Mesoscale Numerical Method for Prediction of Thermal Fluid Flow through Porous Media

Transcription

1 Nor Azwad Che Sdk, Mohd Irwan Mohd Azm Mesoscale Numercal Method for Predcton of Thermal Flud Flow through Porous Meda NOR AZWADI CHE SIDIK and MOHD IRWAN MOHD AZMI Department of Thermofluds Unverst Teknolog Malaysa Skuda Johor MALAYSIA Abstract: - In ths paper, the lattce Boltzmann method, a mesoscale numercal tool based on partcle dstrbuton functon s used to smulate thermal flud flow n porous meda. The key pont s to combne the smplest four and nne lattce velocty model to represent the temperature and densty dstrbuton functons respectvely. Wde range of Raylegh numbers and materal's porosty has been appled to study ther effects on the thermal flud flow n the enclosure. Our numercal experments demonstrate excellent agreements when the computed results are compared wth those predcted by the fnte element soluton to the Brnkmann- Forcchemer uaton and the conventonal lattce Boltzmann scheme. Ths ndcates the applcablty of the present approach n realstc smulaton of thermal flud flow n porous meda. Key-Words: - Lattce Boltzmann, Double populaton, Natural convecton, Porous meda 1 Introducton As The nteracton between flud flow behavor and heat transfer mechansm can be seen not only n almost all ndustral processes such as metal furnace, power plants, jet engne, etc, but also n everyday stuaton such as ventlaton, ar condtonng, har dryer and so on. In some applcatons, such as mcro-electro mechancal systems (MEMS), the detal understandng of the flud flow and heat transfer phenomenon s unrelentngly rured n order to acheve the most effectve method of mcrochps coolng [1]. On the other hand, lack of understandng n ths problem can result n huge cost lost and neffcency repercussons. For nstance, naccurate predcton of heat transfer and flud flow can also leads to loss of human lves n the reentry of space shuttle due to the great heat nvolved n ths actvty. Among the man three types of heat transfer mechansm, the convecton type has a more pronounced effect on flud flow. In fact, the convectve heat transfer domnates the heat transfer mechansm n most cases when nteract wth surroundng flud. Ths mechansm s very dffcult to measure because of the effect on flud flow only appears when dealng at severe condtons such as hgh Raylegh or Grashof numbers. Furthermore, when the contact flud s gas, t becomes dffcult to vsualze ths flow confguraton expermentally. Flow drven by buoyancy force s a knd of flow resulted from convectve heat transfer. Ths type of flow can be found n certan engneerng applcatons wthn nsulaton technologes, n everyday stuaton such as roof ventlaton or n academc research where t may be used as a benchmark problem for testng newly developed numercal methods. A classc example s the case where dfferentally heated walls of the cavty boundares nduce the flow. Two vertcal walls wth constant hot and cold temperature s the most well defned geometry and was studed extensvely n the lterature. A comprehensve revew was presented by Davd [2]. Other examples are the work by Azwad and Tanahash [3], Davs [4] and Trc [5]. The analyses of flow and heat transfer n a dfferentally heated sde walls was extended to the ncluson of porous meda n the enclosure. Cheng [6] provdes an extensve revew of lterature on natural convecton n flud saturated porous meda wth regard to applcatons n geothermal systems. Neld and Bejan [7] gves an excellent summary of the subject. Other works are [8-11] and [12]. Due to the complexty of porous structure and flud nteracton wth the boundares, most of the mentoned researchers preferred numercal approach to understand the flud flow behavor n the system. Interestngly, many of them appled conventonal numercal schemes based on dscretzatons of the E-ISSN: Issue 1, Volume 7, January 2012

2 Nor Azwad Che Sdk, Mohd Irwan Mohd Azm sememprcal models as ther numercal tools. Currently, numercal solutons to the flud flow problem can be dvded nto three scales, whch are macro, meso and mcroscale solutons. Macroscale soluton consders the Naver-Stokes uaton as ts governng uaton and apples one or a combnaton of dscretzaton methods to be solved usng dgtal computer. However, due to the nonlnear nature of the uaton, greater attenton has to be pad durng preprocessor step to determne sutable mesh sze, crtera of computatonal stablty, error propagaton, etc. There are few numercal solutons that smulate the evoluton of flud flow at mcroscopc scale. Among them are drect smulaton Monte Carlo [13] and Molecular Dynamcs methods [14]. In these methods, the trajectores of every partcle together wth ther poston n the system are predcted usng the second Newton's law. But remember, a cup of water contans number of molecules. Even when a gas s beng consdered where there are fewer molecules and a larger tme-step can be used, because of the longer mean free path of the molecules, the number of molecules that can be consdered s stll lmted. However, the queston s, do we really need to know the behavor of each molecule or atom? The answer s no. It s not mportant to know the behavor of each partcle, t s mportant to know the functon that can represent the behavor of many partcles (mesoscale). Therefore, n current study, we brng the so-called lattce Boltzmann method [15] as our numercal tool. The evoluton of two dstrbuton functons s consdered to predct the velocty and temperature felds n the system. After showng how the formulaton of mesoscale partcle fts n to the framework of lattce Boltzmann smulatons, a mathematcal formulaton s developed n order to nvestgate the effect of buoyancy force and the presence of porous meda wthn the soluton manfold. The current study s summarzes as follow: twodmensonal flud flow and heat transfer n porous meda flled n square cavty s nvestgated numercally. The two sdewalls are mantaned at dfferent temperatures whle the top and bottom walls are set as an adabatc wall. Here, we fx the aspect rato to unty. The flow structures and heat transfer mechansm are hghly dependent upon the porosty of the medum. By also adoptng the Raylegh and Darcy numbers as contnuaton parameters, the flow structure and heat flow represented by the streamlnes and sotherms lnes can be dentfed as a functon of porosty. Comparsons of results among those publshed n lterature are carred out n terms of a computed averaged Nusselt number. Secton two of ths paper presents the governng uatons for the case study n hand and ntroduces the numercal method, whch wll be adopted for ts soluton. Meanwhle secton three presents the computed results and provde detaled dscussons. The fnal secton of ths paper concludes the current study. 2 The Governng Equatons Followng Nthsrasu [16], the generalzed model for athermal ncompressble flud flow n porous meda can be expressed by the followng uatons u = 0 (1) u t + u u ε = 1 ρ εp + υ e 2 u + F (2) T t + ut = χ 2 T (3) where υ e s the effectve vscosty, ε s the porosty of the meda and χ s the thermal dffusvty. F represents the total body force due to the presence of a porous medum and buoyancy force resulted from dfferentally heated sdewalls, and s gven by F = ευ K u εK uu + εg (4) where υ s the knematc vscosty and K s the permeablty of whch can be related to nondmensonal parameter of Darcy number Da as follow K = Da H 2 (5) where H s the characterstc length. In present study, the Boussnesq approxmaton s appled to the buoyancy force term where G = βg 0 ( T T m )j (6) Here, β s the thermal expanson coeffcent, g 0 s the acceleraton due to gravty, T m s the averaged temperature and j s the vertcal drecton opposte to that of gravty. E-ISSN: Issue 1, Volume 7, January 2012

3 Nor Azwad Che Sdk, Mohd Irwan Mohd Azm 2.1 The Lattce Boltzmann Method In recent years, the lattce Boltzmann method (LBM) has been developed nto an alternatve and promsng numercal technque of Computatonal Flud Dynamcs (CFD) [17-20]. Unlke conventonal numercal schemes based on the dscretzaton of partal dfferental uatons descrbng macroscopc conservaton laws, the LBM s based on solvng the dscrete-velocty Boltzmann uaton n statstcal physcs. In ths work, the governng uatons of thermal flud flow n porous meda are solved ndrectly,.e., by usng LBM wth second-order accuracy. Our lterature study found that there are several nvestgatons have been conducted usng the LBM to understand the problem n hand [21-23]. However, most of them appled the same lattce model to predct the evoluton of velocty and temperature felds n the system. Combnaton of nne-lattce model for the densty and also the same model for the temperature dstrbuton functons s the most common approach by the prevous researchers. Currently, one of present authors has developed the smplest lattce model to predct the evoluton of temperature feld [24]. Unfortunately, the developed model was found not n good agreement wth the lterature studes when predctng thermal flud flow at hgh Raylegh numbers. Ths was due to the lmtaton of the model where unable to capture hgh speed of flud flow n the system [24]. The presence of the porous medum s expected to decelerate the flow although depended on the magntude of the porosty. Therefore, the objectve of present paper s to reconsder the newly developed model and predct the flud and thermal flow n an enclosure flled wth porous medum at hgh Raylegh numbers. To see ths, we start wth the evoluton uatons of densty and temperature dstrbuton functons, gven as [24] f g ( x + c Δx, t + Δt) f x, t ( x + c Δx,t + Δt) g x, t [ ] + F = 1 τ v f f = 1 τ g (7) [ g g ] (8) where densty dstrbuton functon f s used to calculate the densty and velocty felds and temperature dstrbuton functon g s used to calculate the macroscopc temperature feld. Note that Bhatnagar-Gross-Krook (BGK) collson model [25] wth a sngle relaxaton tme s used for the collson term. For the D2Q9 model, the dscrete lattce veloctes are defned by c 0 = 0,0 c 1 4 = ( ±1,0 ), 0,±1 c 5 8 = ±1,±1 (9) The ulbrum functon for the densty dstrbuton functon f for the D2Q9 model s gven by f = ρω 1+ 3c u + 9 c u 2ε 2 3u2 2ε (10) where the weghts are ω 0 = 4 9, ω =1 9 for =1 4 and ω =1 36 for = 5 8. Accordng to Azwad and Tanahash [24], the smplest four-lattce velocty model can be appled to represent the temperature dstrbuton functon of g where =1 4. The uvalent ulbrum functon for temperature dstrbuton functon s gven as g 1,2,3,4 = T [ 4 1+ c u ] (11) The effectve vscosty υ e and the dffusvty χ are determned by υ e =1 3( τ v 1 2) and χ = τ c 1 2, respectvely. In order to obtan the correct macroscopc governng uatons, the forcng term F must be expressed n terms of medum porosty and buoyancy force as follows [16] F = ω ρ 1 1 2τ v 3c F + 9 uf : c c ε 3u F ε (12) The macroscopc varables, densty ρ, and temperature T can be evaluated as the moment to the dstrbuton functon ρ = T = f g (13) In order to consder the effect of porous meda, the flud velocty u must be calculated as follow E-ISSN: Issue 1, Volume 7, January 2012

4 Nor Azwad Che Sdk, Mohd Irwan Mohd Azm u = c 0 + v c c 1 v (14) where v = c f ρ + εg 2 s the temporal velocty, c 0 = 1+ ευ 2κ 2 and c 1 =1.75ε 2 150ε 3 κ. It s noted that, f we set ε =1, the lattce Boltzmann uaton reduces to the standard uaton for free flud flows. 3 Problem Soluton In ths secton, we begn wth the valdaton of the thermal lattce Boltzmann model by settng ε 1. Table 1 shows the average Nusselt number computed by the LBM for ε = , Da =10 7 and Ra =10 3 to 10 4 and comparsons wth those by Davs [4] and Ntharasu et al. [16]. Table 1. Comparson among Naver Stokes solver, fnte element method and present model Raylegh Davs [4] Ntharasu Present Number et. al [16] As can be seen from the table, the results predcted by current model agree well wth the prevous studes. Ths gves us confdence to apply the proposed method for smulaton of thermal flud flow n porous meda. We next extend our smulaton study to predct the thermal and flud flow characterstcs wth the presence of porous meda at three values of porosty. For the sake of comparson, we brng the results predcted by LBM scheme usng the combnaton of D2Q9 and D2Q9 [21], and soluton to Brnkman-Forchhemer uaton usng fnte element method [16]. The Darcy number and Prandtl number are set at constant value of 0.01 and 1.0 respectvely. As can be seen from the table, the results predcted by present LBM model are n excellent agreement wth those from FEM and conventonal LBM. There are two nterestng characterstcs whch can be drawn from the table; (1) for a fx porosty, the Nusselt number ncreases as the Raylegh number ncreases and (2) for a fx Raylegh number, the Nusselt number decrease lnearly as the porosty decrease. These ndcate the applcablty of the combnaton of D2Q9 and D2Q4 lattce model for smulatng thermal flud flow n porous medum. Table 2. Comparson of average Nusselt number among Naver Stokes solver usng fnte element method, LBM and present model. FEM[16] LBM[21] Present ε = Ra =10 3 ε = ε = ε = Ra =10 4 ε = ε = ε = Ra =10 5 ε = ε = Concluson In ths paper, the smplest combnaton of twodmensonal thermal lattce Boltzmann method s brought to predct natural convecton n a square cavty flled wth porous medum. We found that the present method correctly predcted the flow feature for dfferent Raylegh number and porosty, and gves excellent agreement wth the results of prevous studes. The results obtaned demonstrate that ths proposed approach n the thermal lattce Boltzmann model s very effcent procedure to study flow and heat transfer n a dfferentally heated square enclosure wth the presence of porous medum. Computaton at lower value of Darcy numbers to nvestgate the behavor of flud flow at non-darcan regon wll be our near future research topc. References: [1] Y. Zhang, N. Bao, X.D. Yu, J.J. Xu, H.Y. Chen, Improvement of heat dsspaton for polydmethylsloxane mcrochp electrophoress, Journal of Chromatography, Vol. 1057, 2004, pp [2] N. Davd, H.O. Patrck, Introducton to Convectve Heat Transfer Analyss, McGraw Hll, [3] C.S. Nor Azwad, T. Tanahash, Threedmensonal thermal lattce Boltzmann smulaton of natural convecton n a cubc cavty, Internatonal Journal of Modern Physcs B, Vol. 21, 2007, pp [4] D.V. Davs, Natural convecton of ar n a square cavty: A benchmark numercal soluton, Internatonal Journal for Numercal Methods n Fluds, Vol. 3, 1983, pp [5] E. Trc, G. Labrosse, M. Betroun, A frst ncurson nto the 3D structure of natural convecton of ar n a dfferentally heated E-ISSN: Issue 1, Volume 7, January 2012

5 Nor Azwad Che Sdk, Mohd Irwan Mohd Azm cubc cavty, from accurate numercal soluton, Internatonal Journal of Heat and Mass Transfer, Vol. 43, 2000, pp [6] P. Cheng, Heat transfer n geothermal systems, Advanced Heat Transfer, Vol. 4, 1978, pp [7] A. Bejan, D.A. Neld, Convecton n Porous Meda, Sprnger, New York, [8] H.C. Chan, W.C. Huang, J.M. Leu, C.J La, Macroscopc modelng of turbulent flow over a porous medum, Internatonal Journal of Heat and Flud Flow, Vol. 28, 2007, pp [9] K.S. Chem, Y. Zhao, Numercal study of steady/unsteady flow and heat transfer n porous meda usng a characterstcs-based matrx-free mplct FV method on unstructured grds, Internatonal Journal Heat and Flud Flow, Vol. 25, 2004, pp [10] M.A. Seddeek, Effects of non-darcan on forced convecton heat transfer over a flat plate n a porous medum-wth temperature dependent vscosty, Internatonal Communcaton n Heat and Mass Transfer, Vol. 32, 2005, pp [11] D.Y. Lee, J.S. Jn, B.H. Kang, Momentum boundary layer and ts nfluence on the convectve heat transfer n porous meda, Internatonal Journal n Heat and Mass Transfer, Vol. 45, 2002, pp [12] C.Y. Cheng, Non-Darcy natural convecton heat and mass transfer from a vertcal wavy surface n saturated porous meda, Appled Mathematcs and Computaton, Vol. 182, 2006, pp [13] G.A. Brd, Approach to translatonal ulbrum n a rgd sphere gas, Physcs of Flud, Vol. 9, 1963, pp [14] B.J. Alder, T.E. Wanwrght, Phase transton for a hard sphere system, Journal of Chemcal Physcs, Vol. 27, 1957, pp [15] Y.H. Qan, D. Humeres, P. Lallemand, Lattce BGK models for Naver-Stokes uaton, Europhyscs Letters, Vol. 17, 1992, pp [16] P. Ntharasu, K.N. Seetharamu, T. Sundarajan, Natural convectve heat transfer n a flud saturated varable porosty medum, Internatonal Journal of Heat and Mass Transfer, Vol. 40, 1997, pp [17] M. Bernsdorf, G. Brenner, F. Durst, Numercal analyss of the pressure drop n porous meda wth lattce Boltzmann (BGK) automata, Journal of Computatonal Physcs, Vol. 129, 2000, pp [18] X. He, S. Chen, G.D. Doolen, A novel thermal model for the lattce Boltzmann method n ncompressble lmt, Journal of Computatonal Physcs, Vol. 146, 1998, pp [19] G. McNamara, B. Alder, Stablzaton of thermal lattce Boltzmann models, Journal of Statstcal Physcs, Vol. 81, 1995, pp [20] Y. Peng, C. Shu, Y.T. Chew, Smplfed thermal lattce Boltzmann model for ncompressble thermal flows, Physcal Revew E, Vol. 68, 2003, pp [21] T. Seta, E. Takegosh, K. Oku, Lattce Boltzmann smulaton of natural convecton n porous meda, Mathematcs and Computers n Smulaton, Vol. 72, 2006, pp [22] T. Seta, E. Takegosh, K. Ktano, K. Oku, Thermal lattce Boltzmann model for ncompressble flows through porous meda, Journal of Thermal Scence and Technology, Vol. 1, 2006, pp [23] Z. Guo, T.S. Zhao, Lattce Boltzmann model for ncompressble flow through porous meda, Physcal Revew E, Vol. 66, 2002, pp / /9. [24] C.S. Nor Azwad, T. Tanahash, Smplfed thermal lattce Boltzmann n ncompressble lmt, Internatonal Journal of Modern Physcs B, Vol. 20, 2006, pp [25] P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collson processes n gasses. 1. Small ampltude n charged and neutral onecomponent systems, Physcal Revew, Vol. 94, 1954, pp E-ISSN: Issue 1, Volume 7, January 2012