9 an Analysis of Iso Thermal Phase Change of PCM

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003

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AN ANALYSIS OF ISOTHERMAL PHASE CHANGE OF PHASE CHANGE

MATERIAL WITHIN RECTANGULAR AND CYLINDRICAL CONTAINERS

†, †, 11

B. ZIVKOVIC and I. FUJII B. ZIVKOVIC and I. FUJII

Meiji University, Department of Mechanical Engineering, 1-1-1 Higashi-Mita, Tama-ku, 214-8571 Meiji University, Department of Mechanical Engineering, 1-1-1 Higashi-Mita, Tama-ku, 214-8571

Kawasaki, Japan Kawasaki, Japan

Received 10 January 2000; revised version accepted 7 June 2000 Received 10 January 2000; revised version accepted 7 June 2000

Communicated by ERICH HAHNE Communicated by ERICH HAHNE

Abstract

Abstract—In this paper, a simple computational model for isothermal phase change of phase change material—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 (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 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 theoretical model was veriﬁed with a test problem and an experiment performed in order to assess the validity of

of the the assassumpumptitions ons of of the the mamathethematmaticaical l modmodel. Witel. With h ververy y googood d agragreemeement ent betbetweeween n expexperierimenmental tal andand computational data, it can be concluded that conduction within the PCM in the direction of heat transfer fluid 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 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 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 and cylindrical containers was then performed using the computational model presented in this paper. Results show that

show that the rectangthe rectangular containular container requires nearly half er requires nearly half of of the melting time as the melting time as for the for the cylicylindricndrical containeal container r of of the same volume and heat transfer area.

the same volume and heat transfer area. ©© _{2001 Published by Elsevier Science Ltd.}_{2001 Published by Elsevier Science Ltd.}

1.

1. INTROINTRODUCTIDUCTIONON Ghoneim, 1989). Using the shell-and-tube modelGhoneim, 1989). Using the shell-and-tube model for

for the the LHELHES S uniunit, t, MorMorrisrison on and and AbdAbdel-el-KhaKhalik lik Effective and economic thermal energy storage of

Effective and economic thermal energy storage of

(197

(1978) 8) perperforformed med a a lonlong-tg-term erm ananalyalysis sis of of air air--a

a daidaily ly sursurpluplus s of of irrirradadiatiated ed solsolar ar eneenergy is rgy is anan

bas

based ed ((air air as as HTF) HTF) and and liqliquiduid-b-baseased d ((watwater er asas unavoidable necessity for the efﬁcient use of solar

unavoidable necessity for the efﬁcient use of solar

HTF)

HTF) solsolar ar heaheatinting g syssystemtems, s, assassumiuming ng thathat t thethe ene

energy rgy for for heaheatinting g pupurporposes ses (Duf(Dufﬁe ﬁe anand d BeBeck-

ck-thermal conductivity of the PCM in the direction thermal conductivity of the PCM in the direction ma

man, n, 19199191). ). AmAmonong g ththe e vavaririouous s memeththodods s of of

of the HTF ﬂow, as well as the thermal resistance of the HTF ﬂow, as well as the thermal resistance energy storage, latent heat thermal energy storage

energy storage, latent heat thermal energy storage

of the PCM in the direction normal to the HTF of the PCM in the direction normal to the HTF is particularly attractive. The motivation for using

is particularly attractive. The motivation for using ﬂo

ﬂow, w, cacan n boboth th be be igignonorered. d. In In a a lalateter r ananalalysysisis,, ph

phasase e chchanange ge mamateteririalals s (PC(PCM) M) is is ththeieir r hihighgh Gh

Ghononeieim m (1(198989) 9) totook ok ininto to acaccocoununt t boboth th ththee energy storage density and their ability to provide

energy storage density and their ability to provide

conduction within the PCM in the direction of the conduction within the PCM in the direction of the heat at a constant temperature (Abhat, 1983).

heat at a constant temperature (Abhat, 1983).

HTF ﬂow and the direction normal to the HTF HTF ﬂow and the direction normal to the HTF In

In ororder der to to perperforform m a a lonlong-tg-term erm peperforformarmancence ﬂo

ﬂow w anand d shshowowed ed ththat at a a susubsbstatantntiaial l ererroror r inin ana

analyslysis is of of a a spespeciﬁciﬁed ed solsolar ar heaheatinting g syssystemtem, , anan es

estitimamatiting ng ththe e sosolalar r frfracactition on ((amamouount nt of of tototatall adequate model of the heat storage unit is needed

adequate model of the heat storage unit is needed

heating load supplied by solar energy) was heating load supplied by solar energy) was intro-(Klein

(Klein et alet al.., 1976) which, naturally, depends on, 1976) which, naturally, depends on

duc

duced ed by by negnegleclectinting g the the conconduductiction on witwithin hin thethe its design. A survey of the previously published

its design. A survey of the previously published

PCM

PCM. . HowHoweveever, r, the the comcommon mon conconcluclusiosion n frofromm papers dealing with latent heat storage reveals that

papers dealing with latent heat storage reveals that

both works is that the reduction in storage volume both works is that the reduction in storage volume the most intensively analyzed latent heat energy

the most intensively analyzed latent heat energy

by using PCM is not as nearly as pronounced for by using PCM is not as nearly as pronounced for storage (LHES) unit is the shell-and-tube LHES

storage (LHES) unit is the shell-and-tube LHES

liquid-based systems as it is for air-based systems. liquid-based systems as it is for air-based systems. unit with the PCM ﬁlling the shell and the heat

unit with the PCM ﬁlling the shell and the heat It

It isis, , ththererefeforore, e, prprefefererabable le to to ususe e a a LHLHES ES ununitit tra

transfnsfer er fluifluid d (HTF) (HTF) flowflowing ing thrthrougough h the the tubtubeses

coupled with an air-based system. For that reason, coupled with an air-based system. For that reason, (Lacroix, 1993; Bansal and Buddhi, 1992; Esen

(Lacroix, 1993; Bansal and Buddhi, 1992; Esen et et

air is

air is conconsidsidereered d as as the HTF the HTF in in the subsethe subseququentent

al

al.., , 19199898; ; SoSoma ma anand d DuDutttta, a, 19199393; ; IsIsmamail il anandd

analysis. analysis. Alves, 1986) or, vice versa, PCM ﬁlling the tube

Alves, 1986) or, vice versa, PCM ﬁlling the tube

The problem of the phase change of PCMs falls The problem of the phase change of PCMs falls and HTF ﬂowing parallel to it (Esen

and HTF ﬂowing parallel to it (Esen et alet al.., 1998;, 1998;

into the category of moving boundary problems. into the category of moving boundary problems. Wh

When en ththe e PCPCM M chchanangeges s ststatate, e, boboth th liliququid id anandd

†

† solid phases are present and they are separated bysolid phases are present and they are separated by

Author to whom correspondence should be addressed. Tel./ Author to whom correspondence should be addressed. Tel./

the

the movimoving ng interinterface between them. face between them. TherThere e havehave

fax:

fax: 11_{81-44-943-7395; e-mail: fujii@isc.meiji.ac.jp}_{81-44-943-7395; e-mail: fujii@isc.meiji.ac.jp}

1 1

ISES member.

ISES member. bebeen en mamany ny didiffffererenent t nunumemeriricacal l memeththodods s de de--51

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velo

veloped for deaped for dealing witling with the phase chah the phase change pronge prob- b- accoaccounted unted for for (V(Volleroller et alet al.., 1987; Lacroix, 1993;, 1987; Lacroix, 1993; lem

lem (Ban(Bansal asal and Bnd Bududdhidhi, 19, 1992; S92; Soma aoma and Dnd Duttutta, a, SomSoma a and and DutDutta, ta, 1991993). Whi3). While le in in momost st worworks,ks, 19

199393; ; MuMurrrray and Lanay and Landidis, 195s, 1959) 9) of whiof which the ch the coconvnvecectitive ve heheat at trtranansfsfer er frfrom om ththe e HTHTF F to to ththee mo

most st attattracractivtive e and and cocommommonly nly useused d are are the the so- so- PCM iPCM is cals calculculateated thrd througough the mh the mean vean valualue of the of thee cal

called led ententhalhalpy mpy methethods ods (V(Volloller aer and Cnd Crosross, 1s, 1981981; ; heaheat t tratransfnsfer er coecoefﬁcfﬁcienient t ((a a _conv_conv), ), sosome me auauththororss V

Volleroller, 1985; , 1985; VVolleroller, 1990). The , 1990). The majomajor reason for r reason for solvesolved the probld the problem of phase chaem of phase change counge coupled withpled with thi

this is s is thathat the t the metmethohod does not requd does not require expire expliclicit it tratransinsient ent conconvecvectivtive e heaheat t tratransfnsfer er betbetweeween n HTFHTF tre

treatmatment of ent of the conthe conditditionions s on the on the phphase chanase change ge and PCMand PCM, , i.ei.e. . witwith the h the comcompleplete Oberte Oberbecbeck’s setk’s set bo

boundundary ary ((cf. cf. CaCarslrslaw aw and and JaeJaegerger, , 191959), 59), i.ei.e. . thethere re of of equequatiations ons solsolved ved for for the the HTF HTF (Bel(Bellecleci i andand is no

is no need need for tfor trackiracking thng the phae phase chse change ange bounboundary dary ContConti, 1993; Trpi, 1993; Trp et alet al.., 1999)., 1999). thr

thrououghoghout ut the phase the phase chachangnge e domdomainain. . HoHowevwever, er, In the In the prepresensent t anaanalyslysis, is, the the matmathemhematiatical cal momodeldel be

besiside de ththe e fafact ct ththat at imimplplicicit it ﬁnﬁnitite e didiffffererenence ce fofor r phphasase e chchanange of ge of ththe e enencacapspsululatated ed PCPCM M isis dis

discrecretiztizatiation on resresultults s in in a a set set of of nonnonlinlinear ear equequa- a- derderiveived unded under the follor the followinwing assumg assumptiptionsons:: ti

tionons, the meths, the method has sevod has severeral othal other draer drawbwbacacksks. . ((AA) The) Thermrmal coal condnducuctitivivity of the PCty of the PCM in theM in the The

These se are are the the ququite ite cumcumberbersomsome e calcalculculatiation on of of dirdirectection ion of thof the HTF ﬂe HTF ﬂow iow is igns ignoreored.d. li

liququid id frfracactition on upupdadatetes s anand d ththe e fafact ct ththat at ththe e (B) (B) ThThe e efeffefectcts s of of nanatuturaral l coconvnvecectition on wiwiththin in ththee temp

temperatuerature ﬁere ﬁeld wild within thin the Pthe PCM iCM is not s not calcucalculated lated melt are melt are neglinegligiblgible e and can and can be ignoredbe ignored.. ex

explplicicititly ly bubut t vivia a enenththalalpypy-t-temempeperaratuture re cocorrrrelela- a- (C) The PC(C) The PCM behM behavaves ides ideaealllly, i.y, i.e. sue. such phch phe- e-tio

tion. n. In In thithis s pappaper, er, a a sligslightlhtly y momodiﬁdiﬁed ed ententhalhalpy py nomnomena aena as pros properperty degty degradradatiation anon and supd supercercooloolinging meth

method, wod, which ehich enablenables decos decoupliupling of ng of tempetemperaturrature e are not accouare not accounted fornted for.. an

and ld liqiquiuid fd fraractctioion fin fieleldsds, i, is ps preresesentnteded. D. Dererivivatatioion n (D) (D) ThThe e PCPCM M is is asassusumemed d to to hahave ve a a dedefinfinititee of discret

of discretizatiization equatioon equations is ns is straistraightfghtforwarorward and d and meltimelting poinng point (isothermt (isothermal phase chaal phase changenge).). th

the e memeththod od ititseself lf is is vevery ry eaeasy sy to to imimplplememenent. t. (E) (E) ThTherermomophphysysicical al prpropoperertities es of of ththe e PCPCM M araree different for the solid and liquid phases but are different for the solid and liquid phases but are independent of temperature.

independent of temperature.

2.

2. MATHEMATHEMATICMATICAL MODELAL MODEL

(F) The PCM is homogeneous and isotropic. (F) The PCM is homogeneous and isotropic. Th

The e susubjbjecect t of the of the prpresesenent t ininveveststigigatatioion n is is a a (G) Th(G) Therermamal rel resisiststanance ace acrcrososs ths the wae wall oll of thf thee sing

single cole containntainer ﬁller ﬁlled wed with PCith PCM (Fig. M (Fig. 1). Pac1). Packing king contcontainer is neglecteainer is neglected.d. th

the e PCPCM M in in sisingngle le cocontntaiaineners rs enenabableles s momodudulalar r (H) La(H) Lateteraral sidl sides of tes of the rehe rectctanangugulalar conr contataininerer con

constrstructuction ion of of the the LHELHES S uniunit t and and is is alsalso o ververy y are are welwell inl insulsulateated, d, i.ei.e. he. heat tat tranransfesfer ocr occurcurs ons only oly onn econ

economic omic from from the vthe viewpoiewpoint oint of mass f mass prodproductiouction. n. sidessides xx55_{0 and}_{0 and} _xx55_d_d_{(cf. Fig. 1).}_{(cf. Fig. 1).}

More

Moreoverover, co, complemplete mte meltinelting og of thf the Pe PCM, CM, whicwhich is h is In oIn order rder to vto validaalidate thte the ﬁre ﬁrst assst assumptumption ion (A(A), th), thee abso

absolutellutely necessary necessary for long-tey for long-term (searm (seasonal) heat sonal) heat inﬂueinﬂuence of the connce of the conductduction withion within the PCM in thein the PCM in the sto

storagrage (Zivke (Zivkoviovic and Fujc and Fujii, 19ii, 1999), is di99), is difficfficult to ult to dirdirectection ion of of HTF HTF flow flow was was numnumeriericalcally ly invinvestesti- i-ob

obtatain usiin using a ng a shshelell-l-anand-d-tutube LHEbe LHES S ununit wheit where re gagateted d in in ththe e fofollllowowining g waway. y. ReRetataininining g alall l ththee la

largrge e mamasssses es of of ththe e PCPCM M arare e ininvovolvlveded. . ababovove e asassusumpmptitionons s anand d neneglglecectiting ng ththe e cocondnducuctitionon The

There re are are sevseveraeral apl approproachaches es to tto the he matmathemhematiati- - witwithin hin the the PCM PCM in in the the dirdirectection ion nonormarmal l to to thethe ca

cal l momodedeliling of LHEng of LHES S ununitits. In soms. In some e momodedelsls, , HTHTF F ﬂoﬂow, w, ththe e gogovevernrnining g eqequauatitionons s fofor r heheatat con

conducductiotion within within the n the PCM in both the direPCM in both the directiction on tratransfnsfer and phaser and phase change change e can be writcan be written as:ten as: of the HTF ﬂow and the direction normal to the

of the HTF ﬂow and the direction normal to the

≠

≠_H_H ≠≠ _k_k ≠≠_T_T

HTF ﬂow are taken into account (Lacroix, 1993; HTF ﬂow are taken into account (Lacroix, 1993;

]

_≠_≠_t_t 55

]

_≠_≠_z_z

S

S DD

]

_r_r

_]

_≠_≠_z_z (1)(1)

Gh

Ghononeieim, m, 191989)89), , whwhilile e in in otothehers rs ththe e efeffefect ct of of natural convection within the molten PCM is also

natural convection within the molten PCM is also _≠_≠_T_T

ai ai rr

]

m m cc 55_a_a _A_{A T}_s_s_T22_T_T _dd ₍₂₎₍₂₎ f f p f p f _≠_≠_z_z convconv ht ht aiai rr

where all the variables are deﬁned in the where all the variables are deﬁned in the nomen-clature table. Detailed description of the clature table. Detailed description of the numeri-ca

cal l sosolulutition on of of ththe e ababovove e seset t of of eqequauatitionons s isis omitted as it is described later in the paper. Here, omitted as it is described later in the paper. Here, only the results obtained are discussed brieﬂy. only the results obtained are discussed brieﬂy.

Even though conduction within the PCM in the Even though conduction within the PCM in the direction of HTF flow plays an important role for direction of HTF flow plays an important role for long LHES units (containers) and relatively high long LHES units (containers) and relatively high convective heat transfer coefficients (Fig. 2), its convective heat transfer coefficients (Fig. 2), its

Fig. 1.

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convec-2 2

Fig. 2.

Fig. 2. TempTemperaterature variature variation within the PCM ion within the PCM in the in the diredirectioction n of the of the HTF ﬂow forHTF ﬂow for ll55₃₃ _m_m _and_{and a}_a 55_{300 W/m K.}_{300 W/m K.}

con convv

tiv

tive he heat eat tratransfnsfer er coecoefﬁcfﬁcienients ts (s(such uch are are thothose se whewhen n WitWith h the the forforegoegoing ing assassumpumptiotions, ns, the the ententhalhalpypy ai

air r is is usused ed as as ththe e HTHTF) F) anand d rerelalatitivevely ly shshorort t foformrmululatatioion n fofor r ththe e cocondnducuctitionon-c-conontrtrololleled d phphasasee con

contaitainerners s (fro(from m 20200 0 to to 400 400 mmmm). ). FurFurthethermormore, re, chachangnge e can be writcan be written as (Vten as (Volloller, 199er, 1990; 0; VVollollerer et et

com

comparparisoison n of of the the solsolutiution on obobtaitained ned usiusing ng thethe alal.., 1987):, 1987): mathematical model deﬁned with the set of Eqs.

mathematical model deﬁned with the set of Eqs.

≠

≠_H_H _k_k

(1) and (2) and the solution obtained solving the (1) and (2) and the solution obtained solving the

]

_≠_≠_t_t 55ddiiv v

S

]

_r_rggrraaddT T

DD

(3)(3)

sa

same me prproboblelem m ususining g ththe e lulumpmped ed mamass ss memeththodod sh

showows s ththat at no no sisigngniﬁiﬁcacant nt imimprprovovememenent t of of ththee

An alternative form of Eq. (3) can be obtained by An alternative form of Eq. (3) can be obtained by res

resultults s is is obtobtainained ed if if the the conconducductiotion n witwithin hin thethe

split

splitting ting the the total enthalpytotal enthalpy H H intinto o sensensibsible le andand PCM is accounted for (Fig. 4). These results are

PCM is accounted for (Fig. 4). These results are

latent heat components: latent heat components: in

in acaccocordrdanance ce wiwith th ththe e reresusultlts s obobtatainined ed byby Ghoneim (1989), who concluded that there is no

Ghoneim (1989), who concluded that there is no _H_H55_h_h11_L_L??_f_f ₍₄₎₍₄₎ ll

sig

signiﬁniﬁcancant t chachange nge in in the the prepredicdictiotion n of of the the solsolarar fraction with reduction of

fraction with reduction of DD_z_z _to_{to les}_less_{s tha}_than_{n 20}₂₀₀_{0 mm}_{mm. .} _whe_where_re

2 2

Fig. 3.

Fig. 3. TempTemperaterature variature variation within the PCM ion within the PCM in the in the diredirectioction n of the of the HTF ﬂow forHTF ﬂow for ll55_0.314_0.314 _m_m _and_{and a}_a 55_{15.5 W/m K.}_{15.5 W/m K.}

con convv

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Fig. 4.

Fig. 4. CompaComparison betwerison between the en the lumpelumped d mass methomass method and d and the case when the the case when the axiaaxial conductiol conduction is n is accoaccounted for.unted for. T T ≠ ≠_f_f ≠ ≠_h_h ≠≠ _k_k≠≠_T_T _ll

]

_]

]

h h55

EE

_cc_d_d_T_T ₍₅₎₍₅₎ 5 5

S DD

S

22_L_L ₍₈₎₍₈₎ ≠ ≠_t_t ≠≠_x_x _r_r ≠≠_x_x ≠≠_t_t T T _m_m The fully

The fully impliimplicit cit discrdiscretizaetization equation for tion equation for anan and

and T T _m_m is the melting temperature of the PCM.is the melting temperature of the PCM.

internal node ‘

internal node ‘ii’ can be written as (Fig. 5):’ can be written as (Fig. 5): For the problem of isothermal phase change, the

For the problem of isothermal phase change, the local liquid fraction

local liquid fraction f f ll is deﬁned as:is deﬁned as: _≠_≠_h_h _k_k ≠≠ f f _ll

ii ii

]

_≠_≠_t_t 55

]

22 s s T T ii22112222T T i i 11T T ii1111dd22 L L

_]

_≠_≠_t_t (9)(9) 1 1 if if T T .._T_T _r_r_D_D_x_x m m f f _ll((T T ))55

HH

₍₆₎₍₆₎ 0 0 if if T T ,,_T_T m m wh

wherere e ththe e sesensnsibible le enenththalalpy py teterm rm anand d liliququidid Substituting Eq. (4) into Eq. (3) gives:

Substituting Eq. (4) into Eq. (3) gives: fraction term are deliberately left in the differen-fraction term are deliberately left in the differen-ti

tial al foform rm fofor r ththe e coconvnvenenieiencnce e of of susubsbseqequeuentnt

≠ ≠_f_f ≠

≠_h_h _k_k _ll

]

_≠_≠_t_t55ddiiv v

S

]

_r_rggrraaddT T

DD

22 L L

_]

_≠_≠_t_t (7)(7) numerical computation. It should be noted that thenumerical computation. It should be noted that the

discr

discretizaetization tion equaequations tions for for bounboundary dary nodenodes s de- de-Eq.

Eq. (7) r(7) reprepreseesentsnts, t, togogethether er witwith h EqsEqs. (5) . (5) and and (6) (6) pend pend on on spespeciﬁciﬁc c bouboundandary ry conconditditionions s and and areare an

and d ththe e apapprpropopririatate e ininititiaial l anand d bobounundadary ry cocon- n- dedeririveved d frfrom om an an enenerergy gy babalalancnce e on on bobounundadaryry dit

ditionions, s, the the matmathemhematiatical cal modmodel el of of conconducductiotion n concontrotrol l vovolumlumes.es.

cconontrtroolllleed d isisooththerermamal l pphahase se chchanangge. e. TThhe e kkey ey ffeaeatuturre e oof f ththe e pproroppososed ed mmetethhod od is is ththee fact that for an isothermal phase change (which is fact that for an isothermal phase change (which is the case for most salt hydrates), the temperature the case for most salt hydrates), the temperature

3.

3. NUMERNUMERICAL SOLUTIICAL SOLUTIONON

o

of f ththe e PCPCM M wiwiththin in a a gigivven en coconntrtrool l vovolulummee For the problem of one-dim

For the problem of one-dimensioensional isothenal isothermal rmal remaremains conins constant astant and eqnd equal to itual to its meltis melting temng tempera- pera-ph

phase chanase change of ge of the PCM encathe PCM encapsupsulatlated withied within n a a turture e untuntil the PCM il the PCM has melhas melted comted complepleteltely. Con-y. Con-re

rectctanangugulalar r cocontntaiainener r (Fi(Fig. g. 1), 1), ththe e gogovevernrnining g sisideder ﬁrst thr ﬁrst the case whe case when coen contntrorol volul volume ‘me ‘‘i‘i‘ is‘ is eq

equauatition on fofor tr the he PCPCM fM folollolows fws frorom Em Eq. q. (7): (7): fufulllly soy solilid or fd or fulully lly liqiquiuid. Id. In thn that cat casase, fe, frorom thm thee

Fig. 5.

(5)

deﬁnition of sensible enthalpy, Eq. (5), and the

deﬁnition of sensible enthalpy, Eq. (5), and the _k_k_D_D_t_t

ol ol dd

]

liquid fraction, Eq. (6), it follows that:

liquid fraction, Eq. (6), it follows that: f f _l_l 55_f_f_ll 11 ₂₂ _s_s_T_T_ii 22₂₂_T_T 11_T_T _dd ₍₁₈₎₍₁₈₎

2 211 m m ii1111 i i ii _r_r L_L _D_D_x_x ≠ ≠_h_h_i_i ≠≠_T_T_ii wh

whicich h is is ththe e eqequauatition on fofor r upupdadatiting ng ththe e liliququidid

]

≠≠_t_t ;; cc

]

≠≠_t_t (10)(10)

fra

fractiction on ﬁelﬁeld d witwithin hin the the concontrotrol l volvolumume e thathat t isis undergoing a phase change. As Eq. (15) shows, undergoing a phase change. As Eq. (15) shows, and

and

liquid fractions are updated from the temperature liquid fractions are updated from the temperature

≠ ≠_f_f

ll_ii field field anand d nonot t frfrom om ththe e sesensnsibible le enenththalalpy py fiefieldld

]

≠≠_t_t ;; 00 ((1111)) _(V_(Voller_{oller, , 1990}_{1990; ; Lacr}_Lacroix,_{oix, 1993).}_{1993). The}_{The temp}_temperatu_erature_re

and

and the the liqliquid uid frafractiction on ﬁelﬁeld d are are decdecououplepled, d, thethe where

where cc is the speciﬁc heat of the solid or liquidis the speciﬁc heat of the solid or liquid

temperature ﬁeld within the PCM being calculated temperature ﬁeld within the PCM being calculated phase, depending on state of the control volume.

phase, depending on state of the control volume. in

indedepependndenentltly y frfrom om EqEq. . (13) (13) foforcrciningg aa 55 ii2211

Af

Afteter r inintrtrododucucining g EqEqs. s. (10) (10) anand d (11)(11), , EqEq. . (9)(9)

a

a 55_0,_0, _a_a 55₁_{1 aan}_nd_d _b_b 55_T_T _fo_for_{r th}_the_{e co}_cont_ntro_roll ii1111 i i i i mm

reduces to ordinary heat diffusion equation: reduces to ordinary heat diffusion equation:

volumes which are undergoing the phase change. volumes which are undergoing the phase change.

≠

≠_T_T _k_k _{Furthermore, on the basis of Eq. (18), it can be}_{Furthermore, on the basis of Eq. (18), it can be} ii

]

_≠_≠_t_t 55

]

22 s s T T ii22112222T T i i 11T T ii1111dd (12)(12) inferred that for an isothermal phase change, allinferred that for an isothermal phase change, all

r r ccDD_x_x

the heat supplied to the control volume the heat supplied to the control volume undergo-Backward differencing of the left side term gives,

Backward differencing of the left side term gives, ining g a a phphasase e chchanange ge is is usused ed fofor r chchanangiging ng ththee after rearran

after rearrangingging, , the the fully implicit ﬁnite fully implicit ﬁnite diffdiffer-er- amamouount nt of of lalatetent nt heheat at cocontntenent t of of ththat at cocontntroroll ence equation of the form:

ence equation of the form: volume.volume.

At this point it is worthwhile to describe the At this point it is worthwhile to describe the

a

a ??_T_T 11_a_a ??_T_T 11_a_a ??_T_T 55_b_b ₍₁₃₎₍₁₃₎

ii2211 ii2211 i i i i ii1111 ii1111 ii _im_impl_plem_emen_enta_tati_tion_{on of}_{of th}_the_{e co}_comp_mput_utat_atio_iona_nal_{l mo}_mode_del,_l,

which is as follows: which is as follows: where coefﬁcients

where coefﬁcients aa 55_a_a 55 22_Fo,_Fo, _a_a 55₁₁11 ii2211 ii1111 ii

ol

ol dd _{(a) Coefﬁcients ‘}_{(a) Coefﬁcients ‘}_a_a_{’ of Eq. (13) are formed. For}_{’ of Eq. (13) are formed. For}

2

233_{Fo and}_{Fo and} _b_b 55_T_T _{are introduced for the sake}_{are introduced for the sake} i

i ii

nodal points where the liquid fraction

nodal points where the liquid fraction f f is strictlyis strictly of

of comcompuputattationional al simsimpliplicitcity. y. SupSuperserscricript pt ’ol’old’d’ lili

in the interval [0,1], 0

in the interval [0,1], 0 ,,_f_f ,,_{1, the coefﬁcients of}_{1, the coefﬁcients of}

refers to the previous time step and Fo is a ﬁnite

refers to the previous time step and Fo is a ﬁnite lili

Eq. (13) are set to:

Eq. (13) are set to: aa 55_a_a 55_0,_0, _a_a 55_{1 and}_{1 and}

difference form Fourier number:

difference form Fourier number: ii2211 ii1111 ii

b

b_i_i55_T_T_m_m_..

a a ?? DD_t_t

(b

(b) ) ThThe e seset t of of lilinenear ar alalgegebrbraiaic c EqEqs. s. (13) (13) isis

]

Fo

Fo55 ₂₂ ₍₁₄₎₍₁₄₎ D

D_x_x _{solved using the Gauss-Seidel iterative procedure.}_{solved using the Gauss-Seidel iterative procedure.}

(c) Liquid fractions are updated from the (c) Liquid fractions are updated from the tem-where

where a a 55_k_k_/_/_r_r cc _{is the thermal diffusivity of the}_{is the thermal diffusivity of the}

perature ﬁeld using Eq. (18). perature ﬁeld using Eq. (18). PCM. The thermophysical properties in Eq. (14)

PCM. The thermophysical properties in Eq. (14)

(d) A check for the ‘start’ and for the ‘end’ of (d) A check for the ‘start’ and for the ‘end’ of depend on the state of the control volume. Those

depend on the state of the control volume. Those

phase change is performed. Explicitly, if the state phase change is performed. Explicitly, if the state of the solid phase should be inserted if the control

of the solid phase should be inserted if the control of

of ththe e liliququid id frfracactition on ﬁeﬁeld ld chchanangeges s wiwiththin in ththee volume is in the solid state and those of the liquid

volume is in the solid state and those of the liquid

given time step, i.e. a ﬁnite volume commences or given time step, i.e. a ﬁnite volume commences or phase if the control volume is in the liquid state.

phase if the control volume is in the liquid state.

terminates with the phase change, the coefﬁcients terminates with the phase change, the coefﬁcients No

Now, w, coconsnsidider er ththe e cacase se whwhen en memeltltining g ((oror of

of EqEq. . (1(13) 3) neneed ed to to be be upupdadateted d anand d ststepeps s ((aa)) freezing) occurs around a certain node ‘

freezing) occurs around a certain node ‘ii’. In that’. In that

through (d) repeated for the same time step. through (d) repeated for the same time step. cas

case, e, the the liqliquid uid frafractictionon f f _li_li lilies es ststririctctly ly in in ththee

In

In prpracactiticece, , fofor r momost st titime me ststepeps s ononly ly ononee interval [0,1]. Recognizing that for an isothermal

interval [0,1]. Recognizing that for an isothermal

iteration is needed per time step. The only time iteration is needed per time step. The only time phase change:

phase change:

when two iterations are needed is when the phase when two iterations are needed is when the phase change boundary moves from one control volume change boundary moves from one control volume

T

T ;i i ; T T mm (15)(15)

to the next one. to the next one. and from Eq. (5):

and from Eq. (5):

3.1.

3.1. ChecChecking king for start for start / / end of the phase changeend of the phase change

≠ ≠_h_h

ii

]

≠≠_t_t ;; 00 ((1166)) At the end of each time step, the check for startAt the end of each time step, the check for start

and/or end of the phase transition is performed and/or end of the phase transition is performed Eq. (9) becomes:

Eq. (9) becomes: _th_thro_roug_ugho_hout_{ut th}_the_{e en}_enti_tire_{re do}_doma_main_{in. . Fo}_For_{r th}_the_{e ca}_case_{se of}_of melting, checking for the ‘start’ and ‘end’ of the melting, checking for the ‘start’ and ‘end’ of the

≠ ≠_f_f

ll_ii k k _ph_phas_ase_{e ch}_chan_ange_{ge is}_{is pe}_perf_rfor_orme_med_{d in}_{in th}_the_{e fo}_foll_llow_owin_ing_g

]

L L 55 _s_s_T_T 22₂₂_T_T 11_T_T _dd ₍₁₇₎₍₁₇₎ 2 2 ii2211 m m ii1111 ≠ ≠_t_t _r_r_D_D_x_x _fashion:_fashion: 3.1.1.

3.1.1. START of meltingSTART of melting. For a given time step,. For a given time step, Backward differencing of the liquid fraction term

Backward differencing of the liquid fraction term

ol ol dd

g

giivveess:: iiffT T $$ _T_T _while_while _T_T ,,_T_T _{, it indicates that within}_{, it indicates that within} i

(6)

this time step, the ﬁnite volume in question begins

this time step, the ﬁnite volume in question begins 3.1.2.3.1.2. END of meltingEND of melting. For a given time step, if . For a given time step, if

ol ol dd

with melting. In that case, the coefﬁcients of Eq.

with melting. In that case, the coefﬁcients of Eq. f f _lli_i$$_{1 while}_{1 while} _f_f_llii ,,_{1, it indicates that within this}_{1, it indicates that within this}

ti

time me ststepep, , ththe e cocontntrorol l vovolulume me in in ququesestition on hahass (13)

(13) arare e upupdadateted d as as dedescscriribebed d ababovove e anand d ththee

melted completely. In that case coefﬁcients of Eq. melted completely. In that case coefﬁcients of Eq. cal

calculculatiation on for for thathat t stestep p is is perperforformed med agaagain. in. ItIt (1

(13) 3) arare e agagaiain n seset t toto aa 55_a_a 55 22_Fo,_Fo, _a_a 55

should be noted, however, that in the time step

should be noted, however, that in the time step ii2211 ii1111 ii

ol ol dd

1

111₂₂33_{Fo and}_{Fo and} _b_b 55_T_T _{and the calculation for}_{and the calculation for}

wh

when en a a cocontntrorol l vovolulume me hahas s jujust st bebegugun n wiwithth i i ii

that time step is performed again. In the time step that time step is performed again. In the time step melting, Eq. (18) has the form:

melting, Eq. (18) has the form:

in which the phase change boundary moves from in which the phase change boundary moves from

k

k DD_t_t _{the control volume in question to the next one,}_{the control volume in question to the next one,}

ol ol dd

]

f f 55_f_f 11 _s_s_T_T 22₂₂_T_T 11_T_T _dd l

l _i_i ll_ii 22 ii2211 m m ii1111 coefﬁcientcoefﬁcient _b_b has the following form:has the following form:

ii r r L L DD_x_x cc olol dd LL olol dd

]

2 2

s s

_T_T 22_T_T

dd

₍₁₉₎₍₁₉₎

]

b b 55_T_T 22 ₁₁22_f_f ₍₂₀₎₍₂₀₎ m m ii i i m m

s s

ll

dd

L L cc ii

where the last term on the right hand side can be where the last term on the right hand side can be The last term on the right-hand side of Eq. (19)

The last term on the right-hand side of Eq. (19)

described as the amount of heat needed to described as the amount of heat needed to com-re

reprpresesenents ts ththe e amamouount nt of of sesensnsibible le heheat at ththat at isis

pletely melt the control volume in question within pletely melt the control volume in question within ne

neededed ed to to raraisise e ththe e tetempmpererataturure e of of ththe e cocontntroroll

the time step and which consequently can not be the time step and which consequently can not be volume from the temperature in the previous time

volume from the temperature in the previous time

ol

ol dd used to raise the temperature of the PCM.used to raise the temperature of the PCM.

sstteep p ((T T _i_i ) ) to to ththe e mmeleltitinng g tetemmppereratatuure re ((T T _m_m).).

The ﬂow chart for the computational procedure The ﬂow chart for the computational procedure Consequently, that amount of heat can not be used

Consequently, that amount of heat can not be used

is given in Fig. 6. is given in Fig. 6. for melting the PCM.

for melting the PCM.

4.

4. VERIFVERIFICATIICATION OF ON OF THE MATHEMATTHE MATHEMATICALICAL MODEL

MODEL

4.1.

4.1. Test problemTest problem

Th

The e perperforformanmance ce of of the the prepresensented ted metmethod hod isis first verified with a one-dimensional phase change first verified with a one-dimensional phase change test problem explained in Voller (1990). A pure test problem explained in Voller (1990). A pure liq

liquid uid iniinitiatially lly at at 2288_C_{C occu}_occupies_{pies the}_{the semi-}_semi-inﬁn_inﬁnite_ite

space

space xx$$_{0. At time}_{0. At time} _t_t55_{0 the surface at}_{0 the surface at} _xx55₀_{0 is}_is

ﬁx

ﬁxed ed at at ththe e tetempmpererataturure e of of 22₁₀₁₀88_C,_{C, wh}_whic_ich_{h is}_is

bel

below ow the the frefreeziezing ng popoint int of of the the subsubstastancence T T 55 m m

0

088_{C. As time proceeds, a solid layer builds up on}_{C. As time proceeds, a solid layer builds up on}

the surface

the surface xx55_{0 and moves out into the liquid.}_{0 and moves out into the liquid.}

Simply stated, the problem is to determine how Simply stated, the problem is to determine how the

the solsolid–liquid–liquid id sursurfacface e momoves ves witwith h timtime. e. ThThee thermal properties of the material in question are thermal properties of the material in question are assumed to be constant and equal for both solid assumed to be constant and equal for both solid

6 6

and

and liqliquid uid phphasease:: k k 55₂_{2 [W}_{[W// mK}_mK],_], _r_r cc55_2.5_2.533₁₀₁₀

3

3 8 8 33

[[J / m J / m KK] a] anndd r r L L55₁₁₀_{0 [J/m}_{[J/m ]].}_{. IIn}_{n n}_nu_ume_meri_riccal_al

4 4

analysis 50 time steps of

analysis 50 time steps of DD_t_t55_4.32_4.3233₁₁₀_{0 [[ss]}_{] (1/2}_(1/2

days) and 20 space increments of

days) and 20 space increments of DD_x_x55_{0.125 [m]}_{0.125 [m]}

were used. were used.

The position of the phase front after 25 days The position of the phase front after 25 days (i.e. 50 time steps) obtained with proposed (i.e. 50 time steps) obtained with proposed meth-od is

od is xx55_{0.8405 [m]. The difference between this}_{0.8405 [m]. The difference between this}

result and the result of

result and the result of xx55_{0.8415 [m] obtained by}_{0.8415 [m] obtained by}

Voller (1990) is merely 0.12%. Therefore, it could Voller (1990) is merely 0.12%. Therefore, it could be

be conconclucluded that ded that the the accaccurauracy cy of of the the proproposposeded comp

computatutational ional modmodel el for for condconductiouction n contcontrollerolledd isothermal phase change is satisfactory.

isothermal phase change is satisfactory.

4.2.

4.2. Experimental veriﬁcationExperimental veriﬁcation

As was discussed in the previous paragraph, the As was discussed in the previous paragraph, the

Fig. 6.

(7)

Tabl

Table e 1. 1. ThermThermophysiophysical propertical properties es of of CaC1CaC1 ??_6H_{6H O}_O

2

2 22 sides. A thermocouple was placed in the centre of sides. A thermocouple was placed in the centre of

Melting point [

Melting point [88_C_C]_] ₂₂₉_9..9₉ _{the container in the manner indicated in Fig. 7.}_{the container in the manner indicated in Fig. 7.}

L

Laatteennt t hheeaat t [[KKJJ // kkgg] ] 118877

3

3 ThThe e cocontntaiainener r wiwith th ththe e sosolilid d PCPCM M wawas s ththenen

D

Deennssiitty y [[kkgg // m m ]]: : SSoolliid d 11771100

placed vertically in the constant temperature bath, placed vertically in the constant temperature bath,

L

Liiqquuiid d 11553300 S

Sppeecciiﬁﬁc c hheeaat t [[kkJJ // kkggKK]]: : SSoolliid d 11..44 _{where the temperature was set to}_{where the temperature was set to} _T_T 55₆₀₆₀88_C._C.

` `

L

Liiqquuiid d 22..22

Th

The e cocompmpututatatioionanal l momodedel l wawas s seset t up up to to re

re--T

Thheerrmmaal l ccoonndduuccttiivviitty y [[WW/ m/ mKK]]: : SSoolliid d 11..0099

prod

produce uce expeexperimerimental ntal condconditionitions s withiwithin n the the con-

con-L

Liiqquuiid d 00..5533

stant temperature bath. The convection heat stant temperature bath. The convection heat trans-ver

very y satsatisfisfactactory ory resresultults s for for the the conconduductiction-on-concon- - fer coeffer coefﬁciﬁcient betwent between the air een the air and the contand the containainerer tro

trollelled d oneone-di-dimenmensiosional nal phaphase se chachangnge e testest t proprob- b- walwall l was detewas determirmined usinned using g the corrthe correlaelatiotion n givgivenen le

lem. m. HoHowewevever, r, memeltltining g of of ththe e PCPCM M in in sesealaled ed in Inin Incrcropoperera and Da and DeeWiWitt (19tt (1985)85), and w, and was caas calclcu-

u-2 2

con

contaitainerners s is is gengeneraerally lly mumultilti-di-dimenmensiosional (nal ( botboth h latlated to bed to bee a a 55₁₆_{16 [W/m}_{[W/m K]. Furth}_{K]. Furtherm}_ermor_ore,_{e, aa}

co co nnvv

radi

radial and al and axial axial condconductiouction exisn exist) and at) and also nalso natural tural time time stepstep DD_t_t55_{5 s and space increment}_{5 s and space increment} DD_x_x55₂_{2 mm}_mm

co

convnvecectition on ococcucurs rs wiwiththin in ththe e memeltlted ed PCPCM. M. ThTherere- e- wewere re usused ed in in ththe e cacalclcululatatioion. n. In In FiFig. g. 8 8 ththee for

fore, e, an an expexperierimenment t was was perperforformed med in in ordorder er to to varvariatiation ion witwith tih time me of nof numeumericrical aal and nd expexperierimenmentaltal inve

investigastigate the inte the inﬂuenﬂuence of thce of the assume assumptionptions of the s of the valuvalues of the es of the tempetemperaturrature at e at the centrthe centre of e of the testthe test ma

maththememataticical al momodedel. l. ThThe e PCPCM M usused ed fofor r exex- - cocontntaiainener r is is shshowown.n. pe

peririmementntal anaal analylysis is casis is calclciuium chlm chlororidide hexe hexahahy- y- FrFrom the reom the resusultlts s shshowown n in Figin Fig. . 8, it can be8, it can be dr

dratate e (Ca(CaClCl ??_{6H O)}_{6H O) wit}_with_{h the}_{the the}_thermo_rmoph_physi_ysical_{cal con}_conclu_cluded_{ded tha}_{that th}_{t the ag}_{e agree}_reemen_{ment be}_{t betwe}_tween_{en nu}_numer_merica_icall

2

2 22

pro

properpertieties s as as lislisted ted in in TabTable le 1 1 (Fuj(Fujii ii anand d YYanoano, , and and expexperierimenmental dtal data iata is wels well witl within ehin expxperierimenmentaltal 1

1999966)). . uunncceerrttaaiinnttiiees s ((ii..ee. . ppoossiittiioonniinng g oof f tthhe e tthheerrmmooccoou u--A re

A rectangctangular ular contacontainer, iner, made made of sof stainltainless sess steel, teel, ple’ple’s s tip exactly in tip exactly in the centre of the centre of the containthe container iser is with dimensions of

with dimensions of ll55_b_b55_{100 mm and}_{100 mm and} _d_d55₂₀_{20 mm}_{mm qu}_quit_ite_{e di}_difﬁ_fﬁcu_cult_{lt an}_and_{d ex}_expe_peri_rime_ment_ntal_{al da}_data_{ta we}_were_{re re}_read_ad

(Fig

(Fig. . 1), 1), was was ﬁllﬁlled ed witwith h the the calcalciucium m chlchlorioride de frofrom cm charharts, ts, whiwhich ch redreduceuces as accuccuracracy). y). ThThe he highigherer hex

hexahyahydradrate te and and welwell l insinsulaulated ted on on the the latlateraeral l sloslope pe of of the the thetheoreoretictical cal curvurve in e in the the liqliquid uid regregionion

Fig. 7.

Fig. 7. Test contaTest container.iner.

Fig. 8.

(8)

is du

is due to the fe to the facact that that the Grt the Grasashohof numf numbeber was r was veveststigigatated ed fofor r ththe e rerectctanangugulalar r anand d cycylilindndriricacall calculated for the temperature difference of

calculated for the temperature difference of T T 22 _containers._containers.

` `

T

T ¯mm¯30 K, whil30 K, while e in rein realalitity y ththis diis diffffererenence bece be- - ThThe e mamaththememataticical al momodedel l fofor r ththe e isisotothehermrmalal com

comes es smasmalleller r as as the tempethe temperatraturure e of of the liquithe liquid d phaphase chanse change of the PCM ﬁllge of the PCM ﬁlling thing the cyline cylindridricalcal PCM

PCM incincreareasesses. C. Consonsequequentently, ly, the the conconvevectictive ve heaheat t concontaitainer ner (Fig(Fig. . 9) 9) was was derderiveived d undunder er the the samsamee tr

tranansfsfer er cocoefefﬁcﬁcieient nt bebetwtweeeen n ththe e aiair r anand d ththe e asassusumpmptitionons s as in as in ththe e cacase of se of ththe e rerectctanangugulalarr con

contaitainer wall used in ner wall used in the calcthe calculaulatiotion n is is highigher her concontaitainerner. . HowHoweveever, r, the the gogoververninning g equequatiation on forfor tha

than n the the ‘re‘real’ al’ onone. e. FurFurthethermormore, re, it it can can be be obob- - the the two two momodeldels is s is difdifferferenent ant and fd for or the the cascase oe of thf thee serv

served froed from Fig. 8 thm Fig. 8 that in theat in theory PCory PCM reacM reaches its hes its cylincylindricadrical l contcontainer it has ainer it has the followthe following form:ing form: melting temperature faster than in the experiment.

melting temperature faster than in the experiment.

≠

≠_H_H ₁₁ ≠≠ _k_k ≠≠_T_T

This is assumed to be due to the basic assumption This is assumed to be due to the basic assumption

]

_≠_≠_t_t 55

]

_r_r

_]

_≠_≠_r_r

S DD

S

]

_r_r

_]

_≠_≠_r_r (21)(21)

made in the mathematical model that the made in the mathematical model that the conduc-tion resistance of the container wall is neglected. tion resistance of the container wall is neglected.

In

In ththe e ababovove e eqequauatitionon,, H H reprepresresentents s the the tottotalal Mo

Morereovoverer, , frfrom om ththe e sasame me ﬁgﬁgurure e it it cacan n be be ob

ob--enthalpy, which can be split into its sensible and enthalpy, which can be split into its sensible and served that the calculated PCM’s melting time is

served that the calculated PCM’s melting time is la

latetent nt cocompmpononenent t ((cfcf. . EqEq. . (4)(4)). ). ThThe e sosolulutitionon slightly longer than the experimental one, which

slightly longer than the experimental one, which

met

methohodoldology ogy and and numnumerierical cal proprocedcedurure e for for EqEq.. may be due to the fact that the natural convection

may be due to the fact that the natural convection

(21) is the same as for Eq. (3). (21) is the same as for Eq. (3). within the liquid PCM is ignored. However, it can

within the liquid PCM is ignored. However, it can be seen that neglecting both the natural be seen that neglecting both the natural

convec-5.1.

5.1. Results and discussionResults and discussion

tio

tion n witwithin hin the the liqliquid uid PCM PCM and the and the conconducductiotionn wit

within hin the the PCM PCM in tin the dhe direirectiction oon of thf the HTe HTF ﬂoF ﬂow w ThThe dimee dimensinsionons of the cos of the contantaineiners werrs were choe chosensen do n

do not inot introtroducduce sige signiﬁniﬁcancant errt error in tor in the prhe prediedictiction on in such in such a a manmanner that the ner that the vovolumlume e as as welwell l as as thethe of the

of the PCM’PCM’s s temptemperatuerature re variavariation during melt- tion during melt- convconvectivective heae heat trant transfer sfer area area for bfor both toth the cyhe cylindlindri- ri-iinngg. . ccaal l aannd d rreeccttaanngguullaar r ccoonnttaaiinneerrs s wweerre e eeqquuaall. . TThhee ﬁxed dimensions were chosen to be the length of ﬁxed dimensions were chosen to be the length of the cylindrical container

the cylindrical container ll 55_{0.2 m (cf. Fig. 9) and}_{0.2 m (cf. Fig. 9) and} c

c

5.

5. COMPCOMPARISON OF THE ARISON OF THE MELTIMELTING TIMENG TIME

the width of the rectangular container

the width of the rectangular container bb55_{0.1 m}_{0.1 m} FOR RECTANGULAR AND CYLINDRICAL

FOR RECTANGULAR AND CYLINDRICAL

(cf. Fig. 1). The air velocity was assumed to be (cf. Fig. 1). The air velocity was assumed to be

CONTAINERS CONTAINERS

w

w55₅_{5 m/}_{m/ s}_{s an}_and_{d it}_its_{s te}_temp_mper_erat_atur_uree _T_T 55₆₀₆₀88_C._{C. Th}_Thee

ai ai rr

The

The melmeltinting g timtime e of of the the incincapsapsulaulated ted PCM PCM is is thethermormophyphysicsical pal proproperterties oies of thf the aire air, as w, as well aell as ths thee on

one e of of the the essessentential ial parparameameterters s for deterfor determinmining ing conconvecvectivtive e heaheat t tratransfnsfer er coecoefﬁcfﬁcienient t betbetweeween n thethe the

the sizsize ane and thd the she shapape of e of the the concontaitainerner, as , as it mit must ust air air and and the the concontaitainerner’s ’s walwalll a a _co_{co n}_nvv, , wewere re tatakekenn cor

corresrespopond nd to to the the tottotal al amoamoununt ot of df dailaily iy insonsolatlationion. . frofrom m InIncrocroperpera a and and DeDeWitWitt t (198(1985). 5). The The PCMPCM To

To be be spspececifiific, c, ththe e cocontntaiainener r cocontntaiainining ng PCPCM M filfilliling the conng the contataininerers s wawas s chchososen to en to be calbe calciciumum sho

should uld be dbe desiesignegned in sd in such uch a waa way thy that aat at tht the ene end d chlchlorioride de hexhexahyahydrdrateate, , witwith h the the thethermormophphysiysicalcal of

of ththe e daday, y, cocompmplelete te memeltltining g of of ththe e PCPCM M is is prpropoperertities es as as liliststed ed in in TaTablble 1e 1.. ac

achihieveveded. I. In tn thahat wt wayay, t, the he mamaxiximumum em efﬁfﬁcicienency cy of of FiFig. g. 10 10 shshowows s ththe e vavaririatatioion n wiwith th titime me of of ththee the

the LHELHES S uniunit t is is achachievieved. ed. FurFurthethermormore, re, comcompleplete te PCMPCM’s ’s temtemperperatuature re in in the the cencentretres s of of boboth th thethe mel

meltinting og of thf the PCe PCM is M is a nea necescessarsary coy condinditiotion fn for or recrectantangugular lar and and cylcylindindricrical al concontaitaineners rs for for thethe lon

long-tg-term erm (s(seaseasononal) al) thethermarmal l eneenergy rgy stostoragrage. e. In In difdifferferent diment dimensensionions s of the contof the containainersers. For small. For small li

lighght t of of ththatat, , ththe e ininﬂuﬂuenence ce of of ththe e cocontntaiainener’r’s s vavalulues oes of f d d (cf. Fig. 1) and(cf. Fig. 1) and r r _o_o (cf. Fig. 9), the(cf. Fig. 9), the dim

dimensensionions s and and its its shashape pe werwere e numnumeriericalcally ly inin- difdifferferencence in e in melmeltinting tg time ime betbetweeween tn the rhe rectectangangulaularr

Fig. 9.

(9)

Fig

Fig. . 10. 10. ComComparparisoison n of of the variathe variatiotion n witwith h timtime e of of the PCM’s the PCM’s temtemperperatuature re at at the centrthe centre e of of the rectathe rectangungular and lar and cylcylindindricricalal containers.

containers.

and cylin

and cylindridrical cal concontaitainerners s is is not not so so propronounouncenced. d. PCM ﬁllPCM ﬁlling thing the contae containeiner on the meltir on the melting time of ng time of How

Howeveever, r, on on incincreareasinsing g the the masmass s of of the the PCM PCM PCMPCM. I. It cat can bn be oe obsebserverved td that hat for for larlarger ger ququantantitiitieses ﬁlling the container, i.e. with increasing

filling the container, i.e. with increasing d d for thfor the e of thof the matee material filrial filling ling the cothe containntainer, ther, the dife differeferencence recta

rectangulngular ar contacontainer iner andand r r _o_o for for the the cylcylindindricrical al in the mein the meltilting timng time betwe between theen the recte rectangangulaular andr and co

contntaiainener, r, ththe e didiffffererenence ce in in ththe e memeltltining g titime me cycylilindndriricacal l cocontntaiaineners rs is is vevery prory prononoununceced, withd, with inc

increareases ses conconsidsideraerablybly, , witwith h the the recrectantangulgular ar the the melmeltinting g timtime e of of the the cylcylindindricrical al concontaitainer ner beibeingng con

contaitainer ner shoshowinwing g a a mucmuch h shoshorterter r melmeltinting g timtime e neanearly trly twicwice thae that of tht of the rece rectantangugular olar one. Ine. It shot shoulduld tha

than thn the cye cylinlindrdricaical col contantaineiner of r of the the samsame ve voluolume me be be popointinted ed ouout t thathat t in in ordorder er to to makmake e the the com com--aannd d hheeaat t ttrraannssffeer r aarreeaa. . ppaarriissoon n bbeettwweeeen n tthhe e ttwwo o ggeeoommeettrriiees s ssiiggnniiﬁﬁccaanntt,,

A

A serseries ies of of numnumerierical cal expexperierimenments ts werwere e perper- - the numthe numerierical anacal analyslysis was perfois was performermed under thed under the for

formed and med and thetheir ir resresultults s are summaare summarizrized ed in in FigFig. . conconditdition of equion of equal volual volume and heame and heat transt transfer arefer areaa 11

11 which shows which shows the the inﬂuinﬂuence of ence of the the amouamount nt of of for for both both the the rectarectangulngular ar and and cylincylindricadrical l contcontainerainers.s.

Fig. 11.