2.5D Green s functions for transient heat transfer by conduction and convection

Transcription

1 .5D Grn s functions for transint hat transfr by conduction and convction A. Tadu & N. Simõs Dpartmnt of Civil Enginring, Univrsity of Coimbra, Portugal Abstract This papr prsnts fundamntal solutions for computing th hat fild gnratd by spatially sinusoidal harmonic hat lin sourcs placd in layrd solid formations. Ths xprssions modl th transint hat transfr by conduction and convction in thr-dimnsional mdia. Th tmpraturs and th hat fluxs at som point in th layrd mdium can both b computd whn a hat lin sourc producs nrgy. Th simulation is first prformd in th frquncy domain, whil tim rsponss ar computd using invrs Fourir transforms. Th authors bliv that ths solutions can b usful as bnchmark modls for numrical applications, such as th Boundary Elmnt Mthod. Thy ar also of grat valu if usd togthr with numrical modls, sinc th full discrtiation of th solid layr intrfacs is thn unncssary. Kywords: transint hat transfr, conduction, convction,.5d Grn s Functions, layrd formation. Introduction Most of th known tchniqus to solv transint convction-diffusion hat problms hav bn formulatd in th tim domain or using Laplac transforms. In th tim marching approach th solution is assssd stp by stp at conscutiv tim intrvals aftr an initially spcifid stat has bn assumd. Using th Laplac transform, a numrical transform invrsion is rquird to calculat th physical variabls in th ral spac, aftr th solution has bn obtaind for a squnc of valus of th transform paramtr. Th Laplac transform tchniqu has bn broadly usd for solving diffusion problms, but small truncation rrors can b magnifid in th invrsion procss and so th 4 WIT Prss, ISBN

2 44 Boundary Elmnts XXVI accuracy dpnds on an fficint and prcis invrs transform. Diffrnt invrsion mthods hav bn proposd ovr th yars, such as th Sthfst algorithm []. This work prsnts Grn s functions for calculating th transint hat transfr wav fild in layrd solid formations, in th prsnc of conduction and convction. Th problm is formulatd in th frquncy domain using tim Fourir transforms. Th proposd tchniqu allows th us of any typ of hat sourc, and dals with th unavoidabl static rspons. This work xtnds th work prformd by th authors for th dfinition of th stady-stat rspons of layrd solid mdia subctd to a spatially sinusoidal harmonic hat conduction lin sourc (.g. Tadu t al. []). Th problm is now solvd incorporating convction phnomna. Th proposd fundamntal solutions, rlat th hat fild variabls (fluxs or tmpraturs) at som position in th solid domain causd by a hat sourc placd lswhr in th mdia, in th prsnc of both conduction and convction phnomna. As in th prvious work, th tchniqu rquirs th knowldg of th Grn s function for th unboundd mdia, which is writtn first as a suprposition of cylindrical hat wavs along on horiontal dirction ( ) and thn as a suprposition of plan wavs, following a tchniqu similar to th on usd first by Lamb [3] for th propagation of lastodynamic wavs in two-dimnsional mdia. Othr authors, such as Bouchon [4] and Tadu and António [5], hav usd an quivalnt approach to calculat thr-dimnsional lastodynamic filds using a discrt wav numbr rprsntation. Th Grn s functions for a solid layrd formation ar formulatd as th sum of th hat sourc trms qual to thos in th full-spac and th surfac trms rquird to satisfy th boundary conditions at th intrfacs i.. continuity of tmpraturs and normal fluxs btwn solid layrs, and null normal fluxs or null tmpraturs at th outr surfac. Th total hat fild is achivd by adding th hat sourc trms, qual to thos in th unboundd spac, to that st of surfac trms, arising within ach solid layr and at ach intrfac. Th Grn s functions for th cas of a spatially sinusoidal, harmonic hat lin sourc placd in an unboundd mdium ar dvlopd by first applying a tim Fourir transform to th tim convction-diffusion quation for a hat point sourc and thn a spatial Fourir transform to th rsulting Hlmholt quation, along th dirction, in th frquncy domain. This mthodology is vrifid comparing its rsults with th xact tim solutions for on, two and thr-dimnsional point hat sourcs placd in an unboundd mdium. Bsids this vrification, th rsponss of th Grn s functions for a solid layrd formation ar compard with solutions providd by th Boundary Elmnt Mthod. Th xprssions prsntd in this work allow th hat fild insid a layrd solid mdium to b computd, without ithr th discrtiation of th intrior domain, which is ncssary whn using numrical tchniqus, such as th finit diffrncs mthod, or vn th discrtiation of th solid intrfacs using boundary lmnts tchniqus. 4 WIT Prss, ISBN

3 3D problm formulation and Grn s functions in an unboundd mdium Th transint convction-conduction hat transfr in solids with constant vlocitis along th x, y and dirctions is xprssd by th quation T T T T + + T V +V +V x y, () x y K x y K t in which V x, V y and V ar th vlocity componnts in th dirction x, y and rspctivly, t is tim, Ttxy (,,, ) is tmpratur, K k ( ρ c) is th thrmal diffusivity, k is th thrmal conductivity, ρ is th dnsity and c is th spcific hat. Applying a Fourir transform in th tim domain, on obtains iω + + V V V ˆ(,,, ) x + y + + T ω x y x y K x y K, () whr i and ω is th frquncy. Eqn () diffrs from th Hlmholt quation by th inclusion of a convctiv trm. Th fundamntal solution of qn () for a hat point sourc in an unboundd mdium, locatd at (,, ), can b xprssd as ˆ Tf ( ω, x, y, ) k x + y + Vxx+ Vyy+ V i Vx + Vy + V iω x + y + 4K K. (3) Whn th gomtry of th problm rmains constant along on dirction () th full 3D problm can b xprssd as a summation of simplr D solutions. This rquirs th application of a Fourir transformation along that dirction, writing this as a summation of D solutions with diffrnt spatial wavnumbrs k (Tadu & Kausl [6]). Th application of a spatial Fourir transformation to Vx + Vy + V iω i x + y + 4K K, (4) x + y + along th dirction, lads to Vxx+ Vyy+ V i Vx + Vy + V iω f( ω,,, ) ( ) T x y k H k r, (5) 4k 4K K H ar Hankl functions of th scond kind and ordr, and whr ( ) Boundary Elmnts XXVI 45 r ( x x) + ( y y). Th full thr-dimnsional solution is thn achivd by applying an invrs Fourir transform along th k domain. This invrs Fourir transformation can b xprssd as a discrt summation if w assum th xistnc of virtual sourcs, qually spacd at L, along, which nabls th solution to b obtaind by solving a limitd numbr of two-dimnsional problms, 4 WIT Prss, ISBN

4 46 Boundary Elmnts XXVI π T x y T x y k, (6) M i ˆ(,,, ) (,,, ) km ω ω m L m M with k m bing th axial wavnumbr givn by k m π m. Th distanc L is L chosn so as to prvnt spatial contamination from th virtual sourcs (Bouchon & Aki [7]). This tchniqu rflcts th adaptation of othr mathmatical and numrical formulations applid to solv problms such as wav propagation (Tadu t al [8]). Th application of a spatial Fourir transformation along th dirction in qn () lads to th following quation iω + V V x + y + ( k) T ( ω, x, y, k) x y K x y K, (7) whn V. Th fundamntal solution of this quation is givn by qn (5) ascribing V. Ths sam quations can b writtn as a continuous suprposition of hat plan phnomna. Eqn 5, which rsults whn a spatially sinusoidal harmonic hat lin sourc is applid at th point ( x, y ) along th dirction, subct to convction vlocitis V x, V y and V, is thn givn by th xprssion, Vxx+ Vyy+ V + iν y y i ikx ( x x ) f( ω,,, ) x 4π k ν T x y k dk, (8) V + V + V i x y ω whr ν ( k ) kx with ( Im( ν ) ), and th K intgration is prformd with rspct to th horiontal wav numbr ( k x ) along th x dirction. Th intgral in th abov quation can b transformd into a summation if an infinit numbr of such sourcs ar distributd along th x dirction, at qual intrvals L x. Th abov quation can thn b writtn as Vxx+ Vyy+ V n+ i E T f( ω, x, y, k) E Ed, (9) 4k n νn i i n whr E, E ν y y ik ( ), xn x x Ed, kl ν x V + V + V iω n xn x y ( k ) k with ( Im( n ) ) 4K K ν, k xn π n, which can L in turn b approximatd by a finit sum of quations ( N ). Notic that k corrsponds to th two-dimnsional cas. Th hat in th spatial-tmporal domain is calculatd by applying a numrical invrs fast Fourir transform in k, in th frquncy domain. Th computations ar prformd using complx frquncis with a small imaginary part of th form x 4 WIT Prss, ISBN

5 ωc ω iη (with η.7 ω, and ω bing th frquncy stp) to prvnt intrfrnc from aliasing phnomna. In th tim domain, this ffct is rmovd t by rscaling th rspons with an xponntial window of th form η. Th tim variation of th sourc can b arbitrary. Th tim Fourir transformation of th incidnt hat fild dfins th frquncy domain whr th BEM solution nds to b computd. Th rspons may nd to b computd from.h to vry high frquncis. Howvr, as th hat rsponss dcay vry fast with incrasing frquncy, w may limit th uppr frquncy whr th solution is rquird. Th frquncy.h corrsponds to th static rspons that can b computd whn th frquncy is ro. Th us of complx frquncis allows th solution to b obtaind bcaus, whn ωc iξ (for.h ), th argumnts of th quations ar othr than ro.. Vrification of th solution In ordr to vrify th formulation prsntd abov, th rsults ar compard with th analytical rspons in th tim domain. Th xact solution of th convctiv diffusion, xprssd by qn (), in an unboundd mdium subctd to a unit hat sourc is wll known and it allows th computation of th hat fild givn by both conduction and convction phnomna in th prsnc of thr, two or ondimnsional problms. Whn th hat sourc is applid at point (,, ) at tim t t, th tmpratur at ( x, y, ) is givn by th xprssion ( τvx + x) ( τvy + y) ( τv + ) 4Kτ Ttxy (,,, ) d / ρc( 4πKτ), if t > t, () whr τ t t, r is th distanc btwn th sourc point and th fild point ( x, y, ), and d 3, d and d whn in th prsnc of a thr, two and on-dimnsional problm, rspctivly (.g. Carslaw & Jagr []). In th vrification procdur, a homognous unboundd mdium with th - concrt thrmal proprtis ( k.4w.m. o C, Boundary Elmnts XXVI 47 c J.Kg. o C and -3 ρ 3Kg.m ) was xcitd at t 77.8h, by a unit hat sourc placd at ( x. m, y.m,. m ). Th convction vlocitis applid in th x, y 6 and dirction wr qual to m/s. Figur shows th tmpratur calculatd by qn () along a lin of 4 rcivrs placd from y.5 m to y.5 m, for cylindrical ( d ) and sphrical ( d 3 ) unit hat sourc, at diffrnt tims. Th tmpratur rsponss along ths rcivrs wr computd using th Grn s functions formulation. Th calculations wr first prformd in th frquncy domain in th frquncy rang [, 4x H ] with a frquncy incrmnt of ω H, which dfins a tim window of T 777.8h. Th rspons for a sphrical unit hat sourc has bn computd dividing th rsults givn by qn (3) by π. Th solution for th two-dimnsional cas (cylindrical 4 WIT Prss, ISBN

6 48 Boundary Elmnts XXVI unit hat sourc) has bn found with qn (5), whil th rsults for a plan unit hat sourc propagating along th y axis has bn modlld ascribing k and k xn to qn (9), multiplid by L x. Complx frquncis of th form ωc ω i.7 ω hav bn usd to avoid th aliasing phnomnon. Th spatial priod has bn st as Lx L k ( ρ c f ). In Figur, th solid lin rprsnts th xact tim solution givn by qn () whil th marks show th rspons obtaind using th proposd Grn s functions. Thr is good agrmnt btwn ths two solutions h 45 h 55 h 65 h.3 35 h 45 h 55 h 65 h Tmpratur (ºC).3. Tmpratur (ºC) X (m) X (m) a) b) Figur : Tmpratur along a lin of 4 rcivrs, at diffrnt tims (35 h, 45 h, 55 h and 65 h): a) for a cylindrical ( d ) unit hat sourc; b) for a sphrical ( d 3 ) unit hat sourc. 3 Grn s functions in a layrd formation Th Grn s functions for a multi-solid layr ar stablishd using th rquird boundary conditions at th solid-solid intrfacs. Considr a systm built from a st of m solid plan layrs of infinit xtnt boundd by two flat, smi-infinit, solid mdia. Th top smi-infinit mdium is dnominatd mdium, whil th bottom smi-infinit mdium is assumd to b th mdium m +. Th thrmal matrial proprtis and thicknss of th diffrnt layrs may diffr. Diffrnt vrtical convction vlocitis can b ascribd to ach solid layr. Th convction is computd assuming that th origin of convction coincids with th conduction sourc. Th systm of quations is achivd considring that th multi-solid layr is xcitd by a spatially sinusoidal hat sourc locatd in th first solid layr (mdium ). Th hat fild at som position in th solid domain is computd taking into account both th surfac hat trms gnratd at ach intrfac and th contribution of th hat sourc trm. For th solid layr, th hat surfac trms on th uppr and lowr intrfacs can b xprssd as 4 WIT Prss, ISBN

7 Boundary Elmnts XXVI 49 Vy ( y y ) n+ E t ( ω,,, ) n d n ν n T x y k E A E, Vy ( y y ) n+ E b ( ω,,, ) n d n ν n T x y k E A E, () whr E i, kl ν V iω + k k y n xn K x E layr l. Manwhil, K k ( ρ c) iν n y hl iν n y l l, E, with ( n ) Im ν and h l is th thicknss of th is th thrmal diffusivity in th solid mdium ( k, ρ and c rprsnt th thrmal conductivity, th dnsity and th spcific hat of th matrial in th solid mdium,, rspctivly). Th hat surfac trms producd at intrfacs and m +, govrning th hat that propagats through th top and bottom smi-infinit mdia, ar rspctivly xprssd by Vy ( y y) n+ E b ( ω,,, ) n d n ν n T x y k E A E, Vy( m+ ) ( y y) n+ K E ( m+ ) ( m+ ) t ( m+ )( ω,,, ) ( m+ ) n( m+ ) d n ν nm ( + ) T x y k E A E. () A systm of ( m + ) quations is assmbld, nsuring th continuity of tmpraturs and hat fluxs along th m + solid intrfacs btwn layrs. Each quation taks into account th contribution of th surfac trms and th involvmnt of th incidnt fild. All th trms ar organid according to th form Fa b cc3 cc4 cc4c5... c3 c4 c4c iνny 5... cc 4 k k k c4 i ny cc3c5 cc3... b ν A k n c3c5 c t iν 3 nh y... An cc3 k k b An c iνn h y... k... cmc4m c mc4mc5m t A nm... c4m c4c b 5m... Anm km km t Anm ( + )... cmc3mc5m c mc3m c ( m ) c + 4( m+ ) c c 3mc5m c3m 4( m+ )... km km k ( m+ ) (3) hl 4 WIT Prss, ISBN

8 4 Boundary Elmnts XXVI V y whr c + iν n i nh 5 ν. c, c Vy -i n ν, c 3 Vy hl y l, c ν n 4 Vy hl y l and ν Th rsolution of th systm givs th amplitud of th surfac trms in ach solid intrfac. Th tmpratur fild for ach solid layr formation is obtaind by adding ths surfac trms to th contribution of th incidnt fild, lading to th following quations top smi-infinit mdium (mdium ) Vy ( y y) n+ E b n d n ν n T ( ω, x, y, k ) E A E, if y < ; solid layr (sourc position) Vy( y y) Vy( y y) n+ i E t E b ( ) + n+ n d 4k n νn νn T ( ω, x, y, k ) H K r E A A E, if < y < h ; solid layr ( ) Vy ( y y ) n+ E E t b n + n d n νn ν n T ( ω, x, y, k ) E A A E if bottom smi-infinit mdium (mdium m + ) Vy( m+ ) ( y y) n+ E m+ ( m+ ) t ( m+ )( ω,,, ) ( m+ ) n( m+ ) d n ν n( m+ ) l l l l n h < y< h ; T x y k E A E. (4) Notic that whn th position of th hat sourc is changd, th matrix F rmains th sam, whil th indpndnt trms of b ar diffrnt. Howvr, as th quations can b asily manipulatd to considr anothr position for th sourc, thy ar not includd hr. 3. Vrification of th solution Th accuracy of th formulation prsntd in this papr is vrifid by comparing its rsults with th solution of th BEM modl for a crtain problm. Th BEM cod, which involvs th discrtiation of all solid intrfacs, maks us of th Grn s Functions for an unboundd mdium. Th BEM cod has bn validatd by applying it to a cylindrical circular ring cor, sinc th analytical solutions hav bn drivd for this particular cas. In ordr to avoid th unlimitd discrtiation of th solid intrfacs in th BEM modl a damping factor is considrd. This factor uss complx frquncis with a small imaginary part of th form ωc ω iη (with η.7 ω ) [Bouchon and Aki [9], Phinny []]. In th prsnt cas th lmnts ar distributd along th surfac up to 4 WIT Prss, ISBN

9 Boundary Elmnts XXVI 4 ( ρ ) L k c f, using th thrmal matrial proprtis from th solid dist mdium that lad to th largst spatial distanc. A flat concrt layr, 3mthick, boundd by two smi-infinit stl mdia, as displayd in Figur, is usd to valuat th accuracy of th proposd formulation. Th convction vlocitis applid to th thr mdia wr 6 5 m/s, 8 m/s and m/s for th top mdium, concrt layr and bottom mdium, rspctivly. Th thrmal matrial proprtis usd ar prsntd in Tabl. Th calculations hav bn prformd in th frquncy domain from H to 3 H, with a frquncy incrmnt of ω H and considring a singl valu of k qual to.4rad/m. Th amplitud of th rspons for two rcivrs placd in two diffrnt mdia was computd for a hat point sourc applid at ( x.m, y. m ). Th ral and imaginary parts of th rspons at rcivr ( x.m, y.5 m ) and rcivr ( x.m, y 3.5 m ) ar displayd in Figur, whn th imaginary part of th frquncy has bn st to η.7 ω. Th solid lins rprsnt th analytical rsponss, whil th markd points corrspond to th BEM solution. Th squar and th round marks dsignat th ral and imaginary parts of th rsponss, rspctivly. Th two solutions sm to b in vry clos agrmnt Tabl : Thrmal matrial proprtis. Thrmal conductivity Solid layr (concrt) Lowr solid mdium (stl) Top solid mdium (stl) - k.4 W.m. o C - k 63.9 W.m. o C - k 63.9 W.m. o C Dnsity ρ 3 Kg.m ρ 783 Kg.m ρ 783 Kg.m Spcific hat - c 88 J.Kg. o C - c 434 J.Kg. o C c 434 J.Kg. o C Amplitud (ºC) Amplitud (ºC) Figur : x -6 x -6 3x -6 Frquncy (H) -. x -6 x -6 3x -6 Frquncy (H) a) b) On solid layr boundd by two smi-infinit solid mdia: a) Rcivr. b) Rcivr. 4 WIT Prss, ISBN

10 4 Boundary Elmnts XXVI 4 Conclusions.5 Grn s functions, for computing th transint hat transfr by conduction and convction in an unboundd mdium and layrd mdia, hav bn prsntd. In this approach th calculations ar first prformd in th frquncy domain. Th rsults for a layrd formation ar obtaind adding th hat sourc trm and th surfac trms, rquird to satisfy th intrfac boundary conditions (tmpratur and hat fluxs continuity). Th vrification of th unboundd mdium formulation was obtaind comparing its tim rsponss and th xact tim solutions. In turn, th analytical solutions usd in th solid layrd mdia formulation wr vrifid using a BEM algorithm. Using ths two approachs togthr can b usful in th rsolution of nginring problms, such as inclusions placd in layrd formations. Rfrncs [] Sthfst, H., Algorithm 368: Numrical invrsion of Laplac transform. Communications of th Association for Computing Machinry, 3(), pp , 97. [] Tadu A., António J. & Simõs N.,.5D Grn s functions in th frquncy domain for hat conduction problms in unboundd, half-spac, slab and layrd mdia, accptd for publication in Computr Modling in Enginring & Scincs-CMES. [3] Lamb, H., On th propagation of trmors at th surfac of an lastic solid. Phil. Trans. Roy. Soc. London, A3, pp. -4, 94. [4] Bouchon, M., Discrt wav numbr rprsntation of lastic wav filds in thr-spac dimnsions, J. of Gophysical Rsarch 84, pp , 979. [5] Tadu, A. & António, J.,.5D Grn s functions for lastodynamic problms in layrd acoustic and lastic formations, Journal of Computr Modling in Enginring and Scincs-CMES, (4), pp ,. [6] Tadu, A. & Kausl, E., Grn s functions for two-and-a-half dimnsional lastodynamic problms, Journal of Enginring Mchanics ASCE, 6(), pp ,. [7] Bouchon, M. & Aki, K., Discrt wav-numbr rprsntation of sismicsourc wav fild. Bulltin of th Sismological Socity of Amrica, 67, pp , 977. [8] Tadu, A., Godinho, L., & Santos P. Wav motion btwn two fluid filld borhols in an lastic mdium, Enginring Analysis with Boundary Elmnts-EABE, 6(), pp. -7,. [9] Bouchon M., & Aki, K., Tim-domain transint lastodynamic analysis of 3D solids by BEM. Int. J. Numr. Mthods in Eng., 6, pp.79-78, 977. [] Carslaw, H. S., & Jagr, J. C., Conduction of hat in solids, scond dition, Oxford Univrsity Prss, 959. [] Phinny, R. A., Thortical calculation of th spctrum of first arrivals in th layrd lastic mdium, J. Gophysics Rs., 7, pp , WIT Prss, ISBN