } and we descie tempeatue field V (, z, ) in the wall ϕ with the equation: V V V () z ϕ The cylindical fin of length L occupies the domain { % [ R, R], z% [, H], ϕ [, ] } and the tempeatue field V (, z, ϕ ) fulfills the equation: V V V () z ϕ And we have following ounday conditions in ϕ diection (othes needed ounday conditions will e added in non-dimensional fom in next su-section): V% V% q% (,), % z% q% (,), % z% ϕ ϕ ϕ ϕ (3) V% V% q% (, % z% ), q% (, % z% ) ϕ ϕ ϕ ϕ Intoducing following aveage integal values fo agument ϕ we can educe equations () and () fom 3-D to -D: U% ( %, z% ) V% ( %, z%, ϕ) dϕ, (4) U% ( %, z% ) V% (, ϕ) dϕ % % Integation the equation () fo the wall ove ϕ [, ] gives following equation (exact consequence of 3-D patial diffeential equation ()): % U% % % % % z% V% V% % ϕ ϕ ϕ ϕ The fist pai of ounday conditions (3) allows ewites the last equality in fom of twodimensional non-homogeneous equation:

2 % U% % Q% (,) % z%, % % % z% q (,) z q(,) z Q% % % % % % % (,) % z% % % U% % Q% (, % z% ), % % % z% q(,) z q(, Qz % % % % % % % (,) % % % (5) (6) Dimensionless Tempeatue Field in the Wall We will use following dimensionless aguments, paametes to tansfom ou polem to dimensionless polem: z R %, z % R R,,,, H H H H H H hh hh hh, β, β, β, H k k k β β β γ, γ, γ And tempeatues: U% (, Ta U% (, Ta Uz (, ), U(, T Ta T Ta Hee k( k ) - heat conductivity coefficient fo the fin (wall), h( h ) - heat exchange coefficient fo the fin (wall), H - width (thickness) of the fin, L - length of the fin, H - thickness of the wall, T - the suounding tempeatue on the left (hot) side (the heat souce side) of the wall, T a - the suounding tempeatue on the ight (cold - the heat sink side) side of the wall and the fin One element of the wall (ase) placed in the domain now is { [, ], z [, ]} and we can descie the dimensionless tempeatue field U (, in the wall with the equation: U Q (, (7) z We add needed ounday conditions as follow: γ ( U),, z [,], (8) U, γ, z [,] (9) And in contadiction with ou pevious papes we assume geneal non-homogeneous ounday conditions on top and the ottom of the wall (the same genealization will e assumed fo the fin): q (), [, ], z () βu q() z We assume the conjugations conditions on the suface etween the wall and the fin as ideal themal contact - thee is no contact esistance: U U, γ γ () The Tempeatue Field in the Fin The cylindical fin of length l occupies the domain,, z, and the tempeatue field { [ ] [ ]} U (, fulfills the equation: U U Q(, () z We have following ounday conditions fo the fin: U β U, [, ], z, (3) z γu,, z [, ] (4) And non-homogeneous ounday conditions: q (), [, ] (5) z 3 Exact Solution of -D Polem To cleae explain the main idea we will stat with the case of homogeneous equations fo the cylindical wall and fin and fo simplicity in this section we assume additionally homogeneous ounday conditions (), (5) The geneal case (with non-homogeneous diffeential equations and non- homogeneous ounday conditions) will e consideed in next section 3 The Sepaate -D Polem fo the Wall We ewite the ounday condition (9) togethe with the conjugation conditions () in following common fom:

3 γ F(, < z <, γ U, z, < < (6) F ( U γ The solution fo the wall can e witten in well known fom y means of Geen function, see, eg []-[]: U (, G z d πγ (,,, η ) η πγ U G (, z,, η) d η γ ξ ξ V The epesentation () is not the solution ecause V of unknown function F ( η ) (, πγ δ G z d V G z (,, δ, η ) η πγ δ (,, δ, η) γ ξ ξ δ (7) The two-dimensional Geen function has following fom: ( ) ( G(, ξ, η) G (, ξ) G (, z η), (8) The one-dimensional Geen functions have such epesentations: ( ) π λk G (, ξ ) Hk() Hk() ξ 4 k Bk (9) γ J ( λ ) λ J ( λ ), [ k k k ] Bk ( k ( ) ) J k kj k ( λk ( γ ) )[ γ J λ ] k λ kj λ k λ γ γ ( λ ) λ ( λ ) ( ) ( ), Hk() γy( λk) λky( λk) J ( λ ) γ J ( λ ) λ J ( λ ) Y ( λ ); k k k k k ( (, η) ϕl() ϕl(), η l G z z () ϕl( cos( μl, μl ( n / ) π Hee λk( μl) ae positive oots of following tanscendental equations: γj( λ) λj( λ) [ γy( λ) λy( λ) ] γ J ( λ ) λj ( λ ) [ ] Y Y γ ( λ ) λ ( λ ) V Repesentation (7) in notations (6) can e ewitten in shote fom (with known function (, ): U(, (, () πγ F ( η ) G (,, η ) V (, (, πγ δ F ( η) G (, δ, η) This function is given y expession: (, πγ G(,, η ) 3 The Sepaate -D Polem fo the Fin V (, πδ V G(, δ, η) γ ξ ξ δ We ewite the conjugations conditions () in following fom: γu γf(, () F( U γ Simila as in case of wall the solution fo the fin can e epesent in following fom: Uz (, ) πγ U G (, z,, η) d η γ ξ ξ V (, πδ F( η) G(, δ, η) The shote fom looks as follow: (3) Uz (, ) πγ F() η Gz (,,, η) (4) The second Geen function with small modifications has a same fom as expession (8):

4 π λ (, ) () () ξ ( ) n G ξ Hn Hn 4 n Bn [ γj( λn) λnj( λn) ], Bn ( λn ( γ) )[ γj( λn) λnj( λn) ] ( λn ( γ) )[ γj( λn) λnj( λn) ], Hn( ) [ γy( λ n ) λny( λ n ) ] J( λn) [ γj ( λ ) λ J ( λ )] Y ( λ ); G n n n n ϕ ( ϕ ( η) (, ), ( m m z η m ϕm ( μm γ ) ϕm γ, ϕ ( μ cos( μ γ sin μ z ( ) l m m m Hee λn( μm) ae positive oots of following tanscendental equations: [ γ J( λ) λj( λ) ][ γy( λ) λy( λ) ] γj ( λ ) λj ( λ ) γy ( λ ) λy ( λ ), [ ][ ] μ γ cot ( μ) 33 The Conjugation of Two Sepaate Polems We otain easy fom epesentations (), (4) following two equalities: F( % (, π F ( η ) G% (,, η ), F ( πγ F( η) G% (,, η) Hee % (,, γ (5) G G% (, ξη, ) γg, (6) G Gz % (,,, η) γg The system (5) allows witing out following second kind Fedholm integal equation: F( % (, F( η) Γ( η) (7) F( ( δ, F( η) Γ( η) Hee Γ (, z η) π γ ( ) G % (,, η) G % (8) (,, η) This second kind Fedholm integal equation (7) y the given kenel Γ(, z η) has exact one solution Knowing F( we can find fom epesentation (4) the solution fo the fin In simila way we constuct integal equation fo the function F ( ) z and find solution fo the wall 4 Exact Solution y Non-homogeneous Envionment Tempeatue Now we will conside the case of non-homogeneous equations and non-homogeneous ounday conditions 4 The Statement of the Full Mathematical Polem As the main equations fo the wall and the fin we take diffeential equations (7), (): U Q (,, z (9) U Q(, z The ounday conditions fo the wall ae as follow: γu γϑ(, γu (,, z [,], γ ϑ (3) q(), [, ], z βu q() z Simila ae the ounday conditions fo the fin:

5 γu z γϑ(, ), [, ], γu γϑ(,, (3) z [, ], z q (), [, ] To complete the full statement of genealized polem, we must add the conjugations conditions to equations (9)-(3): U U, γ γ (3) 4 The Sepaate Solutions fo the Wall and the Fin In the same way as in the su-section we intoduce the notation (6) Then the solution in the wall again can e pesented in the same fom (): U(, (, (33) πγ F ( η ) G (,, η ) Now the expession fo the fist tem of ight hand side has significant moe complicate fom: (, πβ ϑ(, η ) G (,, η ) G z d πβ ϑ ( η ) (,,, η ) η π ξ q ( ξ ) G (, ξ,) dξ π ξ q ( ξ ) G (, ξ,) dξ π ξ dξ Q ( ξ, η ) G (,, η ) (34) The solution fo the wall with ounday conditions () in simila way as fo wall can e pesented in the fom: Uz (, ) (, πβ F( η) G(,, η) (35) The expession fo the fist tem of ight hand side has the fom: (, πγ ϑ(, η) Gz (,,, η) πβ ξϑ( ξ, Gz ) (,, ξ, d ) ξ π ξq( ξ) G(, ξ,) dξ π ξdξ Q( ξη, ) G(, ξη, ) (36) 43 The Junction of Solutions fo the Wall and the Fin We intoduce following notations: % (,, γ % (,, γ G G% (, ξη, ) γg, G Gz % (,,, η) γg Then the epesentations (33) and (35) allow otain easy following two equations: F( % (, π F ( η ) G% (,, η ), F ( % (, πγ F( η) G% (,, η) (37) Fom this system (37) we otain following second kind Fedholm integal equation: F( Ψ (, F( η) Γ( η) (38) Hee the kenel of the Fedholm integal equation is given y the same fomula (8):

6 ( ) Γ (, z η) π γ G % (,, η) G % (,, η) In its tun the fist tem in the ight hand side has moe complicate expession: Ψ (, % (, π % (, η) G% (,, η) Evidently this second kind Fedholm integal equation (38) has exact one solution Again, y known F( z ) the epesentation (35) allows find the solution fo the fin In simila way we can constuct integal equation fo the function F ( ) z and find solution fo the wall 5 Conclusions We have constucted two exact two-dimensional analytical solutions (in oth cases: homogeneous and non-homogeneous envionment) fo a system with cylindical fin when the wall and the fin consist of mateials, which have diffeent themal popeties Acknowledgements: Reseach was suppoted y Euopean Social Fund and Council of Sciences of Latvia (gant 555) [6] Buikis A, Buike M Some analytical 3-D steadystate solutions fo systems with ectangula fin IASME Tansactions Issue 7, Vol, Septeme 5, pp -9 [7] ABuvee, ABuikis Some appoximate analytical steady-state solutions fo cylindical fin Poceedings of 4th IASME/WSEAS Intenational Confeence on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT, Elounda, Cete Island, Geece, August -3, 6, p [8] ABuvee, ABuikis Analytical 3-D steadystate statement fo cylindical fin and some of it s appoximate solutions WSEAS TRANSACTIONS on HEAT and MASS TRANSFER, Issue 4, vol, 6, p 45-4 [9] Buikis A, Buike M, Guseinov S Analytical two-dimensional solutions fo heat tansfe in a system with ectangula fin Advanced Computational Methods in Heat Tansfe, VIII WIT pess, 4, pp [] Őzişik, M Necati Bounday Value Polems of Heat Conduction Dove Pulications, Inc, Mineola, New Yok, 989 [] Caslaw, HS, Jaege, CJ Conduction of Heat in Solids Oxfod, Claendon Pess, 959 [] Polyanin, AD Handook of Linea Patial Diffeential Equations fo Enginees and Scientists Chapman&Hall/CRC, Refeences: [] Ken, DQ, Kaus, AD Extended Suface Heat Tansfe McGaw-Hill Book Company 97 [] Kaus, AD, Analysis and evaluation of extended suface themal systems Hemisphee pulishing Copoation, 98 [3] Madassa Manzoo Lectue Notes in Engineeing Heat Flow Though Extended Sufaces Heat Exchanges Edited y CA Beia and SA Oszag Spinge-Velag, Belin, Heideleg, New Yok, Tokyo, 984 [4] Wood AS, Tupholme GE, Bhatti MIH, Heggs PJ, Steady-state heat tansfe though extended plane sufaces, Int Commun in Heat and Mass Tansfe,, No, 995, pp99-9 [5] Buikis A, Buike M, Closed two-dimensional solution fo heat tansfe in a peiodical system with a fin Poceedings of the Latvian Academy of Sciences Section B, Vol5, N5, 998, pp8-