A Porous Media Approach for Complex Heat and Fluid Flow System

Transcription

1 A Poros Meda Approach or Complex Heat and Fld Flow System Ara Naayama, Dept. o Mechancal Engneerng, Shzoa Unversty, Hamamats, Japan Insttte o Promechancs, Whan Polytechnc Unversty, Hbe, Whan, Chna Abstract A poros meda approach based on local volme averagng has been proposed to attac a varety o engneerng problems related to complex heat and ld low system consstng o both small-scale and large-scale elements. Hot and cold ld passages n a compact heat exchanger are treated as dstnct poros meda havng hghly ansotropc permeabltes. A general set o macroscopc governng eqatons has been derved sng the volme averagng theory. Possble sb-scale modelng and mcroscopc treatments or ts closre are proposed on the bass o the ansotropc poros meda theory. Key words Poros meda, Volme averagng, Heat exchanger, CFD, Thermal non-eqlbrm Introdcton In ths poros meda approach, we se a grd system, whch s st ne enogh to resolve macroscopc low and temperatre elds. We do not grd small-scale elements bt model them as a poros medm. We note, CFD calclatons o engne nacelles [] may become easble only we appeal to a volme-averaged set o macroscopc eqatons along wth some sbscale model to accont or low resstance resltng rom small scale elements sch as bndles o hydralc tbes, rbs and some other obstrctons. ewse, hot and cold ld passages n a heat exchanger, as llstrated n Fgre, are to be treated as two dstnct poros meda o hghly ansotropc permeabltes, separated by sold walls. In order to apply these macroscopc governng eqatons based on VAT (volme averagng theory) sch as ntrodced by Naayama et al. [] or analyzng the complex heat and ld low eqpment, we mst model the low resstance assocated wth sbscale sold elements (modeled as an ansotropc poros medm) and also the heat transer between the lowng ld and the sbscale elements, beorehand. Sch sbscale modelng can be made perormng mcroscopc (pore scale) nmercal calclatons on ndvdal small elements (sch as ns and bndles o hydralc tbes). The mcroscopc nmercal reslts ths obtaned at a pore scale can be processed to extract the macroscopc hydrodynamc and thermal characterstcs n terms o the volme-averaged qanttes. A great deal o eort has been drected towards ths endeavor [3]. In the ndamental stdy o trblent low throgh an sotropc poros medm, Naayama and Kwahara [4] ponted ot that the smaller eddes (.e. trblent mxng rather than mechancal dsperson) mst be modeled rst, as n the case o ES, and started wth the Reynolds averaged set o the governng eqatons and ntegrated them over a representatve control volme to obtan the set o macroscopc eqatons, namely, the eqatons o contnty, momentm, energy, trblence netc energy and ts rate o dsspaton. The nnown model parameters, ncldng the nteracal heat transer coecent, permeablty and Forchhemer constants were determned by carryng ot exhastve nmercal experments sng a perodc array o sqare rods and then ntegratng these mcroscopc reslts over a nt poros strctre [4]. Ther stdy can be generalzed to treat ansotropc

2 poros meda, whch represent complex heat and ld low eqpment consstng o synthetc strctral elements. Possble steps based on the theory o poros meda or analyzng complex heat transer eqpment may be smmarzed as ollows: st step: Derve a set o volme averaged governng eqatons by ntegratng the set o Reynolds averaged conservaton eqatons or mass, momentm, energy and trblence qanttes over a local control-volme. nd step: Establsh possble sbscale models or nnown terms assocated wth devatons rom averaged vales, namely, permeablty tensor, Forchemer tensor, and nteracal heat transer coecent. 3 rd step: Determne all model constants by condctng mcroscopc nmercal smlatons over ndvdal elements. 4 th step: Solve the set o volme averaged governng eqatons wth the sbscale model eqatons and ther model constants st determned. Hot ld(=) Cold ld(=) Sold strctre (s) Fgre Schematc vew o heat exchanger Volme averaged governng eqatons In most applcatons assocated wth heat exchangers, the lds do not mx bt transer heat throgh a separatng wall, as llstrated n Fgre. Thereore, we mst consder the two sets o the mcroscopc governng eqatons or both hot and cold lds. Each set o the mcroscopc governng eqatons conssts o the Reynolds averaged contnty eqaton, Naver-Stoes eqaton, energy eqatons, the transport eqaton o trblence netc energy and that o ts dsspaton rate. We ntegrate the two sets o Reynolds averaged eqatons or the hot and cold ld phases and the heat condcton eqaton or the sold phase over a control volme V, whch s mch larger than a mcroscopc (cltter) characterstc sze bt mch smaller than a macroscopc characterstc sze. The resltng macroscopc eqatons natrally yeld a nmber o correlatons among the spatally devatng components. Naayama and Kwahara [4] mathematcally modeled these nnown terms (correlatons) to close the set o macroscopc eqatons or the case o sotropc poros meda. Ther mathematcal treatment can be generalzed or the case o ansotropc poros meda as ollows: x =0 () t x = p ρ x ρ 3 x ( ) t x x φ K b φ () φρ T φρ ρ c φδ T ( ) ( ) ( ) p t s cp cp T = tor ds ah T T t x x σ x (3)

3 t x ( ) = x e c x ( ) ds t δ t s s φ b n σ φρ p 3 (4) t x ( ) ds t = δ φ x σ e c x K φ ρ t c φ ( c s s c ) c b ( ) D t n p n = c D ( ), s and = x x a (5) = adv (6) V V The sb- and sper-scrpts (= and ) and s reer to hot ld (=), cold ld (=) and sold (s), respectvely. (Note that s the ntrnsc average o trblence netc energy, whch shold not be consed wth the thermal condctvty o ld,.) Moreover, φ = V V s the ld volme racton whle e and e are the ews nmbers or the mechancal dspersons. Moreover, K, b and h are the permeablty tensor, Forchhemer tensor and nteracal heat transer coecent, respectvely, and K n = K / and bn = b / are the drectonal permeablty and Forchhemer constant. The resltng macroscopc eqatons, namely, (), (), (3), (4) and (5) or hot and cold lds (.e. = and ), and the heat condcton eqaton or the sold phase, consttte a set o the three-energy eqaton model or analyzng complex trblent low and heat transer encontered n heat exchangers. (For the case o lamnar low, only eqatons (), (), (3) and the heat condcton eqaton or the sold phase may be consdered wth t = 0 everywhere.) These macroscopc eqatons may be tlzed to analyze low and temperatre elds wthn any other complex assembles consstng o small-scale elements n a cltter, whch one does not want to grd. Note that the same set o the eqatons can be sed also or clear ld lows wth one o the porostyφ = and the otherφ = 0, and or the condcton wthn a sold wthφ = φ = 0. In other words, the poros meda concept allows s to cover not only convecton n poros meda bt all other heat traser modes rom pre condcton to pre convecton, smply by prescrbng the spatal dstrbton o porosty (and that o the correspondng permeablty). 3 Sbscale modelng In order to se the oregong set o the macroscopc eqatons or analyzng heat transer n a tblar heat exchanger, or example, we mst establsh the sbscale models or nnown terms sch as an nteracal heat transer coecent. Accordng to Kwahara et al. [5], the nteracal heat transer coecent may be obtaned by sbstttng the mcroscopc temperatre reslts nto the ollowng eqaton: h T dp P T da nt V A Ald n = s s nt ( T T ) ( T T ) where A s the total nterace between the ld and sold, whle nt da s ts vector element pontng otward rom the ld sde to sold sde. In Fgre, the heat transer reslts obtaned at α =0 and π / 4 or the cross lows (.e. γ = π / ) are presented n terms o the nteracal Nsselt nmber N = h / aganst the Reynolds nmber Re. The lower Reynolds nmber data stay constant or 0. 6 the gven array and low angle, whereas the hgh Reynolds nmber data vary n proporton to Re. Thereore, we model the drectonal Nsselt nmber as ollows: h 0.6 / 3 N = c d Re Pr (8) The present reslts ollow closely along the expermental correlatons o Zasas [6] and Grmson (7)

4 [7] as ncreasng Re. These correlatons are beleved to hold or a comparatvely wde Reynolds nmber range, coverng rom a predomnantly lamnar low regme to trblent low regme. 0 α=0 [deg] α=45 [deg] Zasas Grmson N Re Fgre Eect o Reynolds nmber on drectonal Nsselt nmber (Pr=) m m m 0.m D H =0mm H=40mm D=4mm Fgre 3 Cross low tblar heat exchanger 4 Applcaton to complex heat transer eqpment To llstrate possble engneerng applcatons, the oregong macroscopc eqatons and ther sbscale models are exploted to analyze varos heat and ld low systems sch as a plate n heat exchanger and a cross low tblar heat exchanger as shown n Fgre 3. The hot gas at 30C enters rom the pper let nlet at the norm velocty ( ( m c p ) =4700W/K) and goes ot rom the lower rght ext, whle the water ( ( c p ) C passes throgh a m =4700W/K) at 30 coled tbe o nner dameter 4mm to be heated p. It s assmed that the gas low s lamnar whereas the water low n the tbe s trblent. Only the macroscopc eqatons (), () and (3) or the gas

5 phase (=) are solved nmercally along wth the sbscale models establshed beorehand or the permeablty, Forchemer tensors and nteracal heat transer coecent or the rod arrangement shown n Fgre 3. The water low, on the other hand, s assmed to ollow Dtts-Boelter ormla, and ts bl temperatre along the tbe s obtaned solvng the one-dmensonal energy balance eqaton sng the srrondng gas temperatre avalable n that nstant. Both gas and water temperatres are copled to get converged temperatre elds. The macroscopc temperatre elds o the gas and water on the center plane are dsplayed n Fgres 4(a) and (b), respectvely. The gres clearly llstrate heat transer rom the hot gas to cold water. Natrally, heat transer n the regon close to the gas entrance s most sgncant (a) Gas temperatre (b) Water temperatre Fgre 4 Cross low tblar heat exchanger 5 Conclsons A systematc modellng procedre based on the theory o poros meda s proposed to attac heat and ld low n complex manmade strctres consstng small elements. A general set o macroscopc governng eqatons s derved explotng the volme averagng theory. The sb-scale modelng and mcroscopc nmercal treatments or ts closre are proposed on the bass o the ansotropc poros meda theory. As or llstratve applcatons, extensve nmercal experments are condcted or three-dmensonal heat and ld low throgh a ban o cylnders n yaw, whch represents a nmercal model or manmade strctres sch as plate n heat exchangers, and also or gas and water lows n a cross low tblar heat exchanger. The poros meda concept dscssed here s not lmted to complex heat transer eqpment, bt has a great potental or rther advances n complex heat transer systems and or rther exploraton o emergng new technologes. Its applcatons to compostng systems, boheat transer and cryoablaton treatments are nderway. Reerences [] P. E. DesJardn, Prvate commncaton (00) [] A. Naayama, PC-Aded Nmercal heat transer and convectve low, Boca Raton: CRC Press (995). [3] A. Naayama and F. Kwahara, Nmercal modelng o convectve heat transer n poros meda sng mcroscopc strctres, Vaa, K. ed. Handboo o Poros Meda, Marcel Deer, Inc., New Yor, pp (000). [4] A. Naayama and F. Kwahara, A macroscopc trblence model or low n a poros medm, J. Flds Engneerng, Vol., pp (999). [5] F. Kwahara, M. Shrota and A. Naayama, A Nmercal stdy o nteracal convectve heat transer coecent n two-energy eqaton model or convecton n poros meda, Int. J. Heat Mass Transer, Vol. 44, pp (000). [6] A. Zasas, Heat transer rom tbes n crosslow, Advances n Heat Transer, Vol. 8, pp (987). [7] E. D. Grmson, Heat transer and low resstance o gases over tbe bans, Transacton o ASME, Vol. 58, pp (983).