Some Appoxmate Analytcal Steady-State Solutons fo Cylndcal Fn ANITA BRUVERE ANDRIS BUIIS Insttute of Mathematcs and Compute Scence Unvesty of Latva Rana ulv 9 Rga LV459 LATVIA Astact: - In ths pape we constuct some appoxmate analytcal thee-dmensonal solutons fo one element of cylndcal wall and fn We assume that the heat tansfe pocess n the wall and the fn s statonay These solutons ae otaned y the ognal method of consevatve aveagng and they ae compaed to some one-dmensonal solutons whch ae well known n lteatue We gve some cteons when t s possle to eplace thee-dmensonal fomulaton of polem wth two- o one-dmensonal statement ey-wods: - steady-state thee-dmensonal heat exchange cylndcal fn analytcal solutons consevatve aveagng Intoducton Otanng effcent coolng fo the components of devces s a dffcult challenge n moden ndusty It s elated to efgeatos adatos engnes and moden electoncs etc Usually ts mathematcal modelng s ealed y one dmensonal steady-state assumptons [][5] In ou pevous papes we have constucted two dmensonal analytcal appoxmate []-[4] and exact [3] solutons In ths pape we otan few new appoxmate analytcal thee dmensonal solutons y the ognal method of consevatve aveagng and some ts smplfcatons (specal cases In [] the so-called Muay Gadne assumptons ae fomulated They ae: The heat flow n the fn and the tempeatue at any pont on the fn eman constant wth tme; The fn mateal s homogeneous; ts themal conductvty s the same n all dectons and emans constant; 3 The heat tansfe coeffcent etween the fn and the suoundng medum s unfom and constant ove the ente suface of the fn; 4 The tempeatue of the medum suoundng the fn s unfom; 5 The fn wdth s so small compaed wth ts heght that tempeatue gadents acoss the fn wdth may e neglected; 6 The tempeatue at the ase of the fn s unfom; 7 Thee ae no heat souces wthn the fn tself; 8 Heat tansfe to o fom the fn s popotonal to the tempeatue excess etween the fn and the suoundng medum; 9 Thee s no contact esstance etween fns n the confguaton o etween the fn at the ase of the confguaton and the pme suface; The heat tansfeed though the outmost edge of the fn (the fn tp s neglgle compaed to that though the lateal sufaces (faces of the fn Mathematcal Fomulaton of 3D Polem and educton to D We wll stat wth accuate thee-dmensonal fomulaton of steady-state polem fo one element of peodcal system fo cylndcal wall and fn The one element of the wall (ase s placed n the doman { [ R ] R [ Z] ϕ [ φ ]} and we desce tempeatue feld V ( ϕ n the wall wth the equaton: V ( The cylndcal fn of length L occupes the doman { % [ R R L] % [ Z] ϕ [ φ ]} and the tempeatue feld V ( ϕ fulflls the equaton:
V ( And we have followng ounday condtons n accodance wth M-G pont 5 n ϕ decton % % ; (3 ϕ ϕ φ ϕ ϕ φ We can educe polem ( and ( fom 3D to D usng followng aveage ntegal fo agument ϕ φ U ( V ( ϕ dϕ φ and U% ( % % φ V% ( ϕ dϕ φ % % (4 Descpton of Tempeatue Feld n the Wall We wll use followng dmensonless aguments paametes to tansfom ou polem to dmensonless polem: R R L R l Z Z Z Z Z Z hz hz Z h Z k k k And tempeatues: U% ( Ta U% ( Ta U ( U( T Ta T Ta Hee k( k - heat conductvty coeffcent fo the fn (wall h( h - heat exchange coeffcent fo the fn (wall Z - wdth (thckness of the fn L - length of the fn Z - thckness of the wall T - the suoundng tempeatue on the left (hot sde (the heat souce sde of the wall T a - the suoundng tempeatue on the ght (cold - the heat snk sde sde of the wall and the fn and U % ( ( U % ( One element of the wall (ase placed n the doman now s { [ ] [ ] } and we can desce the dmensonless tempeatue feld U ( n the wall wth the equaton: U U (5 We add needed ounday condtons as follow: ( U [] (6 U [] (7 And homogeneous ounday condtons [ ] (8 We assume the conjugatons condtons on the suface etween the wall and the fn as deal themal contact - thee s no contact esstance: U U (9 Descpton of Tempeatue Feld n the Fn The cylndcal fn of length l occupes the doman { [ l] [ ] } and the tempeatue feld U ( fulflls the equaton U U ( We have followng ounday condtons fo the fn: U [ l ] ( U l [ ] ( And homogeneous ounday condtons [ l] (3 3 Appoxmate Soluton of D Polem fo Peodcal System We wll use the ognal method of consevatve aveagng We wll stat wth the case of peodcal system wth cylndcal fns 3 Reducton of the D Polem fo the Fn Smlaly as n papes [][4] we wll use ognal method of consevatve aveagng and appoxmate the D tempeatue feld U ( fo the fn n followng fom: ρ U( f( ( e f( (4 ρ - ( e f ( ρ wth thee unknown functons f ( Fo ths pupose we ntoduce the ntegal aveage value of functon U ( n the - decton: u( ρ U( d (5
Ths equalty togethe wth two ounday condtons (at and allow us to exclude all the unknown functons f ( fom the epesentaton (4 The ounday condton (3 fo the functon U ( at gves mmedately the equalty f( f ( The susttuton of epesentaton (4 n (5 gves expesson: u( f( f ( (6 ( snh( and epesentaton (4 takes fom: cosh( ρ U( u( snh( (7 snh( cosh( ρ ( snh f ( Fnally y the use of the ounday condton ( we can exclude f ( fom last expesson and epesent the D soluton U ( fo the fn n followng fom: U ( u ( Φ ( (8 It s easy to check that the functon Φ( looks lke ( ρ snh( cosh( cosh( Φ ( (9 snh( (cosh( snh( The second stage fo the method of consevatve aveagng s to tansfom the patal dffeental equaton (5 fo the functon U ( to the dffeental equaton fo one aguments functon u ( To eale ths goal we ntegate the man dffeental equaton ( n the - decton and usng (5 get: d du ( d d By usng the ounday condton (3 at fo the functon U ( and expessng fom the ounday condton ( at the fst devatve tough the functon U ( we otan followng dffeental equaton: d du U ( d d It emans to expess n dffeental equaton ( the functon U ( though the functon u ( wth the help of the equalty (8 and we eceve the new dffeental equaton whch desces the D dmensonal tempeatue feld u ( n the fn: d du d d μ u( ( Hee μ ( Φ (3 Solvng dffeental equaton ( we gan soluton though Bessel s modfed functons I : u( C( μ I ( μ ( μ (4 whee μ( μ ( μ μ l (5 μi( μ I ( μ l And fom (8 and (4 we get soluton fo fn whch ncludes only one unknown constant C U C μ I ( μ ( μ Φ( (6 ( ( 3 Reducton of the D Polem fo the Wall We wll use same method of consevatve aveagng and appoxmate the D tempeatue feld U ( fo the fn n followng fom: U ( g( g( (7 g( wth thee unknown functons ( g Fo ths pupose we ntoduce the ntegal aveage value of functon U ( n the - decton: u ( U ( d (8 Ths equalty togethe wth equalty (7: u g ( ϕ g ( ϕ g ( (9 ( whee ϕ ϕ ( 3( Fndng devaton of (7 and usng ounday condton (6 we get ( ( g g( (3 ( g( Expess functon g ( fom equaton (9 and put to expesson (3 gves g( A g( Bu ( D (3 whee
( ϕ(( ϕ ( ( A ϕ ( ( B D ϕ ( Expess functon g ( fom equaton (9 and put to expesson (3 gves g ( A g ( B u ( D (3 whee ( ϕ (( ϕ A B ( ϕ ( ϕ D Equatons (3 and (3 gve g( ag ( u ( d (33 g( ag( u ( d A B D whee a d Puttng (33 to (7 we gan a a U( g( ( ( u( (34 d d ( Now we stll have two unknown functons g ( and u ( Theefoe we wll use dffeent ounday and conjugatons condtons on the wall to exclude these functons 33 Soluton fo uppe Wall Puttng (34 to ounday condton (7 we get g ( u ( d (35 B D whee d B a a Puttng (35 n (33 we get U ( Φ ( u ( ψ ( (36 a a whee Φ ( ( ( da d d da and ψ ( ( d Now ntegatng patal dffeental equaton (5 and takng account equaton (8 get d u (37 d Usng ounday condtons ( (3 and (33 d u k u Q (38 d whee ( Φ ( Φ( k Q ( ( ψ( ψ( Soluton of dffeental equaton (38 usng ounday condton ( looks as follows Q u ( C cosh( k( k (39 As we can see we have solved functon u ( and now polem educes to the polem of fndng constant C 33 Soluton fo lowe Wall Usng equaton (34 and (6 and puttng them n to conjugaton condton (9 at value g( C( μ I ( μ ( μ Φ( (4 Equaton (9 we can ewte as Φ ( F F cosh( ρ (4 whee F and snh( cosh( snh( snh( F cosh( F We can contnue wth equaton (4 and (4 and get g( C( F F cosh( ρ whee (4 F ( μ I ( μ ( μ F Usng (6 and (4 we get followng devaton at value C ( ( μ μi μ ( μ Φ( (43
But fom (9 (4 and (43 we get C ( μ μi μ ( μ Φ( ( ( F F cosh( ρ C whee (44 F μ μ I ( μ ( μ F ( Usng equaton (4 and devaton of equaton (34 at value we get a a C ( F% F% cosh( ρ d d u( (45 Now sumttng equatons (4 and (4 n equaton (37 we get dffeental equaton du ug FC H ECcosh( ρ (46 d whee G C a a E F % F % a a F F % F% d d H Soluton of dffeental equaton looks followng G G u( C3e C4e EC FC H (47 cosh( ρ ρ G G Fom condton (8 and (47 follows that constants C theefoe u ( can ewte n the 3 C 4 followng way u( C3cosh( G EC FC H (48 cosh( ρ ρ G G puttng togethe (34 (4 and (48 we get a a U( Cu ( Φ( ( ( (49 o EC FC H C G ρ G G 3 cosh( cosh( ρ d d ( Now soluton fo wall contans only two unknown constants C and C 3 34 Soluton We have two addtonal condtons on functon u ( espectvely ( u( u (5 and u u Also n pont ( (5 values of functons U ( and U ( must e equvalent U ( U( (5 To satsfy equaton (5 we need to use (39 and (48 Q H C cosh( k( k G (53 E F C3cosh( G C cosh( ρ G G ut to satsfy equaton (5 - devatve of (39 and (48 Eρ GC3snh( G C snh( ρ G (54 Ck snh( k ( And to satsfy equaton (5 we wll use equatons (6 and (36 Cu ( Φ ( d ECcosh( FC H (55 C3cosh( G ρ G G And now polem educes to soluton of thee lnea equatons (53 (54 and (55 fo thee unknown constants C 3
Cu ( Φ ( d ECcosh( FC H C3cosh( G ρ G G Eρ GC3snh( G C snh( ρ G (56 Ck snh( k ( Q H C cosh( k( k G E F C3cosh( G C cosh( ρ G G Afte solvng system (56 we can put constants C 3 to equaton (7 o (49 and calculate value of tempeatue at any pont of ou D doman 4 D Soluton as the Smple Case of the 3D Soluton Usng followng ntegal values fo equatons (5 and (8 v ( U ( d (57 and fo equaton ( v( U ( d (58 we get new D polem d dv ( d d (59 d dv v ( l d d (6 wth followng ounday condtons dv ( v d (6 dv v l d (6 v v (63 dv d dv d (64 dv dv v ( (65 d d The soluton of polem (59-(65 can e wtten n followng fom: v( Cln C (66 v ( CI 3 C 4 whee I s Bessels modfed functons Hee the fou unknown constants can e easy detemned fom the fou ounday and conjugatons condtons (6-(64: ( C C ln ( l ( l ( l C3 C 4 ( l I ( l I ( l C ( ( ln C( C3 I C4 Cln C C3I C4 (67 5 Concluson We have constucted some appoxmate thee dmensonal analytcal solutons fo a peodcal system wth cylndcal fn when the wall and the fn consst of mateals whch have dffeent themal popetes Acknowledgements: Reseach was suppoted y Euopean Socal Fund and Councl of Scences of Latva (gant 555 Refeences: [] aus AD Analyss and evaluaton of extended suface themal systems Hemsphee pulshng Copoaton 98 [] Buks A Buke M Closed two-dmensonal soluton fo heat tansfe n a peodcal system wth a fn Poceedngs of the Latvan Academy of Scences Secton B Vol5 N5 998 pp8- [3] Buks A Buke M Gusenov S Analytcal two-dmensonal solutons fo heat tansfe n a system wth ectangula fn Advanced Computatonal Methods n Heat Tansfe VIII WIT pess 4 pp 35-44 [4] Buks A Buke M Some analytcal 3-D steady-state solutons fo systems wth ectangula fn IASME Tansactons Issue 7 Vol Septeme 5 pp -9 [5] Wood AS Tupholme GE Bhatt MIH Heggs PJ Pefomance ndcatos fo steady-state heat tansfe though fn assemles Tans ASME Jounal of Heat Tansfe 8 996 pp 3-36