Solar Energy Vol. 70, No. 1, pp. 51–61, 2001
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AN ANALYSIS OF ISOTHERMAL PHASE CHANGE OF PHASE CHANGE MATERIAL WITHIN RECTANGULAR AND CYLINDRICAL CONTAINERS B. ZIVKOVIC and I. FUJII
†, 1
Meiji University, Department of Mechanical Engineering, 1-1-1 Higashi-Mita, Tama-ku, 214-8571 Kawasaki, Japan Received 10 January 2000; revised version accepted 7 June 2000 Communicated by ERICH HAHNE
Abstract—In this paper, a simple computational model for isothermal phase change of phase change material (PCM) encapsulated in a single container is presented. The mathematical model was based on an enthalpy formulation with equations cast in such a form that the only unknown variable is the PCM’s temperature. The theoretical model was verified with a test problem and an experiment performed in order to assess the validity of the assump assumpti tions ons of the mathe mathemat matica icall model. model. With With very very good good agreem agreement ent betwee between n experi experimen mental tal and computational data, it can be concluded that conduction within the PCM in the direction of heat transfer fluid flow, thermal resistance of the container’s wall, and the effects of natural convection within the melt can be ignored for the conditions investigated in this study. The numerical analysis of the melting time for rectangular and cylindrical containers was then performed using the computational model presented in this paper. Results show that the rectangular rectangular container container requires nearly half of the melting time as for the cylindric cylindrical al container container of the same volume and heat transfer area. © 2001 Published by Elsevier Science Ltd.
Ghoneim, 1989). Using the shell-and-tube model for the LHES LHES unit, unit, Morris Morrison on and AbdelAbdel-Kha Khalik lik Effective and economic thermal energy storage of (1978) (1978) perfor performed med a long-t long-term erm analy analysis sis of airaira daily daily surplu surpluss of irrad irradiat iated ed solar solar energy energy is an based based (air as HTF) and liquid liquid-b -base ased d (water water as unavoidable necessity for the efficient use of solar HTF) solar solar heatin heating g system systems, s, assumi assuming ng that that the energy energy for heatin heating g purpo purposes ses (Duffie (Duffie and and BeckBeckthermal conductivity of the PCM in the direction man, man, 1991 1991). ). Amon Among g the the vari variou ouss meth method odss of of the HTF flow, as well as the thermal resistance energy storage, latent heat thermal energy storage of the PCM in the direction normal to the HTF is particularly attractive. The motivation for using flow, flow, can can both both be igno ignore red. d. In a late laterr anal analys ysis is,, phas phasee chan change ge mate materi rial alss (PCM) (PCM) is thei theirr high high Ghon Ghonei eim m (198 (1989) 9) took took into into acco accoun untt both both the the energy storage density and their ability to provide conduction within the PCM in the direction of the heat at a constant temperature (Abhat, 1983). HTF flow and the direction normal to the HTF In order order to perfor perform m a long-t long-term erm perfo performa rmance nce flow flow and and show showed ed that that a subs substa tant ntia iall erro errorr in analys analysis is of a specifi specified ed solar solar heatin heating g system system,, an esti estima mati ting ng the the sola solarr frac fracti tion on (amou amount nt of tota totall adequate model of the heat storage unit is needed heating load supplied by solar energy) was intro(Klein et al., 1976) which, naturally, depends on duced duced by neglec neglectin ting g the condu conducti ction on within within the its design. A survey of the previously published PCM. PCM. Howeve However, r, the common common conclu conclusio sion n from from papers dealing with latent heat storage reveals that both works is that the reduction in storage volume the most intensively analyzed latent heat energy by using PCM is not as nearly as pronounced for storage (LHES) unit is the shell-and-tube LHES liquid-based systems as it is for air-based systems. unit with the PCM filling the shell and the heat It is, is, ther theref efor ore, e, pref prefer erab able le to use use a LHES LHES unit unit transf transfer er fluid fluid (HTF) flowing flowing throug through h the tubes tubes coupled with an air-based system. For that reason, (Lacroix, 1993; Bansal and Buddhi, 1992; Esen et air is consid considere ered d as the HTF in the subsequ subsequent ent al., 1998 1998;; Soma Soma and and Dutt Dutta, a, 1993 1993;; Isma Ismail il and and analysis. Alves, 1986) or, vice versa, PCM filling the tube The problem of the phase change of PCMs falls and HTF flowing parallel to it (Esen et al., 1998; into the category of moving boundary problems. When Wh en the the PCM PCM chan change gess stat state, e, both both liqu liquid id and and † solid phases are present and they are separated by Author to whom correspondence should be addressed. Tel./ the moving moving interface interface between them. There There have fax: 1 81-44-943-7395; e-mail:
[email protected] 1 ISES member. been been many many diff differ eren entt nume numeri rica call meth method odss dede1. INTRODUCTI INTRODUCTION ON
51
52
B. Zivkovic and I. Fujii
veloped veloped for dealing dealing with with the phase change change probproblem (Bansal (Bansal and and Bud Buddhi dhi,, 1992; 1992; Soma Soma and and Dutt Dutta, a, 1993 1993;; Murr Murray ay and Landi Landis, s, 1959) 1959) of which which the most most attrac attractiv tivee and commo commonly nly used used are the socalled called enthal enthalpy py meth methods ods (Voll (Voller er and and Cros Cross, s, 1981 1981;; Voller, oller, 1985; Voller, oller, 1990). The major major reason for this this is that that the metho method d does not require require explic explicit it treatm treatment ent of the condit condition ionss on the phase phase change change bound boundary ary (cf. Carsl Carslaw aw and Jaeger Jaeger,, 1959), 1959), i.e. i.e. there there is no need for tracki tracking ng the the phase phase change change boundary boundary throu througho ghout ut the phase chang changee domain domain.. Howev However, er, besi beside de the the fact fact that that impl implic icit it finit finitee diff differ eren ence ce discre discretiz tizati ation on result resultss in a set of nonlin nonlinear ear equaequation tions, s, the method method has sever several al other other drawb drawbac acks ks.. These These are the quite quite cumber cumbersom somee calcul calculati ation on of liqu liquid id frac fracti tion on upda update tess and and the the fact fact that that the the temperatu temperature re field field within within the PCM PCM is is not calculated calculated expl explic icit itly ly but but via via enth enthal alpy py-t -tem empe pera ratu ture re corr correl elaation. tion. In this this paper, paper, a slightl slightly y modifi modified ed enthal enthalpy py method, method, which which enable enabless decoupli decoupling ng of temperatur temperaturee and and liq liqui uid d fra fract ctio ion n fiel fields ds,, is is pre prese sent nted ed.. Der Deriv ivat atio ion n of discretizati discretization on equations equations is straightf straightforwar orward d and the the meth method od itse itself lf is very very easy easy to impl implem emen ent. t. 2. MATHEMATIC MATHEMATICAL AL MODEL
The The subj subjec ectt of the pres presen entt inve invest stig igat atio ion n is a single single contain container er filled filled with with PCM PCM (Fig. 1). Packing Packing the the PCM PCM in sing single le cont contai aine ners rs enab enable less modu modula larr constr construct uction ion of the LHES LHES unit unit and is also also very very economic economic from the viewpo viewpoint int of of mass productio production. n. Moreover Moreover,, comple complete te meltin melting g of of the the PCM, PCM, which which is absolutel absolutely y necessary necessary for long-term long-term (seasonal) (seasonal) heat storag storagee (Zivkovi (Zivkovicc and Fujii, Fujii, 1999), 1999), is diffic difficult ult to obta obtain in using using a shel shelll-an andd-tu tube be LHES LHES unit unit where where larg largee mass masses es of the the PCM PCM are are invo involv lved ed.. There There are severa severall appro approach aches es to the the mathem mathemati ati-cal cal mode modeli ling ng of LHES LHES unit units. s. In some some mode models ls,, conduc conductio tion n within within the PCM in both the directi direction on of the HTF flow and the direction normal to the HTF flow are taken into account (Lacroix, 1993; Ghon Ghonei eim, m, 1989) 1989),, whil whilee in othe others rs the the effe effect ct of natural convection within the molten PCM is also
accounted accounted for (Voller (Voller et al., 1987; Lacroix, 1993; Soma Soma and Dutta, Dutta, 1993). 1993). While While in most most works, works, conv convec ecti tive ve heat heat tran transf sfer er from from the the HTF HTF to the the PCM is is calcul calculate ated d throug through h the mean mean valu valuee of the the heat heat transf transfer er coeffic coefficien ientt (a conv ), some some auth author orss solved solved the problem problem of phase change change coupled coupled with transi transient ent convec convectiv tivee heat heat transf transfer er betwee between n HTF and PCM, PCM, i.e. i.e. with with the comple complete te Oberbec Oberbeck’s k’s set of equati equations ons solved solved for the HTF (Bellec (Bellecii and Conti, Conti, 1993; Trp et al., 1999). In the presen presentt analys analysis, is, the mathem mathemati atical cal model model for for phas phasee chan change ge of the the enca encaps psul ulat ated ed PCM PCM is derive derived d under under the followin following g assumpti assumptions ons:: (A) Therm Thermal al cond conduc ucti tivi vity ty of the PCM PCM in the direct direction ion of the the HTF flow flow is is ignore ignored. d. (B) The The effe effect ctss of natu natura rall conv convec ecti tion on with within in the the melt are negligibl negligiblee and can be ignored. ignored. (C) The PCM PCM behav behaves es idea ideall lly, y, i.e. i.e. such such phephenomena nomena as as proper property ty degrad degradati ation on and and superc supercool ooling ing are not accounted accounted for. for. (D) The The PCM PCM is assu assume med d to have have a defin definit itee melting melting point point (isothermal (isothermal phase change change). ). (E) Ther Thermo moph phys ysic ical al prop proper erti ties es of the the PCM PCM are are different for the solid and liquid phases but are independent of temperature. (F) The PCM is homogeneous and isotropic. (G) Ther Therma mall resi resist stan ance ce acr acros osss the the wall wall of of the the container container is neglected. neglected. (H) Late Latera rall sides sides of the the rect rectan angu gula larr conta contain iner er are well well insul insulate ated, d, i.e. i.e. heat heat tran transfe sferr occur occurss only only on on sides x 5 0 and x 5 d (cf. Fig. 1). In order order to valida validate te the the first first assumpt assumption ion (A), (A), the the influence influence of the conduct conduction ion within within the PCM in the direct direction ion of HTF flow was numeri numerical cally ly invest investiigate gated d in the the foll follow owin ing g way. way. Reta Retain inin ing g all all the the abov abovee assu assump mpti tion onss and and negl neglec ecti ting ng the the cond conduc ucti tion on within within the PCM in the direct direction ion norma normall to the HTF HTF flow, flow, the the gove govern rnin ing g equa equati tion onss for for heat heat transf transfer er and phase phase change change can be written written as: H k T ] ] ] ] S t z r z D ≠
≠
5
≠
≠
≠
≠
]T z]
m fc p f
Fig. 1. Geometry Geometry of the rectangular rectangular container. container.
≠
≠
ai r
5 a conv A ht sT 2 T ai rd
(1) (2)
where all the variables are defined in the nomenclature table. Detailed description of the numerical cal solu soluti tion on of the the abov abovee set set of equa equati tion onss is omitted as it is described later in the paper. Here, only the results obtained are discussed briefly. Even though conduction within the PCM in the direction of HTF flow plays an important role for long LHES units (containers) and relatively high convective heat transfer coefficients (Fig. 2), its influence can be ignored (Fig. 3) for low convec-
An analysis analysis of isother isothermal mal phase phase change change of phase phase change change materia materiall within within rectangul rectangular ar and cylindr cylindrical ical contai containers ners
Fig. 2. Temperat Temperature ure variation variation within the PCM in the directio direction n of the HTF flow for l 5 3 m and a con
tive tive heat heat transf transfer er coeffic coefficien ients ts (such (such are those those when when air air is used used as the the HTF) HTF) and and rela relati tive vely ly shor shortt contai container nerss (from (from 200 200 to 400 mm mm). ). Furthe Furthermo rmore, re, compar compariso ison n of the soluti solution on obtai obtained ned using using the mathematical model defined with the set of Eqs. (1) and (2) and the solution obtained solving the same same prob proble lem m usin using g the the lump lumped ed mass mass meth method od show showss that that no sign signifi ifica cant nt impr improv ovem emen entt of the the result resultss is obtain obtained ed if the conduc conductio tion n within within the PCM is accounted for (Fig. 4). These results are in acco accord rdan ance ce with with the the resu result ltss obta obtain ined ed by Ghoneim (1989), who concluded that there is no signifi significan cantt change change in the predic predictio tion n of the solar solar fraction with reduction of D z to less less than than 200 200 mm mm..
v
5 300
53
2
W/m K.
With With the forego foregoing ing assump assumptio tions, ns, the enthal enthalpy py form formul ulat atio ion n for for the the cond conduc ucti tion on-c -con ontr trol olle led d phas phasee chang changee can be written written as (Voll (Voller, er, 1990; 1990; Voller oller et al., 1987): ≠
H ] t ≠
S]r k grad T D
5 div
(3)
An alternative form of Eq. (3) can be obtained by splitting splitting the total enthalpy H into into sensib sensible le and latent heat components: (4)
H 5 h 1 L ? f l
where where
Fig. 3. Temperat Temperature ure variation variation within the PCM in the directio direction n of the HTF flow for l 5 0.314 m and a con
v
5 15.5
2
W/m K.
54
B. Zivkovic and I. Fujii
Fig. 4. Comparison Comparison between between the lumped lumped mass method method and the case when the axial axial conduction conduction is accounted accounted for. T
h5
E c dT
(5)
≠ f k ≠T ≠h ≠ ] 5 ] S] ]D 2 L ] r ≠ x ≠t ≠ x ≠t l
(8)
T m
and T m is the melting temperature of the PCM. For the problem of isothermal phase change, the local liquid fraction f l is defined as: 1 0
H
f l (T ) 5
The fully implicit implicit discretiza discretization tion equation for an internal node ‘ i ’ can be written as (Fig. 5): ≠
k ] sT ]ht ] r x ≠
if T . T m if T , T m
(6)
Substituting Eq. (4) into Eq. (3) gives: ≠ f ≠h k ] 5 divS] grad T D 2 L ] r ≠t ≠t l
(7)
Eq. (7) repr represe esents nts,, tog togeth ether er with with Eqs. Eqs. (5) and (6) and and the the appr approp opri riat atee init initia iall and and boun bounda dary ry concondition ditions, s, the mathem mathemati atical cal model model of conduc conductio tion n contr ontro olle lled iso isother therma mall phase hase chan chang ge. 3. NUMERICAL NUMERICAL SOLUTION SOLUTION
For the problem of one-dimensio one-dimensional nal isothermal isothermal phase phase change change of the PCM encapsu encapsulat lated ed within within a rect rectan angu gula larr cont contai aine nerr (Fig. (Fig. 1), the the gove govern rnin ing g equa equati tion on for for the the PCM PCM fol follo lows ws fro from m Eq. Eq. (7):
i
5
D
2
i 2 1 2 2T i 1 T i 11d 2 L
≠ f l
i
] ≠t
(9)
wher wheree the the sens sensib ible le enth enthal alpy py term term and and liqu liquid id fraction term are deliberately left in the differential tial form form for for the the conv conven enie ienc ncee of subs subseq eque uent nt numerical computation. It should be noted that the discretiza discretization tion equations equations for boundary boundary nodes nodes depend pend on specifi specificc bounda boundary ry condit condition ionss and are deri derive ved d from from an ener energy gy bala balanc ncee on boun bounda dary ry contro controll volum volumes. es. The key featu eaturre of the the prop roposed osed meth ethod is the the fact that for an isothermal phase change (which is the case for most salt hydrates), the temperature of the the PCM PCM with within in a giv given con contro trol volu volum me remains remains constant constant and and equal equal to its its melting melting temperatemperature ture until until the PCM has melted melted comple completel tely. y. Conside siderr first the the case when when cont contro roll volume volume ‘‘i ‘‘i‘‘ is full fully y soli solid d or ful fully ly liq liqui uid. d. In In that that cas case, e, fro from m the the
Fig. 5. Discreti Discretizati zation on domain domain for a one-dimen one-dimensiona sionall phase change problem. problem.
An analysis analysis of isother isothermal mal phase phase change change of phase phase change change materia materiall within within rectangul rectangular ar and cylindr cylindrical ical contai containers ners
definition of sensible enthalpy, Eq. (5), and the liquid fraction, Eq. (6), it follows that: ≠
≠
]ht ; c ]T t i
≠
i
(10)
≠
and ≠ f l
i
] ; 0 ≠t
( 11)
where c is the specific heat of the solid or liquid phase, depending on state of the control volume. Afte Afterr intr introd oduc ucin ing g Eqs. Eqs. (10) and and (11), (11), Eq. Eq. (9) reduces to ordinary heat diffusion equation: ≠
k ]T t ] ] sT r c x i
≠
5
D
2
i 21
2 2T i 1 T i 1 1d
(12)
Backward differencing of the left side term gives, after rearranging rearranging,, the fully implicit finite differdifference equation of the form: a i 21 ? T i 21 1 a i ? T i 1 ai 11 ? T i 11 5 b i
(13)
where coefficients a i 1 5 a i 1 5 2 Fo, a i 5 1 1 ol d 2 3 Fo and b i 5 T i are introduced for the sake of compu computat tation ional al simpli simplicit city. y. Supers Superscri cript pt ’old’ ’old’ refers to the previous time step and Fo is a finite difference form Fourier number: 2
Fo 5
1
?D
a t ] ] x D
(14)
2
where a 5 k / / r r c is the thermal diffusivity of the PCM. The thermophysical properties in Eq. (14) depend on the state of the control volume. Those of the solid phase should be inserted if the control volume is in the solid state and those of the liquid phase if the control volume is in the liquid state. Now, Now, cons consid ider er the the case case when when melt meltin ing g (or freezing) occurs around a certain node ‘ i ’. In that case, case, the liquid liquid fracti fraction on f li lies lies stri strict ctly ly in the the interval [0,1]. Recognizing that for an isothermal phase change: (15)
T i ; T m
and from Eq. (5):
D
k t ] ] sT r L x D
2
i 21
2 2T m 1 T i 1 1d
(18)
whic which h is the the equa equati tion on for for upda updati ting ng the the liqu liquid id fracti fraction on field field within within the contro controll volum volumee that that is undergoing a phase change. As Eq. (15) shows, liquid fractions are updated from the temperature field field and and not not from from the the sens sensib ible le enth enthal alpy py field field (Voller (Voller,, 1990; 1990; Lacroix, Lacroix, 1993). The temperatu temperature re and the liquid liquid fracti fraction on field field are decou decouple pled, d, the temperature field within the PCM being calculated inde indepe pend nden entl tly y from from Eq. Eq. (13) forc forcin ing g ai 1 5 a i 1 5 0, a i 5 1 and b i 5 T m for for the the cont contro roll volumes which are undergoing the phase change. Furthermore, on the basis of Eq. (18), it can be inferred that for an isothermal phase change, all the heat supplied to the control volume undergoing ing a phas phasee chan change ge is used used for for chan changi ging ng the the amou amount nt of late latent nt heat heat cont conten entt of that that cont contro roll volume. At this point it is worthwhile to describe the impl implem emen enta tati tion on of the the comp comput utat atio iona nall mode model, l, which is as follows: (a) Coefficients ‘a ’ of Eq. (13) are formed. For nodal points where the liquid fraction f li is strictly in the interval [0,1], 0 , f li , 1, the coefficients of Eq. (13) are set to: a i 1 5 a i 1 5 0, a i 5 1 and b i 5 T m . (b) (b) The The set set of line linear ar alge algebr brai aicc Eqs. Eqs. (13) is solved using the Gauss-Seidel iterative procedure. (c) Liquid fractions are updated from the temperature field using Eq. (18). (d) A check for the ‘start’ and for the ‘end’ of phase change is performed. Explicitly, if the state of the the liqu liquid id frac fracti tion on field field chan change gess with within in the the given time step, i.e. a finite volume commences or terminates with the phase change, the coefficients of Eq. Eq. (13) (13) need need to be upda update ted d and and step stepss (a) through (d) repeated for the same time step. In prac practi tice ce,, for for most most time time step stepss only only one one iteration is needed per time step. The only time when two iterations are needed is when the phase change boundary moves from one control volume to the next one. 2
1
2
1
Checking for start / end of the phase change 3.1. Checking / end
≠h i
]t ; 0
( 16)
≠
Eq. (9) becomes: ≠ f l
ol d
f l 5 f l 1 i i
55
k L ] 5 ]] sT ≠t r D x i
2
i 21
2 2T m 1 T i 11d
(17)
Backward differencing of the liquid fraction term gives:
At the end of each time step, the check for start and/or end of the phase transition is performed thro throug ugho hout ut the the enti entire re doma domain in.. For For the the case case of melting, checking for the ‘start’ and ‘end’ of the phas phasee chan change ge is perf perfor orme med d in the the foll follow owin ing g fashion: 3.1.1. START of melting. For a given time step, ol d
if T i $ T m while T i
, T m , it indicates that within
56
B. Zivkovic and I. Fujii
this time step, the finite volume in question begins with melting. In that case, the coefficients of Eq. (13) are are upda update ted d as desc descri ribe bed d abov abovee and and the the calcul calculati ation on for that that step step is perfor performed med again. again. It should be noted, however, that in the time step when when a cont contro roll volu volume me has has just just begu begun n with with melting, Eq. (18) has the form: ol d
f l 5 f l i
i
2
c
1
D
k t ] sT ] r L x D
2
i 21
2 2T m 1 T i 1 1d
] sT 2 T i d L m ol d
(19)
The last term on the right-hand side of Eq. (19) repr repres esen ents ts the the amou amount nt of sens sensib ible le heat heat that that is need needed ed to rais raisee the the temp temper erat atur uree of the the cont contro roll volume from the temperature in the previous time ol d step (T i ) to the the melti eltin ng tem temperat eratu ure (T m ). Consequently, that amount of heat can not be used for melting the PCM.
3.1.2. END of melting. For a given time step, if ol d
f l i $ 1 while f l i , 1, it indicates that within this
time time step step,, the the cont contro roll volu volume me in ques questi tion on has has melted completely. In that case coefficients of Eq. (13) (13) are are agai again n set set to a i 1 5 a i 1 5 2 Fo, a i 5 ol d 1 1 2 3 Fo and b i 5 T i and the calculation for that time step is performed again. In the time step in which the phase change boundary moves from the control volume in question to the next one, coefficient b i has the following form: 2
L ol d b i 5 T m 2 ] s1 2 f l d i c
1
(20)
where the last term on the right hand side can be described as the amount of heat needed to completely melt the control volume in question within the time step and which consequently can not be used to raise the temperature of the PCM. The flow chart for the computational procedure is given in Fig. 6. 4. VERIFICATI VERIFICATION ON OF THE MATHEMATICAL MATHEMATICAL MODEL
4.1. Test problem
The The perfor performan mance ce of the presen presented ted method method is first verified with a one-dimensional phase change test problem explained in Voller (1990). A pure liquid liquid initia initially lly at 28C occupies occupies the semi-infin semi-infinite ite space x $ 0. At time t 5 0 the surface at x 5 0 is fixed fixed at the the temp temper erat atur uree of 2 108C, whic which h is below below the freezi freezing ng point point of the substa substance nce T m 5 08C. As time proceeds, a solid layer builds up on the surface x 5 0 and moves out into the liquid. Simply stated, the problem is to determine how the solid–liqu solid–liquid id surfac surfacee moves moves with with time. time. The The thermal properties of the material in question are assumed to be constant and equal for both solid 6 and liquid liquid phase phase:: k 5 2 [W/ [W/ mK], mK], r c 5 2.5 3 10 3 8 3 [J / m K] an and r L 5 10 [J/m ]. In numeri mericcal 4 analysis 50 time steps of Dt 5 4.32 3 10 [s] (1/2 days) and 20 space increments of D x 5 0.125 [m] were used. The position of the phase front after 25 days (i.e. 50 time steps) obtained with proposed method is x 5 0.8405 [m]. The difference between this result and the result of x 5 0.8415 [m] obtained by Voller (1990) is merely 0.12%. Therefore, it could be conclu concluded ded that the accura accuracy cy of the propos proposed ed computat computational ional model model for conductio conduction n controlle controlled d isothermal phase change is satisfactory. 4.2. Experimental verification Fig. 6. Flow chart. chart.
As was discussed in the previous paragraph, the solution obtained with the proposed method gives
An analysis analysis of isother isothermal mal phase phase change change of phase phase change change materia materiall within within rectangul rectangular ar and cylindr cylindrical ical contai containers ners Table Table 1. Thermophysi Thermophysical cal properties properties of CaC1 2 ? 6H 2 O Melting point [ 8C] Latent heat [KJ / kg] 3 Density [kg / m ]: Specific heat [kJ / kgK]: Thermal conductivity [W/ mK mK]:
Solid Liquid Solid Liquid Solid Liquid
29.9 18 7 1 7 10 15 3 0 1.4 2.2 1.09 0.53
very very satisf satisfact actory ory result resultss for the condu conducti ctionon-con con-trolle trolled d one-di one-dimen mensio sional nal phase phase chang changee test test probproblem. lem. Howe Howeve ver, r, melt meltin ing g of the the PCM PCM in seal sealed ed contai container nerss is genera generally lly multi multi-di -dimen mensio sional nal ( both both radial radial and axial conductio conduction n exist) exist) and also also natural natural conv convec ecti tion on occu occurs rs with within in the the melt melted ed PCM. PCM. Ther Thereefore, fore, an experi experimen mentt was perfor performed med in order order to investiga investigate te the influen influence ce of the the assumption assumptionss of the math mathem emat atic ical al mode model. l. The The PCM PCM used used for for exexperi perime ment ntal al analy analysis sis is calc calciu ium m chlor chlorid idee hexah hexahyydrat dratee (CaCl (CaCl2 ? 6H 2 O) with with the thermo thermoph physi ysical cal proper propertie tiess as listed listed in Table Table 1 (Fujii (Fujii and and Yano, ano, 1996). A rectang rectangular ular container, container, made of stainl stainless ess steel, steel, with dimensions of l 5 b 5 100 mm and d 5 20 mm (Fig. (Fig. 1), was filled filled with with the calciu calcium m chlori chloride de hexahy hexahydra drate te and well well insula insulated ted on the latera laterall
57
sides. A thermocouple was placed in the centre of the container in the manner indicated in Fig. 7. The The cont contai aine nerr with with the the soli solid d PCM PCM was was then then placed vertically in the constant temperature bath, where the temperature was set to T 5 608C. The The comp comput utat atio iona nall mode modell was was set set up to rereproduce produce experime experimental ntal condition conditionss within within the constant temperature bath. The convection heat transfer coeffici coefficient ent between between the air and the contain container er wall wall was determi determined ned using using the correla correlatio tion n given given in Incr Incrop oper eraa and De DeWitt Witt (1985) (1985),, and was was calc calcuu2 lated lated to be be a co n 5 16 [W/m K]. Furtherm Furthermor ore, e, a time step Dt 5 5 s and space increment D x 5 2 mm were were used used in the the calc calcul ulat atio ion. n. In Fig. Fig. 8 the the variat variation ion with with time time of nume numeric rical al and and experi experimen mental tal values values of the temperatur temperaturee at the centre centre of the test cont contai aine nerr is show shown. n. From From the resu result ltss show shown n in Fig. Fig. 8, it can be conclu concluded ded that that the the agree agreemen mentt betwe between en numer numerica icall and experi experimen mental tal data data is is well well within within exp experi erimen mental tal uncertainties (i.e. positioning of the thermocouple’s ple’s tip exactly in the centre of the container container is quit quitee diffi difficu cult lt and and expe experi rime ment ntal al data data were were read read from from char charts, ts, which which reduce reducess accu accurac racy). y). The The high higher er slope slope of the theore theoretic tical al curv curvee in the liquid liquid region region `
v
Fig. 7. Test container. container.
Fig. 8. Variation ariation with time of PCM’s temperat temperature ure at the centre of the rectangular rectangular container container (cf. Fig. 7.). 7.).
58
B. Zivkovic and I. Fujii
is due due to the fac factt that that the Gras Grasho hoff numbe numberr was calculated for the temperature difference of T 2 T m ¯ 30 K, whil whilee in real realit ity y this this diff differ eren ence ce bebecomes comes smalle smallerr as the temperat temperatur uree of the liquid liquid PCM increa increases ses.. Cons Consequ equent ently, ly, the conve convecti ctive ve heat heat tran transf sfer er coef coeffic ficie ient nt betw betwee een n the the air air and and the the contai container ner wall used in the calcula calculatio tion n is higher higher than than the ‘real’ ‘real’ one. one. Furthe Furthermo rmore, re, it can be obobserved served from from Fig. 8 that that in theory theory PCM PCM reaches reaches its melting temperature faster than in the experiment. This is assumed to be due to the basic assumption made in the mathematical model that the conduction resistance of the container wall is neglected. More Moreov over er,, from from the the same same figur figuree it can can be obobserved that the calculated PCM’s melting time is slightly longer than the experimental one, which may be due to the fact that the natural convection within the liquid PCM is ignored. However, it can be seen that neglecting both the natural convection tion within within the liquid liquid PCM and the conduc conductio tion n within within the PCM in the the dire directi ction on of of the the HTF HTF flow flow do not not intro introduc ducee signifi significan cantt error error in the the predi predicti ction on of the PCM’s PCM’s temperatu temperature re variation variation during melting. `
5. COMPARISON COMPARISON OF THE MELTING MELTING TIME FOR RECTANGULAR AND CYLINDRICAL CONTAINERS
The meltin melting g time time of the incaps incapsula ulated ted PCM is one one of the essent essential ial parame parameter terss for determin determining ing the size size and and the the shap shapee of the contai container ner,, as it must must corres correspo pond nd to the total total amoun amountt of of dail daily y inso insolat lation ion.. To be spec specifi ific, c, the the cont contai aine nerr cont contai aini ning ng PCM PCM should should be desi designe gned d in such such a way way that that at at the the end end of the the day, day, comp comple lete te melt meltin ing g of the the PCM PCM is achi achiev eved ed.. In In tha thatt way way,, the the maxi maximu mum m effi effici cien ency cy of the LHES LHES unit unit is achiev achieved. ed. Furthe Furthermo rmore, re, comple complete te meltin melting g of of the the PCM PCM is a neces necessar sary y condi conditio tion n for for long-t long-term erm (seas (season onal) al) therma thermall energy energy storag storage. e. In ligh lightt of that that,, the the influ influen ence ce of the the cont contai aine ner’ r’ss dimens dimension ionss and its shape shape were were numeri numerical cally ly in-
vest vestig igat ated ed for for the the rect rectan angu gula larr and and cyli cylind ndri rica call containers. The The math mathem emat atic ical al mode modell for for the the isot isothe herm rmal al phase phase change change of the PCM filling filling the the cylindri cylindrical cal contai container ner (Fig. (Fig. 9) was derive derived d under under the same same assu assump mpti tion onss as in the the case case of the the rect rectan angu gula larr contai container ner.. Howeve However, r, the gover governin ning g equati equation on for the two model modelss is differ differen entt and and for for the case case of of the the cylindrica cylindricall container container it has the following following form: ≠
H ] t ≠
5
1
≠
S k T D ≠
(21)
]r ] ]] ≠r r ≠r
In the the abov abovee equa equati tion on,, H repres represent entss the total total enthalpy, which can be split into its sensible and late latent nt comp compon onen entt (cf. cf. Eq. Eq. (4)). (4)). The The solu soluti tion on metho methodol dology ogy and numeri numerical cal proced procedur uree for Eq. Eq. (21) is the same as for Eq. (3). 5.1. Results and discussion
The The dimensi dimension onss of the conta containe iners rs were were chosen chosen in such a manner manner that the volum volumee as well well as the convectiv convectivee heat heat transfer transfer area for both both the the cylind cylindrirical and rectangular containers were equal. The fixed dimensions were chosen to be the length of the cylindrical container lc 5 0.2 m (cf. Fig. 9) and the width of the rectangular container b 5 0.1 m (cf. Fig. 1). The air velocity was assumed to be w 5 5 m/ s and and its its temp temper erat atur uree T ai r 5 608C. The The thermo thermophy physic sical al prop propert erties ies of of the the air, air, as well well as as the the convec convectiv tivee heat heat transf transfer er coeffic coefficien ientt betwee between n the air and the contai container ner’s ’s wall wall a co n , were were take taken n from from Incro Incroper peraa and DeWit DeWittt (1985). (1985). The PCM filli filling ng the conta contain iner erss was was chos chosen en to be calci calcium um chlori chloride de hexahy hexahydr drate ate,, with with the thermo thermoph physi ysical cal prop proper erti ties es as list listed ed in Tabl Tablee 1. 1. Fig. Fig. 10 show showss the the vari variat atio ion n with with time time of the the PCM’s PCM’s temper temperatu ature re in the centre centress of both both the rectan rectangu gular lar and cylind cylindric rical al contai containe ners rs for the differ different ent dimens dimension ionss of the contain containers ers.. For small valu values es of of d (cf. Fig. 1) and r o (cf. Fig. 9), the differ differenc encee in meltin melting g time time betwee between n the the rect rectang angula ularr
Fig. 9. Geometry Geometry of the cylindrical cylindrical container. container.
v
An analysis analysis of isother isothermal mal phase phase change change of phase phase change change materia materiall within within rectangul rectangular ar and cylindr cylindrical ical contai containers ners
59
Fig. Fig. 10. Compar Compariso ison n of the variatio variation n with with time time of the PCM’s temper temperatu ature re at the centre centre of the rectangu rectangular lar and cylind cylindric rical al containers.
and cylindri cylindrical cal contai container nerss is not so pronou pronounce nced. d. Howeve However, r, on increa increasin sing g the mass mass of the PCM filling the container, i.e. with increasing d for the the rectangul rectangular ar container container and r o for the cylind cylindric rical al cont contai aine ner, r, the the diff differ eren ence ce in the the melt meltin ing g time time increa increases ses consid considera erably bly,, with with the rectan rectangul gular ar contai container ner showin showing g a much much shorte shorterr meltin melting g time time than than the the cylin cylindr drica icall conta containe inerr of the same same volu volume me and heat transfer area. A series series of numeri numerical cal experi experimen ments ts were were perperformed formed and their their result resultss are summariz summarized ed in Fig. Fig. 11 which shows the influence influence of the amount amount of
PCM filling filling the the containe containerr on the melting melting time of PCM. PCM. It It can can be be obse observe rved d that that for larger larger quant quantiti ities es of the the material material filling filling the contain container, er, the the differe difference nce in the melti melting ng time time between between the the rectang rectangula ularr and cyli cylind ndri rica call cont contai aine ners rs is very very prono pronoun unce ced, d, with the meltin melting g time time of the cylind cylindric rical al contai container ner being being nearly nearly twic twicee that that of the the rectan rectangu gular lar one. one. It It should should be point pointed ed out out that that in order order to make make the comcomparison between the two geometries significant, the numeri numerical cal analys analysis is was performe performed d under the condit condition ion of equal equal volume volume and heat heat transfer transfer area area for both the rectangul rectangular ar and cylindrica cylindricall container containers. s.
Fig. 11. Comparison Comparison of the melting time for rectangula rectangularr and cylindrical cylindrical containers containers of equal equal volume volume and heat transfer area.
60
B. Zivkovic and I. Fujii
It is intere interesti sting ng to note note that that the correlat correlation ion between the melting time and the mass of the PCM is nearly linear. 6. CONCLUSION CONCLUSION
m f r T T , T air T m t a a con d D x Dt r `
v
A simple and easily implemented computational model model simula simulatin ting g transi transient ent behavi behaviou ourr of the isot isothe herm rmal al phas phasee chan change ge is deve develo lope ped. d. A test test problem was solved and results obtained with the proposed method show very close agreement with the result resultss obtai obtained ned from from other other comput computati ation onal al model modelss (cf. (cf. Voll Voller, er, 1990). 1990). An experi experimen mentt was perfor performed med and numer numerica icall result resultss verifie verified d with with experimental data. Taking into consideration the basic assumptions of the mathematical model and recognizing very good good agreem agreement ent betwee between n the numeri numerical cal and experimental results, the following conclusions can be made: 1. Heat conduction conduction resistance resistance of the container’s container’s wall can be neglected without losing accuracy in the computational data. 2. Conductio Conduction n within the PCM in the direction of the HTF flow can be ignored as its influence was found to be negligible. 3. For flat thin thin conta containe iners, rs, the effects effects of natur natural al conv convec ecti tion on with within in the the liqu liquid id PCM PCM can can be ignored without introducing a significant error in the prediction prediction of the temperature temperature variation with time within the PCM. Moreov Moreover, er, compar compariso ison n betwee between n the meltin melting g time for the rectangular and cylindrical containers was was perf perfor orme med d and and the the resu result ltss show show that that the the rectangular container requires half of the melting time as for the cylindrical container of the same volume (i.e. equal mass of the PCM filling the container) and the same heat transfer area between HTF HTF and and the the cont contai aine ner’ r’ss wall wall.. It is, is, ther theref efor ore, e, prefer preferabl ablee to use rectan rectangul gular ar contai container nerss for encapsulating the PCM. NOMENCLATURE
A ht b c pf c f l
Fo
H h k l L
2
Heat transfer surface [m ] Widt Width h of rect rectan angu gula larr cont contai aine nerr [m] [m] Specific heat of air [J/kgK] Specific heat [J / kgK] Liquid fraction Fourier number Total enthalpy [J / kg] Sensible enthalpy [J/kg] J/kg] Conductivity [W/mK /mK] Length of container [m] Latent heat of fusion [J/kg]
Mass flow rate of air [kg/s] Radius of cylindrical container [m] Temperature of PCM [ 8C] Air temperature [ 8C] Melting point of PCM [ 8C] Time [h] 2 Thermal Thermal diffusivity diffusivity of PCM [m / s] 2 Heat transf transfer er coefficie coefficient nt [W/m K] Thickness of rectangular container [m] Space increment [m] Time step [h] 3 Density of PCM [kg/m ]
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An analysis analysis of isother isothermal mal phase phase change change of phase phase change change materia materiall within within rectangul rectangular ar and cylindr cylindrical ical contai containers ners Belleci Belleci C. and Conti M. (1993) Transient Transient behaviour behaviour analysis analysis of a latent heat thermal storage module. Int . J . Heat Mass Transfer 36(15), 3851–3857. Trp A., Franko Frankovic vic B. and Lenic Lenic K. (1999) An analys analysis is of phase change change heat heat transfer transfer in a solar thermal thermal energy energy store. store. Proceedings of ISES World Congress, July, Jerusalem, In Proceedings Jerusalem, Israel.
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Fujii I. and Yano Yano N. (1996) A conside considerati ration on on phase phase change change Proceedings of behaviour of latent heat storage material. In Proceedings the the Socie Society ty of Heat Heating ing, Air -conditi conditionin oning g and Sanitar Sanitaryy Engineers Engineers of Japan (in Japanese). Incropera Incropera F. P. P. and DeWitt DeWitt D. P. P. (1985) Introduction to heat transfer , John Wiley & Sons, New York, pp. 288–308.
