Effect of foam geometry on heat absorption characteristics of PCM-metal foam composite thermal energy storage systems

International Journal of Heat and Mass Transfer 134 (2019) 866–883

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effect of foam geometry on heat absorption characteristics of PCM-metal foam composite thermal energy storage systems Battula Venkata Sai Dinesh, Anirban Bhattacharya ⇑ Indian Institute of Technology Bhubaneswar, Odisha 752050, India

a r t i c l e

i n f o

Article history: Received 28 August 2018 Received in revised form 13 January 2019 Accepted 19 January 2019

Keywords: Energy storage PCM Thermal enhancement Metal foam

a b s t r a c t This paper presents a numerical study of the energy absorption characteristics of phase change material (PCM) based energy storage systems enhanced with metal foam structure. In particular, the effect of metal foam geometry on total energy absorption and melting of PCM is analyzed. A generalized geometry creation model is developed to create the foam geometry by using random distribution of overlapping spheres of constant or variable sizes. Different foam geometries are created by varying parameters such as pore radius, pore overlap distance and overall porosity. The main feature of the model is that instead of considering volume-averaged PCM-metal foam composite structure, individual pores of the metal foam are resolved, thus leading to more accurate representation of the heat transfer between the PCM and metal foam. The model is coupled with an enthalpy-based phase change model to simulate the effect of high temperature boundary condition on melting and heat absorption. Parametric studies reveal that smaller pore size is always preferable irrespective of the heat transfer duration. However, the optimum overlap and porosity is dependent on the heat transfer duration. Thus for developing an optimized system, these parameters should be judiciously selected based on the transient characteristics of the overall system. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction As the demand for energy skyrocket and the sources of conventional energy decrease, renewable sources of energy are becoming more popular. Efficient storage of this energy is very important. Latent heat based thermal energy storage using phase change materials (PCM), which provides much higher storage density with a smaller temperature difference, has been an active area of research for the past 20 years [1,2]. However, phase change materials have a very low thermal conductivity of around 0.2–0.5 W/ m K which leads to low heat transfer rate. As a result, the rate of heat storage is very slow and the full energy storage potential of such a system cannot be utilized. To counter this, heat transfer enhancements such as metallic fins [3–14], finned tubes [15–20], composites of PCM [21–26], metal matrix structures [27–31], non-metallic foams [32–34], nanoparticles [35–37] and encapsulation of PCM [38–41] have been used. Fan and Khodadadi [42] and Lin et al. [43] have classified the effect of thermal conductivity enhancements in PCM. These systems can be used mainly for solar ⇑ Corresponding author at: School of Mechanical Sciences, IIT Bhubaneswar, Argul, Jatni, Dist – Khurda, Odisha 752050, India. E-mail address: [email protected] (A. Bhattacharya). https://doi.org/10.1016/j.ijheatmasstransfer.2019.01.095 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

energy storage [44], cooling of electronics [6] and latent heat storage in buildings [45,46]. Open cell metal foam, which has low density and high surface area, has been proven to be an effective heat transfer enhancement material. Siahpush et al. [47] performed both analytical and experimental study to find out the effect of copper foam in PCM during melting as well as solidification. It was seen that when copper foam was used, the effective thermal conductivity increased from 0.423 W/m K to 3.06 W/m K. Zhao et al. [48] experimentally studied the effect of metal foam in PCM and compared it with a pure PCM system. It was seen that the metal foam improved the effective thermal conductivity significantly depending on the foam material and structure. Xiao et al. [49,50] investigated the effect of nickel and copper foam in paraffin. These PCM-metal foam composite structures were prepared using vacuum impregnation method. It was found that the effective thermal conductivity of these composites were nearly 3 times more than that of pure paraffin. Lafdi et al. [51] conducted experiments to investigate the influence of geometrical parameters, such as porosity and pore size, of aluminum foams. It was observed that for low porosity foam heat conduction rate was higher and the heater temperature was lower. On the other hand, for higher porosity, convection was higher and steady state temperature was achieved faster.

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Nomenclature Cp fl f l;new H k L m r sðtÞ Ste t T Tm T0 Ti T x;t T sx;t T lx;t T0 To x; y; z

specific heat (J/kg K) liquid fraction new liquid fraction total enthalpy (sensible and latent) (J/kg) thermal conductivity (W/m K) latent heat of fusion (J/kg) distance between two pores radius of the pore (cm) distance of interface from the boundary (cm) Stefan number time ðsÞ temperature ðKÞ melting temperature of PCM ðKÞ boundary temperature ðKÞ initial temperature ðKÞ temperature at distance x and time t ðKÞ temperature of solid PCM at distance x and time t ðKÞ temperature of liquid PCM at distance x and time t ðKÞ temperature at present time step ðKÞ temperature at previous time step ðKÞ coordinates

Although there has been significant experimental work in this area, the number of numerical studies is relatively limited. For numerical analysis of PCM enhanced with metal foam, the model used for simulating the system plays a critical role. The numerical model proposed must capture the effect of heat transfer between the metal foam and PCM accurately. Tong et al. [52] numerically investigated the heat transfer characteristics of aluminum foam filled with water. It was seen that the heat transfer rate was enhanced by an order of magnitude when compared with the base case without metal. Chen et al. [53] used thermal latticeBoltzmann method to predict the melting of PCM with metal foam. The two-dimensional model was validated with experimental results. It was found that for metal foam-PCM composite structure, the melting rate was higher as compared to that for pure PCM system. It was also observed that, for metal-foam PCM system the effect of natural convection was not significant while for pure PCM system convection considerably affected the melting rate. Tian and Zhao [54] compared the heat exchange rate and exergy efficiency of PCM and composite PCM-metal foam and found that the heat transfer rate is increased by 2–7 times for composite PCM-metal foam structures. Jourabian and coauthors [55–60] extensively studied the effect of porous matrix on the melting of ice and other phase-change materials in annular regions through the use of enthalpy-lattice-Boltzmann method. They found that the melting rate increased significantly with lower porosity due to increased effective conductivity. However, the effect of natural convection decreased and was insignificant at lower porosity. For most of the numerical models, the volume averaging technique has been used to investigate the effect of porous medium in PCM [29,61–63]. Mesalhy et al. [61] considered a volume averaged model for studying the influence of porous medium in PCM. The model used separate energy equations for PCM and metal foam and coupled them through appropriate interface heat flux source term. It was found that decrease in porosity leads to increased rate of melting and reduction in convection in the molten PCM. Srivatsa et al. [62] studied the effect of porosity and pore density of PCM-based heat sinks with inserted metal foam using a similar two-equation non-equilibrium model. Tian and Zhao [29] used a

Greek symbols thermal diffusivity (m2/s) d distance between adjacent grid points (cm) D distance between control volume faces (cm) Dt time step ðsÞ x fraction of pore overlap u volume fraction of metal for each control volume q density (kg/m3) n interpolation factor

a

Subscripts a; b; c; d; e; f center point of faces average av g l liquid phase m metal new present time step old previous time step P; A; B; C; D; E; F grid points pcm phase change material s solid phase

two-equation non-equilibrium heat transfer model for studying heat transfer in PCM embedded in porous metal structure. Their results showed that the heat conduction rate is increased but the convection is reduced due to flow obstruction. As conduction is the dominant mechanism, the overall heat transfer is increased when metal foam is introduced into the PCM. Further, it was seen that the metal foams with smaller pores and larger pore densities enhance heat transfer rate. Giorgio et al. [63] modelled open-cell Kelvin foam metal structure and paraffin composite using a similar local thermal non-equilibrium model. A new approach for the calculation of interface heat transfer coefficient was used in their model. Zhang et al. [64] modelled a 3D structured metal foam created using body-centered cubic (BCC) and face-centered cubic (FCC) structures. The model was used for analyzing PCMstructured metal foam composite for energy storage. Sundarram and Li [65] studied the effect of pore size and porosity on the heat transfer characteristics of a paraffin wax-aluminum foam system by considering an FCC unit cell structure. It was seen that both pore size and porosity significantly influence the heat transfer rate. The study also showed that the flow velocity due to natural convection was minimal and the effect of natural convection was not significant for small pore sizes. Yang et al. [66] studied the effect of structured metal foam in PCM using both volume averaged method and direct numerical simulation with and without convection. Their study showed that for a high porosity metal foam structure, the effect of natural convection is significant. Numerical study on PCM with unstructured metal foam is limited. Deng et al. [67] used fractal Brownian motion for the pore distribution for creating an unstructured 2D metal foam and studied the melting characteristics of PCM combined with this foam. Abishek et al. [68] studied the melting characteristics in a highly porous aluminum metal foam-n-eicosane composite by developing the foam structure using open-source software Gmsh [69]. The study showed that the metal foam surface area has a significant influence on the melt front evolution pattern. Recently, Ren et al. [70,71] have performed pore-scale modelling of melting in PCM-metal foam systems using the enthalpy-lattice Boltzmann method. They found that the presence of metal foam strongly influences the melting rate and its

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effect on enhancement of heat transfer in PCM is higher than that of nanoparticles. It was also seen that reduction in pore size increases the heat transfer rate [70]. From the literature review, it is seen that most of the numerical models represent the PCM-metal foam composite structure using volume averaging techniques instead of resolving the individual pores of the metal foam. However, the metal foam geometry may have a strong influence on the heat absorption characteristics of the composite system and thus it is important to resolve the foam geometry for accurate representation of heat transfer at the PCMmetal foam interface. The aim of this paper is to present a generalized model which can create different 3D foam geometries – both structured and unstructured – based on specified parameters such as foam pore size, pore to pore overlap distance and overall porosity and then analyze their effectiveness in increasing the rate of heat transfer for PCM-metal foam thermal energy storage systems. Unlike majority of the previous numerical models, the present model resolves the individual pores of the metal foam structure and thus can capture the thermal interaction between the metal and the PCM more accurately. This is particularly useful for studying the effect of intricate geometric details on the heat transfer characteristics of a system. As a model problem, a 3D paraffin wax-aluminum foam composite is considered with heating from one side, and the melting and energy storage characteristics are studied. The developed model is subsequently used for analyzing the effect of geometrical parameters such as the overall porosity, the individual pore size and the pore to pore overlap distance on transient thermal energy storage performance. 2. Model description The physical model consists of a 3-dimensional cuboidal domain containing a porous metal foam structure filled with PCM. The pattern of porosity of the metal foam is defined by assuming that each individual pore is a partial or a complete sphere which may or may not overlap with each other depending on the specified parameters. The PCM-metal foam composite system is initially at a temperature which is below the melting temperature of the PCM. It is assumed that, at the onset of simulation, one of the boundaries of the system is subjected to a temperature higher than the PCM melting temperature while all the other boundaries are kept adiabatic. Due to heat flux from the high temperature wall the system initially gains sensible heat followed by melting of PCM and energy storage as latent heat. The total energy increase of the system from the onset of the simulation indicates the thermal energy stored within that time per-

Fig. 1. Schematic diagram of the problem domain.

iod. The schematic diagram of the problem domain, showing the boundary conditions, is presented in Fig. 1. The model can be divided primarily into two parts: (a) Geometry creation model and (b) Heat transfer and melting-solidification model.

2.1. Geometry creation model To capture PCM-metal foam heat exchange accurately, a foam geometry must be created which would replicate realistic foam geometry. The foam geometry depends on the method used for manufacturing the foam [72,73]. If the metal foam is generated through the evolution of gas in liquid metal during solidification or by using a foaming agent, the resulting structure will have overlapping spherical pores as a result of formation of spherical gas bubbles. Only, if the porosity is very high, the metal foam structure will deviate from this and become more mesh-like. Gas-releasing particle decomposition in semi-solids can create metal foams with non-overlapping pores [72]. As the porosity values considered for our simulations are relatively small, we have assumed that the metal foam structure can be modelled by overlapping or nonoverlapping spheres. The basic mechanism for creating the foam geometry is to fill the entire volume of the foam, which is a cuboid, with spherical pores. The volume of the pores represents the volume of PCM. By subtracting the volume of the pores from the total volume, the metal foam structure is generated. To do this, at first, the center of each pore is randomly generated at any point inside the domain. A radius value is also assigned to the pore. Based on given values, one pore can overlap another pore to a specified extent. A pore overlap parameter is defined to specify the extent of pore overlap, as shown in Fig. 2. Suppose a new sphere with radius r is inserted into the geometry and there be a sphere with radius ro already present. x, y and z are coordinates of the center of the new sphere and xo ; yo and zo are coordinates of the center of the existing sphere. In this case, the pore overlap is given by Eq. (1).

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx  xo Þ2 þ ðy  yo Þ2 þ ðz  zo Þ2 P r o þ r  minðxr; xro Þ

ð1Þ

The overlap distance is given by minðxr; xr o Þ where x is the pore overlap fraction with a value of less than 1. If a fixed overlap is used, the equality sign holds true. In general, ro þ r  minðxr; xr o Þ is the minimum distance between the centers of the two spheres. For spheres of equal sizes, the pore overlap distance is xr where r is the common radius of the spheres. A negative value of pore overlap represents a closed cell metal foam structure while a positive value represents an open cell structure.

Fig. 2. Pore overlap criterion.

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For a pore to be inserted at a particular random point the overlap condition of that pore is checked with all the existing pores previously inserted. Now we can follow two methods depending on the type of foam geometry required. The first one involves filling pores of same size inside the volume until no other pore can be inserted for that particular overlapping condition. A count variable is maintained. When a pore cannot be inserted at a certain random point then the value of the count variable increases by one. If a pore is inserted, the count variable again goes to zero. If the count variable reaches a certain limiting value, the algorithm ends. This is to ensure that the program does not get stuck in an unending loop during geometry creation. The second method involves developing a foam structure with variable pore size and pore overlap but with a specified porosity. By changing the pore size and pore overlap we can obtain a particular porosity as desired. When pores of different sizes are overlapped the overlapping condition is checked for the smallest pore with all the other pores because if the overlapping condition is checked for the bigger pore the entire smaller pore can get merged inside the bigger pore. After insertion of each pore the porosity is checked and when it reaches the limiting value the

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insertion of the pores stop and the algorithm ends. Examples of the domain filled with spheres and the resultant foam structures are shown in Fig. 3 and the algorithm for the geometry creation model is presented in Fig. 4. It should be mentioned here that the main aim of the geometry creation model is to develop a foam structure with spherical pores, similar to those manufactured using a foaming agent [72,73]. There are other advanced geometry construction methods for foam structures, such as surface minimization technique [74], stochastic distribution of fibers [68,69] and random generation growth method [75]. After the geometry creation is complete, we have the coordinates of centers of the generated pores and the corresponding radii. The geometry is resolved in the numerical scheme for solidification and melting by defining the volume fraction of metal, u. To calculate the volume fraction field, a mapping technique is used. At first, the entire domain is divided into a large number of grid points. Initially, all the points are assigned a value of one as their default value. If a point lies inside a pore its value is changed to zero. Because of large number of grid points used for defining each sphere (around 106 nodes per sphere), this binary representation

Fig. 3. (a) Geometry created using method 1 – With constant sphere size (b) Geometry created using method 2 – With specified porosity and variable sphere size.

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Fig. 4. Geometry creation algorithm (a) Method 1 – With constant sphere size (b) Method 2 – With specified porosity and variable sphere size.

of metal fraction is sufficient to resolve the geometrical intricacies. The grid is now mapped to a coarser grid with about 103 nodes per sphere. For calculating the value of u, the number of original grid points corresponding to a single grid point in the final coarse mesh

with a value of one is counted and divided by the total number of original grid points within a single final grid point. To increase numerical efficiency, for each pore, only points in the volume around its vicinity are checked. Finally, we are left with an array

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of u where values of zero represent PCM, values of one represent metal and the values between zero and one represent metal-PCM interface. This mapping allows good accuracy of representation of the pore geometry, particularly the metal-PCM interface, even with a relatively coarse mesh. Subsequently, only the final mesh is used for calculation of heat transfer, melting and solidification. The initial fine mesh is discarded after the geometry creation part is finished. The cross-section of a typical pore with the final mesh is presented in Fig. 5 to show the relative size of the pore with respect to the grid spacing. 2.2. Melting and solidification model The pores are assumed to be completely filled with PCM. The remaining volume acts as the metal foam. For simulating melting and solidification of PCM, the enthalpy method [76] is used. The following assumptions are made to develop the model. 1. As the metal foam structure consists of small pores, the effect of convective heat transfer is neglected and it is assumed that the dominant heat transfer mechanism is conduction. It has been shown from experiments [53] that although natural convection significantly affects the phase change process for a pure PCM system, its effect for hybrid metal foam-PCM systems is negligible. Mesalhy et al. [61] has shown that the effect of convection is insignificant for metal foam with porosity of less than 90%. Sundarram and Li [65] has shown that for metal foam with small pore size, the flow velocity due to natural convection is minimal. The small size of the pores hinder natural convection and thus heat transfer due to conduction remains the dominant mechanism for hybrid systems. We have considered 3D simulations for a large domain. Solving momentum equations to simulate the effect of convection requires significantly higher computational resources. Hence, to increase simulation speed we have ignored the effect of convection in our model. As the main aim of the paper is to study metal foam systems with different geometries, the model will give reasonably accurate results with fast simulation times. Comparison with experimental results given in [53] (Section 3.2) show that the temperature evolution predicted by the present conduction-based model matches well with those given in [53]. We should also mention that, the porosity values used for most of our simula-

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tions are relatively small (less than 0.9). It has been shown that at lower porosity, the effect of natural convection is negligible [53,61]. 2. There is no change in density of PCM during phase change and hence no shrinkage driven convection. 3. Thermal conductivity and specific heat of PCM is same for both solid and liquid phases. The governing equations are formulated using average thermal conductivity, specific heat and density. The average thermal conductivity, specific heat and density are calculated at each grid point using the volume fraction of metal at each point as given in Eq. (2).

qav g ¼ qpcm ð1  uÞ þ qm ðuÞ kav g ¼ kpcm ð1  uÞ þ km ðuÞ C pav g ¼ C ppcm ð1  uÞ þ C pm ðuÞ

ð2Þ

In Eq. (3), qav g ; kav g ; C pav g are the average density, thermal con-

ductivity and specific heat, qpcm ; kpcm ; C ppcm ; qm ; km ; C pm are the respective properties of PCM and metal and u is the volume fraction of metal at each grid point. Heat transfer is computed using the following energy conservation equation [77].

qav g

      @H @ @T @ @T @ @T ¼ kx þ ky þ kz @t @x @x @y @y @z @z

ð3Þ

The volume averaged enthalpy H is given by

h i H ¼ C ppcm ð1  uÞ þ C pm u T þ ð1  uÞf l L

ð4Þ

In Eq. (4), f l is the liquid fraction of PCM and L is the latent heat of melting. The control volume interface conductivities are calculated based on the harmonic mean of the adjacent grid point thermal conductivities, as given in Appendix A. In a time step, at first, the new value of H is calculated by solving the energy conservation equation (Eq. (3)) using the previous time step values of T. Subsequently, f l at each grid point is updated using the following steps.  At first, the new value of liquid fraction (f l;new ) is calculated from h   i the updated H by using f l;new ¼ H  C ppcm ð1  uÞ þ C pm u T m = ð1  uÞL  If 0 < f l;new < 1, T is fixed and equal to the melting temperature T m . Therefore the new value of liquid fraction is given by h   i f l;new ¼ H  C ppcm ð1  uÞ þ C pm u T m =ð1  uÞL  If the calculated value of f l;new is less than 0, f l;new is set equal to 0. In this case the temperature T is less than the melting temperature T m . New value of T is calculated using   T ¼ H= C ppcm ð1  uÞ þ C pm u  If the calculated value of f l;new is greater than 1, f l;new is set equal to 1. In this case T is greater than T m . New value of T is calcu  lated using T ¼ ðH  ð1  uÞLÞ= C ppcm ð1  uÞ þ C pm u

Fig. 5. Cross-section of a typical pore with the final grid for heat transfer calculation showing the relative size of the pore with respect to the grid spacing.

After solving the energy equation in each time step, parameters such as latent heat, sensible heat taken by PCM, sensible heat absorbed by metal and total heat absorbed are calculated. The governing equations are discretized using the control volume approach and solved using an explicit scheme. For spatial discretization second order central difference scheme is used and for temporal discretization Euler’s time integration is used. Details about the discretization scheme is presented in Appendix A. For all the simulations presented in this paper, paraffinaluminum foam systems are considered. The dimensions and

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material properties of the PCM and metal foam are given in Table 1. All the domain faces are maintained adiabatic except one face which is maintained at a higher temperature of 353 K. Initially the temperature of both PCM and metal foam are taken as room temperature (298 K). 3. Results and discussion At first, the model is validated for melting and solidification with analytical solutions for 1D phase change problems and with experimental results from literature. Subsequently, the validated model is used to perform a case study with specified values of pore size and pore overlap and find heat transfer and phase change characteristics of a PCM-metal foam system. This is followed by parametric analysis of the effect of geometrical parameters on energy storage characteristics of such a system. 3.1. Validation of melting and solidification model Exact solutions of phase change problems are limited and can be obtained for only few simple cases [77]. To validate the model, a single-phase melting problem and a two-phase solidification problem are considered. For the first validation, a single-phase melting problem is taken. Initially, the PCM is in solid state at the melting temperature (T m ) at time t ¼ 0. The left boundary is maintained at high temperature (T 0 ) while all the other boundaries are adiabatic. At time t > 0, melting of PCM starts from the left boundary and the solid-liquid interface moves forward with time. This problem is similar to 1D melting in a semi-infinite domain with heating from one side for which analytical solution exists and is given by the following equation [77]. Temperature distribution:

h i 0:5 T x;t  T 0 erf x=2ðat Þ ¼ Tm  T0 erfðbÞ

ð5Þ

where b can be calculated using: 2

beb erf ðbÞ ¼

C p ðT 0  T m Þ Ste pffiffiffiffi ¼ pffiffiffiffi p L p

ð6Þ

In Eqs. (5) and (6), Ste is Stefan number, T m is the melting temperature, T 0 is the boundary temperature, x is the distance from the boundary, sðt Þ is the distance of interface from the boundary, k; q; L; C p are the thermal conductivity, density, latent heat and specific heat of the PCM and a is the thermal diffusivity. This problem is known as a single-phase problem because temperature variation occurs only in the liquid phase while the temperature of the solid phase remains constant. Fig. 6(a) and (c) shows the comparison of the temperature variation with x at time t = 1000 s, and the variation of interface position with time for the numerical prediction and the analytical

Table 1 Thermo-physical properties of PCM (paraffin) and metal (aluminum). Parameters

Units

Magnitude

Dimensions Specific Heat of metal ðC pm Þ Specific Heat of PCM ðC ppcm Þ

cm J=ðkg KÞ J=ðkg KÞ ðW=m KÞ ðW=m KÞ

25  25  25 910 2100

Thermal Conductivity of Metal ðkm Þ Thermal Conductivity of PCM ðkpcm Þ Density of Metal ðqm Þ Density of PCM ðqpcm Þ Latent Heat of PCM ðLÞ Melting Temperature of PCM ðT m Þ

ðkg=m3 Þ ðkg=m3 Þ kJ=kg K

205 0:2 2830 880 169 335

solution. It is observed that they are very closely matched. This can also be seen from the relative differences between the numerical prediction and the analytical solution as shown in Fig. 6(b) and (d). For this study, the initial temperature is equal to the melting temperature (335 K for paraffin) and the left boundary temperature is taken as 573 K. For the second validation a two-phase solidification problem is considered, where initially the entire PCM is in liquid state maintained at a temperature T i which is higher than the melting temperature T m . At time t > 0, the left boundary temperature is decreased to T 0 which is lower than the melting temperature T m and PCM starts solidifying. The solid–liquid interface moves forward with time. For this problem, the temperature of both the solid and liquid phases vary and the analytical solution for variation of temperature with distance for the solid and liquid phases are given by [77]: Temperature distribution:

h i h i 0:5 0:5 T lx;t  T i erfc x=2ðal t Þ T sx;t  T 0 erf x=2ðas tÞ h i and ¼ ¼ erfðbÞ Tm  T0 T m  T i erfc bðas =a Þ0:5 l

ð7Þ

where b can be calculated using:

pffiffiffiffi  0:5 2 2 eb kl as Tm  Ti eb ðas =al Þ bL p h i¼ þ C ps ðT m  T 0 Þ erf ðbÞ ks al T m  T 0 erfc bðas =a Þ0:5 l

ð8Þ

In Eqs. (7) and (8), T m is the melting temperature, T 0 is the boundary temperature, T i is the initial temperature, x is the distance from the boundary and sðtÞ is the distance of the interface from the boundary. q; ks ; C ps ; L; kl are the density, thermal conductivity of solid PCM, specific heat of the PCM, latent heat and thermal conductivity of liquid PCM, respectively. as and al are the thermal diffusivities of the solid and liquid PCM. The comparison of the numerical prediction with the analytical solution for the temperature distribution and interface position is shown in Fig. 7(a) and (c). For this study, the initial temperature of the liquid is taken as 353 K while the boundary temperature is taken as 273 K. The relative differences between the numerical prediction and the analytical solution are also shown (Fig. 7b and d). In this case also, very good agreement is observed, thus validating the melting and solidification schemes used in the model. 3.2. Validation with experimental results To validate the model for PCM-metal foam system, numerical prediction for melting of PCM is compared with the experimental results given in Chen et al. [53]. For the simulation, the same domain dimensions, material properties and boundary conditions are taken as that for the experimental study given in [53]. Similar to that given in [53], metal foam porosity is taken as 91.37%. For the numerical simulation, the metal foam geometry is created with random generation of pores using the second method described in Section 2.1. It should be noted here that although the porosity is same for both the cases, the foam structure may not match exactly. Fig. 8 shows the comparison of temperature contours at t = 90 min predicted by the numerical simulation with the experimental temperature profile given in Chen et al. [53]. It is observed that the temperature variation is similar in both the cases with gradual decrease in temperature from the heated left face to the right face. In Fig. 9, the temperature variation along the central line is plotted at 90 min for both the cases and it is found that they have similar pattern. Random fluctuations are present in both cases due to the random presence of pores in the metal foam structure. This comparison shows that the present model works correctly for the

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Fig. 6. Comparison of numerical prediction with analytical solution for a single-phase melting problem (a) Temperature variation at t = 1000 s (b) Relative difference in temperature variation (c) Variation of interface position with time (d) Relative difference in interface position.

PCM-metal foam hybrid system. It also indicates that the effect of natural convection in PCM with metal foam is minimal and the heat transfer is mainly dominated by conduction. 3.3. Case study: energy storage characteristics of a PCM-metal foam structure At first, a general case is studied so that it can used for comparison with all the other cases. For the general case the pore size is taken as 2 cm and the maximum pore overlap is taken as 25%. The dimensions of the system are 25 cm  25 cm  25 cm. The bottom face is maintained at 353 K while all the other faces are kept adiabatic. The initial temperature of the system is 298 K. Based on properties of paraffin and the specified temperature difference, the Stefan number for the simulation is Ste = 0.683. Initial grid spacing for geometry creation is taken as 0.25 mm while for solving the energy equation a final grid spacing of 2.5 mm is taken. Time step is taken as 0.01 s. The variations of temperature and liquid fraction of PCM with time are shown in Figs. 10 and 11. From Fig. 10, it is seen that a temperature gradient is established in the metal foam due to the high temperature boundary condition at the bottom boundary. The temperature of the metal foam increases relatively quickly within the entire domain. The low temperature circular regions are the PCM filled pores which gradually reach the melting temperature of PCM. From Fig. 11, it is observed that due to rapid heat transfer through the metal

foam the PCM starts melting in most of the pores. The localized melt front in each pore progresses in a spherical manner from the side of the pore towards the center. Due to the presence of the metal foam heat can be transferred to the PCM at the far end at a faster rate. The total energy absorbed by the system is the sum of the latent heat taken by the PCM, sensible heat taken by the metal foam, sensible heat taken by the PCM before melting and sensible heat taken by the PCM after melting. The variation of these parameters with time is shown in Fig. 12(a). It is observed that during initial heat transfer the dominant factor is the sensible heat increase in the metal and PCM. As the temperature reaches the melting temperature of PCM, sensible heat increase becomes less important and latent heat absorption becomes the dominant mechanism of heat storage. Sensible heat taken by PCM after melting is significantly less because it can only increase when liquid PCM absorbs heat. Fig. 12(b) shows the variation of overall percentage of molten PCM in the domain with time. It is seen that the melting occurs at a steady rate even after lot of PCM has melted and gradually slows down only when more than 90% of PCM gets melted. This high rate of melting can be sustained because of the intricate network of high thermal conductivity metal present in the PCM. As a result, most of the PCM is close to the metal foam structure thus enhancing the heat transfer rate. The variations of average temperature and average liquid fraction along the length of the domain at different times are shown

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Fig. 7. Comparison of numerical prediction with analytical solution for a two-phase solidification problem (a) Temperature variation at t = 1000 s (b) Relative difference in temperature variation (c) Variation of interface position with time (d) Relative difference in interface position.

Fig. 8. Temperature contours at t = 90 min (a) Numerical prediction using the present model (b) Experimental result taken from Chen et al. [53].

in Fig. 13. Averaging is done along a line from the bottom of the domain to the top. The values at all the points on a plane are averaged to a single point on the line. It is seen that the temperature and liquid fraction variations shown in Fig. 13 are not uniform

because the pores are randomly distributed. So, there can be more metal in one plane as compared to another plane. As the time increases the average temperature and the average liquid fraction increases.

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Fig. 9. Variation of temperature across the domain along the central line (Experimental data taken from Chen et al. [53]).

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Fig. 11. Liquid fraction contours with pore overlap of 25% and pore size of 2 cm at (a) 50 s (b) 500 s (c) 1000 s (d) 4000 s.

ulations which has been calculated based on the selected grid spacing using stability condition. 3.5. Parametric study Different combinations of pore size, pore overlap and porosity will have different heat absorption rate and different heat storage capacity. So, it is important to optimize these parameters to get the required heat absorption rate and energy storage within a specified cycle time. This makes the parametric study of these variables important and critical for designing a system with optimized energy storage performance. In this study, the pore overlap, pore size and porosity are taken as the variable parameters and their effect on the energy storage rate is analyzed. The effect of each parameter is studied separately by keeping all the other parameters constant during the geometry creation phase. The generalized nature of the geometry creation model makes it possible to create different geometries with varying values of pore size, overlap and porosity.

Fig. 10. Temperature contours for a PCM-metal foam system with pore overlap of 25% and pore size of 2 cm at (a) 50 s (b) 500 s (c) 1000 s (d) 4000 s.

3.4. Grid independence test To observe the effect of grid spacing, three different grid spacing are considered 1.25 mm, 2.5 mm and 5 mm. Simulations are performed for aluminum foam-paraffin system with a domain length of 10 cm for each side. All the other parameters except grid spacing are taken same as that given in Section 3.3. Fig. 14 shows the variation of temperature and liquid fraction for the 3 cases. It is observed that the predictions match well for grid spacing of 2.5 mm and 1.25 mm. For 5 mm spacing, although the overall pattern is similar, the pores are not resolved accurately. Fig. 15 shows the variation of average temperature in the domain and the total heat absorbed with time. It is seen that the predicted variation for all the 3 cases are similar and the results are very closely matched for grid spacing of 2.5 mm and 1.25 mm. Based on the grid independence study, a grid spacing of 2.5 mm is chosen for all the simulations. Time step of 0.01 s has been used for the sim-

3.5.1. Effect of pore overlap At first, the effect of pore overlap is considered. As mentioned in Section 2.1, pore overlap is defined as the maximum fraction of radius by which a pore can overlap another pore. As the value of pore overlap is changed the geometry changes. For this analysis, the pore size (radius) is taken as constant and equal to 2 cm and four different values of pore overlap are considered: 25% (closed foam with no overlap and the minimum distance between the pores equal to 0.25 times the pore radius), 0% (case where two pores touch each other), 25% and 37.5%. The first method of geometry creation, described in Section 2.1, is used to create the geometry. Due to the specified conditions, the volume of PCM and metal can be variable. The overall dimensions, boundary conditions and numerical parameters such as the time step and grid spacing are same as that given in Section 3.3. The total heat absorbed, the latent heat absorbed and percentage of melting of PCM for the four cases are compared in Fig. 16. It is seen that initially, the rate of total heat absorption is higher for the system with least overlap. This is because higher fraction of metal is present in this case leading to higher heat transfer rate. However, as the volume of PCM is lower in this case, for longer duration, the total energy absorbed is less as compared to the other

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Fig. 12. Variation of (a) heat absorbed and (b) liquid fraction of PCM with time for a PCM-metal foam system with pore overlap of 25% and pore size of 2 cm.

Fig. 13. Comparison of average temperature and average liquid fraction along the length of the domain at different time.

cases. If the duration of heat absorption is sufficiently large, the system with the highest overlap stores the maximum energy. This is clear from Fig. 16 which show that although the case with 25% overlap reaches a very high melt fraction fastest, the latent heat increase is less because of the lower volume of PCM present. The temperature and liquid fraction contours at time t = 2500 s for the four cases are shown in Figs. 17 and 18. It is observed that the temperature for 25% overlap case is highest. This is because the number of pores and the amount of PCM is less in this case and thus most of the energy goes towards the increase of sensible heat. From the liquid fraction contours in Fig. 18 it is seen that the melting percentage of 25% pore overlap is higher. But at the same time (t = 2500 s), the latent heat absorbed by the 25% pore overlap system is higher. This is because of the higher volume of PCM in the 25% pore overlap system as compared to the 25% overlap system. From this study, we can say that depending on the required energy storage and energy transfer duration the amount of pore overlap has to be considered. For example, for the system described here, if the cycle time for energy absorption is 1000 s, then the 25% overlap is best. However, for 5000 s cycle time, the 25% overlap case is the best. For very long durations, the system with the highest overlap will be more suitable.

3.5.2. Effect of pore size When the pore radius of the metal foam is varied, the heat absorption rate and heat capacity of the system change. In this study, the effect of pore size (radius) is considered while keeping the pore overlap constant. The overall dimensions, boundary conditions and numerical parameters are same as that given in Section 3.3. The pore overlap is kept constant at 25% while pores of radius 1 cm, 2 cm, 3 cm and 4 cm are considered. For geometry creation, method 1 described in Section 2.1 is used. The total heat absorbed, the latent heat absorbed and the percentage of melting of PCM for the four cases are compared in Fig. 19. The temperature and liquid fraction contours for the four cases at time t = 1000 s are shown in Figs. 20 and 21. It is seen that as the pore size increases the total heat absorbed decreases. This is because there is more melting for smaller pore size which is evident from the Fig. 21. The latent heat absorbed by the PCM also follows the same trend i.e. the system with smaller pores absorb more latent heat. The smaller pore has higher rate of heat absorption because as the pores are smaller the entire amount of PCM in a pore can be melted quickly. In case of larger pores the central regions of the pore are far away from the metal-PCM interface and hence get affected by the lower thermal conductivity of PCM. From the liquid fraction contours shown in Fig. 21, it can

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Fig. 14. Temperature and melt fraction contours at t = 1000 s with different grid spacing (a) and (d) Dx = 1.25 mm (b) and (e) Dx = 2.5 mm (c) and (f) Dx = 5 mm.

Fig. 15. Comparison of average temperature and total heat absorbed with different grid spacing.

be clearly seen that the melting is more for pore size of 1 cm and 2 cm as compared to pore size of 3 cm and 4 cm. This result shows that the melting of bigger pores takes more time even if they have more surface area per pore because the overall contact area between the PCM and the metal foam is higher for the smaller pore size. The results are only shown for up to t = 1000 s because the same trend is followed for longer durations. 3.5.3. Effect of overall porosity To study the effect of overall porosity, the metal foam geometry needs to be created with different specified porosity values. This is not possible if both pore size and pore overlap are kept fixed. To construct a metal foam structure with a specified overall porosity, a different geometry construction algorithm is necessary. This has been described as the second method in Section 2.1. Instead of tak-

ing a constant pore size, variable pore sizes subjected to upper and lower bounds of 0.5 cm and 3 cm are used. Similarly, random pore overlap values within specified limits of 0 and 0.5 are used. Using such geometry, the effect of porosity can be compared. This type of geometry is very similar to the real-world foam geometry as the pores are not always of same size and they do not always have the same overlap. For the comparison, four values of porosity are considered: 60%, 70%, 80%, and 90%. The overall dimensions, boundary conditions and numerical parameters are same as that given in Section 3.3. The total heat absorbed, the liquid fraction of PCM, and the latent heat absorbed for the four cases are compared in Fig. 22. It is seen that, initially, the rate of heat transfer is higher for lower porosity. After a certain period, the higher porosity system can store more energy. This is expected because, for lower porosity, larger fraction

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Fig. 16. Comparison of total heat absorbed, liquid fraction of PCM and latent heat absorbed for different pore overlap.

Fig. 17. Temperature contours for different pore overlap values at time t = 2500 s (a) pore overlap = -25% (b) pore overlap = 0% (c) pore overlap = 25% (d) pore overlap = 37.5%

Fig. 18. Liquid fraction contours for different pore overlap values at time t = 2500 s (a) pore overlap = -25% (b) pore overlap = 0% (c) pore overlap = 25% (d) pore overlap = 37.5%

of metal is present in the system resulting in faster heat transfer. However, for longer energy absorption time, the higher latent heat capacity of the high porosity system becomes the dominant factor. The temperature and the liquid fraction contours for these four

cases at t = 5000 s are shown in Figs. 23 and 24. It is seen that the temperature is higher for lower porosity. This is because less heat is absorbed for melting of PCM and the rate of sensible heating is more for this case.

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Fig. 19. Comparison of total heat absorbed, liquid fraction of PCM and latent heat absorbed for different pore size.

Fig. 20. Temperature contours for different pore size at time t = 1000 s (a) pore size = 1 cm (b) pore size = 2 cm (c) pore size = 3 cm (d) pore size = 4 cm.

Fig. 21. Liquid fraction contours for different pore size at time t = 1000 s (a) pore size = 1 cm (b) pore size = 2 cm (c) pore size = 3 cm (d) pore size = 4 cm.

This study shows that there is no unique optimum porosity value for a given system for all time scales. Depending on the energy absorption duration of a given system, which is fixed by the energy source and the overall system characteristics, different

porosity metal foam structures may be suitable. For example, for an aluminum-paraffin composite system with the dimensions used in this case study, for a time up to about 8000 s, porosity of 60% is more suitable but for time greater than 8000 s 70% porosity is better. At higher cycle times, even higher porosity will be better.

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Fig. 22. Comparison of total heat absorbed, liquid fraction of PCM and latent heat absorbed for different porosity.

Fig. 23. Temperature contours for different porosity at time t = 5000 s (a) porosity = 60% (b) porosity = 70% (c) porosity = 80% (d) porosity = 90%

Fig. 24. Liquid fraction contours for different porosity at time t = 5000 s (a) porosity = 60% (b) porosity = 70% (c) porosity = 80% (d) porosity = 90%

4. Conclusion

model was developed to construct 3-dimensional foam geometry for specified values of important control parameters such as foam pore size, pore-to-pore overlap and overall porosity. The model is used for analyzing the effect of these parameters on the rate of melting of PCM, temperature evolution and total energy

We have presented a study of the effect of geometrical parameters on the energy storage characteristics of a PCM-metal foam composite energy storage system. A generalized geometry creation

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absorption. Simulations reveal that the rate of energy storage increases with decreasing pore size due to the higher total surface area for heat transfer between the metal foam and PCM. However, the effect of pore overlap and overall porosity depends on the energy absorption duration. For shorter duration, lower porosity and pore overlap are preferable while the opposite is true for longer duration. Thus depending on the required transient characteristics, the geometrical parameters need to be selected to obtain optimum energy absorption. This study gives a thorough understanding of the effect of foam geometry on the thermal enhancement of PCM-metal foam energy storage systems and can be used to optimize the design of such systems based on specific requirements. The developed model can also be used to compare the energy storage characteristics of PCM-metal foam systems with other thermal enhancement mechanisms used for PCM based energy storage systems such as metal fins, encapsulation of PCM, etc. to find the relative effectiveness of metal foam among all the different thermal enhancement mechanisms available. Funding

the control volume P with respect to x, y, z and t, and considering an explicit scheme, we obtain

R tþDt R f R d R b t

e

@ a @x

c



 R tþDt hkb ðT B T P Þ ka ðT P T A Þi kx @T  ðdxÞ DyDzdt dx dy dz dt ¼ t @x ðdxÞb a o o

kb ðT B T P Þ ka ðT oP T oA Þ ¼  ðdxÞ DyDzDt ðdxÞ b

a

ðA1Þ Similarly,

@ @y



 and ky @T @y

@ @z

 @T  kz @z are discretized. Adding all the dis-

cretized diffusion terms and equating with the discretized temporal term, we get

qav g DxDyDz Dt

ðHnew  Hold Þ ¼ ab T oB þ aa T oA þ ac T oC þ ad T oD þ ae T oE þ af T oF    aa þ ab þ ac þ ad þ ae þ af T oP ðA2Þ

DyDz DxDz DyDz DxDy DxDz In Eq. (A2), aa ¼ kaðdx , ab ¼ kbðdx , ac ¼ kcðdy , ad ¼ kðddy , ae ¼ keðdz Þ Þ Þ Þ Þ a

b

c

d

e

k DxDy

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

and af ¼ fðdzÞ . The interface conductivity at a is given by f  1 1na na . Similarly, the interface conductivities kb , kc , kd , ka ¼ kP þ kA

Declarations of interest

ke , kf are obtained. For our model, n ¼ 0:5. The grid point values of thermal conductivity are obtained by volume averaging based on the metal fraction and conductivities of metal and PCM. This can be written as

None. Appendix A.

kP ¼ km ðuP Þ þ kpcm ð1  uP Þ and kA ¼ km ðuA Þ þ kpcm ð1  uA Þ

The discretization scheme for Eq. (3) is presented in this section. A single control volume and its neighboring grid points are consid @T  @ ered, as shown in Fig. A1. Integrating the diffusion term @x kx @x for

Similarly, the thermal conductivities of the other neighboring grid points B, C, D, E and F are defined. Detailed discussion about similar discretization technique is given in [78].

Fig. A1. Geometrical parameters for discretization.

ðA3Þ

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