A CFD STUDY OF NATURAL CONVECTION HEAT AND MASS TRANSFER IN RESPIRING HYGROSCOPIC POROUS MEDIA

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1 Seond Inernaonal Conerene on CFD n he Mnerals and Proess Indsres CSIRO, Melborne, Asrala 6-8 Deember 1999 A CFD SUDY OF NAURAL CONVECION HEA AND MASS RANSFER IN RESPIRING HYGROSCOPIC POROUS MEDIA Mahesh PRAKASH 1, Özden F. URAN 1, Ygo LI and Graham R. HORPE 1 1 Vora Unversy, Faly o Engneerng and Sene, Vora, AUSRALIA CSIRO Dvson o Bldng Consron and Engneerng, Hghe, Vora, AUSRALIA ABSRAC A model s presened o pred hea and mass ranser n a sysem onssng o a rblen low overlyng a saraed hygrosop poros medm. Comparsons wh epermenal and nmeral smlaons have been arred o o he he aray o omponens o he model. A ase sdy sng sla gel as a represenave hygrosop poros medm s smlaed o llsrae an applaon o he model. he wor s progressng owards a mahemaal desrpon o resprng agrlral prods. KEYWORDS Hygrosop, naral onveon, poros medm, rblene. NOMENCLAURE spe hea a onsan pressre [Jg -1 K -1 ] e1, e, e3 oeens assoaed wh he dsspaon o rblene ne energy, eqaon (5) C F Forhhemer oeen, eqaon () dmensonless oeen, eqaons (a) and (11) D wdh o he sysem [m] g aeleraon de o gravy [ms - ] h p hegh o poros layer [m] h s hea o sorpon o waer [Jg -1 ], eqaon (3) H hegh o he sysem [m] H w negral hea o weng [Jg -1 ], eqaon (3) rblene ne energy [m s - ] K permeably [m ] p pressre [Pa] P am amospher pressre [Pa], eqaon (10) Pr Prandl nmber r relave hmdy S Shmd nmber me [s] emperare [K] ld veloy [ms -1 ] * ron veloy [ms -1 ], eqaon (1) w ar mosre onen on dry bass, [g o mosre/g o dry ar] W poros medm mosre onen on dry bass, [g o mosre/g o dry sold] y spae o-ordnae [m] spae o-ordnae or normal dsane rom he [m], eqaon (11) Gree symbols β oeen o hermal epanson [K -1 ] δ Kroneer dela operaor dsspaon o rblene ne energy [m s -3 ] φ porosy ld vsosy [gm -1 s -1 ] eddy vsosy [gm -1 s -1 ] densy [gm -3 ] rblen analoge or Prandl and Shmd nmbers, eqaons (3), (4), (5) and (7) τ shear sress [gm -1 s - ] Sbsrps 0 reerene abs absole vale, eqaon (9) e eeve vale or poros medm ld phase (ar), dreon o veor omponen (, 1, ) rblene ne energy m mosre, eqaon (7) s sold phase sa saraon vale, eqaon (9) emperare w waer dsspaon o rblene ne energy INRODUCION Epermenal and nmeral nvesgaons o naral onveon lows n a lear ld overlyng a saraed poros medm have been onned o sdes n he lamnar regme. In many praal saons, however, one enoners sysems n whh he hea and mass ranser s sgnanly aeed by rblene a leas n he ld regon. Sh a wo layer sysem an be enonered n sorage o poros maerals sh as grans, rs and aalyss. hermal nslaon n bldngs and geohermal reservors are also modelled as wo-layer sysems. In all eamples menoned above, he ld (whh s generally ar), an oen be rblen. I s well esablshed rom naral onveon sdes n lear lds, (he epermenal sdy o Krpar and Bohn (1986) and nmeral sdy o Henes (1990)) ha he enlosre dmensons deermne wheher he low s lamnar or rblen. For large enlosres, he low s rblen de o he large Raylegh nmbers enonered. hs, one anno gnore rblene as a aor whh enhanes hea ranser and hene mosre mgraon n sh wo layer sysems, here s a sgnanly large ld layer. In he poros medm, rblene wll no perss has a senly low permeably. hs, one old assme a lamnar regme n he poros medm. 157

2 158 Flow vsalsaons arred o wh synhe relaed oams o deren permeably vales (Praash e al. 1999a) have onrmed he valdy o hs assmpon. hs paper presens he mporan eares o a general mehod o handlng sysems wh a ld overlyng a hygrosop poros medm where rblene n he ld layer s mporan. A dealed repor o he same sdy has been prepared (Praash e al., 1999). A hygrosop poros medm has been hosen bease s envsaged ha he presen model wold be mos sel n modellng sored agrlral prods sh as grans ha are hygrosop n nare. Componens o he model have been valdaed wh prevos epermenal and nmeral resls ha were ond o be smlar o he presen sysem. An applaon onernng mosre mgraon hrogh a represenave hygrosop poros medm s smlaed o emphasse he poenal applaon o he model. A wo dmensonal sysem has been sed n he enre paper or smply. MODEL DESCRIPION Governng eqaons or ld low and hea ranser Eqaon o onny 0 ) ( (1) Eqaon o moon ( ) ( ) [ ] 1 / 1 / F 0 K C K g p φ φ β δ φ () where he eddy vsosy s represened as n whh (a) I s assmed ha rblene persss only n he ld layer and s rapdly aenaed n he poros medm. hs he eddy vsosy,, apples only or he ld layer. For he poros medm, s vale s aen o be zero. he hrd erm on he rgh hand sde o Eqaon () s he boyany erm whh ses he Bossnesq appromaon. he orh erm represens he Daran ressane o low. he h erm represens he Forhhemer ressane, and s nlded o aon or neral ees n he poros medm. I shold be noed ha he dson erm n he momenm eqaon or he ld layer beomes he Brnman orreon or he poros medm. he eeve vsosy assoaed wh he Brnman erm or he poros medm s assmed o be he ld vsosy. hs assmpon s sed de o he analyss o Neale and Nader (1974), and has been sed sesslly by several ahors nldng Sngh e al. (1993), Song and Vsana (1994) and Chen e al. (1998). he Dary and Forhhemer erms vansh n he ld layer, as he ld has nne permeably. he porosy o he ld layer s aen o be eqal o one, and a sep hange n porosy and permeably s assmed a he nerae beween he ld layer and poros medm. hermal energy balane De o he presene o he hygrosop poros medm, he hermal energy balane assmes deren orms or he ld layer and he poros medm. he paral derenal eqaons or hea, mass and momenm ranser n sored agrlral prods have been derved n horpe e al. (199). For he presen problem one an wre, ( ) W h Pr b s b e (3) where ( ) ( ) H W 1 w w s s e φ φ and 0.9. he erm W w arses de o he presene o mosre n he poros medm. he erm H w represens he negral hea o weng deals o whh an be ond n Sherland e al. (1971). he las erm on he rgh hand sde represens he hea o sorpon. I shold be noed ha or he ld layer he negral hea o weng and he hea o sorpon wold vansh. Agan, or he poros medm he eddy vsosy erm s zero, as rblene s absen n he poros medm. rblene ne energy balane β g (4) where 1.0 Balane or dsspaon o rblene ne energy g 3 1 β (5) where ( ) κ 1, κ von Karman s onsan 0.41, , 1.9.

3 For he oeen 3, he orm sggesed by Henes (1990) s sed. hs, 3 anh v /. I shold be noed ha Eqaons (4) and (5) apply only o he ld layer. Governng eqaons or mosre mgraon n poros meda Iner-granlar mosre balane he eqaon ha governs he onenraon o mosre n he ar has a slghly deren orm n he ld and poros layers de o he onrbon o he solds o he ar mosre balane n he poros medm. he ar mosre balane or he poros medm an be represened by w W w φ w b (6) S he ar mosre balane or he ld layer an be wren as w w w φ (7) S m where, m Noe agan ha he eddy vsosy,, has been nvoed only n he eqaon or he ld layer. An order o magnde analyss arred o on Eqaon (6) showed ha he adveon erm s balaned by he hange n he solds mosre onen, and ha he rae o hange n he ar mosre onen s neglgble n omparson. he molelar dson erm wll be sgnan a he nerae beween he poros medm and lear ld and lose o he s. hs, anno be negleed. he resl enables one o negle he rs erm on he le-hand sde o Eqaon (6) hs allowng he ompaon o W eplly n he poros medm. One problem enonered n a prevos sdy by Chen e al. (1999) was wh he ransen ompaon o sh sysems. An mpl solon proedre was sed by nldng all erms n Eqaon (6). I s epeed ha hs problem an be overome by sng he above epl proedre. he presen ormlaon also allows one o ae large me seps o sdy mosre mgraon one he emperare and veloy elds have evolved almos o a seady sae, bease he mosre eld does no sgnanly ae he emperare and veloy elds. he ably o ae large me seps drng he proess o ompaons s mporan de o he large me sales (days or monhs) enonered n sh sysems. Callaons are sared wh an nal vale or mosre onen o grans n he poros medm. hs vale s hen sed o allae he mosre onen o ar n he poros medm. he mosre onen o ar n he ld layer s allaed by sng Eqaon (7). Sla gel s sed as he represenave hygrosop poros medm or he sdy. Alhogh any hygrosop medm sh as grans old be hosen, sla gel s sed de o he a ha has a lnear sorpon soherm, and also bease s negral hea o weng s zero. For sla gel, W 0.5r (8) where r p / psa represens relave hmdy o ar. he saraon pressre s gven by Eqaon (9), (Hner (1987)) p sa ep 5 (9) abs abs he mosre onen o ar n he poros medm s allaed by sng he relaonshp, 0.6 p w 0.6Pam (10) p Problem denon, bondary and nal ondons or he ase sdy As a ase sdy or he proposed model, mosre mgraon n a reanglar avy o aspe rao wh hal he avy lled wh sla gel s hosen. A Raylegh nmber, Ra10 10 and a Dary nmber, Da 10-8 s hosen or he sdy. hese vales orrespond o a oal sysem hegh, H o abo meres wh a wdh, D o abo 1 mere. he Dary nmber hosen s ypal or a sorage sysem. Fgre 1 shows a shema dagram o he smple wo-dmensonal geomery onsdered. Fgre 1: Shema dagram o he sysem or ase he sdy. Bondary ondons ha are ypal o a sorage vessel are onsdered or he presen sdy. he poros medm s onsdered o be a an elevaed emperare o h 30 0 C wh he sde s and he op a 10 0 C. he boom s onsdered o be adaba. No slp and mpermeable bondary ondons are assmed or all or s. Zero graden bondary ondons are appled or ar mosre onen and mosre onen o he poros medm. Hene, he sysem s over speed. he nerae ondons or mosre and emperare are sased aomaally as hese reqre only a onnos hange n mosre and emperare a he nerae. Sne he Brnman eended Dary low model s mplemened, he nerae ondons or veloy are 159

4 also sased aomaally hrogh he onny o momenm. No nons are sed or eher veloy or emperare as hese are solved rgh p o he. For he rblen ne energy and energy dsspaon, sne rblene s assmed o be absen a he nerae, ondons a he nerae are aen o be denal o he bondary ondon. hs and a he rs nner grd pon rom he op, he pper porons o he laeral s and he nerae are aen as, ( * ) ( * ), 3 (11) κy where * s ron veloy dened by * τ where τ s he shear sress allaed rom τ y, and y s he normal dsane rom he. he above modaon o he sandard - model has been sggesed or naral onveon lows bease he nons ha normally apply or ored onveon lows are no applable or naral onveon lows, see or eample Henes (1990). Valdaon o he model For he presen problem, wo ongraons were sed o valdae omponens o he model. he smlaons and epermenal wor o Song and Vsana (1994) were sed o valdae he Brnman Forhhemer eended Dary low model or a sysem onssng o a ld layer adaen o a poros layer. he rblene model was valdaed wh he epermenal resls o med avy naral onveon repored by Krpar and Bohn (1986) or her HCCC ongraon, bease he bondary ondons sed losely resembled he presen bondary ondons. Mehod o solon he model eqaons are dsresed sng he onrol volme ormlaon, and hey are solved sng he SIMPLE algorhm (Paanar, 1980). he pwnd/enral hybrd sheme (Paanar, 1980) s sed or dsresng he onveon erms. he dson erms are dsresed sng he seond order enral sheme. he nondmensonal sheme employed by Praash e al. (1999b) has been sed or he ompaons. Non-norm grd spang was sed o aon or he large gradens n veloy and emperare wherever applable. Fgres and 3 show shema dagrams o he ongraons o Song and Vsana (1994) and Krpar and Bohn (1986). For he smlaons o he epermens o Song and Vsana (1994), an 8080 grd was ond o be senly arae aer arryng o a grd renemen sdy. he sreamlnes and soherms were n agreemen b are no shown here. hey an be ond n Praash e al. (1999). he omparson beween he measred and smlaed emperare proles s smlar o he resls o Song and Vsana (1994). hs omparson s shown n Fgre 4. 1 θ Presen predons y/h0.9 y/h0.5 y/h0.48 y/h0.1 S & V (1994), Ep. y/h0.1 y/h0.5 y/h /H Fgre 4: emperare proles (omparson wh epermen 3 o Song and Vsana, 1994) he maor dsrepany beween he measred and preded emperare proles s ond a y/h0.1. Song and Vsana (1994) arbed hs dsrepany o he onnm model sed. he modelled and epermenal proles agree n general, as poned o by Song and Vsana (1994). For he smlaons o he epermens o Krpar and Bohn (1986) 6060, 1010 and 4040 non-norm grds were sed n order o sdy he ee o grd renemen. For hese smlaons, a Raylegh nmber o was sed wh he worng ld as waer. All he physal properes were aen a he bl ld emperare. he ompaons were arred o nl he derene beween he ho and old Nssel nmbers was less han 4%. he derene beween he Nssel nmbers o he veral s was less han 1%. I an be seen ha wh a hange n he grd sze, he mamm varaon n he Nssel nmber ors a he veral s. In gong rom he 1010 grd o he 4040grd he op and boom Nssel nmbers vary by less han 1%. However, he veral Nssel nmbers vary by almos 10%. hs resl ndaes ha he solon s sll no grd ndependen or he nes grd omped n he presen sdy. Krpar and Bohn (1986) sed he average o he or-ho/old emperares as he bl emperare. he o bl emperare derene was sed o evalae he Nssel nmbers. For he HCCC ongraon hey ond ha he relaonshp beween Nssel nmber o Raylegh nmber ollapsed o a sngle orrelaon or all or s, namely 160

5 N 0.346Ra 0.85 (1) Usng he same denon or he Nssel nmbers, he orrelaon or all or s a he nes grd s gven n able. he mamm error n he predon o hea ranser ors or he op (an over predon o appromaely 78%). Sdes arred o or he heang rom he sde ongraon (. Henes; 1990) show ha he hea ranser resls or naral onveon lows depend sgnanly on he rblene model sed as well as he near reamen. However, s no he obeve o he presen paper o evalae deren rblene models. Insead, he omparson eerse gves an esmae o he error n he hea ranser predon gven by he presen model. phenomenon has mporan mplaons, as leads o he growh o molds n hese regons whh evenally desroy he grans. I was ond ha lose o he sdes and n he regon lose o he enral rsng plme n he poros medm, he mosre onen o he poros medm nreases. hs nrease beomes sgnan as me progresses. I s evden rom Fgre 5 ha hese are regons o relave oolness n he poros medm. Grd Le Rgh op Boom able : Coeen o he relaonshp beween Nssel nmber o Raylegh nmber. Case sdy: Dssson o resls 10 hrs 30 hrs 50 hrs Fgre 5: Change n soherms or he ase sdy he geomery or he ase sdy s shown n Fgre 1. A 6060 non-norm grd was sed as a ompromse beween aray and ompng me. For he nseady allaons, a me sep o 0.1 s was sed nl he emperare and veloy elds were almos onverged n he ld layer. he onvergene was monored hrogh hanges n he Nssel nmbers. Appromaely 3000 me seps were reqred. he me sep was hen nreased o 10 s, or monorng hanges n he mosre onen and hanges n emperare n he poros layer. Any rher nrease n he me sep lead o dvergene. De o he very small nmber o nernal eraons beween wo me eraons, (ypally hree o or nernal eraons are reqred), ompaons or sgnanly large real mes an be arred o n relavely shor mahne mes. For he larges real me allaed n he presen ase sdy, 50 hors, a CPU me o arond 5 hors was reqred on a Penm II (350 MHz). However, nreasng he me sep s desrable or shorer ompaonal mes. he hanges n he soherms and mosre onen o ar are shown n Fgres 5 and 6 respevely aer 10, 30 and 50 hors. Changes n mosre onen o he poros medm have no been shown here, bease hey were ond o be nsgnan n hs perod o me. he hea and mosre rom he poros medm s drven owards he ld layer hrogh he enral rsng plme as seen n Fgres 5 and 6. he hermal plme s epeed o enhane he ranspor o hea, and hene, mosre rom he poros medm. hs he arae predon o hea ranser by he rblene model s very mporan. De o he hea ranser, he ore o he poros medm beomes progressvely ooler, as an be seen rom Fgre 5. Smlarly, rom Fgre 6, one an see ha he mosre onen o he ar n he poros medm dereases as me progresses. As he ar ges ooler, s ably o hold mosre dereases, and hs era mosre s adsorbed by he hygrosop poros medm. In ase o grans hs 10 hrs 30 hrs 50 hrs CONCLUSION Fgre 6: Change n ar mosre onen or he ase sdy A general proedre o smlang sysems wh a rblen low overlyng a saraed hygrosop poros medm has been presened. Emphass has been plaed on sng hs model or smlang mosre mgraon n sored agrlral prods. Sh a model s apable o predng he mosre mgraon proess more araely by aonng or rblene n he ld layer. he level o sophsaon o he rblene model s lmed by he omparavely large rn mes enonered o smlae sh sysems. he model needs o be valdaed rher wh reals epermenal daa. he presen model s apable o smlang lows only when rblene n he poros medm an be onsdered o be neglgble. However, hs ondon wold no be re or poros meda o hgh permeably. In order o overome hs lmaon, a rblene model or he poros medm needs o be norporaed no he esng model. 161

6 REFERENCES CHEN, L. and LI, Y., (1999), Modellng hgh Raylegh nmber naral onveon lows n gran sorage, CSIRO Researh Repor, BCE DOC. 99/0331 (M). CHEN, L., LI, Y. and HORPE, G.R., (1998), Hgh- Raylegh-nmber naral onveon n an enlosre onanng a poros layer, 11 h In. Hea ranser Con., Seol, Korea. HENKES, R.A.W.M., (1990), Naral onveon bondary layers, Ph.D. hess, Del Unv. eh., he Neherlands. HUNER, A.J., (1987), An sosere eqaon or some ommon seeds, J. Agr. Eng. Res., 37, KIRKPARICK, A.. and BOHN, M., (1986), An epermenal nvesgaon o med avy naral onveon n he hgh Raylegh nmber regme, In. J. Hea and Mass ranser, 9, NEALE, G. and NADER, W., (1974), Praal sgnane o Brnman s eenson o Dary s law: Copled parallel lows whn a hannel and a bondng poros medm, Canadan J. Chem. Eng., 5, PAANKAR, S.V., (1980), Nmeral Hea ranser and Fld Flow, Hemsphere Pblshng Corporaon, USA. PRAKASH, M., MAHONEY, J., L, Y., URAN, Ö.F. and HORPE, G.R., (1999a), Flow vsalsaon or an enlosed a-symmer mpngng rond e wh and who poros medm (n preparaon). PRAKASH, M., URAN Ö.F. and HORPE G.R., (1999b), Program Naon: or he nmeral solon o naral onveon n a reanglar avy, Vora Unv. Researh Repor. PRAKASH, M., URAN Ö.F. and HORPE G.R., (1999), Naral onveon hea and mass ranser n hygrosop poros meda o low permeably, Vora Unv. Researh Repor. SINGH, A.K., LEONARDI, E. and HORPE, G.R., (1993), hree-dmensonal naral onveon n a onned ld overlyng a poros layer, J. Hea ranser, 115, SONG, M. and VISKANA, R., (1994), Naral onveon low and hea ranser whn a reanglar enlosre onanng a veral poros layer, In. J. Hea and Mass ranser, 37, SUHERLAND, J.W., BANKS P.J. and GRIFFIHS, H.J., (1971), Eqlbrm hea and mosre ranser n ar low hrogh gran, J. Agr. Eng. Res., 16, HORPE, G.R., MOORE, G.A. and SINGH, A.K., (199), Hea, mass and momenm ranser n hreedmensonal bls o sored grans, Con. on Eng. n Agr., Albry, NSW. 16