Numerical model and experimental validation of heat storage with phase change materials

Transcription

1 Energy and Buldngs 39 (2007) Numercal model and expermental valdaton of heat storage wth phase change materals Jacques Bony *, Stéphane Ctherlet Laboratory of Solar Energetcs and Buldng Physcs (LESBAT), Appled Unversty of West-Swtzerland (HES-SO/HEIG-VD), CH-1401 Yverdon-les-Bans, Swtzerland Receved 27 June 2006; receved n revsed form 10 October 2006; accepted 20 October 2006 Abstract Ths paper descrbes the numerc model developed to smulate heat transfer n phase change materals (PCM) plunged n water tank storage. Ths model, based on the enthalpy approach, takes nto account the conducton and the convecton nto PCM as well as at the nterface between PCM and water of the storage. Furthermore, hysterss and subcoolng are also ncluded. Ths model has been mplemented n an exstng TRNSYS type of water tank storage. It allows the smulaton of a water storage tank flled wth PCM modules made of dfferent materals and dfferent shapes such as cylnders, plates or spheres bed. Comparsons between measurements and smulatons has been undertake to evaluate the potental of ths model. # 2007 Elsever B.V. All rghts reserved. Keywords: PCM; TRNSYS; Heat transfer; Latent heat; Solar energy 1. Introducton For several decades, dfferent numercal models of storage tanks usng PCM for latent energy storage have been developed; a few of these models have been elaborated to work wth the TRNSYS smulaton package [1]. Unfortunately, none of them gave enough satsfacton to allow a large dffuson, ether due to a lack of tme to test ts relablty [2],or a lack of flexblty whch does not allow the modellng of dfferent types of contaners for PCM [3]. In the framework of the IEA Task 32, whch nvestgates advanced storage solutons n thermal solar systems for buldngs, the potental of new PCMs s nvestgated to ncrease the energy densty of small szed water storage tanks. Ths approach should have the advantage of reducng solar store volume for a gven solar fracton as well as the store s heat losses. It should also ncrease the solar fracton for a gven avalable volume. To fulfll the Task s requrements, we have developed a numercal model and compared data measurements wth smulatons results. * Correspondng author. E-mal address: jacques@leg-vd.ch (J. Bony). 2. Numercal model The developed model s an extenson of the exstng TRNSYS Type 60, for stratfed flud tanks based on sensble energy storage only [4]. The tank can be made up to 100 fully mxed stacked volume segments (Fg. 1a). Ths model has been adapted to be able to take nto account the PCM calculaton. The standard Type 60 ncludes nternal heat exchangers, two drect nput output and auxlary heaters. The water tank can be consdered vertcal or horzontal. Ths model allows the smulaton of most of water storage tanks. The number of horzontal segments called also node (Fg. 1a) determnes the degree of calculaton accuracy that can be mproved wth the ncrease of the node number. The heght and the thermal losses of every layer can be defned separately. So t s possble to take nto account the losses by thermal brdge as for example a ppe juncton. Fg. 1b and Eq. (1) show the dfferent heat fluxes of the energy balance consdered n each th node (segment or layer) of the water tank [4]. For each node, the temperature s assumed to be unform. The energy balance for each storage node s gven by the followng equaton: Q ðmedumþ ¼ Q ðflowþ þ Q ðhxþ þ Q ðauxþ þ Q ðcondþ þ Q ðlossþ þ Q ðmodulesþ (1) /$ see front matter # 2007 Elsever B.V. All rghts reserved. do: /j.enbuld

2 1066 J. Bony, S. Ctherlet / Energy and Buldngs 39 (2007) Fg. 1. Energy balance for each th water node. wth Q (medum) s the energy of the storage medum of node ; Q (flow) the chargng or dschargng energy va drect (nlet/ outlet) flow ncludng the flow upward/downward n the tank; Q (hx) the heat flux through nternal heat exchanger; Q (aux) the auxlary energy; Q (cond) the thermal conducton to neghbourng nodes; Q (loss) the thermal losses through the tank envelope to the ambent; Q (modules) s the energy exchange between the storage medum and PCM modules. The energy exchange between the storage medum and the PCM modules s governed by the followng equaton [5]: Q ðmodulesþ ¼ N ðmodulesþ fu A PCM ½T T PCM ðh PCM ÞŠg (2) wth N (modules) s the number of PCM contaners; U the heat transfer coeffcent water/pcm; A the surface between water and PCM contaner; T the storage medum temperatures (node ); T PCM s the surface temperatures of the PCM contaner. The calculaton of heat transfer through the PCM uses the enthalpy method, whch means that for a gven volume and a materal, a contnuous and reversble functon can be calculated whch wll return the temperature T dependng on the calculated enthalpy h. Ths temperature s used durng the smulaton to determne the node temperature, accordng to the enthalpy of the system at tme t. Fg. 2 shows ths functon consttuted of a successon of fve straght lnes: two for the sensble heat n sold or lqud phase and three straght lnes n the phase change part. Thus, the accuracy s enough for the calculatons Numercal approach The numercal resoluton of the set of equatons can be done by an explct or mplct method. (A) The explct method s smple to program but s condtonally steady. It needs to have a tme step smaller than a lmt value n order to avod any dvergence. On the other hand t ncreases smulaton tme. (B) The mplct method s more complex to program but t s uncondtonally steady. There s no lmt for the tme step except f we would lke good calculaton accuracy. We have chosen the explct method, so t s necessary to pay attenton to the tme step n order to avod a calculaton dvergence. The crtera of convergence are calculated wth the followng equatons [6]: for a surface node : Foð2 þ BÞ 1 2 for a node nsde materal : Fo 1 4 Fo ¼ lt r Cp x 2 and B ¼ ax (5) l wth Fo s the Fourer number; B the Bot number; l the PCM thermal conductvty (W/(m K)); t the tme step smulaton (s); r the PCM densty (kg/m 3 ); Cp the PCM specfc heat (J/ (kg K)); x the dstance between two nodes (m); a s the convectve coeffcent between water and PCM (W/(m 2 K)). From Eqs. (6) and (7), we get the maxmum tme step possble for calculaton. It takes nto account the heat transfer coeffcents (convectve and conductve) as well as the thermal capacty of every node and the poston of the node consdered [6]. (3) (4) Fg. 2. Example of an enthalpy curve for a partcular volume and PCM type. for nterface node water=pcm; t for a node nsde materal; t r Cp x2 4l r Cp x 2 2lð2 þðax=lþþ (6) (7)

3 J. Bony, S. Ctherlet / Energy and Buldngs 39 (2007) Fg. 3. Representaton of dfferent shapes avalable PCM meshng The nternal calculaton model n the PCM s b-dmensonal, whch allows the smulaton of dfferent PCM shapes: cylnder, sphere or plate. An onon peel approach has been used. It conssts of representng a PCM element by a constant thckness layer successon whose shape depends on the object, as shown n Fg. 3. For each node, we calculate the energy balance whle supposng a unform temperature n the volume of the correspondng node (Fg. 4), whch gves an enthalpy varaton gven by the Eq. (8): Dh t1 ;k ¼ Q t1 ;k 1! ;k Dt þ Q t1 ;kþ1! ;k þ Q t1 1;k! ;k þ Q t1 þ1;k! ;k (8) where the heat transfer between two nodes s: Q t1 ;k 1! ;k ¼ l;k x ;k þ l ;k 1 x ;k 1 A ;k 1! ;k ðt;k 1 t0 Tt0 ;k Þ (9) wth s the vertcal axe (depend of number of water nodes); k the horzontal axe (PCM meshng); l the thermal conductvty; x the dstance between two nodes; A the exchange surface between two nodes; t 0 the ntal tme; t 1 s the fnal tme. The enthalpy at t 1 tme s 2.3. Hysteress H t1 ;k ¼ Ht0 ;k þ Dht1 ;k (10) The hysteress phenomenon appears durng coolng of materals. It results n a delay of the phase change. It does not Fg. 4. PCM mesh. depend on the sold phase presence n the surroundngs, and therefore ths process can be calculated ndependently for every PCM node. In a frst tme, the model dd not take nto account ths hysteress phenomenon and the heatng or the coolng follow the same curve. Followng measurements made for valdaton purpose, the modelsaton of ths physcal behavour was necessary. Fg. 5 llustrates ths new functon. There s a smple shft of the enthalpy curve accordng to a dfferental temperature defned wth one parameter. Durng a heatng or coolng step nsde the phase change zone, the slope of transton s the same as the sold phase one n the bottom part of the phase change. It s also dentcal to the slope of the lqud phase n the superor part of the phase change. It avods dscontnuty of the enthalpy curve when the transton pont s close to the complete phase change (lqud or sold) (Fg. 5) Subcoolng Fg. 5. Hysteress model. Contrary to the prevous phenomenon, the subcoolng depends on sold phase presence. The determnaton of ths process appearance takes nto account the global state of the PCM module. It s necessary that the whole PCM s n lqud phase to obtan subcoolng. Durng the coolng mode, as soon as a PCM part reaches the pont of crystallzaton, the whole PCM wll go nto sold phase (full lne of Fg. 6). Ths phenomenon can start at a lower temperature than the soldfcaton temperature. Thus, for cylnders or plates, the subcoolng wll be able to appear only one tme durng the coolng process. Indeed, when crystallzaton starts around a condensaton core, the soldfcaton step grows n the entre element at a speed that s supposed nstantaneous n the numercal model (Fg. 7). In fact, t s not the case. But t s dffcult to know the crystallzaton propagaton speed n a PCM element. The followng fgure shows dfferences between the model and realty. The measurements are made on sodum acetate wth graphte durng the coolng process. In the sphercal module case, ths phenomenon wll appear at every layer of spheres. Indeed durng the thermal dscharge, the

4 1068 J. Bony, S. Ctherlet / Energy and Buldngs 39 (2007) Fg. 6. Subcoolng model. Fg. 7. Comparson between smulaton and measurement about crystallzaton propagaton speed. lower part of the tank storage cools down n frst. Thus, the frst lower layer of spheres wll be n subcoolng process whereas the other ones should wat as the tank storage contnues to cool down (Fg. 8). The phenomena of hysteress and subcoolng can be used together as Fg. 9 shows. In the model, each of these phenomena s treated ndependently by a specfc state ndcator Heat transfer Thermal conducton nsde PCM n sold or lqud phase In order to take nto account the thermal conducton dfference between the sold and lqud states of a materal, the model allows two dstnct values for the conducton coeffcent; one for the sold and one for the lqud phase. At the tme of the phase change, the thermal conductvty value s calculated by lnear nterpolaton of the enthalpy (Fg. 10). Below the enthalpy value H1, the thermal conductvty l s constant and equal to l sol. Above the enthalpy value H2, the thermal conductvty s constant and s equal to l lq. Between H1 and H2, the conductvty s gven by lnear nterpolaton: l sol=lq ¼ l sol þ l lq l sol H2 H1 ðht H1Þ (11) Fg. 8. Crystallzaton propagaton process accordng to the contaner type. where, H t s the enthalpy value at tme step t and gven by Eq. (10).

5 J. Bony, S. Ctherlet / Energy and Buldngs 39 (2007) convecton are gven by [6]: Nu mxed ¼ðNu 3 free þ Nu3 forced Þ1=3 (12) a ¼ Nu mxedl ; ðw=m 2 KÞ (13) x 2.6. Expermental data Fg. 9. Combnaton subcoolng and hyteress. Fg. 10. Calculaton of the thermal conductvty l n accordance to the enthalpy H Water/PCM convecton The convectve coeffcent between the water of the tank storage and the PCM contaner s calculated for every node and each tme step, accordng to the contaner shape chosen: Plate and cylnder! vertcal plate convecton [6]. Sphere! convecton around a sphere n free convecton and n a sphere bed n forced convecton [8]. Table 1 gves the dfferent equatons of the convectve coeffcent used accordng to the shape of the PCM module as well as the type of flud flow around these modules [6,8]. Mxed Nusselt number and convectve coeffcent calculaton wth free and forced (water flow nto tank storage) The numercal model takes nto account the hysteress and subcoolng phenomena that can be observed wth some phase change materals. Although the developed model takes these two aspects nto account, n the case here of paraffn, these two phenomena are neglgble. So we focus on model valdaton of heat transfer by convecton and conducton. To do ths, some temperature measurements have been performed. The temperature evoluton nsde the PCM durng the chargng and dschargng cycle has been montored wth thermocouples placed nsde a PCM module (paraffn). The tme evoluton of these temperatures has been compared wth the smulaton. For these measurements, we have used an alumnum contaner whose dameter s 88 (mm), the heght 150 (mm) and the thckness 0.3 (mm). To keep a constant dstance between the sensors, a grd and a cross n plastc have been used as shown n Fg. 11a. Then the lqud paraffn s poured n the contaner as shown n Fg. 11b. The bottom part of the contaner remans open to allow sensor cables to be connected wth the acquston equpment. After the PCM has soldfed, ths contaner s plunged n the water tank wth the sensor cables gone downwards (Fg. 12). Ths method s possble because the lqud paraffn remans n the top of the contaner for the followng reasons: The densty of sold and lqud paraffn s lower than that of water. The paraffn s non mscble wth water. No hgh velocty of water flow nsde the tank Measurements versus smulatons A frst step has conssted to compare the data measurements and the smulaton results done wth a model whch dd not take nto account the convecton nsde PCM module. In ths case, the temperature measurements done n the PCM module are Table 1 Equatons for water/pcm convecton Convecton Vertcal plate of cylnder Sphere bed Lamnar free Turbulent free 2 1=4; 0:387 Ra Nu ¼ 0:825 þ 1=6 Pr Nu ¼ 2 þ 0:56 ðra < Þ ½1þð0:492=PrÞ 9=16 Š 8=27 0:846þPr Ra Lamnar forced Nu x ¼ 0:332 Rex 1=2 Pr 1=3 ; ðre < 5:10 5 Þ Nu lamnar ¼ 0:664 Re 1=2 e Pr 1=3 ; Nu turbulent ¼ Turbulent forced Nu x ¼ 0:0296 Rex 4=5 Pr 1=3 ; ð5:10 5 < Re < 10 7 Þ 0:037ðRe=eÞ 0:8 Pr ; 1þ2:443ðRe=eÞ 0:1 ðpr 2=3 1Þ Nu global ¼ 2 þðnu 2 lumnar þ Nu2 turbulent Þ1=2 ; Nu ¼ð1 þ 1:5ð1 eþþnu global ; ðe ¼ vod fracton n sphere bedþ

6 1070 J. Bony, S. Ctherlet / Energy and Buldngs 39 (2007) Fg. 11. Temperature measurement devce nto paraffn. (a) Support for thermocouples. (b) Alumnum contaner. very dfferent to the results obtaned by smulaton, as the comparson between Fgs. 13 and 14 shows. We can notce that the phase change s complete after 8 h n ths smulaton nstead of about 3.5 h for the measurements Convecton heat transfer In order to mprove the modellng of the heat transfer nto the PCM module, we ntroduced an effectve thermal conductvty whch supports the convecton n lqud phase of the PCM. It s gven by: l effectve ¼ l Nu (14) where Nu s the Nusselt number for nternal convecton. Two dfferent equatons descrbng the convecton nsde cavty have been compared manually [6,7]. As the results are smlar, we have mplemented the easest equatons n the model (Eqs. (15) and (16)). These two equatons do not use heght noton for the convectve cell, whch smplfes ts mplementaton n the code. Indeed, they requre only the thckness of the PCM s lqud layer to determne the Nusselt number at each node. Besdes, durng a thermal cycle, t s possble to have several lqud layers separated by a sold PCM layer. The calculaton of Nusselt number s gven by: To a rectangular cavty wth: 10 6 < Ra L < 10 9 [6] Nu L ¼ 0:046 Ra 1=3 L (15) Fg. 12. PCM contaner plunged n a water tank.

7 J. Bony, S. Ctherlet / Energy and Buldngs 39 (2007) Fg. 13. Laboratory measurements. Fg. 16. Smulaton takng nto account nternal convecton nsde PCM (80 nodes). To a sphercal cavty wth: 10 2 < Ra < 10 9 [7] Nu ¼ 0:228 Ra 0:226 (16) Fg. 14. Smulaton wthout convecton nsde PCM. Fg. 15 shows results whle takng nto account the nternal convecton nsde PCM. The small oscllatons on the curves are generated by the non-contnuty of the smulaton model (meshng). It should be notced that these oscllatons have nothng to do wth numercal nstablty. Between two spatal nodes, the change from sold to lqud s nstantaneous for each layer dependng on the temperature node. So, the effectve conducton coeffcent gets suddenly a strong varaton between two tme steps. Whle ncreasng the node number for the PCM module calculatons, t s possble to reduce the temperature oscllatons as Fg. 16 shows due to a reducton of spatal meshng. On the other hand, smulaton tme ncreases also strongly wth the ncrease of the node number. In the example of Fg. 16, ths smulaton tme s multpled by a factor 30 for an ncrease of the calculaton nodes of a factor 4 (20 80 nodes) as obtaned n Fg. 15. In future studes, requrng yearly smulatons, t wll be necessary to lmt the node number n order to reduce calculaton tme. 3. Concluson Fg. 15. Smulaton takng nto account nternal convecton nsde PCM (20 nodes). The smulaton of the heat transfer between water and PCM module s often dffcult to solve. Indeed, the nternal convecton nsde PCM module s often dsregarded to smplfy the model. Ths approach s only foreseeable for PCM havng a very bg vscosty. For the other PCM, such as paraffn, t s necessary to take nto account the nternal convecton. The model presented here, uses the effectve conducton coeffcent approach. The comparson between montored data and smulaton results has shown a good agreement. Ths method has an nterestng potental and seem promsng. However, t remans to confrm ts qualtes by performng other measurements and by reducng the

8 1072 J. Bony, S. Ctherlet / Energy and Buldngs 39 (2007) temperature oscllatons observed when the phase change occurs, wthout ncreasng sgnfcantly the smulaton tme. Acknowledgements We would lke to thank our natonal government (Federal Offce of Energy (OFEN/BFE)). We also would lke to deeply thank Jean-Chrstophe Hadorn, representatve of the Internatonal Energy Agency (IEA) for havng ntated the Task 32. References [1] J. Joksalo, et al., Thermal Smulaton of PCM Structures wth TRNSYS Terrastock 2000, n: Proceedngs of the Eghth Internatonal Conference on Thermal Energy Storage, Stuttgart, Germany, [2] H. Vsser, Energy storage n phase-change materals development of a component model compatble wth the TRNSYS transent smulaton program, Delft Unversty of Technology (1986). [3] P. Egolf, Project Latentwärmespecher für de Sonnenenergenutzung: Lade-und Entladevorgänge, EMPA [4] S.A. Klen, TRNSYS reference manual [5] J. Bony et al., Three dfferent approaches to smulate PCM bulk elements n solar storage tank, PCM2005, Yverdon-les-Bans, June [6] F.P. Incropera, D.P. De Wtt, Fundamentals of Heat and Mass Transfer (1990). [7] Y.A. Cengel, Heat Transfer: A Practcal Approach, Internatonal Edton [8] E. Achenbach, Heat and flow characterstcs of packed beds, Expermental Thermal and Flud Scence 10 (1995)