Modelling of heat transfer in phase change materials (PCMs) for thermal energy storage systems

Modelling of heat transfer in phase change materials (PCMs) for thermal energy storage systems

Modelling of heat transfer in phase change materials (PCMs) for thermal energy storage systems 12 G. Ziskind Ben-Gurion University of the Negev, Isr...

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Modelling of heat transfer in phase change materials (PCMs) for thermal energy storage systems

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G. Ziskind Ben-Gurion University of the Negev, Israel

12.1 Introduction Thermal energy storage based on the use of latent heat is linked inherently to the processes of solid-liquid phase change during which the heat is alternately charged into the system and discharged from it. These phenomena – melting and solidification – have unique physical characteristics. They involve a moving boundary that separates two different phases which themselves may have considerably different transport properties. The behaviour of the moving interface is such that its instantaneous position is unknown a priori and must be determined as a part of the solution. The other common features include convection in the melt, volume change due to phase change, melting/solidification extending over a temperature range, and subcooling at solidification. This chapter aims to provide an overview of the modern approaches to phase change material (PCM) modelling. Specifically, we focus on the socalled macro-level modelling, in which there exist distinct solid and liquid phases separated by a region defined as the solid–liquid interface, or melting/solidification front. Sections 12.2 and 12.3 address, in a compact but comprehensive fashion, the main methods of heat transfer analysis. Basic relevant heat transfer phenomena and mechanisms are described, with special attention paid to the description of phase change and moving boundary problems. The limitations of analytical approaches are outlined, and numerical modelling is introduced. The main methods of multidimensional numerical simulation are discussed. Then, we present some examples, invoking the basic configurations used in or suggested for the latent-heat based thermal energy storage systems. In particular, we address such basic macro-encapsulation and bulk storage configurations as spherical and cylindrical shells. Furthermore, modelling of storage units, suitable for, for example, solar thermal power plants and other renewable energy applications, is discussed. Another purpose of this chapter is to present issues on which the current research is focusing, with special attention to unresolved problems that will require further consideration by researchers. These issues include such concerns as heat transfer in novel materials, e.g. composites, modelling of emerging Advances in Thermal Energy Storage Systems. http://dx.doi.org/10.1533/9781782420965.2.307 Copyright © 2015 Elsevier Ltd. All rights reserved.

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storage configurations, and simultaneous numerical simulation of various relevant phenomena. The chapter is concluded by a section dedicated to sources of further information and advice, giving a concise description of the major sources that readers might wish to consult at various stages of their work, from getting started with basic modelling to advanced simulation methods and approaches. A number of books and key journal articles/book chapters, not cited directly in the text because of their generality, are presented with a clear definition of their scope. Finally, a list of references, addressing the specific problems used in the examples, is included, including both well-known and the most recent works on heat transfer in PCMs.

12.2 Inherent physical phenomena in phase change materials (PCMs) Here we present, in brief, the most important features that commonly appear in PCM melting and solidification and thus should be accounted for in a model or, as may be possible with at least some of them under certain circumstances, discarded after careful consideration.

12.2.1 Moving solid-liquid interface In fact, an accurate prediction of interface shape reflects the quantity and location of the material that has already melted or solidified by a certain instant, thus reflecting the amount of heat stored in the system or discharged from it until a certain moment. Therefore, from a practical perspective, it is probably the most important feature of the process and thus its prediction must be as accurate as possible. Still, in many cases simplification is achievable, for example by assuming that the front is onedimensional or at least remains parallel to itself, or by applying such approaches as the effective heat capacity method, in which the latent heat is incorporated into the sensible heat, and thus the phase-change problem is replaced with a much simpler heat conduction problem.

12.2.2 Buoyancy effects in the melt Due to volumetric expansion and temperature differences in the liquid phase, natural convection commences. Moreover, in the course of melting more and more liquid is produced, making the resulting flow field even more complicated. This flow may affect the melting considerably, for example by changing the shape of the melting front and thus the melting rate. This point is schematically illustrated in Figure 12.1. There, the problem itself appears to be one-dimensional because the temperature difference exists in the horizontal direction only, whereas in the vertical direction both

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Insulated

Thot

Liquid

Tcold Solid Tmelt

Insulated

Figure 12.1 Melting from side in a rectangular enclosure.

boundaries are insulated. However, due to convection in the melt which unavoidably develops once the width of the molten region becomes large enough, the melting front becomes skewed, and in reality it is also far from being flat (for the details, see e.g. Gadgil and Gobin, 1984). The process of melting is thus thoroughly affected by this effect. Fortunately, the role of convection in solidification is much less important and in many cases may be discarded. As for melting, natural convection, the strength of which depends strongly on the dimensions of the enclosure, is almost entirely suppressed in many practical situations, for example inside PCM-soaked building panels or porous matrices, where it may be neglected without a considerable impact on the resulting prediction.

12.2.3 Volume change at phase change Most of the common phase change materials are characterized by a significant density change that accompanies the phase change. Usually, the solid density is higher, like in most organic and inorganic materials, whereas water is a notable exception. In practical terms, this change, which may be as large as about 10% in both organic (paraffins) and inorganic (salts) PCMs, must be accommodated by a proper design of the system. On the other hand, its role in heat transfer and phase change is comparatively minor in many situations. For instance, this is definitely the case in modelling of PCM-soaked panels. In such systems as PCM-based heat exchangers for renewable energy installations, it is possible to discard the volume increase in melting while accounting for density change that can affect the performance. On the other hand, volume decrease due to solidification in these systems may lead to creation of voids and separation between the PCM and its container, thus affecting the performance considerably.

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12.2.4 Phase change over an extended temperature range For a pure substance, melting and solidification occur at a certain sharply defined temperature, below and above which only the solid or liquid phase is found, accordingly. However, due to their high cost, pure substances are hardly suitable for practical heat storage applications. Rather, commercial-grade materials and mixtures are used. Those are characterized by the phase change which extends over a range of temperatures, sometimes reaching an order of 10°C or more. If this feature is not reflected in the model, the results may be considerably flawed: for instance, the predicted complete melting or solidification, which is essential for ensuring proper performance, would not occur in practice. This problem is less pronounced when the difference between the temperature of the heat transfer surface and the phase change temperature of the PCM is large. However, in many applications, e.g. buildings, this difference may be a matter of several degrees, making any simplification of this sort unsafe. It is thus recommended to incorporate the phase change range as accurately as possible in the model, for example by using a measured enthalpy–temperature curve.

12.2.5 Enthalpy hysteresis Many prospective materials are characterized by the effects of subcooling or superheating, due to which the phase change does not occur at the nominal, or expected, temperature. This means that, for instance, when a liquid PCM is cooled down to its assumed solidification temperature, no actual phase change is encountered when reaching the latter and the liquid state persists until the liquid is cooled further to some other temperature. This problem is less important in such traditional fields of phase change as metal casting, where the mould is usually much cooler than the actual solidification temperature in any case. However, in PCM applications in buildings and renewable energy systems the temperature differences are small, and careful consideration of this phenomenon is required, similar to the phase change over an extended temperature range, as discussed above.

12.3 Modelling methods and approaches for the simulation of heat transfer in PCMs for thermal energy storage Analytical solutions of melting/solidification problems require a great deal of simplification and are generally one-dimensional, limit heat transfer to conduction, have simple boundary and initial conditions and involve constant thermophysical properties. Moreover, analytical solutions can be strictly valid for semi-infinite domains only. Therefore, the practical role of these solutions is in providing basic understanding of the phenomena as well as the basis for validation of more complicated theoretical and numerical schemes. Most situations of practical interest fall outside

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of this set of simplifying assumptions and thus require numerical methods to obtain sufficiently accurate solutions of the problem. This involves creation of purpose-built codes for specific situations, or adaptation of much broader commercial software packages to the specific needs related to PCM behaviour and involving melting and solidification. To be appropriate for a description of a system in question, the specific model should be chosen carefully. First and foremost, it is obvious that the model must correspond to the physical system as accurately as possible. However, simplifications are usually unavoidable due to the complexity of the system, absence of detailed property data and inability to simulate the performance in a reasonable timeframe. It is thus essential to understand the physical characteristics of the system and adopt simplifications which still allow, under the existing computational constraints, for catching the essential behaviour properly. Several basic methodologies have been developed and are broadly used for modelling of solid-liquid phase change, including the approaches based on effective heat capacity, additional heat source, front tracking, etc. Let us discuss the basic modelling first and then proceed to more elaborate approaches.

12.3.1 Basic formulation The most basic problem accounts for heat conduction only and thus involves the energy equation only. In the latter, the temperature is the only dependent variable, and the equation form is identical to that commonly used in heat conduction texts (and termed ‘heat equation’ in mathematical texts):

rc p

∂T = — · (k—T ) ∂t

(12.1)

where r is the density, cp is the specific heat, k is the thermal conductivity, and T is temperature. For a case where the phase change is one-dimensional in Cartesian coordinates and the thermal conductivity is constant, Eq. (12.1) may be rewritten as

rc p

∂T ∂2T =k 2 ∂t ∂x

(12.2)

Equation (12.1) must be written separately for the solid (s) and liquid (l) regions, involving the corresponding thermal properties, while the density may be constant or different between the regions. Unlike other applications of the heat equation, in order to ensure energy conservation, heat balance at the solid–liquid moving interface where the heat is absorbed or released at the phase change temperature, Tm, in melting and solidification, respectively, is given by: ks —Ts · n – kl—Tl · n = rLv · n

(12.3)

where n is the unit normal on the phase interface, v is the velocity vector of the interface and L is the latent heat per unit mass. In fact, this condition means that the

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heat, absorbed/released at the melting/solidification front, is conducted through the phases themselves. For the particular case presented by Eq. (12.2), the condition of Eq. (12.3) becomes one-dimensional too: ks

∂Ts ∂T – kl l = r Lu ∂x ∂x

(12.4)

where u is the front velocity normal to its plane, i.e. in the x-direction. The problem defined by Eqs (12.2) and (12.4) is known as the Stefan problem. However, it is quite common to use this name also for the formulation that uses Eqs (12.1) and (12.3), notwithstanding its multidimensionality. We note that such basic dimensionless groups as the Fourier and Stefan numbers may be derived directly from Eqs (12.2) and (12.4). It is important to note that Eqs (12.3) and (12.4) assume that the density, r, is the same for the solid and liquid phases. Thus, a problem that involves solid/liquid density difference will require an additional term accounting for advection. Note that the simplest way to address phase change using Eq. (12.1) is a socalled ‘effective-heat capacity approach’, where the latent heat is substituted using an equivalent sensible heat capacity that includes the latent capacity as well. This may be done, for instance, using the following relation: Ï c Ô p Ô ÔÔ L efff ef c p (T ) = Ì + cp Ô 2DT Ô T Ô cp ÔÓ

@ T < Tm – DT T @ Tm – DT < T < Tm + DT

(12.5)

@ T > Tm + DT T

Obviously, this approach does not require any additional conditions, but it is suitable for approximate calculations only.

12.3.1.1 Enthalpy formulation An alternative formulation is based on the enthalpy that is included in the energy equation along with the temperature:

r

∂h = — · (k—T ) ∂t

(12.6)

where h is the enthalpy per unit mass, defined by the following relation: Ï c T @ T < Tm ÔÔ ps h=Ì Ô c pl T + (cc ps – c pl )Tm + L @ T ≥ Tm ÔÓ

(12.7)

In fact, in this conserved formulation, Eqs (12.1) and (12.3) are reduced to a single relation, Eq. (12.6), because it can be demonstrated that the latter includes, implicitly but strictly, the heat balance condition at the interface, Eq (12.3).

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For numerical solutions, the enthalpy formulation is advantageous as, unlike the Stefan formulation, it does not require an explicit tracking of the phase interface. As a result, it also allows for using fixed-grid numerical schemes, whereas the Stefan formulation requires deforming grids. It may be noted that the front tracking methods allow prediction of the melting/solidification front in exchange for much higher complexity and calculation times. The enthalpy methods are suitable for developing in-house numerical codes when the convection may be neglected and thus even a multi-dimensional problem is relatively straightforward. Moreover, the enthalpy formulation does not require a ‘sharp’, i.e. that of zero width, solid–liquid interface, being thus applicable to more general situations where phase change occurs over a range of temperatures and a so-called ‘mushy zone’, which is neither completely solid nor liquid, exists. The latent part of enthalpy within this range may be a general function of temperature, Dh = F(T). If we define the temperatures above which no solid exists and below which no liquid exists as TL and TS, respectively, the general form of F(T) may be written as Ï L @ T > TL Ô ÔÔ F (T ) = Ì L (1 – FS ) @ TL ≥ T ≥ TS Ô Ô 0 @ T < TS ÔÓ

(12.8)

where FS is the solid mass fraction. It is now quite common to define the liquid, rather than solid, fraction, f = 1 – FS, within the phase change temperature range by a ‘lever’ rule, i.e. putting it in a direct proportion to the local temperature. Generally, fixed-grid enthalpy-based methods, extended to treat also motion in the liquid phase as discussed below, are better suited for handling mushy regions, cyclic problems with multiple fronts and problems with several heated/cooled surfaces.

12.3.2 Motion in the liquid phase The modelling presented above is applicable when motion in the liquid phase is neglected and thus conduction is the only heat transfer mechanism in both phases. However, if the liquid motion takes place, usually caused by natural convection and/or squeezing, the problem becomes much more complicated and requires a simultaneous solution of the continuity, momentum and energy equations. The momentum equation may have the following form:

r

Dv = – —p + m— 2 v + rg + S Dt

(12.9)

where r is the density, m is the dynamic viscosity, S is the momentum source term, and v is the velocity vector.

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12.3.3 Enthalpy-porosity approach The liquid fraction, f, is also related to a major modification of the enthalpy formulation, usually termed the ‘enthalpy-porosity approach’. The latter is based on an apparent analogy between the partially liquid material in the mushy zone and the fluid flow through a porous medium. Obviously, it is applied when the liquid medium is treated as such, i.e. is allowed to flow. Accordingly, a complete system of governing conservation equations, rather than just Eq. (12.6), is solved. In the momentum equation, Eq. (12.9), the source term is given by S = –A(f ) · v

(12.10)

where A(f ) is the ‘porosity function’ used to reduce gradually the velocities from a finite value in the liquid to zero in the full solid, over the computational cells that are changing phase. The definition of A(f) makes the momentum equation ‘mimic’ the Carman–Kozeny equation for flow in porous media: A (f ) =

C (1 – f )2 (f 3 + e )

(12.11)

where e is a small computational constant used to avoid division by zero, and C is a constant originally supposed to reflect the structure of the melting front. Although it was assumed that in order to play its role in motion suppression and eventual termination this constant should be a large number, its exact values are somewhat vague, and the problem in question should be carefully examined in conjunction with the up-to-date literature. For instance, the shape, dimensions and character of the mushy zone depend considerably on this parameter, thus affecting the patterns and rate of melting or solidification. Although it has become usual to set C at about 105–106, with a common notion of it being dependent on morphology or material, the recent literature argues that the values should be much higher, about C = 108, and that this requirement is rather universal and not material-dependent.

12.3.4 Density/volume change It is worth mentioning two additional recent developments related to the modelling of solid-liquid phase change. The first of them is associated with the necessity to accommodate the density/volume change due to phase change. In the materials commonly suggested for heat storage at relatively low and high temperatures, e.g. paraffins and salts like NaNO3, respectively, this density difference is about 10%, and this fact should not be ignored in the analysis. Since in practice the excessive volume is commonly occupied by a contracting/expanding gaseous medium, a moving boundary between the melting/solidifying material and the adjacent gas should be modelled. In order to describe a material–air system with a moving interface but without interpenetration of the two, a so-called ‘volume-of-fluid’ (VOF) approach, initially developed for liquid-gas multiphase flows and widely used in that field, may be applied (Shatikian et al., 2005). In this approach, if the nth fluid’s volume

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fraction in the computational cell is denoted as an, then the following three conditions are possible: if an = 0 the cell is empty of the nth fluid; if an = 1 the cell is full of the nth fluid alone; and if 0 < an < 1 the cell contains the interface between the nth fluid and one or more other fluids. Thus, the variables and properties in any given cell are either purely representative of one of the media, or representative of a mixture of the media, depending upon the volume fraction values. Following the inclusion of VOF, the most advanced numerical works in the field actually model two moving interfaces simultaneously: an internal one between the solid and the liquid in the PCM itself and an external one between the material and the adjacent gaseous medium. Due to this advancement, such entirely new capabilities as void formation prediction in solidification have been achieved: since common materials like paraffins or salts are contracting due to solidification, the released volume is filled with an expanding gas. On the other hand, a proper prediction of the resulting solid shape presents a considerable challenge.

12.3.5 Solid motion in the liquid Another major development is numerical modelling of melting in configurations where the remaining solid phase eventually loses its contact with any solid boundary. In laterally heated rectangular enclosures or vertical tubes, the solid usually stands firmly on the horizontal base throughout the entire process. In such systems as horizontal tubes or spherical shells heated from outside, however, the melt surrounds the solid phase completely. Thus, the solid is suspended in the liquid and should sink in the latter (or float if the material is anomalous like H2O). This phenomenon does not appear to have been considered by the creators of the enthalpy-porosity approach. However, it has been reproduced numerically in some cases, for example in systems with a horizontal heated base and vertical conducting partitions between which the phase change material is stored. While sinking, the solid, which is completely surrounded by the liquid, approaches the lower part of the wall (heat transfer surface) but cannot reach it because there the phase change occurs and liquid is continuously produced. The advanced numerical models, able to predict this experimentally proven phenomenon, commonly termed ‘close-contact melting’, are under development. The existing results, obtained with the generally accepted value of about C = 105, agree with the experiments quite well in terms of the melt fraction dependence on the time, and fairly well in terms of the predicted and observed melting patterns.

12.3.6 Close-contact melting Close-contact melting (CCM), mentioned above, is a well-known phenomenon related to the motion of the solid phase in the liquid one. CCM occurs when a bulk of solid PCM approaches a hot surface. A thin molten layer is formed between the solid and the hot surface, and is squeezed by the descending bulk solid to the sides. Thus, more PCM is melting, keeping the flow in the thin molten layer. The heat transfer rate in this type of melting is high due to the relatively small thickness of

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the liquid PCM layer. Close-contact melting may occur in various simple and more complex geometries, some of which are directly related to heat storage. Prominent examples include horizontal cylindrical enclosures (tubes) and spherical enclosures with heated walls. Special models are usually applied, in which a set of equations is based on some assumptions. The examples presented in the following section deal with some of the features discussed above. This is done by applying the advanced methods of numerical modelling available today.

12.4 Examples of modelling applications This section shows some examples of modelling which deal mostly with the basic configurations that can be used in latent-heat-based thermal energy storage systems. In particular, we present such basic macro-encapsulation and bulk storage configurations as spherical and cylindrical shells. We also discuss modelling of a storage unit suitable for renewable energy applications, e.g. for solar thermal power plants. Finally, an example of CCM modelling is presented. Note that the references cited in this section and listed at the end of the chapter are representative ones and relate to the corresponding physical problems rather than specifically to simulation results shown below.

12.4.1 Melting in a vertical circular tube Plate V (between pages 316 and 317) presents numerical modelling of melting in vertical cylindrical tubes, when the wall temperature is higher than the melting temperature of the PCM (see, e.g., Sparrow and Broadbent, 1982; Menon et  al., 1983; Jones et al., 2006; Shmueli et al., 2010). From above the PCM is exposed to air, whereas the tube is insulated at the bottom. The solid PCM, liquid PCM and air are shown using different colours. It is assumed that both solid and liquid phases are homogeneous and isotropic, and the melting process is axisymmetric. The molten PCM and the air are incompressible Newtonian fluids, and laminar flow is assumed in both. The numerical approach combines enthalpy-porosity and volume-of-fluid methods, making it possible to calculate simultaneously the processes that occur inside the tube wall (conduction), solid PCM (conduction), liquid PCM (convection), and air (convection). The boundary conditions for the momentum equation are noslip and no-penetration at all solid boundaries. At the upper boundary, open to the atmosphere, the pressure outlet boundary condition is used. For the energy equation, the numerical model follows the physical one, in order to make possible a proper comparison with the experiments. In particular, the outer tube wall is maintained at a constant temperature, whereas the tube and PCM are insulated from below. The ambient temperature is assumed at the open upper boundary. The initial temperature of the air inside the tube is equal to the initial temperature of the PCM. It is interesting to note that the melting front is skewed, reminiscent of that in Figure 12.1. In fact,

Velocity color 1 5.000e+000 4.583e+000 4.167e+000 3.750e+000 3.333e+000 2.917e+000 2.500e+000 2.083e+000 1.867e+000 1.250e+000 8.333e+001 4.167e+001 0.000e+000 [m sA-1]

Plate I (Chapter 4) Velocity field of the air flow in a packed bed storage (left) and in a honeycomb-based storage (right).

Plate II (Chapter 4) Convection cells forming in the regenerator packing during standstill (Zunft, et al., 2012).

Plate III (Chapter 4) Local contact stresses at the contact point of particle and inner insulation (half-turn symmetry): compressive stresses (left, top); tensile stresses (left, bottom) and an example of a calculated spatial force distribution in the packed bed storage (right).

Plate IV (Chapter 7) FEFLOW’s graphical interface in simulation of a BTES system under the influence of groundwater flow.

Air

Liquid

Front

Solid

Plate V (Chapter 12) Melting from side in a vertical tube.

Air

Liquid

Front

Solid

Plate VI (Chapter 12) Melting in a spherical enclosure.

Plate VII (Chapter 12) Solidification in a vertical cylindrical enclosure (tube).

20 min

40 min

60 min

Plate VIII (Chapter 12) Melting in a storage unit with circumferential fins.

80

60

40

30 60

Plate IX (Chapter 17) Stratified storage tank.

(a)

(b)

(c)

(d)

Plate X (Chapter 17) General ground storage types: (a) hot-water store, (b) gravel-water store, (c) earth-tube store, (d) aquifer store.

High temperature user

55° – 35°C

85° – 65°C

Low temperature user

35° – 25°C 90° – 70°C

Combined heat and power unit Warm drilling

Cold drilling

Aquifer

Plate XI (Chapter 17) Diagram of a cascaded heating system applied to an aquifer store. Collector

(C)

Controller

Pump Sensor

Mains Heat exchanger Tank Supply

Plate XII (Chapter 17) Solar-thermal heated warm water storage.

Solar collector Mains water 80°C

Hot outlet Heating circuit

Boiler/heat pump 30°C

Thermal store

Plate XIII (Chapter 17) Stratified store with solar-thermal collector and boiler.

Plate XIV (Chapter 22) Schematic presentation of the PCM application in Berlin Botanical Garden. Courtesy of Rubitherm Technologies GmbH, Germany.

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the liquid motion is characterized by vortices of rising warm and descending colder liquid which apparently cause the conical shape of the solid phase (compare with melting in a tall enclosure, e.g. Pal and Joshi, 2001). Due to the phase change and corresponding increase in the PCM volume, the interface between the PCM and air rises.

12.4.2 Melting in a spherical enclosure Plate VI (between pages 316 and 317) shows an example of melting in a spherical enclosure, which is qualitatively similar to melting in a horizontal circular tube (see, e.g., Moore and Bayazitoglu, 1982; Roy and Sengupta, 1987; Prasad and Sengupta, 1987; Khodadadi and Zhang, 2001; Assis et al., 2007; Ramos Archibold et al., 2012). The case considered is for a material that has a higher density in its solid state than in its liquid state. In such a case, it is expected that the solid bulk, if it is not fixed by some mechanical obstacle, will move vertically downward. The motion of the solid bulk is accompanied by generation of liquid at the melting interface. This liquid is squeezed up through a narrow gap between the melting surface and the wall of the shell, to the space above the solid. The effect of solid phase sinking and appearance of ‘close-contact melting’ is very significant. The simulation is performed using the same method as that described above for the vertical tube in Plate V. The solid phase is in the lower part of the enclosure, and is separated from the wall by a thin liquid layer. In the upper part of the PCM volume, the liquid phase accumulates, and a flow caused both by squeezing and natural convection is observed in it. In the air above the liquid, the velocities are even higher. The interface between the liquid and air goes up because of the volume expansion due to melting.

12.4.3 Solidification in a vertical cylindrical enclosure The geometry considered in Plate VII (between pages 316 and 317) is similar to that of Plate V. However, now the wall temperature is lower than the melting temperature of the PCM. In the case shown, the bottom is insulated. Thus, solidification starts from the envelope and gradually progresses inwards (see, e.g., Sun et  al., 2007; Revankar and Croy, 2007; Assis et  al., 2009). Since, again, the solid is denser than the liquid, the resulting volume must be smaller than the initial volume. This feature expresses itself in the ‘depression’ formed in the upper part of the PCM. This depression is modelled using the volume-of-fluid approach for the PCM–air front, whereas the enthalpy-porosity approach is applied to model the solid–liquid front, which in this case is almost vertical. One can easily see that if the bottom is adiabatic and solidification is ‘radial’, the resulting solid has a deep hollow centre with steep sides. It is important to note that while convection does not play any significant role in heat transfer, downward fluid motion must be taken into account in the modelling, because it is due to this motion that the depression is formed. The cases of Plates V–VII where simulated for the phase change that occurs over a temperature interval rather than for a pure substance. The results indicate that, as

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mentioned above, the enthalpy-porosity method is suitable for this type of materials, which is of significant practical importance. The results agree quite well with the experimental findings reported in the literature.

12.4.4 Melting in a typical storage unit The example of Plate VIII (between pages 316 and 317) deals with a unit in which radial (circumferential) fins are attached to a tube, placed concentrically in a cylindrical shell (see, e.g., Chiu and Martin, 2012; Tay et al., 2012). The shell is filled with a phase change material. Hot heat transfer fluid flows in the tube while the shell is exposed to the ambient temperature, which is lower than the PCM melting temperature. Plate VIII shows a comparison between the numerical model and an experiment for one of the storage device ‘elementary’ volumes between a pair of neighbouring fins. The red colour in the figure is the liquid PCM and the blue colour is the solid PCM. The melting starts near the fins and tube, propagating ‘inwards’ as time passes. A general comparison with the experiment is impossible because the solid PCM near the shell hides the inner parts of the PCM. Even so, the liquid layer between the solid and the fin can be compared, showing quite good agreement. It is interesting that the effect of the Bénard-like convection cells is captured by the simulation, showing that the solid–liquid interface is wavy like in the experiment, although an axisymmetric formulation is used.

12.4.5 Close-contact melting in a rectangular cavity Figure 12.2 shows a typical pattern of CCM in a simple geometry with heating from below (see, e.g., Moallemi et al., 1986; Hirata and Makino, 1991). Initially, the entire volume is filled by a solid PCM. As the main feature to capture here is the motion of the remaining solid, a special CCM model is used. Usually, modelling of this sort is based on the following assumptions: the molten liquid layer thickness is uniform and smaller by an order of magnitude than the size of the enclosure, all the heat

Figure 12.2 Close-contact melting in a rectangular cavity.

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transfer is by CCM from below – heat transfer from other directions is neglected, the sensible heating is neglected, meaning that all the heat is absorbed as latent heat, the solid temperature is constant and equal to the melting temperature, the flow in the thin molten liquid layer is laminar and quasi-steady, the solid bulk downward motion is quasi-steady, natural convection in the melt is neglected, and the PCM properties are constant. In fact, many of these assumptions may be dropped and a more advanced analysis is possible.

12.5 Future trends This section presents issues on which the current research is focusing, with special attention to unresolved problems that will require further consideration by researchers. These issues will include such concerns as multi-level modelling, heat transfer in novel materials, e.g. composites, modelling of emerging storage configurations, and simultaneous numerical simulation of various relevant phenomena. It should be noted that the choice of a proper modelling approach is based on a variety of factors, from the type and character of the process or device to be simulated to the available computing capacity, availability of skilled professionals and cost-effectiveness considerations. In any case, one should bear in mind that although many efforts have been made to improve the modelling techniques for PCM, some drawbacks are yet to be solved. In particular, most of the models are not experimentally validated, casting a doubt on their reliability. Fortunately, this trend started to change in recent years, as demonstrated by a number of examples presented above. Generally, it is essential to find a proper balance between the calculation time vs. accuracy and geometrical complexity.

12.5.1 Computational constraints As stressed above, modelling may have limitations due to poor understanding of some basic phenomena, insufficient information about the properties or difficulties which arise when attempting to incorporate some specific features in a model. There exist, however, also complications related to the computer capacities at hand. Because of nonlinearity of the phase change phenomena, any computation which involves multi-dimensionality and convection in the melt is extremely time-consuming, with a single case taking as long as several weeks or even months to simulate. This situation is obviously unacceptable when a decision is to be made on a certain design configuration, which usually requires a quick evaluation of a number of alternatives. It is thus recommended to separate between the general and specific design stages, meaning that it is desirable to arrive at a certain preferable configuration using a rapid simplified model, whereas for a more advanced step a more elaborate model may be used for fine-tuning. In addition to this, stage-by-stage, separation, there is also a level-by-level one: if, for instance, a PCM is incorporated in some sort of container, e.g. spherical capsules or cylindrical pipes, detailed modelling may be

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possible on the one-container level but would hardly be practical for the system as a whole. Thus, different models might be handy for different stages and levels of the design.

12.5.2 Use of commercial codes Some commercially available software codes are either equipped with modules suitable for modelling of phase change problems or allow for user-defined functions which may be used to add phase change modelling. These codes may include some very elaborate computational fluid dynamics (CFD) and ‘multi-physics’ tools. These packages allow for very detailed modelling. The post-processing, i.e. results presentation, is very convenient. On the other hand, the simulations are usually extremely time-consuming and thus suitable for some basic units only, rather than for the device or system level. Also, it appears that for such cases as, say, PCMimpregnated matrices or boards, these programs would not give any considerable advantage. This is because some effective characteristics should be used anyway, and they, rather than the details of the phase change, determine the result.

12.5.3 Material properties In addition to experimental validation, which is indispensable, many other topics must be considered. In particular, the accuracy of the measurement of the thermophysical properties can be an important source of errors in simulations. Currently, sufficiently accurate data on the density, viscosity, and enthalpy of PCMs as a function of temperature are very rare. For example, for the density usually only one value for the liquid and one for the solid state are available. The consequence is that, even with proven commercial software, the lack of the necessary experimental data causes significant uncertainties in the results. The most critical parameters must be detected for each problem and the acceptable accuracy of the measured data should be defined to ensure numerical reliability. Also, the validity range of the models must be determined, since in many cases empirical expressions, which are valid only for a specific range of parameters, are used to solve the problem.

12.5.4 Hysteresis and subcooling In recent years, such complex physical phenomena as hysteresis and subcooling have started to find their way into the models. However, in order to simulate them, some assumptions have to be made, which have not yet been experimentally demonstrated. This is especially clear for the case of hysteresis, where melting/solidification curves are assumed to have the same slope of the complete melting/solidification when the process is reversed while the material is still within the phase change range. Experimental work is thus required to determine the behaviour of the phase change transition from partial melting/solidification. Moreover, the effect of subcooling is

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almost always neglected. This is highly problematic when the subcooling is of the order of magnitude of the driving temperature difference on discharging the storage, i.e. solidification.

12.6 Sources of further information and advice This section gives a concise description of the major sources that readers might wish to consult at various stages of their work, from getting started with basic modelling to advanced simulation methods and approaches. A number of books and key journal articles/book chapters are presented within a classification based on their scope. This list can by no means be exhaustive, and the interested reader is advised to consult additional sources where necessary. Sources that give a general perspective on latent-heat thermal energy storage systems: İ. Dincer and M.A. Rosen (eds), Thermal Energy Storage: Systems and Applications, John Wiley & Sons, Chichester (2002). İ. Dincer and M.A. Rosen (eds), Thermal Energy Storage: Systems and Applications, 2nd edn, John Wiley & Sons, Chichester (2010). The second edition of the book is very different from the first, thus both editions are cited. B. Zalba, J.M. Marin, L.F. Cabeza and H. Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications, Applied Thermal Engineering 23 (2003) 251–283. Sources that give a mathematical background on the basic Stefan problem and its extensions: J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford (1984). S.C. Gupta, The Classical Stefan Problem: Basic Concepts, Modelling and Analysis, Elsevier Science, Amsterdam (2003). Sources that deal with mathematical modelling of melting and solidification problems: V. Alexiades and A.D. Solomon, Mathematical Modeling of Melting and Freezing Processes, Taylor & Francis, New York (1993). H. Hu and S.A. Argyroropoulos, Mathematical modeling of solidification and melting: a review, Modeling Simul. Mater. Eng. 4 (1996) 371–396. Sources that deal with general approaches to modelling of heat transfer in phase change materials: R. Viskanta, Phase-change heat transfer, in G.A. Lane (ed.), Solar Heat Storage: Latent Heat Materials, Volume I: Background and Scientific Principles, CRC Press, Boca Raton, FL (1983), Chapter 5.

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H. Hu and S.A. Argyroropoulos, Mathematical modeling of solidification and melting: a review, Modeling Simul. Mater. Eng. 4 (1996) 371–396. V.R. Voller, An overview of numerical methods for solving phase-change problems, in W.J. Minkowycz and E.M. Sparrow (eds) Advances in Numerical Heat Transfer 1, Taylor & Francis, London (1996), Chapter 9. O. Bertrand, B. Binet, H. Combeau, S. Couturier, Y. Delannoy, D. Gobin, M. Lacroix, P. Le Quéré, M. Médale, J. Mencinger, H. Sadat and G. Vieira, Melting driven by natural convection. A comparison exercise: first results, Int. J. Therm. Sci. 38 (1999) 5–26. M. Lacroix, Modeling of latent heat storage systems, in İ. Dinçer and M.A. Rosen (eds), Thermal Energy Storage: Systems and Applications, John Wiley & Sons, Chichester (2002), Chapter 7. K.A.R. Ismail, Heat transfer in phase change in simple and complex geometries, in İ. Dinçer and M.A. Rosen, (eds) Thermal Energy Storage Systems and Applications, John Wiley & Sons, Chichester (2002), Chapter 8. Sources that deal with enthalpy and enthalpy-porosity methods of modelling: N. Shamsundar and E.M. Sparrow, Analysis of multidimensional conduction phase change via the enthalpy model, ASME J. Heat Transfer 97 (1975) 333–340. N. Shamsundar and E.M. Sparrow, Effect of density change on multidimensional conduction phase change, ASME J. Heat Transfer 98 (1976) 550–557. V.R. Voller, M. Cross and N.C. Markatos, An enthalpy method for convection/ diffusion phase change, Int. J. Numerical Methods in Engineering 24 (1987) 271–284. V.R. Voller and C. Prakash, A fixed grid numerical modeling methodology for convection-diffusion mushy region phase-change problems, Int. J. Heat and Mass Transfer 30 (1987) 1709–1719. A.D. Brent, V.R. Voller and K.J. Reid, Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal, Numerical Heat Transfer 13 (1988) 297–318. V.R. Voller and C.R. Swaminathan, General source-based method for solidification phase change, Numerical Heat Transfer B 19 (1991) 175. Sources that deal with volume-of-fluid method of modelling for multi-phase systems: A. Prosperetti and G. Tryggvason, Computational Methods for Multiphase Flow, Cambridge University Press, Cambridge (2007), Chapter 3. Sources that deal with close-contact melting in common geometries: M. Bareiss and H. Beer, An analytical solution of the heat transfer process during melting of an unfixed solid phase change material inside a horizontal tube, Int. J. Heat and Mass Transfer 27 (1984) 739–746.

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S.A. Fomin and T.S. Saitoh, Melting inside a spherical capsule with non-isothermal wall, Int. J. Heat and Mass Transfer 42 (1999) 4197–4205.

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D. Sun, S.R. Annapragada, S.V. Garimella and S.K. Singh, Analysis of gap formation in the casting of energetic materials, Numerical Heat Transfer, Part A: Applications 51 (2007) 415–444. N.H.S. Tay, F. Bruno, M. Belusko, A. Castell and L.F. Cabeza, Experimental validation of a CFD model on a vertical finned tube heat exchanger phase change thermal energy storage system, Innostock 2012 – the 12th International Conference on Energy Storage, Lleida, Spain, May 16–18, 2012.