Application of the lattice Boltzmann method for solving conduction problems with heat flux boundary condition.

Transcription

1 November 5-7, 9 - Sousse Tunsa Applcaton of the lattce Boltzmann method for solvng conducton problems wth heat flux boundary condton. Raoudha CHAABANE, Faouz ASKRI, Sass Ben NASRALLAH Laboratore d Etudes des Systèmes Thermques et Energétques Ecole Natonale des Ingéneurs de Monastr Av. Ibn ElJazzar 519 Monastr- Tunse Raoudha.Chaabane@ssatgb.rnu.tn ABSTRACT The lattce Boltzmann method (LBM) has been developed over the last decade as an alternatve promsng tool for flud flows. It has been wdely used n many knds of complex flows such as turbulent flow, solar collectors, multphase flow and mcro flow. Ths artcle deals wth the mplementaton of the lattce Boltzmann method (LBM) for the soluton of conducton problems wth heat flux and temperature boundary condtons. Problems n two dmensonal rectangular geometres have been consdered. In the -D geometry, the south and the north boundary are subjected to constant heat flux condton. The remanng boundares are at prescrbed temperatures. The energy equaton s solved usng the LBM. The results of the LBM have been found to compare very well wth those avalable n the lterature. Index Terms: Lattce Boltzmann method, heat conducton, unform lattces, heat flux, D heat transfer. 1. INTRODUCTION Nowadays, the lattce Boltzmann method (LBM) s beng vewed as a potental computatonal tool to analyze a large class of problems n scence and engneerng [1 11]. Recently, thermal lattce Boltzmann method has attracted much attenton because of ts potental applcatons as well as practcal mportance n engneerng desgns and energy related problems, such as solar collectors, thermal nsulaton, coolng of electronc components, heat exchangers, ar heatng systems for solar dryers, passve solar heatng and storage technology to name just a few. Varous confguratons may be consdered for ths problem. In comparson wth the conventonal computatonal technques based on the fnte dfference method, the fnte element method and the fnte volume method, ths surge n nterest s attrbuted to the bottom-up approach nherent wth the LBM. A smple calculaton procedure, smple and more effcent mplementaton for parallel computaton, straghtforward and effcent handlng of complex geometres and boundary condtons and hgh computatonal performance wth regard to stablty and precson are some of the man advantages of the LBM [1 8]. Owng to the above attrbutes, these days, the LBM s ncreasngly beng appled n the analyss of a large class of flud flow and heat transfer problems [1 11]. Analyss of conducton problems wth flux boundary condtons fnds applcatons n furnace desgn, fre protecton systems, foam nsulatons, soldfcaton/meltng of semtransparent materals, hgh-temperature porous nsulatng materals, glass-fludzed bed, electroncs chp and power plants, etc. [1-13]. A few papers have dscussed the problems wth flux boundary condtons [14-15]. Thus, the present work ams at further extendng the applcaton of the LBM to solve heat conducton problems dealng wth temperature as well as heat flux boundary condtons. We consder two-dmensonal rectangular geometry where one or two boundares can be at prescrbed heat flux condtons. The energy equaton s solved usng the LBM and obtaned results are compared wth those avalable n the lterature.. FORMULATION AND KINETIC EQUATION We have consdered transent heat conducton heat n a -D rectangular geometry. Thermo-physcal propertes of the medum are assumed constant. The system s ntally at temperaturet E. For tme t>, the south and the north boundares are subjected to heat fluxes q T, S and q T, N, respectvely. The east and the west boundares are kept at temperatures T E and T W, respectvely. For the problem under consderaton, and n the absence of convecton and radaton, the energy equaton s gven by T T Q (1) t Where s the thermal dffusvty. The startng pont of the LBM s the knetc equaton satsfes a dscretzed evoluton equaton of the form [16] f e. f, 1,,3,.., b t () - 1 -

2 The collson operator represents the rate of change of fdue to collsons. It ncorporates all the physcs and modellng of any partcular problem at hand. The smplest model for s the Bhatnagar Gross Krook (BGK) model [16] 1 ( ) [ f f ] fs the partcle dstrbuton functon denotng the number of partcles at the lattce node drecton wth (3) r at tme t movng n veloctye along the lattce lnk r et connectng the nearest neghbours. b s the number of drectons n a lattce through whch the nformaton propagates. The bass of the dscrete velocty model s a fnte set of vrtual veloctes e or equvalently, of vrtual fluxes of the consdered scalar feld T whch gven by b T f (4) The observed flux s expressed by b f r, e ( (5) It s a sngle-relaxaton-tme model wth relaxaton constant that can be related, va Chapmann Enskog analyss, to the dffusvty of the medum. f s the equlbrum dstrbuton functon. The relaxaton tme can be related wth the thermal dffusvty, the lattce velocty C and the tme step [17] by the followng relaton 3 t (7) C For the DQ9 model n partcular, the 9 veloctes ther correspondng weghts w are the followng e and e (,) (8) 1 1 e (cos( ),sn( )). C for 1,,3, 4 (9) 1 1 e (cos( ),sn( )). C for 5,6,7, 8 (1) w (11) 9 1 w for 1,,3, 4 (1) 9 1 w for 5,6,7, 8 (13) 36 It s to be noted that n the above equatons, C x / t y / t and the weghts satsfy the relaton b w 1. After dscretzaton, and consderng heat generaton, equaton (6) can be wrtten as t f ( r et, t f [ f f ] wtq * (14) Fg.1: Schematc dagram of the DQ9 lattce. The well-known DQ9 lattce model (Fg.1) wll be consdered here. In that model, the set of e s s such that they connect the pont, on whch the lattce stencl s centred, to ts nearest neghbours on a spatal grd wth unform spacng n both coordnate drectons. Any LBM advances the probablty denstes f n tme and thereby computes the evoluton of the consdered scalar. In the absence of external sources or fluxes for the scalar, the correspondng dscrete evoluton equaton can be wrtten n the followng general form: f 1 e f r t f r t f r t t (6). (, ) [ (, ) (, )] * whereq s the non dmensonal heat generaton and weght n correspondng drecton. w s the To process equaton (8), an equlbrum dstrbuton functon s requred. For heat conducton problems, ths s gven by f wt (15) 3. RESULTS AND DISCUSSION We consder transent heat conducton problems n -D Cartesan geometry wth the followng condtons: Case1: the four boundares are at known temperatures The ntal and the boundary condtons for cases 1 are the followng

3 Intal condton T ( x, y, ) Tref (16) Boundary condtons T x t T (17) (,, ).5 r e f T ( x, Y, T (, y, T ( X, y, Tref (18) Steady state condtons were assumed to have been acheved when the temperature dfference between two consecutve tme levels at each lattce centre dd not 6 exceed1. Non dmensonal tme was defned as t / L where L s the characterstc length. was taken as 4 1. To check the accuracy of the present LBM algorthm, the same problem was solved usng the fnte volume method and the results gven by the two algorthms are compared wth those avalable n the lterature. Fg. 3. Comparson of centrelne (=.5) temperature n the presence and the absence of heat generaton. In fg. 3, the effects of volumetrc heat generaton are shown. The non dmensonal volumetrc heat generaton s taken as unty. Effect of heat generaton s very less n the begnnng compared to steady state because t takes some tme to nfluence the temperature profle. For the -D geometry, the number of teratons for a 5x5 grd s 3719 ( seconds) compared to that cted at the lterature 357 [18]. Case3: The bottom and top boundares are at prescrbed fluxes and remanng two boundares at known temperatures Fg.. Centerlne(=.5) temperature evoluton for dfferent nstants (case 1). In fg., the non dmensonal centrelne ( =.5 ) temperature has been compared at dfferent nstants for Intal condton T ( x, y,) T (19) Boundary condtons q( x,, qs ; q( x, Y, qn ; T (, y, T ( X, y, T () It s seen from the fgure 4 that the steady state results match exactly whch each other. the case 1. Case: Effects of heat generaton and the four boundares are at specfed temperatures - 1 -

4 1. 1. at steady state Fg. 4. Centrelne(=.5) temperature evoluton for dfferent nstants (case3). Fg.5. Isotherms when the bottom and the top boundares are at prescrbed fluxes and remanng two boundares at known temperatures for dfferent. 1. In fg. 5, we present the tme-space evoluton of the sotherms when the bottom and the top boundares are at prescrbed fluxes and remanng two boundares at known temperatures Table 1: CPU tmes (second) and number of teratons of the LBM code (case3) sze Lattces teratons CPU tme (seconds) Temperature at steady state(=.5) 8x x x x Table : Effect of heat generaton on CPU tmes (second) and number of teratons of the LBM code (case3) Lattce sze teratons CPU tme In the absence of heat generaton 5x In the presence of heat generaton 5x To have an dea of the number of teratons for the converged solutons and the CPU tme, tests were performed wth dfferent lattces. The LBM code was found to take slghtly less number of teratons for the lttle lattces (Table 1)

5 The effect of heat generaton on CPU tmes (second) and number of teratons when all boundares at known temperatures, was hghlghted n table. 5. CONCLUSIONS The LBM s used to solve transent heat conducton problems n two dmensonal geometres wth unform lattces havng constant temperature and/or flux boundary condtons. Effect of heat generaton s also studed. The same problems are solved usng the fnte volume method. The results gven by the two numercal approaches are compared wth those avalable n the lterature and good agreement s obtaned. On the other hand, the effect of lattce sze s hghlghted va the number of teratons and the CPU tme. The consdered D geometry s a smple one, to allow smple valdaton. Advecton and radaton are omtted. Thus, t remans to demonstrate the vablty of the LBM as heat dffuson-advecton solver. 4. REFERENCES [1] Chen S, Doolen GD. Lattce Boltzmann method for flud flows. Annual Revew of Flud Mechancs 1998; 3: [] He X, Chen S, Doolen GD. A novel thermal model for the Lattce Boltzmann method n ncompressble lmt. Journal of Computatonal Physcs 1998; 146:8 3. [3] X H, Peng G, Chou S-H. Fnte-volume Lattce Boltzmann schemes n two and three dmensons. Physcal Revew E 1999; 6: [4] Takada N, Msawa M, Tomyama A, Fujwara S. Numercal smulaton of two- and three-dmensonal twophase flud moton by Lattce Boltzmann method. Computer Physcs Communcatons ; 19: [5] Wolf-Gladrow DA. Lattce-Gas Cellular Automata and Lattce Boltzmann Models: An Introducton. Sprnger: Berln-Hedelberg,. [6] Succ S. The Lattce Boltzmann Method for Flud Dynamcs and Beyond. Oxford Unversty Press: New York, 1. [7] Nourgalev RR, Dnh TN, Theofanous TG, Joseph D. The Lattce Boltzmann equaton method: theoretcal nterpretaton, numercs and mplcatons. Internatonal Journal of Multphase Flow 3; 9: [8] Zhu L, Tretheway D, Petzold L, Menhart C. Smulaton of flud slp at 3D hydrophobc mcro channel walls by the Lattce Boltzmann method. Journal of Computatonal Physcs 5; : [9] Ho JR, Kuo C-P, Jaung W-S, Twu C-J. Lattce Boltzmann scheme for hyperbolc heat conducton equaton. Numercal Heat Transfer, Part B ; 41: [1] W.-S Jaung, J.R. Ho, C.-P. Kuo, Lattce Boltzmann Method for heat conducton problem wth phase change, Numercal Heat Transfer, Part B 39, pp ,1. [11] Chatterjee D, Chakraborty S. An enthalpy-based Lattce Boltzmann model for dffuson domnated sold lqud phase transformaton. Physcs Letters A 5; 341:3 33. [1] Segel R, Howell J. Thermal Radaton Heat Transfer (4th edn). Taylor & Francs: New York,. [13] Modest MF. Radatve Heat Transfer (nd edn). Academc Press: New York, 3. [14] Fernandes R, Francs J. Fnte element analyss of planer conductve and radatve heat transfer wth flux boundary. 3rd AIAA/ASME Jont Thermophyscs, Fluds, Plasma and Heat Transfer Conference, Sant Lous, MO, 7 11 June, 199. Paper No [15] Barker C, Sutton WH. The transent radaton and conducton heat transfer n a gray partcpatng medum wth sem-transparent boundares, radaton heat transfer. ASME Journal of Radaton Heat Transfer 1985; 49:5 36. [16] S. Succ, The Lattce Boltzmann Method for Flud Dynamcs and Beyond, Oxford Unversty Press, New York, 1. [17] D.A.Wolf-Gladrow, Lattce Gas Cellular Automata and Lattce Boltzmann Models : An ntroducton, Sprnger Verlag, Berln-Hedelberg,. [18] S.C. Mshra, M. Bttagopal, K. Tanuj, B.S.R. Krshna, Solvng transent heat conducton problems on unform and non unform lattces usng the Lattce Boltzmann Method, Internatonal Communcatons n Heat and Mass Transfer, 36, pp.3-38,