COMPARISON BETWEEN MODELS FOR THE HEAT TRANSFER FLUID IN LATENT HEAT STORAGE SYSTEMS

Transcription

1 COMPARISON BETWEEN MODELS FOR THE HEAT TRANSFER FLUID IN LATENT HEAT STORAGE SYSTEMS Antonio B. Jeu Kamal A. R. Imail UNICAMP, Faculdade de Engenhaia Mecânica, Depto. de Eng. Témica e Fluido Cx. P Campina, SP, Bail Abtact. The poblem o olidiication o a phae-change mateial (PCM) aound a tube caying a heat-tane luid (HTF) inide i olved numeically by to dieent method. The it one employ the enegy equation o the luid, aumed in ully developed lamina lo. The othe pocedue i a impliied one hich conide uniom velocity poile and peom an enegy balance o the HTF uing Nuelt numbe value obtained om a teady tate coelation o eithe a contant all heat lux o a contant all tempeatue bounday condition. Compaion ae made in tem o the poition o the phae-change ont obtained by both pocedue and beteen the calculated Nuelt numbe om the it method ith the value o teady tate coelation. The eult obtained ho that the accuacy o the impliied method and the deciion o hich teady-tate value ae to be ued depend eveely on the atio o themal conductivitie o the HTF, olid PCM and tube all mateial. Keyod: Latent heat toage, Annula geomety, Heat-tane luid, Nuelt numbe. INTRODUCTION The poblem o melting o olidiication o a phae-change mateial (PCM) contained in the annula pace beteen a tube caying a heat tane luid (HTF) and an adiabatic bounding uace, ha eceived much attention in the liteatue due to it application in the deign o latent heat themal enegy toage unit. In ealy pape numeical and emianalytical olution method ee ued and the coupling beteen convection in the HTF and conduction in the PCM a made by peciying a Biot numbe o the luid. Some o thoe o ae evieed in Imail & Jeu (999). Cao & Faghi (99) developed hat eem to be the it tudy in hich the coupling beteen the conduction and convection poblem a done by olving the tanient momentum and enegy equation o the HTF, accounting o the eect o the tube all. They conideed the uion o PCM caued by a lo Pandtl numbe HTF. In thi pape lamina luid lo a aumed, hile in anothe o (Cao & Faghi, 99), HTF lo a tubulent.

2 Bellecci & Conti (993) olved the ame poblem a Cao & Faghi (99) and uggeted that it a not neceay to olve the luid lo poblem i a mean velocity ee peciied and the Nuelt numbe o the HTF evaluated om a teady-tate coelation o contant-all heat lux bounday condition. Thee tatement ee validated by good compaion ith Cao & Faghi (99) eult. Zhang & Faghi (996) olved the poblem o melting o paain aound a metallic tube uing ate a the HTF. The olution pocedue a emi-analytical and adopted the integal method o the PCM and an expeion o evaluating the Nuelt numbe in the ullydeveloped lamina lo o the HTF conideing an abitay all tempeatue vaiation. They hoed that the calculated Nuelt numbe laid beteen the teady-tate value o contant all tempeatue (CWT) and contant-all heat lux (CHF) bounday condition. Accoding to thei concluion, the impliication popoed by Bellecci & Conti (993) ee valid only o a lo Pandtl numbe HTF o a vey long tube ithout pactical inteet. In anothe pape (Zhang & Faghi, 995), they analyzed the poblem uing the ame method but conideing the tubulent ully-developed HTF lo and neglecting the tube all themal eitance. It a hon that, o a lo Pandtl numbe HTF, the atio beteen the HTF and PCM themal conductivitie had a tong inluence in the evolution o the phae-change ont and the Nuelt numbe could even be evaluated om a CHF coelation, i that atio ee uiciently high. They alo hoed that o modeate Pandtl numbe HTF in tubulent lo the Nuelt numbe value could be obtained om the themally developed lo value. At lat, they concluded that thoe tatement ee not valid o lamina lo. In thi o the poblem o olidiication o ate aound a coppe tube i olved numeically. To dieent pocedue, cited above, o teating the HTF convection poblem ae adopted. In the it method the enegy equation i olved conideing a ully developed lamina lo. The econd appoach conide a mean luid velocity and peom a themal enegy balance in the tube adopting Nuelt numbe value obtained om eithe CHF o CWT teady tate condition. The eult obtained o the evolution o the phae-change ont ae plot togethe a ell a the calculated local Nuelt numbe in the it pocedue i diplayed ith the teady tate value. The themal conductivity o the HTF a vaied and compaion ho that the accuacy o the impliied method and the deciion about hich teady tate value ae to be ued depend tongly on thi paamete.. MATHEMATICAL MODEL The ytem on tudy i hon in Fig. belo, e (z,t) l(z,t) V T in in i o z olid phae liquid phae H Figue Setch o the poblem. The mathematical model i baed on the olloing hypothei:

3 To-dimenional and axiymmetic poblem; Lamina, teady-tate, ully-developed HTF lo. Uniom tempeatue, T in, at inlet. The impliied method aume a uniom mean velocity, V in, at inlet. Contant themophyical popetie ith no denity dieence beteen the olid and liquid phae; PCM i bounded by an adiabatic ymmety uace and the end cap o the ytem ae conideed adiabatic a ell; Conduction i the only heat-tane mechanim in the liquid PCM (olidiication poce ith lo initial liquid tempeatue, 4 o C, and adiabatic extenal bounday condition). The poblem i olved by to methodologie. A, o-called, complete method, hee the enegy equation o the HTF i ued, and a impliied method hich conit in peoming an enegy balance in the HTF uing the Nuelt numbe value om CHF o CWT condition. In both cae the conduction equation o the tube all, olid and liquid PCM have to be olved. The enegy conevation equation o the HTF and the tube all ae itten in cylindical axi-ymmetic coodinate and then cat in dimenionle om uing the olloing paamete, V U V z in T T θ T T m VinD ; Re ν ; R ν P α ; ξ ( T T ) ; S α c t p m in α t i Fo ; Ste ; Ste Fo L τ i m in z ; Pe Re P ; Nu hd Ste () hee, U i the dimenionle, ully-developed velocity poile, Re, P, Pe and Nu ae the Reynold, Pandtl, Peclet and Nuelt numbe, epectively. The tube inne diamete D i ued in the deinition and the ubcipt ee to the HTF popetie. Dimenionle adial and axial coodinate and olid phae thicne, R, ξ and S, ae deined in tem o the outide tube adiu,. A dimenionle tempeatue, θ, i deined in tem o the luid inlet tempeatue T in and the phae-change tempeatue T m. Dimenionle time i expeed by mean o a Fouie numbe, Fo, o by the poduct beteen thi, the Stean numbe, Ste, and the quaed atio o the inide and outide tube adii. The latte i gouped in the vaiable τ. The Stean numbe i a quotient beteen the enible and latent heat, L, content o the PCM. The ubcipt mae eeence to the olid phae. The HTF enegy equation o the complete method i deined belo, α θ Ste + U Pe R + () α τ D ξ R ξ In the impliied cheme only the olloing balance equation in tem o the bul tempeatue, θ b, i conideed, α α θ b Ste τ b + Pe D ξ θ ξ b + 4 D Nu ( θ θ ) i b (3)

4 heeθ i i the inne all tempeatue. The conduction equation o the tube all i deined a, α α θ Ste R + τ R ξ (4) The PCM i teated in a dieent ay, a a coodinate tanomation in the adial diection hich immobilize the phae-change ont i ued. The ne adial coodinate ytem i deined a, ( z,t ) η ; ηl (5) ( z,t ) l( z,t ) hee i the adial poition o the olid-liquid inteace and l i the thicne o the liquid phae. Equation (5) ae non a Landau tanom and map the PCM phyical domain to 0 η in the olid egion and 0 η l, o the liquid phae. The phae-change ont become ixed at the poition η ;ηl 0. Witing the conduction equation o the olid and liquid phae in tem o η and η l lead to athe complex expeion in hich appea peudoconvective tem, hich ae multiplied by the phae-change ont velocity, and peudoaniotopic tem ith mixed deivative. Fo aving pace eaon thoe equation ill not be peented hee and all detail about the teatment o the PCM can be ound in Imail & Jeu (999). The bounday condition employed o the luid enegy equation ae a peciied and uniom tempeatue at enty, axial ymmety and zeo tempeatue deivative at the outlet. The continuity o heat luxe i ued o both the luid and the tube all, hich i conideed adiabatic at the end cap. The bounday condition ae hon belo, θ ( R, 0) R R R R. 0 ; i i R 0 R R i R R i ξ Nu D ; H ξ R 0 ( θ θ ) i b R ; ξ H ξ 0, 0 (6) The PCM equie, beide the equality o heat luxe at the tube all and the adiabatic condition at the oute uace and at the end cap, condition concening the phae-change poce. Thee ae a contant tempeatue at the olid-liquid inteace and the heat balance at thi uace, hich account o the latent heat eleaed duing the poce, non a Stean condition. The bounday condition o the PCM ae the olloing, ( ) 0 θ R S,ξ (7a)

5 R R ; 0 ; ξ 0 H R ξ 0, 0 (7b) S θ θ l l S + (7c) ξ η η τ S η l ηl 0 Equation (7c) i the Stean equation and although it i a bounday condition it mut be olved o the dimenionle phae-change ont thicne S ( ξ,τ ). The ytem o equation hon above i itten in dicete om by the inite volume method. The enegy equation ae olved o the HTF, tube all and olid and liquid PCM, uing the coupling condition. The local Nuelt numbe value i calculated by one o Eq. (6). The Stean condition i ued to obtain the phae-change ont poition, and it time deivative ae teated explicitly, hile in the enegy equation the implicit method i ued. Detail about the PCM dicete equation and the teatment o the peudo-convective and peudo-aniotopic tem can be ound in Imail & Jeu (999). It i ound accuate enough eult uing a gid in the adial diection ith 0 contol volume in the HTF, 5 in the tube all and 5 in each phae o the PCM. In the axial diection 00 contol volume ee employed o all egion. In the impliied method thee i only one contol volume in the HTF and the local Nuelt numbe i obtained om the value o a CWT o CHF bounday condition. 3. RESULTS AND DISCUSSION The popoed poblem i olved o the olidiication o ate (PCM), aound a coppe tube ith a eigeant luid loing inide (HTF). The HTF a aumed to have a ixed denity, ρ, and peciic heat, c p, but the themal conductivity,, a vaied. The adopted value o ρ and c p ae typical o econday eigeant luid ound in ice-toage ytem, a ethanol o ethylene-glycol. The HTF themal conductivity ha thee poible value, a loe one, an intemediate, the ame o ethanol, and a highe one. The olution i peomed itly by the complete method, ate by the impliied cheme ith CHF Nuelt numbe value and then by the impliied method adopting CWT Nuelt numbe value. The Peclet numbe i held ixed in all cae. The geometic paamete choen ae the ame ued by Zhang & Faghi (996) and the computational pogam a validated by compaing ith thei eult. In thei o the PCM a paain initially at the phae-change tempeatue and the HTF a ate in lamina ully-developed lo. The tube all a made o a high themal conductivity mateial. The eult choen o compaion ae baed in a numeical tudy o the ame autho ued in (Zhang & Faghi, 996) to validate thei emi-analytical pocedue. The geometic paamete adopted ae decibed belo and Fig. ho the phae-change ont evolution pedicted by them and the complete method o thi o. H o ; 50; E. 304 D D (8) Peent eult ae at maximum 5% highe than Zhang & Faghi (996) due to the dieent method o teating the phae-change and convection pocee.

6 .00 Fo Phae-Change Font Poition - S/E Thi Wo - Complete Method Zhang and Faghi (996) Fo.0 Fo.0 Fo Axial Poition - z/d Figue Compaion ith Zhang & Faghi (996) eult. Table ho the themophyical popetie adopted o the HTF in each cae. Table Popetie o the HTF T in [ o C ] [W/m K] ρ [g/m 3 ] c p [J/g K] Pe Cae Cae Cae Figue 3 and 4 ho, epectively, the phae-change ont evolution and Nuelt numbe in the it cae conideed, a lo themal conductivity HTF. It can be noticed that the calculated Nuelt numbe value ae cloe to the CWT condition and that the impliied method can pedict accuately the phae-change ont poition uing the CWT Nuelt numbe value. It may be obeved that the olid-liquid inteace peent a teep decay and thi i a conequence o the exponential deceae o the heat tane lux that occu in convection poblem in tube unde CWT bounday condition. Figue 5 and 6 ho the eult o the econd cae, an intemediate luid themal conductivity value, equal to the eal ethanol themal conductivity. The Nuelt numbe value hon in Fig. 5 laid appoximately middle ay beteen CWT and CHF value. The impliied method poduce high eo hen tying to captue the phae-change ont poition along the olidiication poce. The olid-liquid inteace continue to have a high to dimenional poile. Figue 7 and 8 ho the ame eult o the thid cae conideed, a.5 W/mK value, geate than the olid PCM themal conductivity, W/mK. A dieent behavio o both the phae-change ont evolution and the Nuelt numbe can be obeved. The olidliquid inteace i nealy uniom along the axial diection hich i chaacteitic o a contant heat lux along the tube. The Nuelt numbe value ae no cloe to the CHF condition teady tate value and the impliied method peom bette hen uing thee Nuelt numbe value.

7 Solid Phae Thicne - S/E t3.0 t.5 PCM: ate - Tube all: coppe Pe ; 0.05 W/mK Complete Method Simpliied Method - Nu - CHF Simpliied Method - Nu - CWT Axial Poition - z/d Figue 3 Phae change ont evolution Cae. Nuelt Numbe PCM: Wate - Tube Wall: Coppe Pe ; 0.05 W/mK Nu - CWT Condition Nu - CHF Condition Nu - t.5 Nu - t.0 Nu - t.5 Nu - t Axial Poition - z/d Figue 4 Nuelt numbe evaluation Cae. The peomance o the impliied cheme compaed to the complete method o dieent HTF themal conductivity value can be analyzed in tem o the extenal themal eitance impoed to the HTF. A the phae change ont i an iothemal uace, neglecting axial conduction eect and auming a quai-teady tate condition, the extenal themal eitance and can be itten a, h ext i ln i + ln (9)

8 and a Biot numbe can be deined, Bi hexti ln i + i ln (0) 0.9 PCM: ate - Tube all: coppe Pe40000 ; 0.5 W/mK Solid Phae Thicne - S/E t.75 t.0 Complete Method Simpliied Method - Nu - CHF Simpliied Method - Nu - CWT Axial Poition - z/d Figue 5 Phae change ont evolution Cae. Nuelt Numbe PCM: Wate - Tube all: Coppe Pe ; 0.5 W/mK Nu - CWT Condition Nu - CHF Condition t.0 t.5 t.5 t Axial Poition - z/d Figue 6 Nuelt numbe evaluation Cae. Spao & Patana (977) hoed that o a themally developed lo inide a tube immeed in an extenal iothemal luid, the CWT and CHF condition ae limiting cae

9 elated to a Biot numbe, a deined in Eq. (0). When thi numbe tend to ininity the Nuelt numbe ill appoach the CWT value and a it goe to zeo the Nuelt numbe i cloe to the CHF value..0 Solid Phae Thicne - S/E t.5 t0.75 PCM: ate - Tube all: coppe Pe40000 ;.5 W/mK Complete Method Simpliied Method - Nu - CHF Simpliied Method - Nu - CWT Axial Poition - z/d Figue 7 Phae change ont evolution Cae Nuelt Numbe PCM: ate - Tube all: coppe Pe ;.5 W/mK Nu - CWT Condition Nu - CHF Condition Nu - t0.5 Nu - t0.75 Nu - t.0 Nu - t Axial Poition - z/d Figue 8 Nuelt numbe evaluation Cae 3. Applying the ame agument the Biot numbe can be calculated o each cae conideed uing Eq. (0). In the beginning o the poce, hen the phae-change ont thicne i mall, it ill achieve a high value, chaacteizing a CWT condition, but deceaing a the phae-change ont move. The velocity o decay ill depend on the all thicne and the atio beteen themal conductivitie o the HTF and the tube all mateial and the HTF and

10 the olid PCM. A imila concluion a tated by Zhang & Faghi (996) o a tubulent and lo Pandtl numbe HTF. Table ho the minimum value o the Biot numbe o each cae conideed. It coepond to a phae-change ont thicne equal to the annula pace. Table Calculated Biot numbe Cae 3 Bi Uing the eult tabulated in Spao and Patana (977) ith the value o Table, it become clea that cae i cloe to a CWT condition, cae ha an intemediate poition and cae 3 ha a nealy CHF bounday condition. The eult above ho that the impliied teatment o the HTF in the deign o latent heat themal enegy toage unit can lead to eoneou eult unle cae i taen to no hethe the ytem unde conideation i cloe to a CHF o a CWT bounday condition, o none o thee. 4. CONCLUSIONS The poblem o olidiication o PCM aound a tube caying a cold, lamina HTF inide a olved by a complete and a impliied method, hich adopted teady tate Nuelt numbe value o eithe a contant all tempeatue o a contant all heat lux condition. The eult obtained ho that the accuacy o the impliied cheme i elated to the themal eitance impoed to the HTF duing the poce, depending pecially on the atio beteen the themal conductivitie o the HTF, olid PCM and tube all mateial. REFERENCES Bellecci, C. & Conti, M., 993, Phae Change Themal Stoage: Tanient Behavio Analyi o a Sola Receive/Stoage Module Uing the Enthalpy Method, Int. Jounal o Heat and Ma Tane, vol. 36, n. 8, pp Cao, Y. & Faghi, A., 99, Peomance Chaacteitic o a Themal Enegy Stoage Module: a Tanient PCM/Foced Convection Conjugate Analyi, Int. Jounal o Heat and Ma Tane, vol. 34, n., pp Cao, Y. and Faghi, A., 99, A Study o Themal Enegy Stoage Sytem ith Conjugate Tubulent Foced Convection, Jounal o Heat Tane, vol. 4, pp Imail, K.A.R. & Jeu, A.B., 999, Modeling and Solution o the Solidiication Poblem o PCM aound a Cold Cylinde, Numeical Heat Tane Pat A: Application, vol. 36, n., pp Spao, E.M. & Patana, S.V., 977, Relationhip among Bounday Condition and Nuelt Numbe o Themally Developed Duct Flo, Jounal o Heat Tane, vol. 99, pp Zhang, Y. & Faghi, A., 995, Analyi o Themal Enegy Stoage Sytem ith Conjugate Tubulent Foced Convection,. Jounal o Themophyic and Heat Tane, vol. 9, n. 4, pp Zhang, Y. & Faghi, A., 996, Semi-Analytical Solution o Themal Enegy Stoage Sytem ith Conjugate Lamina Foced Convection, Int. Jounal o Heat and Ma Tane, vol. 39, n. 4, pp