Analytical versus BEM solutions for transient heat transfer by conduction and convection in 2.5D domains

Transcription

1 Alyticl vesus BEM solutios fo tsiet het tsfe by coductio d covectio i.5d domis N. imões & A. Tdeu Deptmet of Civil Egieeig, Uivesity of Coimb, Potugl Abstct This ppe pesets lyticl d BEM solutios fo tsiet het tsfe by coductio d covectio i the viciity of cylidicl cicul iclusios. Diffeet coditios e pescibed fo modelig iclusios with ull fluxes o tempetues log the boudy, d with elstic iclusios embedded i ubouded homogeeous medium. I the diffeet fomultios developed, the covectio is dil. The solutio is computed i the fequecy domi fo wide ge of fequecies d xil wveumbes, d time seies e the obtied by mes of (fst) ivese Fouie tsfoms ito spce-time. The Gee s fuctios i the fequecy domi, which e to be used i the BEM model, wee vlidted gist time solutios. The BEM models e the vlidted usig the lyticl solutios give hee. Theefte, the BEM model c be used to solve het popgtio though iegul cylidicl stuctues. Keywods: tsiet het tsfe, coductio, covectio, Gee s fuctios, fequecy domi, cylidicl cicul iclusios. Itoductio The het tsfe is subject of study with iteest i sevel es of the egieeig scieces. Most of the poblems coespod to ustedy chges of het betwee diffeet medi. Theefoe, the fomultios fo studyig those systems should cotemplte the tsiet het pheome. Het popgtio though iclusios of specific mteil plced i ubouded homogeeous medium with iegul shpe is fudmetl field of study i this e. Fo isotopic d ihomogeeous medi o iclusios with complex geometies, umeicl techiques such s Fiite Diffeeces, Fiite Elemets Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

2 44 Boudy Elemets XXVI Methods o Boudy Elemet Methods (BEM) e eeded. Appoches usig BEM oly equie the discetiztio of the sufces log the mteil discotiuities, while the othes metioed bove equie full discetiztio of the domi. The BEM techiques epeset lowe computtiol effot, sice the size of the system tht eeds to be solved is smlle. Howeve the theedimesiol tsiet het clcultios emi computtiolly demdig. The solutio becomes much simple if the geomety is two-dimesiol, eve if spheicl het souce is cosideed. I the peset wok, time Fouie tsfom is used to compute the tsiet het tsfe by coductio d covectio oud cylidicl cicul iclusios of ifiite legth. A sptil Fouie tsfom log the diectio i which the geomety does ot chge (the z diectio) is used to compute the solutio. This equies the solutio of sequece of two-dimesiol poblems with diffeet sptil wveumbes, k z, d fequecies. The thee-dimesiol field i the time domi is computed by pplyig ivese spce Fouie tsfoms. The solutio t ech fequecy is expessed equiig the vitio of wveumbe k z, which is the subjected to Fouie tsfomtio ito the sptil domi. The tsfomtio of the wveumbe i discete fom is chieved ssumig ifiite umbe of vitul poit souces plced log the z-xis. These e eqully spced d e f eough pt to void sptil cotmitio (e.g. Boucho d Aki []). This lysis uses complex fequecies to void the lisig pheomeo d to miimize the ifluece of the eighbouig fictitious souces (e.g. Phiey []). I fct, most of the ppoches to modellig het tsfe mke use of the Lplce tsfom of the time domi diffusio equtio, which hs the disdvtge tht ccucy is lost i the ivesio pocess. I this wok bief pesettio of the tsiet het tsfe fomultio by coductio d covectio is give; the bsic expessios used to obti the fequecy domi esposes e peseted. Aftewds, the umeicl (BEM) d lyticl equtios fo solvig het covectio-diffusio i the pesece of cylidicl cicul iclusios withi ubouded medium d subjected to moopole het souces e lso peseted. This wok descibes models fo thee diffeet iclusios, mely, elstic cylidicl cicul iclusio bouded by elstic medium, iclusio whee ull fluxes hve bee pescibed log the boudy, d iclusio fo which ull tempetues hve bee pescibed log its boudy. The veifictio pocedue icluded i this wok coespods to the compiso betwee the BEM d the lyticl model whe cylidicl cicul iclusios with ull fluxes d ull tempetues log the boudy e cosideed. Thee-dimesiol poblem fomultio The tsiet het coductio i solids c be descibed by the diffusio equtio Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

3 Boudy Elemets XXVI 45 T T + ( -V y cos θ +V x si θ) θ z K θ, () T Vz T T ( Vcos x θ+vsi y θ) - = K K z K t i which V x, V y d V z e the velocity compoets i the diectio x, y d z espectively, = x + y, θ is the zimuth, t is time, Tt (,, θ, z) is k tempetue, K = ρ c is the theml diffusivity, k is the theml coductivity, ρ is the desity d c is the specific het. Fouie tsfomtio i the time domi expessio gives the Helmholtz equtio i the fequecy domi T T + ( -V y cos θ +V x si θ) θ z K θ, () T Vz T iω ( Vcos +Vsi ) - ˆ x θ y θ + T( ω, x, y, z) = K K z K whee i= d ω is the fequecy. Eq () c be solved fo het poit souce, locted t the oigi of the coodite system, to give the fudmetl solutio Vx cosθ+ Vysiθ+ Vzz K i Vx + Vy + Vz iω + z 4K K ˆ e Tf ( ω, x, y, z ) = e. (3) k + z I my cses the computtio of 3D poblems equies cosideble compute effot. If the geomety of the poblem does ot chge log oe diectio (z) the solutio becomes simple, sice the full 3D poblem c be computed s summtio of D solutios with diffeet sptil k z wveumbes (e.g. Tdeu d Kusel [3]). This is implemeted pplyig Fouie tsfomtio log the z diectio. Applyig this pocedue to equtio Vx + Vy + Vz iω i + z 4K K e, (4) + z leds to the followig equtio Vx cosθ+ Vysiθ+ Vzz K i e V V V i x + y + z ω f( ω,,, z) = ( z ) T x y k H k, (5) 4k 4K K whee H ( ) e Hkel fuctios of the secod kid d ode. The full thee-dimesiol solutio is sythesized pplyig ivese Fouie tsfom log the k z diectio. The ivese Fouie tsfomtio c be witte s discete summtio, if oe ssumes the existece of vitul souces eqully spced, t distce L, log z. The solutio c thus be obtied by solvig limited umbe of two-dimesiol poblems. Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

4 46 Boudy Elemets XXVI M π ˆ(,,, ) (,,, ) ikzm z T ω θ z = T ω θ kzm e, (6) L m= M π with k zm beig the xil wveumbe give by kzm = m. The distce L must L be lge eough to void sptil cotmitio fom the vitul souces (e.g. Boucho d Aki []). imil techiques hve bee used by the uthos to study wve popgtio (Tdeu et l [4], Godiho et l [5]). 3 Boudy elemet fomultio I ode to compute the full thee-dimesiol het field geeted by het poit souce plced i the viciity of solid cylidicl iclusio with iegul shpe i the pesece of covectio log x d y xis, the BEM is used to solve the followig equtio fo ech vlue of k z, coespodig to idividul D poblems, iω + V V ( ) ˆ x + y + kz T( ω, x, y, kz) = x y K x y K. (7) I homogeeous isotopic solid medium of ifiite extet, bouded by ic sufce, d subjected to icidet het field give by T, the ecipocity theoem my be pplied i ode to wite the elevt boudy itegl equtios [6], log the exteio domi ( ext ) ( ext ) ( ext ) ct ( x, y, k, ω) = q ( xy,, ν, k, ω) G ( xyx,,, y, k, ω) ds z z z ( ext ) ( ext ) H ( x, y, ν, x, y, kz, ω) T ( x, y, kz, ω) ds ( ext ) ( ext) ( ext) G x y x y kz T x y kz ds+ T ic x y kz (,,,,, ω) (,,, ω)v (,,, ω) log the iteio domi (it) (it) (it) ct ( x, y, k, ω) = q ( xy,, ν, k, ω) G ( xyx,,, y, k, ω) ds z z z (it) (it) (,, ν,,, z, ω) (,, z, ω) (it) (it) ( it) (,,,, z, ω) (,, z, ω)v H x y x y k T x y k ds G x y x y k T x y k ds (8). (9) I eqs (8) d (9), supescipts it d ext coespod to the iteio d exteio domi espectively, ν is the uit outwd oml log the boudy, G d H e espectively the fudmetl solutios (Gee s fuctios) fo the tempetue ( T ) d het flux ( q ), t ( xy, ) due to vitul poit het lod t ( x, y ), V = Vxx + Vyy, c is costt defied by the shpe of the boudy, with vlue of / if ( x, y ) whe the boudy is smooth. Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

5 Boudy Elemets XXVI 47 The two-d--hlf dimesiol Gee s fuctios fo tempetue i Ctesi co-odites e those fo ubouded solid medium, V i V iω K Gxyx (,,, y, kz, ω) = e H + ( kz), () 4k 4K K with ( ) ( ) = x x + y y, = x + y is the distce to the covectio souce positio, d H ( ) e Hkel fuctios of the secod kid d ode. The boudy is discetized ito N stight boudy elemets, with oe odl poit i the middle of ech elemet. The itegtios i eqs (8) d (9) e evluted usig Gussi qudtue scheme, whe they e ot pefomed log the loded elemet. Fo the loded elemet, the existig sigul itegds i the souce tems of the Gee s fuctios e clculted i closed fom. The fil itegl equtios e mipulted d combied so s to impose the cotiuity of tempetues d het fluxes log the boudy of the iclusio, to estblish system of equtios. The solutio of this system of equtios gives the odl tempetues d het fluxes, which llow the eflected het field to be defied. 3. Iclusio with ull fluxes log its boudy I this cse the boudy coditios pescibe ull oml het fluxes log the boudy. Thus, eq (9) is simplified to (it) (it) (it) ct ( x, y, kz, ω) = H ( x, y, ν, x, y, kz, ω) T ( x, y, kz, ω) ds. () (it) (it) ( it) G ( x, y, x, y, k, ω) T ( x, y, k, ω)v ds z z The solutio of this itegl fo bity boudy sufce ( ) gi equies the discetiztio of the boudy ito N stight boudy elemets, followig pocedue simil to tht descibed bove. 3. Iclusio with ull tempetues log its boudy Null tempetues e ow pescibed t the sufce of the boudy, which leds to the equtio (it) (it) (it) ct ( x, y, k, ω) q ( x, y, ν, k, ω) G ( x, y, x, y, k, ω) ds. () = z z z The solutio of this equtio is lso obtied s descibed bove. 4 Alyticl solutio Coside solid cylidicl cicul iclusio defied by the dius, bouded by exteio solid medium. This iclusio is heted by hmoic souce with dil covectio, plced i the exteio solid medium (with theml Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

6 48 Boudy Elemets XXVI coductivity k, desity ρ, specific het c d covectio velocity V ). The het geeted by this souce popgtes d hits the sufce of the iclusio. Afte stikig the sufce, pt of the icidet eegy is eflected bck ito the exteio solid medium, d the emiig eegy is tsmitted ito the solid iclusio mteil (with theml coductivity k, desity ρ, specific het c d covectio velocity V ), i the fom of popgtig het wves. This pocess will be epeted util ll the eegy is dissipted. 4. Icidet het field (o fee-field) The thee-dimesiol icidet field geeted by moopole het souce, plced t ( x,,), stisfies eq () scibig V = z, d c be expessed s V K V iω i ' + z 4K K ˆ Ae e Tic( ω,, ', z) = k + z ' with ( ) ' = x x + y, (3) whee the subscipt ic deotes the icidet field, A is the het mplitude, k K = ρ c d ' defies the distce betwee the souce d the eceive. Whe Fouie tsfomtio is pplied log the z diectio, the icidet field c be expessed s summtio of D souces, with diffeet sptil wveumbes, M ˆ ' π ikzm z Tic( ω,, ) = T ic( ω,, ', kzm) e, (4) L m= M ia 4k V K with T ( ω,, k ) = e H ( k '' ), k ( k ) ic ( ) '' = x x + y. zm α V i ω α = + zm d 4K K Eq (4) poses difficulty, howeve, becuse it expesses the icidet field i tems of het wves ceted t the souce poit ( x,, ), d ot t the xis of the cylidicl iclusio. This poblem c be ovecome by expessig the icidet het field i tems of het wves ceted t the oigi, which c be chieved by usig Gf's dditio theoem (Wtso [7]), to give the expessios (i cylidicl coodites): V ia K T ic( ω,, θ, kzm) = e ( ) εj ( kα ) H( kα )cos( θ), whe > 4k = V i K A ic zm = ( ) α α 4k = T ( ω,, θ, k ) e ε H ( k ) J ( k )cos( θ), whe <, (5) i which is the distce fom the souce to the xis of the iclusio, J ( ) if = e Bessel fuctios of ode d ε =. if Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

7 Boudy Elemets XXVI Reflected het field i the exteio egio I the fequecy-xil-wveumbe domi, the eflected het field i the exteio egio c be expessed i fom simil to tht of the icidet field, mely V K ef zm = = T ( ω,, θ, k ) e A H ( k )cos( θ), (6) α i which the subscipt ef deotes the eflected het field, d A is ukow coefficiet to be detemied fom ppopite boudy coditios. 4.3 Tsmitted het field i the iteio egio The tsmitted het field i the iclusio cosists of stdig het wves, which c be expessed s: V K ts zm = = T ( ω,, θ, k ) e B J ( k )cos( θ), (7) α i which the subscipt ts deotes the tsmitted het field, V ω i k kα = + ( k ) zm, K 4K K = ρ c, B is gi ukow coefficiet to be detemied by imposig the ppopite boudy coditios 4.4 Defiitio of A d B Next, ppopite boudy coditios e estblished to obti the eflected d tsmitted het fields withi the solid iclusio; tht is, the cotiuity of tempetues d oml het fluxes t the solid-solid itefce, T ic( ω,, θ, kzm) + T ef ( ω,, θ, kzm) = T ts( ω,, θ, kzm), T ic( ω,, θ, k ) zm T ef ( ω,, θ, kzm) T ts ( ω,, θ, kzm) k + k = k. (8) Combiig eqs (5), (6) d (7) oe obtis system of equtios which is the used to fid the coefficiets A d B, with A ( ) b ε = B b, (9) V K = e H ( k ) ; α V = k H( kα ) + H ( ) ( ) kα k H α + ( kα ) e K V K = e J ( k ) ; α V K ; Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

8 43 Boudy Elemets XXVI V = k J( kα ) + J ( ) ( ) kα k J α + ( kα ) e K V ia K α α 4k b = e H ( k ) J ( k ) ; V K V ia K V = α α + α α + α 4 K b e H ( k ) J ( k ) J ( k ) ( k ) J ( k ). The coefficiets A d B with the seies solutios (eqs (6) d (7)) c ow be used to detemie the eflected d the tsmitted het of the system. The solutio fo het souce excited i the iteio c be obtied i simil wy by chgig the icidet field d gi scibig the cotiuity of tempetues d oml het fluxes t the solid-solid itefce. If the souce is plced i the ie solid, the tems b j e defied s: V ia K = ( α ) ( α ) 4k V ia K V = α α + α α + α 4 K b e J k H k b e J ( k ) H ( k ) H ( k ) ( k ) H ( k ). ; 4.5 Null tempetues log the boudy of the iclusio At the sufce of the iclusio ( = ) the boudy coditio is give by, T ( ω,, θ, k ) + T ( ω,, θ, k ) =. () ic zm ef zm ubstitutig eqs (5) d (6) ito the bove coditio gives the followig, i ( ) ε J( kα ) H( kα ) 4k B =. () J ( k ) α 4.6 Null fluxes log the boudy of the iclusio The icidet het field is ll eflected bck ito the ubouded medium, veifyig the coditio t =, T ic( ω,, θ, kzm) T ef ( ω,, θ, kzm) k + k =. () Ude this coditio, combiig eqs (5) d (6), oe obtis, i V ( ) ε J( kα ) H( kα ) + H ( ) ( ) k k H ( k ) 4k K α α + α B =. (3) V H ( k α ) ( ) ( ) + J k α k α J + ( k α ) K Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

9 Boudy Elemets XXVI 43 5 BEM veifictio The BEM lgoithm ws implemeted d veified by pplyig it to cylidicl cicul iclusio whe ull fluxes o ull tempetues wee pescibed log its boudy. The eflected het esposes fo wide ge of fequecies ws clculted d comped with the solutios of the lyticl ppoch peseted bove. A iclusio, subjected to hmoic het souce with dil covectio 6 ( m/s), plced t ( x =.3m, y =.m ), with dius of.5 m, is - modelled. Its popeties e ssumed to be k =.W.m. o C, - c = 38J.Kg. o C -3 d ρ = 5Kg.m. The clcultios hve bee pefomed i 7 the fequecy domi fom Hz to 64 Hz, with fequecy icemet of 7 ω = Hz d cosideig sigle vlue of the pmete k z equl to.4d/m. The el d imgiy pts of the esposes fo eceive plced t ( x =.m, y =.m ) e illustted i Figue. olid lies epeset the lyticl solutios while mked poits idetify the BEM esults. The cicle d the tigle mks idicte the el d imgiy pts of the BEM esposes, espectively, computed usig 5 costt boudy elemets Amplitude -. Amplitude x -6 4x -6 6x -6 x -6 4x -6 6x -6 Fequecy (Hz) Fequecy (Hz) ) b) Figue : Rel d imgiy pts of the het esposes: ) Cylidicl cicul with ull fluxes log the boudy; b) Cylidicl cicul with ull tempetues log the boudy. All the plots displyed i this ppe evel excellet geemet betwee the two fomultios peseted. Besides the esults illustted, the solutios fo cylidicl cicul iclusio bouded by exteio solid medium, fo het souces d eceives plced t diffeet positios, hve cofimed vey good esults. Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN

10 43 Boudy Elemets XXVI 6 Coclusios The methodology descibed i this wok is bsed o discete itegtio ove wveumbes d fequecies to compute the thee-dimesiol (3D) het field geeted by hmoic het poit souces hetig cylidicl cicul iclusios i ubouded solid medium. The cses fo ull fluxes o ull tempetues pescibed log the boudy iclusio wee lso peseted. The discetiztio of the wveumbe-fequecy itegl tsfom peseted is mthemticlly equivlet to peiodic sequece of souces, pllel to the xis of the cylide, tht e lso peiodic i time. The effects of these peiodicities hve bee emoved by usig complex fequecies. Alyticl d umeicl fomultios wee implemeted i ode to study the tsiet het tsfe cosideig both coductio d covectio pheome. The compiso betwee these two ppoches evels excellet geemet, which suggests tht the BEM lgoithm could coveietly be pplied to poblems whee iegul cylidicl iclusios e peset. Refeeces [] Boucho M., Aki, K., Discete wve-umbe epesettio of seismicsouce wve field, Bulleti of the eismologicl ociety of Ameic, 67, pp , 977. [] Phiey, R. A., Theoeticl clcultio of the spectum of fist ivls i lyeed elstic medium, J. Geophys. Res., 7, pp , 965. [3] Tdeu, A., Kusel, E., Gee s fuctios fo two-d--hlf dimesiol elstodymic poblems. Joul of Egieeig Mechics ACE, 6(), pp ,. [4] Tdeu, A., Godiho, L., tos, P., Wve motio betwee two fluid filled boeholes i elstic medium, EABE Egieeig Alysis with Boudy Elemets, 6(), pp.-7, [5] Godiho, L., Atóio, J., Tdeu, A., 3D soud sctteig by igid bies i the viciity of tll buildigs. Joul of Applied Acoustics, 6(), pp. 9-48, [6] Wobel, L. C., The Boudy Elemet Method Applictio i Themo- Fluids d Acoustics, Joh Wiley d os, LTD.,. [7] Wtso, G. N., A Tetise o the Theoy of Bessel Fuctios, secod editio, Cmbidge Uivesity Pess, 98. Boudy Elemets XXVI, C. A. Bebbi (Edito) 4 WIT Pess, IBN