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10.1016-j.rser.2014.02.032

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Mathematical solutions and numerical models employed for 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, UK b 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 form 9 January 2014 Accept
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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, UK  b 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 2013Received 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 thermalapplications, and has been studied by researchers using theoretical and numerical approaches. The heattransfer 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 phasechangeprocessesiscomplicatedandtheachievedresultsareapproximateaccordingtoanumberofconditions.The 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 theoreticalsolutions according to their researches' purposes. &  2014 Elsevier Ltd. All rights reserved. Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6602. Heat transfer mechanisms during the phase transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.1. Conduction and convection heat transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.2. Stefan problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6612.3. Neumann problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.4. Other possibility mechanisms in phase change process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623. Mathematical solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6633.1. Conduction acting as the only heat transfer mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6633.1.1. Fixed grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6633.1.2. Adaptive grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6653.2. Conduction and convection heat transfer mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6654. Numerical models for various PCMs capsules and packages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6674.1. Macro-encapsulated PCMs numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6674.1.1. Rectangular encapsulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6674.1.2. Cylindrical containment models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6694.1.3. Spherical containment models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6704.2. Microencapsulated PCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6704.2.1. Incorporation with building components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6704.2.2. Microencapsulated phase change slurry (MPCS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.3. Summary of numerical models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6725. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/rser Renewable and Sustainable Energy Reviews http://dx.doi.org/10.1016/j.rser.2014.02.0321364-0321  &  2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.:  þ 44 24 7765 7822. E-mail addresses:  Liu@coventry.ac.uk, liy30@uni.coventry.ac.uk (S. Liu). Renewable and Sustainable Energy Reviews 33 (2014) 659 – 674    1. Introduction Due to the attractive features of latent heat storage, phasechange materials (PCMs) are mainly used to store energy at a  󿬁 xedtemperature (melting/solidi 󿬁 cation 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 1 compares 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 engineering 󿬁 elds. The possibility, feasibility, thermal performance and economicanalysis of using PCMs call a series of theoretical and experientialinvestigations, the mainly two methods used toresearch 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 andadjusted 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/numericalapproaches 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 of mathematical solutions and/or numerical modeling of latent heatthermal energy storage (LHTES) have been published during2002 – 2012. Table 2 summaries these eight relative review papersand comments their mathematical and/or numerical aspects of LHTES. However, few logistic and comprehensive reviews on themathematical solutions and numerical modeling for melting andsolidi 󿬁 cation processes of PCMs were found in published litera-tures even though there are many individual publications in phasechange problems, particularly in heat transfer mechanisms andsimulations. Due to the complexity of phase transformations, Nomenclature f melt fraction c   speci 󿬁 c heat at constant pressure H   total volumetric enthalpy (J/m 3 )h average heat transfer coef  󿬁 cient (J/kg) k  thermal conductivity (W/mK) L  latent heat (J/kg) St   Stefan number t   time (s) T   temperature (K) T  m  melting temperature (K) T  s  initial temperature of solid PCM (K) T  0  constant temperature imposed on  x ¼ 0 (K)P present nodal point δ  ð t  Þ  position of liquid – solid interface  x ;  y  Cartesian coordinates Greek symbols  Δ  difference p  thermal diffusivity of PCM (m 2 /s) η  dimensionless number used in derivations as a tem-porary substitution  λ  dimensionless number in solution to Neumann pro-blem, liquid fraction  ρ  density (kg/m 3 ) Subscriptsl  liquid s  solid i i th spatial step  j j th time step eff   effective  Table 1 Properties comparison of different thermal storage materials.Property Unit MaterialsConcrete (sandand gravel)Water OrganicPCMInorganicPCMDensity [kg = m 3 ] 2240 1000 800 1600Speci 󿬁 c heatcapacity[kJ = kg K] 1.1 4.2 2.0 2.0Latent heat [kJ = kg K]  – –  190 230Storage mass for10 6 kJ, avg[kg] 60,000 16,000 5300 4350Storage volume for10 6 kJ, avg[m 3 ] 26.8 16 6.6 2.7Relative storagemass13.79 4 1.25 1.0Relative storagevolume10 6 2.5 1.01. 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. S. Liu et al. / Renewable and Sustainable Energy Reviews 33 (2014) 659 – 674 660    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 of various PCM packages' geometries in terms of macro-encapsulatedPCMs and micro-encapsulated PCMs are studied. 2. Heat transfer mechanisms during the phasetransformations  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 [2 – 4]. As early as 1831, Lamé 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 simpli 󿬁 ed 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 isthe 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 solidi 󿬁 cation process. Buddhi et al. [8]proposed 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 phase.However, 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 in 󿬂 uenced 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 thesolidi 󿬁 cation 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 transferin 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-phase ’  designatesonly 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 solidi 󿬁 cation in a semi-in 󿬁 nite region is governed by a dimen-sionless number, known as the Stefan number ( St  ), St   ¼½ C  l ð T  l  T  m Þ L  ð 1 Þ where  C  l  is the heat capacity of the liquid PCMs,  L  is the latentheat of fusion, and  T  l  and  T  m  are the surrounding and meltingtemperatures, respectively.  Table 2 Some review papers related to numerical study of phase change problems.Reference Publishedyear Journal CommentZalba et al. [1] 2003 Applied thermalengineeringThe 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 reviewsMathematical 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 theirassumptions used were presented and discussed as well. Jegadheeswaranand Pohekar [3]2009 Renewable and sustainableenergy 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 andmanagementThis paper presented an overview on dynamic characteristics and energy performance of buildingsemploying PCMs by three research methods used, i.e., simulation, experiment, combinedsimulation and experimentAgyenim et al. [5] 2010 Renewable and sustainableenergy reviewsThis 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 reviewsThe 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 reviewsThe review presented models based on the  󿬁 rst lawand on the second lawof 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. S. Liu et al. / Renewable and Sustainable Energy Reviews 33 (2014) 659 – 674  661    The most published solutions are to solve the one-dimensionalcases subjected to an in 󿬁 nite or semi-in 󿬁 nite 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 solid – liquid interface is not  󿬁 xed, but moving with time, the heattransfer mechanisms during a PCM phase transformation processare complex. Therefore the phase change transition is dif  󿬁 cult 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 of scenarios [14]. This view is shared by Kürklü et al. [15] who pointed out that the mathematical models have been proposed ineach research only applied to very speci 󿬁 c 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 oneinterface 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-in 󿬁 nite slab (0 o  x o 1 ), the solid PCM isinitially at a uniform temperature  ð T  s o T  m Þ , and a constanttemperature  ð T  0 Þ  is imposed on the slab surface  x ¼ 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  ρ C  ∂ T  ∂ t   ¼  k ∂ 2 T  ∂  x 2  ð 2 Þ where  ρ  is the density,  C   is the speci 󿬁 c 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:  ρ L ∂  x t  ∂ t   ¼  k l ∂ T  l ∂  x  þ k s ∂ T  s ∂  x  ð 3 Þ where  L  is the latent heat of fusion of the PCM and  ρ  is the densityof liquid PCM.Initial condition T  ð  x 4 0 ; t   ¼ 0 Þ ¼ T  s o T  m  ð 4 Þ Solid – liquid interface temperature T  ð  x ¼ δ  ð t  Þ ;  t  4 0 Þ ¼ T  m  ð 5 Þ With the following boundary conditions: T  ð 0 ;  t  Þ ¼ T  0 4 T  m  for  t  4 0  ð 6 Þ T  ð  x ;  t  Þ ¼ T  s  for  x - 1 ;  t  4 0  ð 7 Þ where  δ  ð t  Þ  is the position of the solid – liquid interface (meltingfront). Fig. 1 clearly illustrates this two-phase Stefan problem.Analytical solution to such a problem was obtained by Neu-mann in term of a similarity variable  ηη ¼  δ  ð t  Þ 2  ffiffiffiffiffiffiffiffi p l p  ð 8 Þ The  󿬁 nal Neumann's solution can be written asInterface position δ  ð t  Þ ¼ 2  λ  ffiffiffiffiffiffiffiffiffiffi p l t  p   ð 9 Þ Temperature of the liquid phase T  ð  x ; t  Þ ¼ T  l ð T  l  T  m Þ erf  ð  x = 2  ffiffiffiffiffiffiffiffiffiffi p l t  p  Þ erf  ð  λ Þ ð 10 Þ Temperature of the solid phase T  ð  x ; t  Þ ¼ T  s þð T  m  T  s Þ  erfc  ð  x = 2  ffiffiffiffiffiffiffiffiffiffi p s t  p  Þ erfc  ð  λ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p l = p s p   Þð 11 Þ The  λ  in Eq. (9) – (11) is the solution to the transcendentalequation St  l exp ð  λ 2 Þ erf  ð  λ Þ  St  s  ffiffiffiffiffiffiffiffi p s p   ffiffiffiffiffiffiffiffi p l p   exp ð p l  λ 2 = p s Þ erfc  ð  λ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p l = p s p   Þ¼  λ  ffiffiffiffi π  p ð 12 Þ where St  l  ¼ C  l ð T  l  T  m Þ L  ð 13 Þ St  s  ¼ C  s ð T  m  T  s Þ L  ð 14 Þ 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, speci 󿬁 c heat capacity, super-cooling, etc. Conse-quently, accurate re 󿬂 ection of the proposed mathematical solutionand numerical model need to consider the dynamic properties of the phase change process. Another major issue with PCMs is thatthey act as self-insulating materials. When PCM solidi 󿬁 cationoccurs from the top of the heat surface, solid insulating layer willbe developed which moves inward during the whole solidi 󿬁 cationprocess. With the increase in the size and thickness of the solidlayer, the heat transfer rate from the liquid PCM to the heatexchanger surface decreases until it becomes so small that will not Fig. 1.  Schematic illustration of the two-phase Stefan problem. S. Liu et al. / Renewable and Sustainable Energy Reviews 33 (2014) 659 – 674 662  

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