Three-dimensional multiple-relaxation-time lattice Boltzmann model. for convection heat transfer in porous media at the REV scale

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1 Thr-dmnsonal multpl-rlaxaton-tm lattc Boltzmann modl for convcton hat transfr n porous mda at th REV scal Q. Lu Y.-L. H Ky Laboratory of Thrmo-Flud Scnc and Engnrng of Mnstry of Educaton School of Enrgy and Powr Engnrng X an Jaotong Unvrsty X an Shaanx 7149 Chna Abstract In ths papr a thr-dmnsonal (D) multpl-rlaxaton-tm (MRT) lattc Boltzmann (LB) modl s prsntd for convcton hat transfr n porous mda at th rprsntatv lmntary volum (REV) scal. Th modl s dvlopd n th framwork of th doubl-dstrbuton-functon (DDF) approach: an MRT-LB modl of th dnsty dstrbuton functon wth th DQ19 lattc (or DQ15 lattc) s proposd to smulat th flow fld basd on th gnralzd non-darcy modl whl an MRT-LB modl of th tmpratur dstrbuton functon wth th DQ7 lattc s proposd to smulat th tmpratur fld. Th prsnt modl s mployd to smulat mxd convcton flow n a porous channl and natural convcton n a cubcal porous cavty. Th numrcal rsults dmonstrat th ffctvnss and accuracy of th prsnt modl n solvng D convcton hat transfr problms n porous mda. Th numrcal rsults also dmonstrat that th prsnt modl s approxmatly scond-ordr accuracy n spac. In addton an nthalpy-basd DDF-MRT modl for D sold-lqud phas chang wth convcton hat transfr n porous mda s also prsntd. Kywords: Lattc Boltzmann mthod; Multpl-rlaxaton-tm (MRT); Porous mda; Convcton hat transfr; Thr-dmnsonal (D); REV scal.

2 1. Introducton Flud flow and hat transfr n porous mda hav attractd consdrabl attnton du to thr fundamntal natur and broad rang of applcatons n many flds of scnc and ngnrng [1-4]. Ovr th last svral dcads varous numrcal mthods hav bn dvlopd to study flud flow and hat transfr n porous mda. Th lattc Boltzmann (LB) mthod [5-14] as a msoscopc numrcal mthod orgnatd from th lattc-gas automata (LGA) mthod [15] has achvd grat succss n smulatng flud flow and hat transfr n porous mda du to ts kntc background [16-5]. Th LB modls for flud flow and hat transfr n porous mda can b gnrally classfd nto two catgors: th por scal mthod [16-1] and th REV scal mthod [-5]. Shortly aftr ts mrgnc n th lat 198s th LB mthod was appld to study D ncomprssbl flows n a random mdum by Succ t al [16]. In th por scal mthod [16-1] flud flow n th pors of th mdum s drctly modld by th standard LB mthod. By usng th no-slp bounc-back rul th ntracton btwn th flud and th sold matrx can b handld ffcntly. Th man advantag of th por scal mthod s that th dtald local nformaton (.g. prmablty) of th flow n th pors can b obtand whch can b usd to nvstgat macroscopc rlatons (.g. th Darcy s law). Morovr som fundamntal ssus such as mdum varablty and scal dpndncy can b assssd quanttatvly [17]. In th REV scal mthod [-5] an addtonal trm accountng for th prsnc of a porous mdum s addd nto th LB uaton basd on som sm-mprcal modls (.g. Darcy modl Brnkman-xtndd Darcy modl and gnralzd non-darcy modl). In ths mthod th statstcal proprts (.g. porosty prmablty nrta coffcnt) of th mdum ar ncorporatd nto th modl drctdly wth th dtald structur bng gnord and thus th dtald local nformaton of

3 th flow n th pors ar usually unavalabl. Howvr by usng approprat modls of th porous mdum rasonabl rsults can b producd by ths mthod. What s mor th REV scal LB mthod can b usd to study flud flow and hat transfr n porous mda systms of larg sz. Basd on th gnralzd non-darcy modl (also calld th Brnkman-Forchhmr-xtndd Darcy modl) [6] Guo and Zhao [5] proposd a gnralzd LB modl for studyng ncomprssbl flows n porous mda. In th gnralzd LB modl th nflunc of th porous matrx s consdrd by ncludng th porosty nto th ulbrum dstrbuton functon and addng a forcng trm to th LB uaton to account for th lnar (Darcy s trm) and nonlnar (Forchhmr s trm) drag forcs of th porous matrx. Subsuntly th gnralzd LB modl was xtndd to smulat convcton hat transfr n porous mda [7 8]. Aftr narly two dcads of dvlopmnt th REV scal LB mthod has bn dvlopd nto an accurat and ffcnt numrcal tool for studyng flud flow and hat transfr problms n porous mda systms. In th ltratur [-5] som fforts hav also bn mad to dvlop thrmal LB modls for studyng sold-lqud phas chang wth convcton hat transfr n porous mda. By combng th nthalpy mthod wth th thrmal LB mthod th movng sold-lqud ntrfac can b tracd by updatng th nthalpy wthout mposng hydrodynamc or thrmal boundary condtons. It s notd that most of th xstng REV scal thrmal LB modls for hat transfr problms [7-5] n porous mda wthout/wth sold-lqud phas chang ar lmtd to two-dmnsonal (D) cass. Lttl attnton has bn pad to study hat transfr problms n D porous mda systms va D REV scal thrmal LB mthod. As a promsng numrcal tool for ngnrng applcatons t s dsrabl to dvlop D REV scal thrmal LB modl to gnralz our undrstandng of flud flow and hat transfr procsss n D porous mda systms to thos n D cass. Hnc th am of ths papr

4 s to dvlop a D REV scal thrmal LB modl usng th MRT approach consdrng that th MRT collson oprator [7 8] s supror ovr ts BGK countrpart [7]. Th modl s dvlopd n th framwork of th DDF approach: a DQ19-MRT modl (or DQ15-MRT modl) s proposd to smulat th flow fld basd on th gnralzd non-darcy modl whl a DQ7-MRT modl s proposd to smulat th tmpratur fld. Numrcal smulatons of mxd convcton flow n a porous channl and natural convcton n a cubcal porous cavty ar carrd out to valdat th D DDF-MRT modl. Th rst of ths papr s organzd as follows. Th macroscopc govrnng uatons ar brfly dscrbd n Scton. In Scton th D DDF-MRT modl for convcton hat transfr n porous mda s prsntd n dtal. Numrcal rsults and som dscussons ar gvn n Scton 4. Fnally a brf concluson s mad n Scton 5.. Macroscopc govrnng uatons For flud flow and convcton hat transfr n an sotropc and rgd porous mdum basd on th gnralzd non-darcy modl th macroscopc govrnng uatons undr local thrmal ulbrum (LTE) condton can b wrttn as follows [4 6]: u (1) u u 1 u p v u F t () T u T T t () whr u T and p ar th volum-avragd flud vlocty tmpratur and prssur rspctvly s th man flud dnsty s th porosty v s th ffctv knmatc vscosty s th hat capacty rato [ c (1 ) c ] / ( c ) (rato btwn man hat capacty of th f p f m pm f pf mxtur and flud hat capacty) and s th ffctv thrmal dffusvty (1 ) f m

5 ( f and m ar thrmal dffusvts of th flud and porous matrx rspctvly). c p s th spcfc hat and th subscrpts f and m rfr to th proprts of flud and sold matrx rspctvly. F F F F dnots th total body forc nducd by porous matrx and xtrnal forc flds x y z whch can b xprssd as [7 9] v F F u u u G (4) K K whr K s th prmablty of th porous mdum v s th knmatc vscosty of th flud ( v s not ncssarly th sam as v ) and u ux uy uz n whch x u u y and u z ar componnts of th flud vlocty u n th x- y- and z-drctons rspctvly. Th frst and scond trms on th rght hand sd of Eq. (4) ar lnar (Darcy s trm) and nonlnar (Forchhmr s trm) drag forcs of th porous matrx. Wthout th nonlnar drag forc trm ( F ) Eq. () rducs to th Brnkman-xtndd Darcy uaton. Accordng to Boussnsq approxmaton th body forc G s gvn by G = g T T a (5) whr g s th gravtatonal acclraton s th thrmal xpanson coffcnt T s th rfrnc tmpratur and a s th acclraton nducd by othr xtrnal forc flds. Basd on Ergun s xprmntal nvstgatons [4] th nrta coffcnt F and th prmablty K can b xprssd as [41] F d p K 151 (6) whr d p s th sold partcl damtr. Convcton hat transfr n porous mda govrnd by Eqs. (1)-() s charactrzd by svral dmnsonlss paramtrs: th Darcy numbr Da th Raylgh numbr Ra th Rynolds numbr R (for mxd convcton flow) th Prandtl numbr Pr th vscosty rato J and th thrmal

6 dffusvty rato whch ar dfnd as follows K Da L gtl Luc Ra R = v v v v Pr J (7) v whr L s th charactrstc lngth u c s th charactrstc vlocty and s th thrmal dffusvty of th flud.. D DDF-MRT modl for convcton hat transfr n porous mda Th MRT mthod was proposd by d Humèrs [7] n 199 whch s an mportant xtnson of th rlaxaton LB mthod dvlopd by Hgura t al. [6]. In th LB communty t has bn wdly accptd that th MRT collson oprator [7 8] s supror ovr ts BGK countrpart [7]. In rcnt yars svral DDF-MRT modls hav bn proposd to smulat hat transfr problms wthout porous mda n two [4-44] and thr [45 46] dmnsons. In our prvous studs [ 4] th DDF-MRT mthod has also bn proposd to smulat hat transfr problms n porous mda. In ths scton a D DDF-MRT modl for convcton hat transfr n porous mda s prsntd whch can b vwd as an xtnson to our prvous studs..1 MRT-LB modl for th flow fld For th flow fld th MRT-LB uaton wth an xplct tratmnt of th forcng trm can b wrttn as [ 47 48].5 x t x t f x t f x t Λ f f S Λ S (8) t t j j j t j j whr f x t s th dnsty dstrbuton functon f t x s th ulbrum dnsty dstrbuton functon Λ= M 1 ΛM s th collson matrx ( M s th transformaton matrx Λ s th rlaxaton matrx) and S s th forcng trm. Through th transformaton matrx M th collson procss of th MRT-LB uaton (8) can b xcutd n th momnt spac :

7 t t Λ Λ (9) x t * m x m x m m t I S Th stramng procss s stll xcutd n th vlocty spac : * f x t f x t (1) t t whr f M m. Th bold-fac symbols m * 1 * m and S dnot b-dmnsonal column vctors: m m M f m m M f S S M S (11) n whch f = f f = f and S S. For brvty th Drac notaton s adoptd to dnot a b-dmnsonal column vctor.g. m m m mb 1 1 T. For th flow fld th DQ15 and DQ19 lattcs can b usd n th MRT-LB modl whch lads to th DQ15-MRT and DQ19-MRT modls. Th transformaton matrcs [49] ar gvn n Appndx A. In what follows th two MRT modls for th flow fld ar ntroducd..1.1 DQ15-MRT modl Th 15 dscrt vlocts 1 14 of th DQ15 lattc ar gvn by (s Fg. 1) c (1) whr c x t s th lattc spd wth t and x bng th dscrt tm stp and lattc spacng n x-drcton rspctvly. Th lattc spd c s st to b 1 ( x t ) n ths work Fg. 1. Dscrt vlocts of th DQ15 lattc. Th momnt vctor m s dfnd as

8 m T jx qx jy qy jz qz pxx pww pxy pyz pzx mxyz (1) whr t jx j t x F x jy j t y F y jz jz F z (14) wth jx ux jy uy and jz uz. For ncomprssbl flows consdrd n ths work th ncomprssbl approxmaton s adoptd.. ( s th dnsty fluctuaton) thn th flow momntum s approxmatd by = jx jy jz J u. Th ulbrum momnts m for th non-consrvd momnts m ( 5 7 ) ar gvn by q z p u 1 7 uz uu p xx x y xy pyz u 7 7 qx ux qy u ux uy uz p ww u u y z uu y z uu z x pzx mxyz (15) y whr 1 and ar fr paramtrs. Th dagonal rlaxaton matrx Λ s gvn by Λ dag s s s s s s s s s s s s s s s (16) j q j q j q v v v v v m Th componnts of th forcng trm S ar gvn as follows: S S1 uf uf 7 7 S 1 S Fx S4 F x S5 Fy S6 F y S 7 F S8 F z 7 z S 9 ux Fx uyfy uzfz S 1 uyfy uzfz S u F u F uyfz uz Fy uxfz uzfx S1 S1 S14 (17) x y y x 11 Th ulbrum dstrbuton functon f n th vlocty spac s gvn by ( ) f w u u u 4 cs cs cs (18) whr w 9 w1~6 19 w7~ and c 1 s th lattc sound spd. s

9 .1. DQ19-MRT modl Th 19 dscrt vlocts 1 18 of th DQ19 lattc ar gvn by (s Fg. ) c (19) Fg.. Dscrt vlocts of th DQ19 lattc. Th momnt vctor m s dfnd as m j q j q j q p p p p p m m m x x y y z z xx xx ww ww xy yz zx x y z T () Th ulbrum momnts m for th non-consrvd momnts m ( 5 7 ) ar gvn by q u 1 uz p y uy qz p xx p xx p uu ww xx u qx u u u u x y z uy uz uu x y ww pww pxy y z z x yz pzx uu m x my mz (1) x whr 1 and ar fr paramtrs. Th dagonal rlaxaton matrx Λ s gvn by Λ dag s s s s s s s s s s s s s s s s s s s () j q j q j q v v v v v m m m Th componnts of th forcng trm S ar gvn as follows: S S1 8 uf uf S 11 S Fx S4 F x

10 S F S6 F y S7 Fz S8 F z 5 y S 9 ux Fx uyfy uzfz S 1 u x F x u y F y u z F z S uyfy uz Fz uyfy uz Fz u F u F S1 S1 11 S y z z y 14 x y y x u F u F uxfz uzfx S15 S16 S17 S18 () Th ulbrum dstrbuton functon f n th vlocty spac s gvn by ( ) f w u u u 4 cs cs cs (4) whr w 1 w1~ w7~ and c 1 s th lattc sound spd. s Th macroscopc flud dnsty and vlocty u of th MRT-LB modl ar dfnd by f (5) t u f F (6) Th macroscopc flud prssur p s dfnd as p c s. Not that Eq. (6) s a nonlnar uaton for th vlocty u. By ntroducng a tmporal vlocty v th macroscopc flud vlocty u can b calculatd xplctly by v u l l l 1 v (7) whr 8 t 1 v f G 1 t v l K F t l1 (8) K Through th Chapman-Enskog analyss [47] of th MRT-LB uaton (8) th followng macroscopc uatons can b obtand: u (9) t

11 u uu p F () t whr v vb v u u T u I (1) s th shar strss tnsor. In th ncomprssbl lmt th macroscopc uatons (9) and () rduc to th gnralzd Navr-Stoks uatons (1) and (). Th ffctv knmatc vscosty v and th bulk vscosty v B ar dfnd as v 1 1 cs t sv v B t 9 s 59cs 1 1 (). MRT-LB modl for th tmpratur fld For th tmpratur fld a nw MRT-LB modl s proposd n ths subscton. th MRT-LB uaton s gvn by 1 x g x t g x t N QN g g () t t j j j t whr g x t s th tmpratur dstrbuton functon g t x s th ulbrum tmpratur dstrbuton functon N s th transformaton matrx and Q s th rlaxaton matrx. Through th transformaton matrx N th collson procss of th MRT-LB uaton () can b xcutd n th momnt spac : whr n n t t x t * n x n x n n Q (4) N g n n N g. Th stramng procss s xcutd n th vlocty spac : * g x t g x t (5) t t whr * 1 * g N n. As th tmpratur s rgardd as a passv scalar th DQ7 lattc can b usd n th MRT-LB modl for th tmpratur fld. Th 7 dscrt vlocts 1 6 of th DQ7 lattc ar gvn n Eq. (1) (s Fg. 1). In th prsnt study th transformaton matrx N of th DQ7-MRT

12 modl s gvn by N 1 1 (6) Th transformaton matrx N s constructd basd on th followng bass vctors: 1 x y z x y z x y x z. Th ulbrum momnts n ar dfnd as n T u u u x y z T (7) whr 1. Th rlaxaton matrx Q s gvn by Q dag (8) T v v n s th only consrvd momnt and th tmpratur T s computd by T n g (9) For convcton hat transfr wthout porous mda th ulbrum momnts n ar dfnd by n T T 1 ux uy uz and th tmpratur T s computd by T n g. Through th Chapman-Enskog analyss of th MRT-LB uaton () th followng macroscopc uaton can b obtand T t 1 ut ζ.5 c T ut t st t 1 (4) whr c s th lattc sound spd of th DQ7 modl. In most cass th dvaton trm st 1 ζ.5 u t t T 1 n Eq. (4) can b nglctd for ncomprssbl thrmal flows thn th tmpratur govrnng uaton () can b rcovrd undr th assumpton that dos not chang wth tm and vars slowly n spac. Th ffctv thrmal dffusvty s dfnd as 1 1 = c st ζ t (41)

13 Th ulbrum tmpratur dstrbuton functon g ( g = N n 1 ) n th vlocty spac s gvn by g T u cst (4) whr 1 1~6 6. Th nvrs of N s gvn by N 6 6 (4) Th DQ7-MRT modl can b xtndd to smulat sold-lqud phas chang wth convcton hat transfr n porous mda. In Appndx B an nthalpy-basd DQ7-MRT modl for sold-lqud phas chang wth convcton hat transfr n porous mda s brfly ntroducd. 4. Numrcal rsults and dscussons In ths scton numrcal smulatons of mxd convcton flow n a porous channl natural convcton n a cubcal porous cavty and two-rgon conducton mltng n a sm-nfnt spac ar carrd out to valdat th prsnt modl. Unlss othrws spcfd w st 1 1 t x y z 1 c 1 J =1 =1 1 and 1 ( c 16). Th fr rlaxaton rats ar chosn as follows: s s 1 s s s 1.1 s 1. (DQ15-MRT modl); s s 1 j q m st j s s sq s 1.1 s m 1. (DQ19-MRT modl); ζt 1 ζ ζv 1. (DQ7-MRT modl). Th non-ulbrum xtrapolaton schm [5] s mployd to trat th vlocty and tmpratur boundary condtons. For natural convcton n a cubcal porous cavty th DQ19-MRT modl s mployd to smulatd th flow fld. 4.1 Mxd convcton flow n a porous channl

14 In ths subscton w frst tst th prsnt modl by smulatng th mxd convcton flow n a porous channl (s Fg. ). Th dstanc btwn th two paralll plats s H th uppr plat s hot ( T Th ) and movs along th x-drcton wth a unform vlocty u whl th bottom plat s cold ( T Tc ). A constant normal flow of flud s njctd (wth a unform vlocty u 1 ) through th bottom plat and s wthdrawn at th sam rat from th uppr plat. Wthout th nonlnar drag forc ( F ) th flow at stady stat s govrnd by th followng uatons [7 1]: uy ux ux v v u y y K 1 p v g T T u y a y K T uy y T x y (44) (45) (46) whr T T T s th rfrnc tmpratur and a y s th xtrnal forc n th y-drcton: h c v xp( yu1 ) 1 ay u1 gt K xp( Hu1 ) 1 (47) Th analytcal solutons of Eqs. (44)-(46) ar gvn by u x y snh( y H) uxp 1 1 H snh u u u (48) y 1 z xp( Pr R y H ) 1 T Tc T xp( Pr R) 1 (49) whr R = Hu1 v s th Rynolds numbr Pr = v s th ffctv Prandtl numbr T Th Tc s th tmpratur dffrnc. Th two paramtrs 1 and n Eq. (48) ar gvn by R 1 J 1 4 J J Da R (5) In smulatons w st Ra 1 Pr 1 ( Pr Pr ).6 and u u 1.1. Prodc boundary condtons ar mposd n th x- and z-drctons and a grd sz of N N N 6 6 x y z s adoptd. Th rlaxaton rats s v and ζ ar dtrmnd by

15 y/h y/h y/h y/h 1 s v JHu 1 csr t ζ cs sv.5 J c Pr st (51) u 1 W T T h u H g y z T T c L u 1 x Fg.. Mxd convcton flow n a porous channl. In Fgs. 4 and 5 th normalzd vlocty and tmpratur profls for dffrnt Rynolds numbrs and Darcy numbrs ar plottd and compard wth th analytcal solutons. As can b obsrvd th prsnt rsults ar n xcllnt agrmnt wth th analytcal solutons =.6Da =.1 R =1 R =5 R =1 Symbols:Prsnt Lns:Analytcal.6.4. =.6Da =.1 R =1 R =5 R =1 Symbols:Prsnt Lns:Analytcal u x (L/ yw/)/u (L/ yw/) (a) vlocty profls (b) tmpratur profls =.6R =5.6 =.6R =5.4. Da =.1 Da =.1 Da =.1 Symbols:Prsnt Lns:Analytcal.4. Da=.1 Da=.1 Da=.1 Symbols:Prsnt Lns:Analytcal u x (L/ yw/)/u (L/ yw/) (c) vlocty profls (d) tmpratur profls

16 y/h y/h y/h y/h Fg. 4. Vlocty and tmpratur profls for dffrnt R and Da (DQ15-MRT modl) =.6Da =.1 R =1 R =5 R =1 Symbols:Prsnt Lns:Analytcal.6.4. =.6Da =.1 R =1 R =5 R =1 Symbols:Prsnt Lns:Analytcal u x (L/ yw/)/u (L/ yw/) (a) vlocty profls (b) tmpratur profls =.6R =5 Da =.1 Da =.1 Da =.1 Symbols:Prsnt Lns:Analytcal.6.4. =.6R =5 Da=.1 Da=.1 Da=.1 Symbols:Prsnt Lns:Analytcal u x (L/ yw/)/u (L/ yw/) (c) vlocty profls (d) tmpratur profls Fg. 5. Vlocty and tmpratur profls for dffrnt R and Da (DQ19-MRT modl). Numrcal smulatons ar also carrd out to valuat th spatal accuracy of th prsnt modl. In smulatons w st Ra 1 Pr 1 R 5.6 and s 1. Th grd numbr N y vars v from to 96. Th rlatv global rror of a varabl ( u or T ) s dfnd by E Φ ΦA x x Φ x Φ A LB x x (5) whr Φ A and Φ LB rprsnt analytcal and numrcal solutons rspctvly and th summaton s ovr th ntr doman. Th rlatv global rrors of th vlocty and tmpratur ar plottd

17 E(u) or E(T) E(u) or E(T) logarthmcally n Fg. 6 whr th symbols dnot prsnt rsults and th lns dnot last-squar fttngs. Th rsults ndcat that th prsnt modl s approxmatly scond-ordr accuracy n spac. 1-1 DQ15-MRT modl 1-1 DQ19-MRT modl 1 - slop= slop= slop= slop= E(u) E(T) x1 - x1-4x1-1/n y (a) DQ15-MRT modl 1-4 E(u) E(T) x1 - x1-4x1-1/n y (b) DQ19-MRT modl Fg. 6. Rlatv global rrors of vlocty and tmpratur at dffrnt grd numbrs ( N ). y 4. Natural convcton n a cubcal porous cavty In ths subscton th prsnt modl s mployd to study convcton hat transfr n a cubcal porous cavty. Th schmatc of ths problm s llustratd n Fg. 7. Th lngth wdth and hght of th cubcal cavty ar L W and H ( L W H ) rspctvly. Th lft and rght walls ar kpt at constant tmpraturs T h and T c ( T h T ) rspctvly whl th othr four walls ar adabatc. Th c buoyancy forc s gvn by G g T T k whr T T T s th rfrnc tmpratur h c and k s th unt vctor n th z-drcton. Th D local Nusslt numbr Nulocal Y Z th z-drcton avragd Nusslt numbr Nu Y at th hot wall and D avrag Nusslt numbr Nu D at th hot wall ar dfnd by Nu local Y Z YZ X X (5) Nu Y Nu Y Z dz (54) 1 local

18 rspctvly whr D local d d d (55) Nu Nu Y Z Y Z Nu Y Y T Tc T T Th Tc and X Y Z x/ L y/ W z/ H Nusslt numbr at th hot wall of th symmtry-plan ( Y.5 ) mp. Th avragd Nu s gvn by Nu Nu mp.5. Th rlaxaton rats s v and ζ ar dtrmnd by 1 1 MaJL Pr s v c Ra t ζ cs sv.5 J c Pr st (56) whr Ma u c c s s th Mach numbr n whch uc g TL s th charactrstc vlocty. For ncomprssbl thrmal flows consdrd n ths work th Mach numbr Ma should b small and s st to b.1. z H g Th y Tc W L x Fg. 7. Natural convcton n a cubcal porous cavty. In smulatons th Prandtl numbr Pr =.71 th porosty.4 and.8 th Raylgh numbr Ra rangs from 5 1 to 6 1 (Darcy-Raylgh numbr Ra * DaRa rangs from 5 to 1 ) and th Darcy numbr Da rangs from 1 to 1 1. Consdrng both th computatonal tm and th accuracy for 6 Ra 1 ( * Ra 1 ) a grd sz of N N N s x y z adoptd for 6 Ra 1 a grd sz of N N N s adoptd. x y z In Tabl 1 th avrag Nusslt numbrs ( Nu D ) obtand by th prsnt modl ar compard wth th boundary lmnt mthod (BEM) rsults [51] of th Brnkman-xtndd Darcy modl ( F ) for

19 dffrnt * Ra and Da wth.8. It can b sn that th prsnt rsults ar n good agrmnt wth th BEM rsults for th whol rang of Darcy-Raylgh and Darcy numbrs. In Tabl th avrag Nusslt numbrs ( Nu D ) obtand by th prsnt modl ar compard wth th numrcal rsults [51 5] of th Brnkman-xtndd Darcy modl for dffrnt Da wth * Ra 1 and.8. Vry good agrmnt btwn ths rsults can b sn from Tabl. Tabl 1. Comparson of th prsnt rsults wth th BEM rsults [51] of th Brnkman-xtndd Darcy * Ra Da 1 1 modl (.8 ). Da 1 Da 1 Rf. [51] Prsnt Rf. [51] Prsnt Rf. [51] Prsnt Tabl. Comparson of th prsnt rsults wth th numrcal rsults [51 5] of th * Brnkman-xtndd Darcy modl ( Ra 1.8 ). Da Rf. [51] Rf. [5] Prsnt In what follows basd on th gnralzd non-darcy modl (th nrta coffcnt F s gvn by Eq. (6)) a paramtrc study has bn conductd for convcton hat transfr n th cubcal porous cavty for varous valus of Ra Da and. In Fg. 8 th sothrms and contour lns of u x for * Ra 1 and.8 n th symmtry-plan ( Y.5 ) ar shown. As can b sn from th fgur a dcras n Da lads to an ntnsfcaton of convcton flow nsd th cubcal porous cavty and as a rsult to thnnr (thrmal and vlocty) boundary layrs nar th hot and cold walls. Th sothrms and flow pattrns n th symmtry-plan ar qualtatvly smlar to thos of th D convcton hat transfr n a squar porous cavty. Howvr th ffct of th sd walls of D convcton hat transfr

20 n a cubcal porous cavty would b notabl whch s to b dscussd latr. Z Z X Y X Y (a) 4 Ra 1 Da 1 1 Z Z X Y X Y (b) 5 Ra 1 Da 1 Z Z X Y X Y (c) 6 Ra 1 Da 1 Fg. 8. Isothrms (lft) and contour lns of u x (rght) for * Ra 1 and.8 n th symmtry-plan ( Y.5 ).

21 Avrag Nusslt Numbr In Tabl th avrag Nusslt numbrs ( Nu D ) at th hot wall ar prsntd for varous * Ra Da and. Th followng trnds can b obsrvd from th prdctd rsults: () for a gvn Da and Nu D ncrass wth th ncras n * Ra wth th lowst valu at * Ra 5 and hghst valu at * Ra 1 ; () for a gvn * Ra and Nu D ncrass wth th dcras n Da and th nflunc of Da s mor pronouncd at hghr valus of * Ra ; () for a gvn * Ra and Da Nu D ncrass wth th ncras n. Th nflunc of on Nu D s shown n Fg. 9. Clarly for hghr valu of th ffct of th nrta and nonlnar drag trms ar lss sgnfcant whch lads to hghr flow vlocts and hghr valus of Nu D. * Ra Tabl. Avrag Nusslt numbrs of th gnralzd non-darcy modl. Da Da 1 Da 1 Da 1 1 Da 1 Da Pr=.71 =.4 =.8 4 Da=.1 1 Da=.1 Da=.1. 4.x1 8.x1 Ra * Fg. 9. Th nflunc of on Nu D of th gnralzd non-darcy modl. Th nflunc of th nonlnar drag forc on Nu D s shown n Fg. 1. In th fgur BD modl rprsnts th Brnkman-xtndd Darcy modl and BFD modl rprsnts th gnralzd non-darcy

22 Avrag Nusslt Numbr modl. Form th fgur w can obsrv that for Da.1 th rsults ar almost th sam for th BD and BFD modls th nflunc of th nonlnar drag forc can b nglctd. Whl for Da.1 and Ra * 1 th nonlnar drag forc bcoms sgnfcant and rducs th ovrall hat transfr whch lads to smallr valus of Nu. For non-darcy flows ( 1 Da 1 D 4 1 ) wth rlatvly larg * Ra t s ssntal to consdr th nflunc of th nonlnar drag forc Pr=.71 =.8 BD modl BFD modl Da=.1 4 Da=.1 1 Da=.1. 4.x1 8.x1 Ra * Fg.1. Th nflunc of th nonlnar drag forc on Nu D. Fnally th ffct of th sd walls of D convcton hat transfr n th cubcal porous cavty s studd. Hr w only consdr th cas wth 5 Ra 1 Da 1 and.8 wthout loss of gnralty. In Fg. 11 th avrag Nusslt numbr along th y-drcton at th hot wall ( Nu Y shown. From th fgur w can obsrv that occurs at.5 Y.. ) s Nu Y ncrass as Y.5 and th maxmum valu Numax Y Nump.777. Th D avrag Nusslt numbr Nu D at th hot wall s.586 (s Tabl ) whl th D avrag Nusslt numbr Nu D at th hot wall of th D squar porous cavty s.754 (obtand by th D DDF-MRT modl [] basd on a grd sz of N N 1 1 ). Owng to th ffct of th sd walls th D rsult ovrstmats th D x y ffctv hat transfr but undrstmats th symmtry-plan ffctv hat transfr. Wth rspct to Nu mp th D hat transfr s always wakr.

23 Nu(Y) 4..8 Ra=1 5 Da=1 = Y Fg.11. Th dstrbuton of th z-drcton avragd Nusslt numbr ( X ). 4. Two-rgon conducton mltng n a sm-nfnt spac In ths subscton numrcal smulatons of two-rgon conducton mltng n a sm-nfnt spac (s Fg. 1) ar carrd out to valdat th nthalpy-basd DQ7-MRT modl. In ths problm th ntal tmpratur ( T ) of th phas chang matral (PCM) s blow th mltng pont of th PCM ( T m ) th tmpraturs of both th lqud and sold phass ar unknown and must b dtrmnd. Ths s th so-calld two-rgon conducton mltng problm. At tm t a constant tmpratur T h ( T T ) s mposd on th lft wall ( x ) and mantand at that tmpratur for t. As a rsults h m th mltng starts at x and th ntrfac movs n th postv x-drcton. Th tmpraturs of th lqud and sold phass ar gvn by [56] T T rf x t T x t T h m l h x xm t rf T x t rspctvly whr rf m s T x x t m T T rfc x t rfc l s l (lqud rgon) (57) (sold rgon) (58) x t t s th locaton of th sold-lqud phas ntrfac d s th rror functon and rfc x 1 rf x m s th complmntary rror functon. Th paramtr can b dtrmnd by th followng transcndntal uaton [56]

24 1 l s ks l T Tm La rf kl s Th Tm rfc cpl Th Tm l s (59) In smulatons th paramtrs ar st as follows: T 1 T T cpl cps 1 St cplt La 1 ( T Tb Tm). A grd sz of Nx N y N z 4 66 s h m s mployd and th thrmal dffusvty rato l s vars from 1 to 1. To smulat such on-dmnsonal conducton mltng problm th prodc boundary condtons ar mposd n th y- and z-drctons and th vlocty fld s st to b zro consstntly ( u ). Th rlaxaton rat ζ s dtrmnd by ζ 1.5 stt c whr f f. In Fg. 1 th tmpratur l l s 1 l profls for dffrnt valus of thrmal dffusvty rato at Fo.1 ar plottd ( Fo t L ). It can s b obsrvd that th prsnt rsults ar n good agrmnt wth th analytcal rsults. Th locatons of th sold-lqud phas ntrfac for dffrnt valus of thrmal dffusvty rato and Fo ar plottd n Fg. 14. Good agrmnt can b obsrvd agan btwn th prsnt rsults and analytcal rsults. T h T m y z x T lqud ntrfac sold x Fg. 1. Schmatc of th two-rgon conducton mltng n a sm-nfnt spac Analytcal l s =1 l s = l s =5 l s =1 T m x/l Fg. 1. Tmpratur profls for dffrnt valus of thrmal dffusvty rato at Fo=.1.

25 x m /L.. Analytcal Prsnt l s =1.1 l s =5 l s = l s = Fo Fg. 14. Locatons of th sold-lqud phas ntrfac for dffrnt valus of thrmal dffusvty rato and Fo. 5. Conclusons In ths papr a D DDF-MRT modl s prsntd for convcton hat transfr n porous mda at th REV scal. Th DDF-MRT modl conssts of two dffrnt MRT-LB modls: an MRT-LB modl of th dnsty dstrbuton functon wth th DQ19 lattc (or DQ15 lattc) s proposd to smulat th flow fld basd on th gnralzd non-darcy modl and an MRT-LB modl of th tmpratur dstrbuton functon wth th DQ7 lattc s proposd to smulat th tmpratur fld. Th ky pont of th modl s to nclud th porosty nto th ulbrum momnts and add a forcng trm to th MRT-LB uaton of th flow fld to account for th lnar (Darcy s trm) and nonlnar (Forchhmr s trm) drag forcs of th porous matrx basd on th gnralzd non-darcy modl. Th prsnt modl s frst mployd to smulat mxd convcton flow n a porous channl. It s found that th prsnt rsults agr wll wth th analytcal solutons and th rsults dmonstrat that th prsnt modl s approxmatly scond-ordr accuracy n spac. Thn th prsnt modl s mployd to smulat convcton hat transfr n a cubc porous cavty n th non-darcy flow rgm. Th rsults dmonstrat that th prsnt modl has th applcablty to solv D convcton hat

26 transfr problms n porous mda. In addton an nthalpy-basd DDF-MRT modl for D sold-lqud phas chang wth convcton hat transfr n porous mda s also prsntd.

27 Appndx A. Transformaton matrcs For th DQ15-MRT modl th transformaton matrx M s gvn by ( c 1) [49] M (A1) For th DQ19-MRT modl th transformaton matrx M s gvn by ( c 1) [49] M (A)

28 Appndx B. Enthalpy-basd DQ7-MRT modl For sold-lqud phas chang wth convcton hat transfr n porous mda undr th LTE condton th nthalpy-basd nrgy govrnng uaton can b wrttn as [5] t fl l Enl 1 fl sens 1 menm l Enl u k T (B1) whr f l s th fracton of th lqud PCM n th por spac En l En s and En m ar nthalpy of th lqud PCM sold PCM and porous matrx rspctvly k s th ffctv thrmal conductvty and L a s th latnt hat of phas chang. Th subscrpts l s and m rfr to th proprts of th lqud PCM sold PCM and porous matrx rspctvly. En l En s and En m ar dfnd by Enl cplt fl La Ens cpst Enm cpmt (B) Substtutng Eq. (B) nto Eq. (B1) aftr a fw stps th followng nrgy govrnng uaton can b drvd k (B) cplt flla cpltu T fllau t l whr l s f s th hat capacty rato (rato btwn man hat capacty of th mxtur and lqud hat capacty) [5] 1 1 fl lcpl fl scps mcpm (B4) c In lqud rgon [ c (1 ) c ]/( c ) ;n sold rgon [ c (1 ) c ]/( c ). l l pl m pm l pl Not that th last trm n Eq. (B).. L f u l a l pl s s ps m pm l pl whch s nducd by th flow n th mushy zon can b nglctd for sothrmal sold-lqud phas chang [5-55]. Thn by ntroducng an ffctv nthalpy En cplt fl La [5] th followng ffctv-nthalpy-basd nrgy govrnng uaton can b obtand En k cpltu T (B5) t l

29 For th nthalpy-basd DQ7-MRT modl th ulbrum momnts n ar dfnd as n En c Tu c Tu c Tu c T pl x pl y pl z prf T (B6) whr 1 cprf s th rfrnc spcfc hat. Th ffctv nthalpy En s computd by En n g (B7) Th rlatonshp btwn ffctv nthalpy En and tmpratur T s gvn by En scpl En En s En En T T T T En En En Tl En En l lcpl En En l s s l s s l En l En s (B8) whr T s and T ( Ts Tl ) ar th soldus and lqudus tmpraturs rspctvly En s l En c T ) and En l ( En lcpltl fl La ) ar th ffctv nthalpy valus corrspondng to ( s s pl s T s and T l rspctvly. Th lqud fracton f l can b calculatd by En En s En En f En En En 1 En En l s l s l En l En s (B9) Through th Chapman-Enskog analyss th followng macroscopc uaton can b obtand En t 1 cpltu t ζ.5 cst cprft t cpltu 1 (B1) 1 For ncomprssbl thrmal flows th dvaton trm t ζ.5 t cpltu 1 can b nglctd thn th ffctv-nthalpy-basd nrgy govrnng uaton (B5) can b rcovrd wth = c 1 1 st t ζ lcprf k (B11) Th ulbrum dstrbuton functon g ( g = N n 1 ) n th vlocty spac s gvn by g En cprft 1 1 cprft ucplt 1 ~ 6 6 (B1) As dd n Rf. [55] th rfrnc spcfc hat cprf s ntroducd nto th modl. Not that th choc

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