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Mathematical solutions and numerical models employedfor the investigations of PCMs' phase transformations

Shuli Liu a,n, Yongcai Li a,b, Yaqin Zhang a

a Department of the Civil Engineering, Architecture and Buildings, Faculty of Engineering and Computing, Coventry University, CV1 5FB, UKb Faculty of the Urban Construction & Environment Engineering, Chongqing University, China

a r t i c l e i n f o

Article history:

Received 5 March 2013

Received in revised form9 January 2014Accepted 22 February 2014Available online 15 March 2014

Keywords:

Stephan problemNeumann problemHeat transfer mechanismsMathematical solutionsNumerical models

a b s t r a c t

Latent heat thermal energy storage (LHTES) has recently attracted increasing interest related to the thermaapplications, and has been studied by researchers using theoretical and numerical approaches. The hea

transfer mechanisms during the charging and discharging periods of the phase change materials (PCMs) andtwo basic problems for phase transformations have been discussed in this paper. 1D, 2D and 3D popularmathematical solutions based on the heat transfer mechanisms of conduction and/or convection have beenanalyzed. Then, various numerical models for encapsulated PCMs in terms of macro-encapsulated PCM andmicro-encapsulated PCM were investigated. The numerical simulation of heat transfer problems in phasechange processes is complicated and the achieved results are approximate according to a number of conditionsThe accuracy of numerical solution depends on the assumptions that are made by the authors. Therefore thisreview will enable the researchers to have an overall view of the mathematical and numerical methods usedfor PCM's phase transformations. It offers the researcher a guideline of selecting the appropriate theoreticasolutions according to their researches' purposes.

&2014 Elsevier Ltd. All rights reserved

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6602. Heat transfer mechanisms during the phase transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

2.1. Conduction and convection heat transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.2. Stefan problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.3. Neumann problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.4. Other possibility mechanisms in phase change process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662

3. Mathematical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6633.1. Conduction acting as the only heat transfer mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

3.1.1. Fixed grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6633.1.2. Adaptive grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

3.2. Conduction and convection heat transfer mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6654. Numerical models for various PCMs capsules and packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

4.1. Macro-encapsulated PCMs numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6674.1.1. Rectangular encapsulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6674.1.2. Cylindrical containment models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6694.1.3. Spherical containment models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

4.2. Microencapsulated PCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6704.2.1. Incorporation with building components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6704.2.2. Microencapsulated phase change slurry (MPCS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

4.3. Summary of numerical models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6725. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672

Contents lists available at ScienceDirect

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http://dx.doi.org/10.1016/j.rser.2014.02.0321364-0321 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.:44 24 7765 7822.E-mail addresses:[email protected],[email protected](S. Liu).

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1. Introduction

Due to the attractive features of latent heat storage, phasechange materials (PCMs) are mainly used to store energy at a xedtemperature (melting/solidication temperature) with highenergy density [1]. PCMs have been applied in various systemsand aspects such as energy storage systems, free ventilation, freeheating and cooling for buildings, spacecraft, food, medicineconservation, smoothing the peaks of exothermic temperature inchemical reactions etc. PCMs are a better choice comparing to thesensible heat storage in applications, because of its nearly iso-thermal storing mechanism and high storage density. PCMs absorbabundant energy through the phase transition and release thestored energy.Table 1compares the typical properties of differentthermal storage materials tested at the room temperature.It indicates that PCMs can save 92.8% of mass and up to 90% spaceto store the same amount of thermal energy comparing to thesensible thermal materials such like concrete and water.

Therefore, PCMs attract the researchers' interesting in practicallyindividual and incorporation applications in different engineeringelds. The possibility, feasibility, thermal performance and economic

analysis of using PCMs call a series of theoretical and experientialinvestigations, the mainly two methods used to research the thermo-physical parameters of PCMs and interrelated systems.

The experimental approaches offer a better indication of theactual PCM behavior and performance in comparison to theore-tical analysis, as the experimental tests can present the PCMs'behaviors more directly, visibly and credibly without any pre-setassumptions however, the experiments are unachievable in somecases, such as the large scale or unsteady around environment, sothere are still a few points need to be considered:

Time and cost consuming will be higher than a theoreticalapproach, hence, the budget and processes needs to be estab-lished and scheduled.

The scales of the experiment test rig have to be considered, andthen suitable laboratory location and space needs to bedetermined to house the rig.

Test rig needs to be constructed and operated properly. Proper environment parameters have to be simulated and

adjusted to imitate the practical environment. Relevant parameters need to be monitored, measuring appara-

tuses need to be calibrated, and the failed data need to beeliminated.

Except these agonizing experimental matters, there are stillsome unavoidable testing errors. However the theoretical methodscan avoid all these weakness and predicate the PCMs performanceconveniently. The major advantage of the theoretical/numerical

approaches is that various conditions can be carried out bychanging the variables in a numerical model. Therefore, moreand more investigators prefer to study the phase change problemsby mathematical solutions and numerical simulations. There areonly eight review papers on PCMs that involves the aspect ofmathematical solutions and/or numerical modeling of latent heatthermal energy storage (LHTES) have been published during20022012.Table 2summaries these eight relative review papersand comments their mathematical and/or numerical aspects ofLHTES. However, few logistic and comprehensive reviews on themathematical solutions and numerical modeling for melting andsolidication processes of PCMs were found in published litera-tures even though there are many individual publications in phasechange problems, particularly in heat transfer mechanisms and

simulations. Due to the complexity of phase transformations,

Nomenclature

f melt fractionc specic heat at constant pressureH total volumetric enthalpy (J/m3)h average heat transfer coefcient (J/kg)k thermal conductivity (W/mK)

L latent heat (J/kg)St Stefan numbert time (s)T temperature (K)Tm melting temperature (K)Ts initial temperature of solid PCM (K)T0 constant temperature imposed onx 0 (K)P present nodal pointt position of liquidsolid interfacex;y Cartesian coordinates

Greek symbols

differencep thermal diffusivity of PCM (m2/s) dimensionless number used in derivations as a tem-

porary substitution dimensionless number in solution to Neumann pro-

blem, liquid fraction density (kg/m3)

Subscriptsl liquids solidi ith spatial stepj jth time stepef f effective

Table 1

Properties comparison of different thermal storage materials.

Property Unit Materials

Concrete (sandand gravel)

Water OrganicPCM

InorganicPCM

Density [kg=m3] 2240 1000 800 1600

Specic heatcapacity

[kJ=kg K] 1.1 4.2 2.0 2.0

Latent heat [kJ=kg K] 190 230Storage mass for

106 kJ, avg[kg] 60,000 16,000 5300 4350

Storage volume for106 kJ, avg

[m3] 26.8 16 6.6 2.7

Relative storagemass

13.79 4 1.25 1.0

Relative storagevolume

10 6 2.5 1.0

1. Storage mass and volume are calculated for storing 106 kJ energy with atemperature rise of 15 K for concrete (sand and gravel) and water.

2. Relative mass and volume are based on the inorganic phase change materials.

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a good understanding of the heat transfer mechanisms, phasechange characteristics and the differences among various mathe-matical and numerical simulation methods is required beforeresearchers starting their theoretical study. The aim of this paperis to comprehensively study the mathematical and simulationmodeling applied for LHTES. Firstly it discusses the issues of theheat transfer mechanisms during the charging and dischargingprocesses and the Stephan problem and Neumann problem.Secondly, it sums the strong and weak aspects of various mathe-matical solutions when conduction and convection heat transferoccurring inside the PCMs by the considering xed grid methodand the adaptive grid method. Finally, numerical modeling ofvarious PCM packages' geometries in terms of macro-encapsulatedPCMs and micro-encapsulated PCMs are studied.

2. Heat transfer mechanisms during the phase

transformations

2.1. Conduction and convection heat transfer

During the charging and discharging processes, the possibleheat transfer mechanisms are conduction and convection. How-ever, the issue of which heat transfers mechanism takes the mainrole in different stages of phase transformation has been arguedfor decades. When PCMs are used to store or release thermalenergy, conduction is usually believed playing the most importantrole on the heat transfer during the phase transformation pro-

cesses[24]. As early as 1831, Lam and Clapeyron have conductedthe rst study on phase change problem by only considering thepure conduction. However, some researchers persist that naturalconvection is a more important mechanism in the phase changeprocess especially in the melt region. Lazaridis [100] proposed astudy of the relative importance of conduction and convection in1970. A pioneer study performed by Sparrow et al. in 1977 [41],they concluded in their study that natural convection could notbe ignored in the analysis of phase change problems. In 1994,Hasan[5] concluded that the convection heat transfer active animportant role in the melting process, and a simplied model byonly considering the conduction heat transfer does not describethe melting process properly. Later, Lacroix et al. [6] reportedthe similar ndings in their research that natural convection is

the main heat transfer mechanism during the melting process.

In 1999, Velraj[7]obtained the same conclusion in their researchand reported that during the melting process natural convectionoccurs in the melt layer which results in the heat transfer rateincrease comparing to the solidication process. Buddhi et al. [8proposed an explanation for this phenomenon that the densitydifferences between the solid and liquid PCM induce the buoy-ancy, which causes the convective motions in the liquid phaseHowever, Zhang and Yi [9] believed that with the solid PCMmelting into liquid, the PCM volume keeps increasing, whichresults in the convection becoming the predominant heat transfermechanism rather than conduction. Sari et al. [10]found that theheat transferred from a heat exchanger to a PCM of stearic acid islargely inuenced by natural convection at the melting layer, inaddition to conduction and forced convection heat transfer.

For the melting process, the PCM changes its phase from solidto a mushy state, and then liquid, which is reversible during thesolidication process. Hence there are six stages to nish acharging and discharging cycle. Therefore it is possible in certainstages of the phase transformation process there are more thanone kind of heat transfer mechanisms acting at work, but how toweight the percentages of conduction and convection heat transfein each stage has been the main challenge for the researcherscurrently. This paper will review the most common researchmethods used for the conduction and convection heat transferinside the PCM packages.

2.2. Stefan problem

Moving boundary problem named Stefan problem is anotherissue to develop a numerical modeling of PCMs [11,12]. Thesimplest of the Stefan problems is the one-phase Stefan problemsince only one-phase involved. The term of one-phasedesignatesonly the liquid phases active in the transformation and the solidphase stay at its melting temperature. Stefan's solution withconstant thermophysical properties shows that the rate of meltingor solidication in a semi-innite region is governed by a dimen-sionless number, known as the Stefan number (St),

StClTl TmL

1

where Cl is the heat capacity of the liquid PCMs, L is the latentheat of fusion, and Tl and Tm are the surrounding and melting

temperatures, respectively.

Table 2

Some review papers related to numerical study of phase change problems.

Reference Publishedyear

Journal Comment

Zalba et al.[1] 2003 Applied thermalengineering

The review was divided into three parts: materials, heat transfer and applications. Heat transferwas considered mainly from a theoretical point of view, considering different simulation techniques.

Verma et al.[2] 2008 Renewable and sustainableenergy reviews

Mathematical models of a LHTES were reviewed to optimize the material selection and to assistin the optimal designing of the systems. Some important characteristics of different models and their

assumptions used were presented and discussed as well.Jegadheeswaran

and Pohekar[3]2009 Renewable and sustainable

energy reviewsVarious thermal conductivity enhancement techniques reviewed in this paper, and the issues relatedto mathematical modeling of LHTES with enhancement techniques are also discussed.

Zhu et al.[4] 2009 Energy conversion andmanagement

This paper presented an overview on dynamic characteristics and energy performance of buildingsemploying PCMs by three research methods used, i.e., simulation, experiment, combinedsimulation and experiment

Agyenim et al.[5] 2010 Renewable and sustainableenergy reviews

This paper provided the formulation of the phase change problem. In terms of problem formulation,it was concluded that the common approach has been the use of enthalpy formulation.

Kuznik[6] 2011 Renewable and sustainableenergy reviews

The review was the rst comprehensive review of the integration of phase change materials in buildingwalls. Various numerical studies concerning the integration were summarized.

Dutil et al.[7] 2011 Renewable and sustainableenergy reviews

The review presented models based on the rst law and on the second law of thermodynamics. This papertried to enable one to start his/her research with an exhaustive overview of the subject.

Zhou[8] 2012 Applied energy This paper reviewed previous works on latent thermal energy storage in building applications, includingPCMs, current building applications and their thermal performance analyses, as well as numericalsimulation of buildings with PCMs.

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The most published solutions are to solve the one-dimensionalcases subjected to an innite or semi-innite region with simpleinitial and boundary conditions. Whereas with the heat transfercontinuing, the interface boundary is constantly moving as theliquid and solid phases shrinking and growing, which disable theprediction of the boundary location[13]. Because of that the solidliquid interface is not xed, but moving with time, the heattransfer mechanisms during a PCM phase transformation process

are complex. Therefore the phase change transition is difcult toanalyze owing to the three reasons: the solid liquid interface ismoving; the interface location is non linear; it consists of thermalconduction and natural convection heat transfer mechanisms.Since these three factors, the non-linearity of the governingequations is introduced to the moving boundary, and the preciseanalytical solutions are only possible for a limited number ofscenarios [14]. This view is shared by Krkl et al. [15] whopointed out that the mathematical models have been proposed ineach research only applied to very specic boundary conditions,hence are not feasible to more complex practical applications. ThisStefan problem is further complicated by the fact that most of themethods previously proposed by researchers only involve onemoving boundary, whereas it actually consist of more than one

interface location[16].

2.3. Neumann problem

The Stefan problem was extended to the two-phase problem,the so-called Neumann problem which is more realistic [17].In Neumann problem, the initial state of the PCM is assumed tobe solid, during the melting process, its initial temperature doesnot equal to the phase change temperature, and the meltingtemperature does not maintain at a constant value. If the meltinghappens in a semi-innite slab (0oxo1), the solid PCM isinitially at a uniform temperature TsoTm, and a constanttemperatureT0 is imposed on the slab surface x 0, with theassumptions of constant thermo-physical properties of the PCM,

the problem can be mathematically expressed as follows:Heat conduction in solid or liquid region

CT

t k

2T

x2 2

where is the density, C is the specic heat, k is the thermalconductivity, and t and x are the time and space coordinatesrespectively.

The heat uxes transferring from the liquid phase to the solid liquid interface, as well as the latent heat absorbing rate by themelting PCM, the movement of the solid/liquid interface can bedetermined through the following energy balance:

Lxtt

klTlx

ksTsx

3

whereLis the latent heat of fusion of the PCM and is the densityof liquid PCM.

Initial condition

Tx40; t0 TsoTm 4Solidliquid interface temperature

Tx t; t40 Tm 5With the following boundary conditions:

T0; t T04Tm for t40 6

Tx; t Ts for x-1; t40 7where

t

is the position of the solidliquid interface (melting

front).Fig. 1clearly illustrates this two-phase Stefan problem.

Analytical solution to such a problem was obtained by Neu-mann in term of a similarity variable

t2 ffiffiffiffiffiffiffiffip l

p 8

The nal Neumann's solution can be written asInterface position

t 2ffiffiffiffiffiffiffiffiffiffip lt

p 9

Temperature of the liquid phase

Tx; t Tl Tl Tmerfx=2 ffiffiffiffiffiffiffiffiffiffip ltp

erf 10

Temperature of the solid phase

Tx; t Ts Tm Ts erfcx=2 ffiffiffiffiffiffiffiffiffiffip stp

erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip l=p s

p 11

The in Eq. (9)(11) is the solution to the transcendentalequation

Stl

exp2erf Sts

ffiffiffiffiffiffiffiffip s

pffiffiffiffiffiffiffiffip l

p expp l2 =p serfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip l=p s

p

ffiffiffiffip

12where

StlClTl Tm

L 13

Sts CsTm Ts

L 14

However, the Neumann's solution is applicable only for movingboundary problems in the rectangular coordinate system.

2.4. Other possibility mechanisms in phase change process

Furthermore, there are several other issues with the use of atheoretical approach in the study of PCMs. Alexiades [18]pointedout that there were many mechanisms involved during a PCMphase transition, such like a change in volume, density, thermalconductivity, specic heat capacity, super-cooling, etc. Conse-quently, accurate reection of the proposed mathematical solutionand numerical model need to consider the dynamic properties ofthe phase change process. Another major issue with PCMs is thatthey act as self-insulating materials. When PCM solidicationoccurs from the top of the heat surface, solid insulating layer willbe developed which moves inward during the whole solidicationprocess. With the increase in the size and thickness of the solidlayer, the heat transfer rate from the liquid PCM to the heat

exchanger surface decreases until it becomes so small that will not

Fig. 1. Schematic illustration of the two-phase Stefan problem.


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be possible to maintain at an acceptable heat transfer rate. Theself-insulating characteristic of the PCM becomes a strong problemwhen bulk containerization is used. For example the large cylind-rical used as containers lled with PCM, the self-insulating issuecan become very serious. Large heat exchange area will acceleratethe formation of the crust and heavily decrease the heat transferrate. Radhakrishnan et al.[14]analyzed that the method to lightenthe crust issue is minimizing the depth of the PCM pack. So that

even a large majority of the PCM performs solidication, thethermal resistance is reduced to allow the removal of the latentheat released from the solidied PCM. The self-insulating char-acteristic of the PCM during discharging process needs to beconsidered in any mathematical model as this can affect thermalenergy discharging time and rate. Furthermore, the self-insulatinghas a direct effect on the phase transition boundary, where thesolid and liquid region meets and co-exists, which is linked back tothe Stephan Problem and Neumann's method again.

Logical heat transfer mechanisms during the charging anddischarging periods of the PCM are the essential elements todevelop any accurate mathematical model. However, as describedthere are quite a few issues to properly describe the heat transfermechanisms: conduction and natural convention during the phase

change process. This paper reviews most of the researchesemploying the mathematical calculation and numerical simulationpublished in the articles and reports, but does not nd anyconsensuses on which one is the dominant heat transfer mechan-ism, which one can be ignored and how to combine these twotransfer mechanisms in each modeling. Different researchers havedeveloped their specic methods to evaluate the thermal energytransfer during the phase transition process for various situations[1923].

3. Mathematical solutions

The analyzes of the heat transfer mechanisms in phase changeprocesses are complex owing to the movable solidliquid bound-

ary, which depends on the latent heat absorbing or releasing rateat the interface. In order to simplify the simulation of the phasechange problems, assumptions have been made by variousresearchers in their studies. In this section, different numericalsimulations in terms of conduction heat transfer and conductionand/or convection heat transfer are discussed and analyzed.

3.1. Conduction acting as the only heat transfer mechanism

To develop a mathematical model, the dominant heat transfermechanism whether conduction or convection, during the char-ging and discharging processes, need to be analyzed and deter-mined. Although there are quite a few issues in the heat transfermechanisms during the phase transformation, some researchers

simplify this process by only considering conduction in pure sub-stances. In this case, the mathematical solutions can be classiedas follows: xed grid and adaptive grid.

3.1.1. Fixed grid

Fixed grid method does not require explicit treatment ofconditions on the phase change boundary, and just requires theuse ofxed-grid schemes able to cope with strong nonlinearities.Therefore this method is able to utilize standard solution proce-dures for the energy equations. They are amenable to physicalinterpretation and easy to implement, especially for multidimen-sional problems and for mufti-interface problems.

In the xed grid method, the heat ow equation is approximatedby nite difference replacements for the derivatives in order to

calculate the values of temperature Ti;j, at any time tnnt, the

moving boundary will be located between two adjacent grid pointsfor instance, between mx andm1xat a location ofx;where 0oo1, as illustrated inFig. 2.

The mathematical solution of the one-phase ice-melting pre-

sents a simple illustration of this method. The instantaneoustemperature at point i;j1 is calculated from the temperaturesof the previous step on the basis of the following formulation:

Ti;j 1 Ti;j t

x2

Ti 1;j 2Ti;j Ti 1;j i 0; 1;m1: 15

In terms of three-point Lagrange interpolation instead oEq.(14), the temperature at x mxis computed as,

Tm;n1 Tm;n 2tx2

1n 1

Tm 1;n 1n

Tm;n

16

The variation of the location of the moving boundary is

n 1n tx2

nn 1

Tm 1;n 1nTm;n

17

As shown, the numerical solution of thisxed grid method iscarried out on a space grid remaining at a xed value throughouthe mathematical calculation. To approximate the Stefan condi-tions both for the moving boundary and the partial differentialequation at the adjacent grid points, various mathematical solu-tions have been proposed. Murray and Landis[24]proposed twoctitious temperatures: one is achieved by quadratic extrapolationfrom the temperatures in the solid PCM and the other one is fromthe temperatures in the liquid PCM. The melting temperature andthe position of the moving interface are incorporated in thectitious temperatures, which are then substituted into a stan-dard approximate to calculate the temperature near the inter

face instead of using special formulae. Ciment and Guenther [25introduced a method of spatial mesh renement on both sides othe moving boundary. With this method, Lazaridis[26]applied theexplicit nite difference approximations to solve two-phase soli-dication problems in both two and three space dimensions.

The major advantage of the xed grid method is that themathematical calculation of the phase transition can be achievedthrough simple modications of existing heat transfer numericamethods and/or software. As such, they can be used for modeling oa variety of complex moving boundary problems. Basu and Date [27and Voller et al. [28] reviewed the applications of the xed gridmethods for phase change problem. The two mainly xed-gridmethods that have been proposed are enthalpy-based methodsand the effective heat capacity method. However, the xed grid

method sometimes cannot work efciently when the boundary

Fig. 2. Position of the moving boundary in a xed grid.


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moves along a distance larger than a space increment in a time step,depending on the velocity of the moving boundary. Therefore, theadaptive grid is developed by researchers and will be discussed in

Section 3.1.2.

3.1.1.1. Enthalpy based method. The existence of nonlinearity of theliquidsolid interface and the unknown location of the movingboundary are the two main challenges in simulating the phasechange problems[29]. To deal with these, a new model formu-lation called the enthalpy method was introduced [30]. In theenthalpy models, the enthalpy is used as a dependent variablevalue along with temperature. By introducing an enthalpy method,the phase-change problem becomes much easier since thegoverning equations are the same for the two phases. Theinterface conditions are automatically achieved and a mushyzone between the two phases is created. This zone avoids sharpdiscontinuities that may create some numerical instability. Hence,

the enthalpy mathematical methods are most attractive andcommonly used to handle phase change problems where asolidliquid mush zone is present between two phases. Hunter[31] conrmed that the enthalpy method is most suitable fortypical applications, the reason for that is this method does notrequire explicit treatment of the conditions on the phase changeboundary[32].

Enthalpy-based method can be illustrated by considering aone-dimensional heat conduction controlled phase transforma-tion. For a phase change process, energy conservation can beexpressed in terms of total volumetric enthalpy and temperature.This relationship between the enthalpy and temperature is usuallyassumed to be a step function for isothermal phase change problems.Fig. 3shows the enthalpytemperature curves for isothermal phase

change.For isothermal phase change, the conservation of energy can be

expressed in terms of temperature and the total volumetricenthalpy[33]

H

t kT 18

where the total volumetric enthalpy His the sum of sensible andlatent heat, and can be expressed in terms of temperature of thePCM as follows:

HT HTlfT 19

The rst term on the right side of Eq. (19)accounts for the sensibleheat of the PCM while the second term accounts for the latent heat

of the PCM.

And where

HZ T

Tm

cdT 20

To be able to calculate the total enthalpy, the liquid fractionisgiven as follows:

0 ToTm solid

0; 1 T Tm mushy1 T4Tm liquid

8>: 21

Integrating the Eqs. (18) and (20), an alternate form for one-dimensional heat transfer in the PCM was obtained:

H

t

x

H

x

l

f

t 22

whereis the thermal diffusivity.

3.1.1.2. Effective heat capacity method (energy based method). Theeffective heat capacity method is also a common method used inmodeling phase change problems except for the enthalpy method.The effective heat capacity method was rst proposed in[34,35]. Inthe effective heat capacity method, the latent heat is approximatedby a large heat insensible form over the phase change temperatureinterval, (T2 T1)[36]. The effective heat capacity of the PCM (cef f)is directly proportional to the energy stored and released during thephase change but inversely proportional to the interval of themelting or solidication temperature range. During the phasechange the heat capacity of the PCM is

cef f L

T2 T1cs 23

where T1 is the temperature at which melting or solidicationbegins andT2is the temperature at which the PCM is totally meltedor solidied. The following is a denition of the effective heat

capacity for each phase change period.

cef fcs ToT1 solid

LT2T1 cs T1 rTrT2 mushycl T4T2 liquid

8>: 24

In terms of the denition of the effective heat capacity, theenergy equation for one dimension can be written as follows:

cef fT

t k

2T

x2 25

Although the enthalpy based method and the effective heatcapacity method achieve much simpler mathematical operations

with reasonable accuracy, it is worth noting that each of themethods has its inherent disadvantages. Enthalpy method isdifcult to handle super-cooling and temperature oscillationproblems. Whilst the effective heat capacity method is difcultin resolving the phase change problems when the phase changetemperature range is small, and is not applicable for the caseswhere phase change occurs at xed temperature.

Two approaches, nite-difference and nite-element techniques,are rstly used to solve the phase change problems numerically.The rst nite-difference studies were carried out by Lam andClapeyron in 1831 and Stefan in 1891, concerning ice formation.In 1912 the results of Neuman were published, establishing the precisesolutions to more general phase change problems. In 1943, Londonand Seban [37] analyzed the process of ice formation for different

geometries (cylinder, sphere, and at plate). However, Shamsundar

Fig. 3. Relationships between enthalpy and temperature for isothermal phasechange.


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and Srinivasan[38]asserted that London's one-dimensional formula-tion led to errors by increasing the progress of the solidicationprocess and proposed a two-dimensional formulation for cylinders. Aone-dimensional formulation was presented by Bonacina et al. toassess the accuracy of the effective heat capacity method[39]. Rabinand Korin presented a multidimensional numerical solution based onthe effective heat capacity method for solving transient phase changeproblems by using the nite-difference method[40]. The proposed

mathematical method showed a good agreement with two exactsolutions and two numerical solutions based on nite-difference andnite-element methods.

Although the nite-difference method has been traditionallyemployed for solving phase change problems in the past, oscilla-tions in the front position and/or temperature occur due to a poorheat balance when the phase change front lies between nodes in anite difference solution [41,42]. Nowadays the nite-elementmethod has been more attractive, because of the ability of thenite-element method to handle complex coupled thermomechanicalmechanisms with various and complex boundary conditions [43].Consequently, nite-element methods can either accurately repre-sent the temperature history or track the phase front, or both. Afew earlier studies based on nite-element formulations were

carried out by Rubinsky and Cravahlo in 1981 [44]and Ettouneyand Brown in 1982[45], concerning on the location of the solidliquid interface. To deal with oscillations, Comini et al. [46] andMcAdie et al. [47] developed nite-element based enthalpyformulations, and applied the three level PredictorCorrectorand Regula-Falsi methods to account for the sudden jumps orsharp changes in the enthalpytemperature relationships. Bhatta-charya et al. developed a new enthalpy based method to studyphase change in a multicomponent material using the Galerkinnite-element method. This method can efciently capture multi-ple thawing fronts which may originate at any spatial locationwithin the sample [48]. Hence, nite-element methods are pre-ferred to be used in enthalpy approaches and the effective heatcapacity method.

Beside the nite-difference and nite-element methods, thenite-volume and control-volume- nite-element methods werealso employed to handle the phase change problems[39].

3.1.2. Adaptive grid

The problem associated with the xed grid method can be dealwith by utilizing the adaptive grid method to improve thecomputational efciency, using the so-called adaptive grid (orfront tracking) numerical schemes to capture the moving bound-ary [49]. In the adaptive grid method, the exact location of themoving boundary is evaluated on a grid at each step. There are twomain approaches used to achieve this: the interface-tting gridsand the variable-space grids. The interface-tting grids (alsoreferred to as the variable time step methods), where a uniformspatial grid but a non-uniform time step are used, has been

repeatedly employed to solve two-phase and one-dimensionalproblems. The another approach is variable-space grids, alsoknown as the dynamic grids, where the number of spatial intervalsis kept constant and the spatial intervals are adjusted in such amanner so that the moving boundary lies on a particular gridpoint. Thus, in these methods the spatial intervals are a function oftime. The applications of the variable grid method to study thephase change problem can be found in[5054].

Lacroix and Voller [55] performed a study on simulation ofdiffusion/convection controlled solidication processes in a rec-tangular cavity by using the nite-difference techniques. Theyconcluded that the xed grid must be ner for solving the meltingproblem of a material with a unique melting temperature. Com-parison between the xed and adaptive grid has also been done by

Bertrand et al.[56]. They found that the adaptive grid method is

better adapted to the melting problem than the xed grid methodHowever the adaptive grid method may fail to simulate thesituations where the transition from the liquid to the solid phaseis not a macroscopic surface.

Table 3 presents some references of the mathematical andnumerical studies undertaken with the only consideration of theconduction heat transfer.

3.2. Conduction and convection heat transfer mechanisms

During the phase transformation especially melting processthe temperature and concentration gradients in the liquid phase oPCM keep varying, which results in the movement of the liquidPCM, named convection occurring under the action of buoyancyforces due to the density gradients. In the previous numericainvestigations, the convection heat ow in the liquid phasereceived less attention than conduction owing to the limitedcomputational capabilities of computer and the mathematicacomplexities to formulate the convection heat transfer duringthe phase transformation. The inuence of the natural convectionon the phase change problems were initially considered in 1977The study of the possible role of the natural convection in the melt

region has been conducted by Sparrow et al. in 1978 [67]. Thendings of this study indicated that the natural convection is rst-order important and has to be accounted in the phase transforma-tion analysis. A number of researchers[6871]have also reportedthat the convection affects not only the rate of melting osolidication but also the resulting of the distribution of the liquidphase of the PCM in a multi-component system. One of thepioneer numerical studies was carried out by Yao and Chen in1980 [72], by using an approximate solution. They studied theeffect of the natural convection on the phase change process andconcluded that it strongly depends on the Rayleigh number. Theequation ofke=kl cRan includes an effective of thermal conduc-tivity was used to describe the inuence of natural convection onthe melting process[73,74].

The inuence of the convection heat transfer on the meltingand freezing processes has been studied by Lamberg et al. [64]they found that when the effect of natural convection is omitted inthe simulation. The numerical result was poor compared with theexperimental result during the melting process, whilst it showed agood estimation during the freezing process. Hence it is essentiato model natural convection in the liquid PCM during the meltingprocess. Besides the numerical methods mentioned inSection 3.1the integral method[75], boundary xing method[76], unstruc-tured nite-element method [77], enthalpy-porosity approach[7880], coordinate transformation method[81,82]and equivalenthermal capacity method [83] have been developed for naturaconvection simulation.

Yao and Cherney[84]studied the effect of the natural convec-tion on the melting of a solid PCM around a hot horizontal cylinder

by using the integral method. The results demonstrated that theintegral solution had surprising accuracy when it was comparedwith the quasi-steady solution when Stefan number was smallRieger et al.[85]investigated numerically the melting process of aPCM inside a heated horizontal circular cylinder. Both heat con-duction and convection have been taken into account to treat thismoving boundary problem, and the complex structure of the time-wise changing physical domain (melt region) have been success-fully overcome by applying body tted coordinate techniqueIsmail and Silva[86]developed a 2D mathematical model to studythe melting of a PCM around a horizontal circular cylindeconsidering the presence of the natural convection in the meltregion. A coordinate transformation technique was used to x themoving front. The numerical predictions were compared with

available results to establish the validity of the model, and a


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Table 3

Numerical studies by only considering the conduction heat transfer.

Reference(s) Dimensional Numericalsimulation method

Assumptions Comment Validation

Silva et al.[57]

1D Enthalpy basedmethod

1-D conduction controlled the phasechange process.

The uid ow was fully developed withtemperature independent properties.

Thermal radiation between the walland uid was neglected.

The model can be used to predict the dynamicperformance of this kind of vertical rectangularheat exchange unit with reasonable accuracy

Yes

Krkl et al.[15]

2D Energy basedequation

The phases were homogeneous. Conduction heat transfer was

only considered. Heat loss or gain from the store was

neglected.

Calculating the temperature of air and PCMat the formerly dened volume nodes.

Yes

Costa et al.[3]

2D Energy basedequation.

The thermophysical properties of the PCMand n were independent of temperature.

The PCM was initially in the solid phase. The PCM was homogenous and isotropic.

Calculations have been made for the melt fractionand energy is stored by conduction only

Numerical[33,58]

Fully implicitnite-differencemethod.

Gong et al.[59]

2D Enthalpy basedmethod.

The heat-transfer uid was incompressibleand viscous dissipation was negligible.

The uid ow was radially uniform andthe axial velocity was an independentparameter.

Thermal losses through the outer wallof the PCM were negligible.

Conduction heat transfer was themainly transfer method.

For mode1, simulation has been done byintroducing

hot and cold uids from the same end of the tube.While for mode 2, simulation has been done byintroducing hot and cold uids from the differentends of the tube.

No

Standard Galerkinnite-element

Sharma et al.[60]


The thermo-physical properties of PCM's andn material were independent oftemperature but different for solid and liquidphases.

The PCM was initially in solid phase. ThePCM was homogeneous and isotropic.

The mode of heat transfer wasconduction only.

Calculations were made to study the effect ofthermal conductivity and thermal capacity of fattyacids, and conductivity of heat exchanger materialsand their effect on melt fraction.

No

Fully implicitnite-differencemethod

Dwarka andKim[61]

3D Implicit enthalpybased method.

Air temperature was constant. Initial temperatures were the same through

the board. There was no energy loss to surroundings.

Conduction heat transfer was onlyconsidered.

All thermo-physical properties wereconstant except for the heat capacity.

Comparison of the thermal performance ofrandomly mixed and laminated PCM drywallsystem

No

Dolado et al.[62]

1D Finite-differencemethod

Only conduction heat transfer wasconsidered inside the PCM plate.

The temperature of the airow was analyzedin a one dimensional way.

A model was developed to simulate theperformanceof a LHTES, and analyze the heat transfer betweenthe air and a commercial macroencapsulated PCM.

Yes

Fukai et al.[63]


The surfaces of the heat exchanger wereadiabatic.

The cross-sections of the tubes was square

The convective heat transfer term wasneglected.

A numerical model predicted well the experimentaloutlet uid temperatures and the localtemperaturesin the composite.

Yes

Control-volumemethod

Lamberg et al.[64]

2D Enthalpy basedmethod and Effectiveheat capacitymethod

The heat conductivity and density of the PCMand container were constant.

The physical properties chosen for the PCMwere the average values of solid andliquid PCM.

The convection heat transfer was negligibleduring solidication process.

Both numerical methods gave good estimations forthe temperature distribution. Particularly, whenthe temperature range was 2 1C, the effective heatcapacity method was the most precise numericalmethod.

Yes

Zivkovic andFujii[65]


The effects of natural convection within themelt were negligible and can be ignored.

The PCM behaved ideally. The PCM was assumed to have a denite

melting temperature.

The cylindrical container required nearly twice ofthe melting time as for the rectangular container ofthe same volume and heat transfer area

Yes


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satisfactory agreement was found. They concluded that thenumerical model was adequate to represent the physical situationof the proposed system.

Furzerland [87] investigated the enthalpy method and thecoordinate transformation method through the comparison ofthe solutions of specic problems of one dimensional heat transferby considering pure convection. One of his conclusions was thatthe enthalpy method is very attractive owing to these: it is easy toprogram and there are no computational overheads associated

with tracking the moving interface within a speci

c range offusion temperatures.A summary for various numerical solutions on phase change

problem considering both conduction and convection is presentedinTable 4.

4. Numerical models for various PCMs capsules and packages

It can be easily found that solidliquid PCMs have been widelyused for studies on phase change problems compared to solid solid, liquidgas and solidgas PCMs. This is due to their large heatstorage capacity and little volume change during the phase changeprocess. Since solidliquid PCMs experience a phase transitionduring the charging process and/or charging process, encapsula-

tion is required for holding the liquid and/or solid phases of thePCM and keeping it isolate from the surrounding. This ensures theexibility in frequent phase change processes, and an increase inheat transfer rate. It is worth noting that the heat transfercharacteristics in the encapsulated PCM would change signi-cantly depending on the parameters of various encapsulations.Hence, vast of numerical studies on phase change problems haveextended to engineering applications. In this section, a compre-hensive review on numerical models for encapsulated PCMs interms of macro-encapsulated PCM and microencapsulated PCM ispresented.

4.1. Macro-encapsulated PCMs numerical models

Macro-encapsulation is a common form of encapsulation usedfor thermal storage applications. The shape of macro-encapsulatevaries from rectangular panels, spheres to cylinders.

4.1.1. Rectangular encapsulation

The most intensely investigated LHS containment is the rec-tangular encapsulation because to its simple boundary conditionsand easy work principle. The rst numerical study on rectangulargeometry has been conducted by Shamsundar and Sparrow[107,108] in 1975 and 1976 by applying the nite-differencemethod to investigate the solidication of a at plate. Shamsundaralso used the same method to the case of a square geometry[109]in 1978.

Hamdan and Elwerr [110] presented a simple 2-D numerical

model to study the melting process of a solid PCM within a

rectangular enclosure. In this model, the rate of melting dependsessentially on the properties of the PCM, such as the thermadiffusivity, viscosity, conductivity, latent heat of fusion and specicheat. Natural convection was treated as the dominant heat transfermode within the melt region. Conduction was only assumed totake place within the layer very close to the solid boundary. Thenumerical results showed a high agreement with previouslypublished experimental results [111,112]. Therefore, it can beeffectively used to predict the energy storage performance of the

PCMs contained within the rectangular encapsulation.Further research was carried out by Lacroix[113]in 2001, whopresented a mathematical model based on the energy conserva-tion equation to study the contact melting of a sub-cooled PCMinside a rectangular cavity including the natural convection effectNumerical results demonstrated that the melted fraction at thebottom of the cavity is larger, by an order of magnitude than thatof the conduction dominated melting at the top. The meltingprocess is essentially governed by the magnitude of the Stefannumber and strongly inuenced by the lateral dimensions of thecavity. Jiji and Gaye [114] also analytically examined one-dimensional solidication and melting of a slab with uniformvolumetric energy generation. They found that the low thermalconductivity of the PCMs present a signicant challenge in thedesign of PCM-based electronic cooling systems.

In order to address the inherent drawback of the low thermaconductivity of PCM, various numerical solutions have beenapplied for PCM with different thermal conductivity enhancer(TCEs). Lacroix and Benmadda [115] numerically analyzed themelting of PCMs in a rectangular enclosure with horizontal nsA 2-D enthalpy model was used to solve the phase changeproblem using a xed-gird method. The effects of the numberand length of ns on the melting rate were considered in thisstudy. It was concluded that a few longer ns were more effectivein the melting rate than a large number of shorter ns. The effecof vertical ns on melting rate was studied by the same team[116by using axed-grid enthalpy model in 1998. It was found that theonset of natural convection in the melted zone was delayed whenthe distance between the ns was decreased. According to their

results, the optimumn spacing decreases as the Rayleigh numbeincreases. Akhilesh et al. [117]numerically studied a rectangularmodule with vertical ns, which was heated from above. Thenumerical model was solved by using nite-difference methodOnly conduction heat transfer was included in this study. Theanalysis showed that more ns increase the rate of energy storedin the melting PCM until a critical value. There is no obviousperformance enhancement by only increasing the number ofnsbeyond the value. Similar results were obtained in anotherresearch of Gharebagi and Sezai [118] in 2005 considering anenclosure with vertical ns added to a horizontal heated wall. Theresults indicated that the heat transfer rate to the melting PCM canbe improved by adding ns. In addition, vertically heated wallswith horizontal ns exhibit better performance than horizontally

heated walls with vertical ns.

Table 3 (continued )

Reference(s) Dimensional Numericalsimulation method

Assumptions Comment Validation

Saman et al.[66]

2D Enthalpy basedequation

The thermo physical properties of PCMcan be assumed constant.

The PCM was initially solid and itstemperature was assumed at a certain valuebelow the melting point.

The natural convection was takeninto account during charging period.

The numerical model was able to predict quiteaccurately the melting time and the heat transferrate during the melting with the geometry beingconsidered

Yes


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Table 4

Numerical studies by considering both the conduction and convection.

Reference(s) Dimensional Numerical solution method Assumptions Comment Validation

Costa et al.[3]

1D Energy based equation The thermophysical properties of thePCM and n material were independentof temperature.

The PCM was initially in the solid phase.

The PCM was homogenous and isotropic.

Calculations have been made for the meltfraction and energy stored under conductionplus convection condition.

Numerical[30,58]Fully implicit nite-

difference method

Halawa et al.[88]

2D Enthalpy based method For the PCM melting process, the PCMwas solid and its temperature wasassumed at a certain value below themelting point.

For freezing process, the PCM wasinitially liquid and its temperature wasassumed at certain value above themelting point.

Typical characteristics of the melting/freezingof PCM slabs were discussed. Considerationsin the design of the LHTES were also given.

Numerical[89]

Wang et al.[90]

2D Energy based method The PCM was pure and homogeneous. The melting front was an isothermal

interface in reality. The thermophysical properties of the

PCM were constant, with the exceptionof the linear densitytemperaturerelation.

The natural convection ow of the liquidphase was incompressible and laminarwith no viscous dissipation.

The density of PCM was constant inthis study.

Calculations were performed to study theeffect of thermal conductivity and thermalcapacity of fatty acids and conductivity ofheat exchanger materials and their effecton melt fraction.

NoFinite-volume method

Jana[91] 2D Enthalpy-porosity basedmethod

The density was considered to beconstant in the unsteady and convectiveterms and was allowed to vary only inthe body-force term,

The phase change occurred at a singletemperature,

The temperatures of liquid and solid atthe interface were equal.

Prediction of solidication and meltingprocesses of pure materialsusing moving grids.

Numerical andExperimental[92]Finite-volume method

Hamdan andAl-Hinti[93]

3D The solidliquid interface wassmooth, plane.

The solid phase was initially at themelting temperature.

The effect of inertia inside the thermalboundary layers was negligible relativeto that of buoyancy and viscosity.

The heat loss from the walls of theenclosure was negligible.

The PCM material was pure, homogenousand isotropic.

Boussinesq approximation was used inthis analysis.

The melting process of a solid PCM containedin a rectangular enclosure heated froma vertical side at a constant heat uxwas investigated analytically.

Experimental[94]

Zhao et al.[95]

1D Energy based equation The heat transfer process inside themolten layer was one-dimensional, andthe tangential temperature gradient wasneglected.

The thermo-physical properties of PCMwere constant.

The process was quasi-steady, and theacting forces on PCM were balanced atevery moment of the melting.

The expression of melting velocity wasobtained by using the lubrication theory.

Numerical andExperimental[96]

Liu et al.[97] 2D E nthalpy porosit y metho d The two-phase mixed region can bedescribed with the porosity.

The effective thermal conductivity of themixture can be calculated as the volumeaverage of the conductivities of porousmatrix material and PCM

The porous media and uid were notassumed to be in thermal equilibrium.

The effects of the structural parameters of theporous media and inlet conditions of HTF onthe thermal performances of LHTES werenumerically analyzed.

Experimental[98]3D

Shmueli et al.[99]

2D E nthalpy porosit y metho d Both solid and liquid phases werehomogeneous and isotropic.

The melting process was axisymmetric.

The ndings of the present study made itpossible to dene the heat transfer

mechanisms from the tube wall to the liquid

Experimental[100]


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Apart from applying ns to PCMs, the effect of metal foams onheat transfer enhancement in rectangular encapsulated PCMswas investigated by Tian and Zhao [98] in 2011. The numericalinvestigation was based on the two-equation non-equilibriumheat transfer model, in which the coupled heat conduction andnatural convection were considered at the phase transition andliquid zones. The numerical results were validated by experimen-tal data. The main ndings of the investigation were that the heatconduction rate increases signicantly by using metal foams, dueto their high thermal conductivities, and that natural convection issuppressed owing to the large ow resistance in metal foams.In spite of this suppression caused by the metal foams, the overallheat transfer performance is improved when metal foams areembedded into PCM. This implies that the enhancement of heatconduction offsets or exceeds the natural convection loss. Theresults also indicated that for different metal foam samples, heattransfer rate can be further increased by using metal foams with

smaller porosities and bigger pore densities.

4.1.2. Cylindrical containment models

Three modes of cylindrical PCM container congurations areclassied. The rst model is the PCM lling the shell and the heattransfer uid (HTF) owing through a single tube (pipe model).In the second mode the PCM lls the tube and the HTF ows parallelto the tube (tube-tube model). The third cylinder model is the PCMencapsulated in a shell and HTF ows through the tube outside theshell to improve the heat transfer of PCMs (tube-shell model).

Ismail and Alves[119]numerically analyzed a pipe model withthe PCM encapsulated in a shell while water as the HTF owingthrough the tube. Radial conduction was assumed to be dominated

during the process of solidication and the energy equation for the

PCM was solved in conjunction with the equation governingthe uid bulk temperature as a function of time by employingthe control-volume method. The effect of Biot number, ratio odiameters and inletuid temperature on the thermal performanceof the storage unit were presented. Lacroix [105] developed anenthalpy based model to predict the transient thermal perfor-mance characteristics of a pipe LHTES with the PCM on the shelside and the HTF circulating inside the tube. Including the axialconduction effect in the PCM and employing an enthalpy basedmethod, the coupled conservation equations governing the heattransfer in the PCM and HTF were solved iteratively using a nitevolume based nite-difference technique. Numerical results werepresented for various thermal and geometric parameters such asshell radius, mass ow rate and inlet temperature of the HTFIt was concluded that for a given type of PCM, these parametersmust be selected carefully in order to optimize the performance ofthe storage unit. Then the tube-tube model was theoretically

studied by Esen and Durmus [120]in 1998. A theoretical modebased on an enthalpy method was developed to predict the effectsof various thermal and geometric parameters, congurations othe cylindrical container on the whole melting time for differentPCMs. The theoretical results showed that the whole melting timeof PCM depends on not only thermal and geometric parameters ofthe container, but also on the thermophysical properties of thePCM. It was found that a thicker PCM pack would result in longermelting time.

During 2006, Regin et al.[121]experimentally and numericallystudied the third model that the melting behavior of parafn waxencapsulated in a cylindrical capsule. The numerical analysiscarried out by using enthalpy method and the results were veriedwith the experimental data. The experiments have been done by

the visualization technique without disturbing the actual process

Table 4 (continued )

Reference(s) Dimensional Numerical solution method Assumptions Comment Validation

The molten PCM and the air wereincompressible Newtonian uids,and laminar ow was assumed in both.

PCM and then to the solid phase at variouslocations and instants.

Ye et al.[101]

2D Enthalpy porosityapproach

Density and dynamic viscosity of thePCM depended on temperature.

Variation of property values in liquid andsolid phase PCM was imposed usingpiecewise linear interpolation.

Constant thermophysical propertieswere specied for aluminum.

The numerical study provided a detailedknowledge regarding interface heat transfer

rate provides a deeper understanding theheattransfer mechanisms.

Experimental[102]and

Numerical[103]

Tao and He[104]

2D Enthalpy method The axial heat conduction and viscousdissipation in the HTF was negligible.

The ow of HTF was treated as onedimensionaluid ow.

The PCM region was treated as anaxisymmetric model.

The effect of natural convection of PCMduring melting was taken into accountwith an effective thermal conductivity ofthe liquid phase of PCM.

The effects of the non-steady inlet conditionsof HTF on melting time, melting fraction, TEScapacity, solidliquid interface, heat ux ontube surface and HTF outlet temperaturewere analyzed.

Experimental[105]

Adine andQarnia[106]

2D Conservation energyequations

The ow of HTF was dynamicallydeveloped.

The axial conduction and viscousdissipation in the uid were negligible.

The thermophysical properties of theHTF and PCMs were independent of thetemperature.

The thermophysical properties of solidand liquid phases of PCM were equal,except their thermal conductivities.

This parametric study could provideguidelinesfor system thermal performance and designoptimization.

Experimental[105]

Control volume approach


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of melting. Numerical results indicated that the melting process ischiey governed by the magnitude of the Stefan number, phasechange temperature range and the capsule radius. The analysisshowed that the agreement between the analytical and experimentalresults is signicantly improved when the results are obtained byconsidering the phase change temperature range and the naturalconvection in the liquid phase instead of only considering theconduction heat transfer.

4.1.3. Spherical containment models

Generally, the PCM in the sphere is subjected to constrained(the solid and liquid phases have the same density) and uncon-strained melting (the solid portion drops out of the melting layerdue to its higher density).

The constrained melting of the PCM within a spherical containerwas investigated by Khodadadi and Zhang in 2001 [4]. Thecomputations were based on an iterative, nite-volume numericalprocedure by using the primitive-dependent variables, whereby thetime-dependent continuity, momentum and energy equations inthe spherical coordinate system were solved. With the buoyancy-driven convection gradually dominated the heat transfer mechan-

ism due to the growth of the melt zone, the solid PCM melted fasterin comparison to the bottom zone. When the buoyancy effectsbecame remarkable, as many as three time-dependent re-circulat-ing vortices were observed from the numerical simulation. Thecomputational ndings were veried through qualitative con-strained melting experiments using a high-Prandtl number wax.

An experimental and computational investigation directed atunderstanding the role of buoyancy-driven convection during con-strained melting of the PCM inside a spherical capsule was reported byTan et al. in 2009 again [122]. The computations were based on aniterative, nite-volume numerical procedure that incorporated asingle-domain enthalpy formulation for simulation of the phasechange phenomenon. A Darcy's Law-type porous media treatmentlinked the effect of the phase change on convection. The computa-tional predictions pointed to the strong thermal stratication in theupper half of the sphere that resulted from rising of the molten PCMalong the inner surface of the sphere thus displaced the colder uid.The measured temperature data and computational results near thebottom indicate the establishment of an unstable uid layer thatpromotes chaotic uctuations and is responsible for waviness of thebottom of the PCM. Regin et al.[123]also discussed an experimentaland numerical study of a spherical capsule lled with a parafn wax(melting temperature range of 52.961.6 1C) as the constrained PCMand placed in a convective environment during the melting process.Time-resolved temperature readings were obtained by using a rake ofnine thermocouples that were placed on a horizontal plane passingthrough the center of the sphere. Keeping the initial temperature ofthe PCM constant, the temperature of the convective heat transferuid varied from 70 to 82 1C corresponding to Stefan numbers of

0.11430.2501. A 1D model for phase change by employing theenthalpy method and nite-difference formulation was developed.The model was then used to investigate the effect of capsule size andthe Stefan number on the instantaneous heat ux, liquid fraction andmelting time.

Considering the spherical geometries, apart from several ana-lytical investigations of diffusion controlled phase change pro-cesses reported before 1980s, the rst study on the unconstrainedmelting of the PCM within spherical container was carried out byMoore and Bayazitoglu[124]in 1982. A mathematical model wasdeveloped by assuming that the liquid remains at the fusiontemperature during the whole period and solved by employingthe perturbation method. The solidliquid interface positions andthe temperature proles were determined for various Stefan and

Fourier numbers. The experimental data agreed well with the

simulation results. During 1987, Roy and Sengupta [125] per-formed a similar study and found an analytical solution for themelting rate at the lower surface of the solid PCM in contact withthe heated surface. Later, Roy and Sengupta [126] performedanother study on the unconstrained melting process inside asphere. Two zones of the melting were identied: a thin meltinglayer at the bottom and a thicker layer at the top of the sphere.They found that a signicant amount of melting took place at the

upper surface due to the signicance of natural convection,whereas in previous research [124,125], the effect of naturalconvection on the melting process was neglected.

4.2. Microencapsulated PCM

Micro-capsulation is a kind of tiny capsules (around 0.11000 m in diameter) [127]. Micro-encapsulation has a higherheat transfer rates as compared to that of macro-encapsulation[128,129] due to a substantially higher surface area to volumeratio. Meanwhile, microencapsulation can tolerate the change involume during phase change process [130]. Microencapsulationhence is a prominent technology for use of PCM in buildingmaterials. With the advent of PCM implemented in gypsum board,

plaster, concrete and/or other wall covering materials, thermalstorage can be part of the building structure even for light weightbuildings.

4.2.1. Incorporation with building components

PCMs can be incorporated with gypsum board, plaster, concreteand/or other wall covering materials for the temperature controlapplication purpose.

In general, storage density and phase change temperatures arevery important parameters, since they determine the storagesystem size and human thermal comfort. Further, accurate estima-tion of the PCM enthalpy variation in the working temperaturerange is essential for correct mathematical modeling of the storagesystem[131].

The rst research on the application of PCMs in buildings hasbeen carried out by Telkes in 1975 [132]. Na2S04 U10 H20 with amelting temperature of 32 1C was incorporated with walls, parti-tions, ceilings and oors to serve as temperature regulators.

Determination criteria for the optimal phase change temperatureand storage density were provided by Zhou[133]as following:

Tm;opt TrQ=htstor 26

Tr tdTd tnTn=td tn 27

Dopt tnhTm;optTn=L 28where Tm;opt is optimal phase change temperature and Dopt isoptimal thickness of the wall; Q represents the heat absorbed by

unit area of the room surface; Tris the average room temperature;tstoris the heat storage time; the subscriptsn andd represent nightand daytime respectively.

Kuznik et al. [134] developed a one-dimensional model tooptimize the thickness of phase change material only consideringpure conduction. The simulation was based on a lightweight walland interior/exterior temperature evolutions within a period of24 h. The results showed that an optimal PCM thickness exists.With the optimal PCM thickness of 1 cm, the thermal inertia of thebuilding was doubled and the PCM can reduce the temperatureuctuations inside rooms. To provide guidelines useful in selectingan optimal PCM for building wallboard, a numerical model wasdeveloped to determine the optimal melting temperature of PCMby Neeper [135]. The results indicated that the optimal melting

temperature depends on the average room temperature. When the


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melting temperature of PCM closed to the average room tempera-ture, the maximum diurnal energy storage was achieved. Xiaoet al.[136]established a simplied theoretical model to optimizean interior PCM for energy storage in a lightweight room. In thispaper energy conservation equations were presented to calculatethe optimal phase change temperature and the total amount oflatent heat capacity. The analytical results agreed with Neeper'snding that the optimal melting temperature depends on the

average indoor air temperature. Further, and it also depends onthe radiation absorbed by the PCM panels.A 1-D enthalpy model for analyzing the thermal performance

of a shape-stabilized PCM oor was developed by Xu et al. [137].By using the model, the inuence of various parameters (meltingtemperature of PCM, thickness of PCM layer, latent heat of fusion,and thermal conductivity of PCM) on the thermal performancewas analyzed by using a fully implicit the nite-difference method.It was found that for a given position or weather condition, thesuitable melting temperature of PCM is roughly equal to theaverage indoor air temperature of sunny winter days, the heat offusion and the thermal conductivity of PCM should be larger than120 kJ/kg and 0.5 W/(m K), respectively, and the thickness of shapestabilized PCM plate used under the oor should not be larger

than 20 mm. Zhou et al. [138]extended the investigation of theinuence of various parameters from oor to walls and ceiling byusing a veried enthalpy model. The model was validated byexperimental work, and simulated results agreed well with themeasured data. The numerical results showed that for the presentconditions, the optimal melting temperature is about 20 1C andthe heat of fusion should not be less than 90 kJ/kg, thin PCM plateswith large areas are advantageous and the effect of PCM plateslocated at the inner surface of interior wall was superior to that ofexterior wall (the south wall). A similar result to Xu et al.[137]wasalso obtained for the condition of the PCM wall and ceiling that thethermal conductivity should be larger than 0.5 W/(mK).

Athienitis et al. [139]experimentally and numerically studiedthe application of PCM in building envelope components forthermal storage. A 1-D non-linear enthalpy model was developedto simulate the transient heat transfer process in the walls usingexplicit nite difference scheme. The numerical result showed thatthe utilization of the PCM gypsum board may reduce the max-imum room temperature by about 4 1C during the day time andcan reduce the heating load at night signicantly.

The numerical simulation of passive heating with PCMs wascarried out by Chen et al. [140] by using a 1-D nonlinearmathematical model. The effective heat capacity method was usedto simulate the PCM heat transfer problem by using the softwareMATLAB. The result indicated that applying proper PCM to theinner surface of the north wall in the ordinary room can not onlyenhance the indoor thermal comfort dramatically, but alsoincrease the utilization rate of the solar radiation. Consequentlythe heating energy consuming is decreased and the goal of saving

energy has been achieved. The energy-saving rate of heatingseason can get to 17% or higher.

4.2.2. Microencapsulated phase change slurry (MPCS)

Microencapsulated phase change slurry (MPCS) as a newtechnique has been getting more and more attention since thistechnique serves as both the heat transfer uids and energystorage media. Consequently it can increase the thermal storagecapacity and improve the energy efciency of the thermal system.Due to the MPCS' increasingly important role, some review paperson the properties, heat transfer characteristics and application ofMPCS have been published in recent years [127,141143]. As akind of suspension, the heat transfer and uidow characteristics

of MPCS are different from those of the marco-encapsulated PCMs.

In accordance to the review of Delgado et al. [142], the heatransfer characteristics of the MPCS can be classied into forcedconvection and natural convection.

4.2.2.1. Laminar forced convection heat transfer. The MPCS used toenhance convective heat transfer is generally laminar due to the highviscosity. Charunyakorn et al. [144] rstly conducted a numerica

simulation of MPCS ow in circular tubes under different boundaryconditions. The energy equation was formulated by taking intoconsideration both the heat absorption (and release) during phasetransition and solved using an implicit nite difference method. Theirparametric study showed that the bulk Stefan number and volumetricparticle concentration are the two dominant parameters. Their modealso showed that the heat transfer enhancement ratio of the uidcould be 24 times of that of water. Zhang and Faghri[145]modiedthe model of Charunyakorn [144] to include the effects of themicrocapsule walls, initial sub-cooling and the melting temperaturerange. A temperature transforming model was used to solve themelting in the microcapsule by instead of a quasi-steady modelThe numerical results showed a very good agreement with theexperimental results of Goel et al. [146]. Their numerical results

suggested that the differences between the experimental results oGoel et al. and numerical predictions of Charunyakom could bedecreased greatly if these additional factors are considered. Theauthors however did not give any correlation or other similarcriteria that could be used for future design.

Complicated source terms are applied to the theoretical modelsabove to account for the phase change effects. In order to simplifythe simulation process, Alisetti and Roy [147]developed a simplereffective specic heat model to study the heat transfer in MPCS. Theresults showed that the exact form of the specic heat function is notcritical as long as the latent heat is incorporated correctly within anite melting temperature range. Based on the concept of effectivespecic heat, Roy and Avanic[148]developed a model to analyze thelaminar forced convection heat transfer to a PCM suspension in acircular duct under constant wall heatux condition. The model hasbeen veried with experimental data. The results demonstrated thatStefan number is the only parameter that has a signicant impact onthe thermal performance. Sub-cooling effect only can affect the heattransfer performance at very low heat uxes or when the inletemperature is much lower than the melting point. A simplecorrelation to predict the wall temperature rise as a function of thetube length was also developed for future design.

Hu and Zhang [149] introduced a model to provide a novel insighfor the forced convective heat transfer enhancement of MPCS owingthrough a circular tube with constant heat ux. The model wasdeveloped based on effective specic heat capacity method and usedto analyze the inuence of various factors on heat transfer enhance-ment. A modied Nusselt number was introduced since the conven-tional Nusselt number correlations for internal ow of single phase

uids are not suitable for accurately describing the heat transfeenhancement with MPCS. The modied local Nusselt number isinversely proportional to the dimensionless wall temperature. Thenumerical results indicated that the exact nature of the phase changeprocess strongly affects the degree of convective heat transfeenhancement. The Stefan number and the PCM concentrationresulted to be the most important parameters on the improvementof heat transfer in MPCS. Zeng et al. [150]analyzed experimentallyand numerically the enhanced convective heat transfer mechanismof MPCS in the thermal fully developed range. An enthalpy modewas proposed and developed for simulating the forced convectiveheat transfer characteristics of laminar ow with MPCS in a circulartube under constant wall heat ux. The results showed that inthe phase change heat transfer region the Stefan number and the

dimensionless phase change temperature range number are the


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most important parameters inuencing the Nusselt number uctua-tion prole and the dimensionless wall temperature. The bulk uidReynolds number, particle diameter and the microcapsule volumetricconcentration also inuence the Nusselt number prole and thedimensionless wall temperature but are independent of the phasechange process.

4.2.2.2. Turbulent forced convection heat transfer. A majority of past

studies have considered laminar forced convection heat transfer toMPCM and only a few numerical investigations on turbulentforced convection heat transfer have been reported. However,the turbulent ows occur in majority of engineering applicationsof MPCS. In order to investigate the turbulent heat transfer to PCMsuspensions, Roy and Avanic[151]presented an effective specicheat capacity model for PCM suspensions in a circular tube withconstant heat ux under turbulent ow condition. The numericalresults were found to agree with previously publishedexperimental data and showed that this effective specic heatmodel can effectively model turbulent heat transfer with PCMsuspensions. The most primary parameter in heat transfer was theStefan number. The melt temperature range and degree of sub-cooling are other two important parameters. Royon and Guiffant

[152] developed a numerical model to describe the thermalbehavior of a slurry of PCM slurries ow in a circular duct underdifferent ow parameters (Reynolds number). The simulationresults demonstrated that the inuence of Reynolds number onthe minimum length of the heat exchanger is relatively weak.

4.2.2.3. Natural convection heat transfer. Inaba et al. [153]developed a 2-D model to study the uid ow and heat-transfercharacteristics for RayleighBnard natural convection of non-Newtonian PCM slurry. They found that Rayleigh number,Prandtl number and aspect ratio could be the primary factors formost of Newtonian and non-Newtonian uids to evaluate thenatural convection. A modied Stefan number was dened in thepaper to evaluate the natural convection in a PCM slurry. Later,

Inaba et al. [154] presented another 2-D model to study thenatural convection heat transfer of a MPCS. It was found that thenatural convection effect and heat transfer enhancement were dueto the contribution of the latent heat transfer.

4.3. Summary of numerical models

It can be seen that the numerical models collected above forencapsulated PCMs all have their limitations. As the practicalphase transformations in different applications are complicated,and the thermal conditions are not ideal. Various assumptionswere set up for each numerical solution according to differentnumerical simulation purposes. There is none of numerical modelscan describe the phase change problem properly. Hence, the

numerical modeling of LHTES only approximately approachesbut not accurately presents the practical thermal performance ofLHTES. Therefore, more investigation should be carried out tostudy the phase change problems so as to provide a more detailedinsight of the actual heat transfer behaviors inside PCMs duringthe phase transformations.

5. Conclusions

This paper presented a comprehensive review of mathematicaland numerical methods applied to the solutions of phase changeproblems. Firstly heat transfer mechanisms during the charging anddischarging processes were discussed in this review. The heattransfer mechanisms play the most important role on the numerical

results. Consequently, it is important to weight the percentages of

conduction and convection heat transfer taken in each stage. Then,it presented the fundamental mathematical descriptions of thephase change phenomenon, the Stephan problem and Neumannproblem. At last, a description of the numerical solutions for phasechange problems based on considering only conduction and con-sidering also convection. The PCMs have been applied in thermalenergy storage applications widely due to their attractive features.Various numerical models for encapsulated PCMs in terms of

macro-encapsulated and micro-encapsulated PCM were also ana-lyzed in this paper. Nevertheless, the incorporation of PCMs in aparticular application calls for an analysis that will enable theresearcher to understand the heat transfer principle inside PCMcorrectly. From this survey, some further works are recommended:

Further investigation should be carried out to obtain adequateknowledge of the phase change problems so as to minimizethe assumptions made in the numerical models.

Development of new numerical model to broaden its applica-tion in phase change problems and improve the accuracy of itsresults.

Assessment of the stability and convergence on the numericalresults is always required, and the numerical predictions should

always be validated by using appropriate experimental data. Unied dimensionless parameters needs to be developed tomake the gained knowledge applicable to other cases.

Life cycle analysis, both economical and energetic feasibility ofPCM application should be performed as LHTES is a moreexpensive form of thermal storage, but few scientic publica-tions covered this matter.

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