Journal o Appled Mathematc and Computatonal Mechanc 7, 6(4), 57-65 www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.4.6 e-issn 353-588 MODELLING OF TRANSIENT HEAT TRANSPORT IN TWO-LAYERED CRYSTALLINE SOLID FILMS USING THE INTERVAL LATTICE BOLTZMANN METHOD Alcja Paecka Belkhayat, Anna Korczak Inttute o Computatonal Mechanc and Engneerng, Slean Unverty o Technology Glwce, Poland alcja.paecka@poll.pl, anna.korczak@poll.pl Receved: 6 December 7; Accepted: 8 December 7 Abtract. In the paper, the numercal modellng o heat traner n one-dmenonal crytallne old lm condered. A generalzed two-layer problem decrbed by the Boltzmann tranport equaton tranormed n the phonon energy denty equaton upplemented by the adequate boundary-ntal condton. Such an approach n whch the parameter appearng n the problem analyed are treated a the contant value wdely ued, but n th paper the nterval value o relaxaton tme and the boundary condton or lcon and damond are taken nto account. The problem ormulated ha been olved by mean o the nterval lattce Boltzmann method ung the rule o drected nterval arthmetc. In the nal part o the paper the reult o numercal computaton are preented. MSC : 65M99, 8A Keyword: Boltzmann tranport equaton, nterval lattce Boltzmann method, drected nterval arthmetc. Introducton In delectrc materal and emconductor, the heat tranport manly realzed by a quanta o lattce vbraton called phonon. The phonon repreent the conducton o heat and electrcty through old. In non-metal, phonon a heat carrer alway move rom the part wth the hgher temperature to the part wth the lower temperature and, durng th move, phonon carry energy. Th knd o phenomena can be decrbed by the Boltzmann tranport equaton (BTE). It hould be ponted out that takng nto account the extremely hort duraton and the doman dmenon expreed n nanometer, the macrocopc heat conducton equaton baed on the Fourer law cannot be ued [, ]. Such an approach n whch the parameter appearng n the mathematcal model are treated a the contant value wdely ued [3, 4]. Here, the nterval value o relaxaton tme and boundary condton
58 A. Paecka Belkhayat, A. Korczak or ucceve ub-doman are taken nto account. The relaxaton tme etmated expermentally, and t actual value tll a ubject o dcuon [5]. In the paper the heat tranport proceedng n a two-layered thn lm condered [6-9]. To olve the problem ormulated, the nterval veron o the lattce Boltzmann method appled ung the rule o drected nterval arthmetc [, ]. In the nal part o the paper the example o numercal computaton are hown.. Boltzmann tranport equaton The unteady BTE n a phonon energy denty ormulaton ung the mplyng aumpton o the Debye model or one-dmenonal two-layered analy [3, ] can be wrtten a + e = + qv τ r e e e v () t where =, correpond to the ucceve layer o the thn lm (lcon, damond), e the phonon energy denty, e the equlbrum phonon energy denty, v the requency-dependent phonon propagaton peed, τ r the requency-dependent phonon relaxaton tme, t denote the tme and q v the external heat generaton rate related to a unt o volume. Ung the Debye model, the dependence between phonon energy and lattce temperature can be calculated rom the ollowng ormula 9η e ( T ) = dz T Θ ΘD / T 3 k b z 4 3 () D exp( z) where Θ D the Debye temperature o the old, k b the Boltzmann contant, T the lattce temperature whle η the number denty o ocllator [3]. The equaton () hould be upplemented by the boundary and ntal condton. 3. Interval lattce Boltzmann equaton The lattce Boltzmann method (LBM) a numercal technque or the mulaton o heat traner. The LBM olve a dcretzed et o the BTE known a the lattce Boltzmann equaton. The phonon energy denty dened a the um e ( x, t) = e ( x, t) + e ( x, t) = e ( x, t) (3) d d=
Modellng o tranent heat tranport n two-layered crytallne old lm 59 where e the phonon energy denty n the potve x drecton or th layer whle e the phonon energy denty n the negatve x drecton and d gne the lattce drecton. The nterval Boltzmann tranport equaton or the one-dmenonal problem take the orm [3] d d d d e e e e + v = + q + t x r, τ τ r v (4) where v = x/ t the component o velocty along the x-ax, x the + lattce dtance rom te to te, t = t t the tme tep needed or a phonon + to travel rom one lattce te to the neghborng lattce te, τ r = r, τ τr the nterval relaxaton tme and e = e ( x, t)/ d. d The et o equaton (4) mut be upplemented by the boundary-ntal condton [6, 9] x = : e (, t) = e( Tb) x = L : e ( L, t) = e ( Tb ) t = : e ( x, ) = e ( T ) (5) where T b and T b are the nterval boundary temperature and T the ntal temperature. Between the ucceve ub-doman the contnuty condton can be taken nto account [9] x= L/ : e ( x, t) = e ( x, t) (6) The nterval LBM algorthm ha been ued to olve the problem analyed [6, ]. The approxmate orm o the equaton (4) o the ollowng orm + ( ) ( )( ) ( ) + + ( ) ( )( ) ( ) e = t τ e + t τ e + t q e = t τ e + t τ e + t q r r v r r v (7) Takng nto account the aumpton that a / D b σ( b) σ( a) σ( b) σ( a) /, /,, = σ( b) σ( b) σ( b) σ( b) /, /, Z, a b a b a b D\ Z a b a b a b D\ Z (8)
6 A. Paecka Belkhayat, A. Korczak and σ( b) σ( a) σ( b) σ( a) a b, a b a, b D\ Z σ( a) τ( b) σ( a) σ( a) τ( b) σ( a) a b, a b, a D\ Z, b Z σ( b) σ( b) τ( a) σ( b) σ( b) τ( a) a b, a b, a Z, b D\ Z a b = + + + + mn ( a b, a b ),max( a b, a b ), a, b Z P + + + + max( a b, a b ),mn( a b, a b ), a, b Z I, ( a ZP, b ZI) or ( a ZI, b ZP) (9) the product t r ( e ) τ ( =, = 3, = 4) calculated ung the rule o drected nterval arthmetc accordng to the ollowng ormula [ ] ( ) [ ] [ ] ( ) t 8 8 r e 5, 5 6.37, 6.69.49,.53 τ = = 8 8 8 8.78,.75.49,.53 =.6,.5 () A a reult, the nterval obtaned mproper. Ater ubequent computaton the nterval lattce temperature determned ung the ormula (ee eq. ()) ΘD / T 3 + 3 z T = 4 e ( T ) Θ D 9η kb dz () exp( z) 4. Reult o computaton A a numercal example, the heat tranport n a lcon-damond lm o the dmenon L = nm ha been analyed. The ollowng nput data have been τ =, ntroduced or a lcon-damond lm repectvely: r [ 6.37, 6.69] p r [.38,.4] p τ =, Θ = 64 K, Θ = K, T = [575, 65]K, D 3 T b = [9.5, 37.5]K, T = 3 K, q v = W/m, x = nm and t = 5p. Fgure llutrate the nterval temperature dtrbuton n the doman condered or the choen tme. Fgure preent the coure o the temperature uncton at the nternal node x = 6 nm () and x = 6 nm () or the lcon and damond layer repectvely. D b
Modellng o tranent heat tranport n two-layered crytallne old lm 6 65 55 3p p 45 p 35 5 5 5 x [nm] Fg.. The nterval temperature dtrbuton In the econd analyed example t aumed that the external heat generaton rate related to an unt o volume the nterval number 8 8 3 q v =.975,.5 W/m. 65 55 45 35 5 t [p] 3 Fg.. The nterval heatng curve at nternal node
6 A. Paecka Belkhayat, A. Korczak Smlar to the prevou example, Fgure 3 llutrate the nterval temperature dtrbuton n the doman condered or the choen tme, and Fgure 4 preent the coure o the temperature uncton at the ame nternal node. 3p 9 p 7 p 5 3 5 5 x [nm] Fg. 3. The nterval temperature dtrbuton 9 7 5 3 t [p] 3 Fg. 4. The nterval heatng curve at nternal node
Modellng o tranent heat tranport n two-layered crytallne old lm 63 3p 9 p 7 p 5 3 5 5 x [nm] Fg. 5. The nterval temperature dtrbuton 9 7 5 3 t [p] 3 Fg. 6. The nterval heatng curve at nternal node In the lat numercal example, accurate boundary temperature T b= 6 K and T b= 3 K have been ntroduced. In Fgure 5, the nterval temperature dtrbuton n the doman condered or the choen tme are hown and Fgure 6 preent the coure o the temperature uncton at the ame nternal node.
64 A. Paecka Belkhayat, A. Korczak It hould be ponted out, that or each node o the doman condered there are two curve repreentng the begnnng and end o temperature nterval. The nterval oluton llutrated by the area between thee two curve. Addtonally one can ee, that or longer calculaton tme, the temperature nterval are wder (ee Fg., 4 and 6). It vble that a greater number o operaton n the et o nterval number mpact on the ncreae o the wdth o the obtaned nterval. 5. Concluon In the paper a nterval veron o the lattce Boltzmann method or olvng D problem n two-layered crytallne old lm ha been preented. A model wth nterval value o relaxaton tme, boundary condton and the external heat generaton rate related to a unt o volume or a lcon-damond lm ha been propoed. The generalzaton o LBM allow one to nd the numercal oluton n the nterval orm, and uch normaton may be mportant, epecally or the parameter that are etmated expermentally, or example the relaxaton tme. The problem analyed can be extended to mult-layered thn lm. Reerence [] Chen G., Borca-Tacuc D., Yang R.G., Nanocale heat traner, [n:] Encycl. o Nanocence and Nanotechnology, CA, Amercan Scentc Publher, Valenca, 7, 4, 49-359. [] Smth A.N., Norr P.M., n Heat Traner Handbook, A. Bejan, D. Krau, Ed. 3, 39. [3] Ecobar R.A., Gha S.S., Jhon M.S., Amon C.H., Mult-length and tme cale thermal tranport ung the lattce Boltzmann method wth applcaton to electronc coolng, Journal o Heat and Ma Traner 6, 49, 97-7. [4] Joh A.A., Majumdar A., Tranent balltc and duve phonon heat tranport n thn lm, Journal o Appled Phyc 993, 74(), 3-39. [5] Narumanch S., Murthy J.Y., Amon C.H., Smulaton o unteady mall heat ource eect n ub-mcron heat conducton, Journal o Heat Traner 3, 3, 896-93. [6] Gha S.S., Km W.T., Amon C.H., Jhon M.S., Tranent thermal modelng o a nanocale hot pot n multlayered lm, Journal o Appled Phyc 6, 99. [7] Majchrzak E., Mochnack B., Suchy J.S., Numercal mulaton o thermal procee proceedng n a mult-layered lm ubjected to ultraat laer heatng, Journal o Theoretcal and Appled Mechanc 9, 47,, 383-396. [8] Majchrzak E., Mochnack B., Greer A.L., Suchy J.S., Numercal modelng o hort pule laer nteracton wth mult-layered thn metal lm, CMES: Computer Modelng n Engneerng and Scence 9, 4,, 3-46. [9] Ppat S., Chen Ch., Geer J., Sammaka B., Murray B.T., Multcale thermal devce modelng ung duon n the Boltzmann Tranport Equaton, Internatonal Journal o Heat and Ma Traner 3, 64, 86-33. [] Markov S.M., On drected nterval arthmetc and t applcaton, Journal o Unveral Computer Scence 995,, 54-56.
Modellng o tranent heat tranport n two-layered crytallne old lm 65 [] Paecka-Belkhayat A., Interval boundary element method or tranent duon problem n two layered doman, Journal o Theoretcal and Appled Mechanc, 49,, 65-76. [] Ecobar R.A., Smth B., Amon C.H., Lattce Boltzmann modelng o ubcontnuum energy tranport n crytallne and amorphou mcroelectronc devce, Journal o Electronc Packagng 6, 8(). [3] Paecka-Belkhayat A., Korczak A., Modellng o tranent heat tranport n one-dmenonal crytallne old ung the nterval lattce Boltzmann method, Recent Advance n Computatonal Mechanc, Taylor & Franc Group, A Balkema Book, London 4, 363-368.