Issue 
Eur. Phys. J. Appl. Phys.
Volume 83, Number 1, July 2018
Materials for Energy harvesting, conversion and storage (Icome 2017)



Article Number  10904  
Number of page(s)  11  
Section  Physics of Energy Transfer, Conversion and Storage  
DOI  https://doi.org/10.1051/epjap/2018180060  
Published online  18 October 2018 
https://doi.org/10.1051/epjap/2018180060
Regular Article
Experimental and numerical thermal analysis of opencell metal foams developed through a topological optimization and 3D printing process^{★}
^{1}
GRESPI, University of Reims ChampagneArdenne, Campus Moulin de la Housse,
51687
Reims Cedex, France
^{2}
EPF, School of Engineering,
2 rue Fernand Sastre,
10430
RosièresPrèsTroyes, France
^{3}
MICADODINCCS, 7 Boulevard Jean Delautre,
08000
CharlevilleMézières, France
^{4}
ICDLASMIS, UMR CNRS 6281, University of Technology of Troyes,
12 rue Marie Curie,
BP2060
10010 Troyes, France
^{*} email: abdelatif.merabtine@epf.fr
Received:
4
February
2018
Received in final form:
17
July
2018
Accepted:
18
July
2018
Published online: 18 October 2018
This study focuses on the thermal analysis and comparing a lattice model and an optimized model of opencell metal foams manufactured thanks to a metal casting process. The topological optimization defines the complex geometry through thermal criteria and a plaster mold reproduces it in 3D printing to be used in casting. The study of the thermal behavior conducted on the two open foam metal structures is performed based on several measurements, as well as numerical simulations. It is observed that the optimized metal foam presented less and nonhomogenous local temperature than the lattice model with the gap of about 10 °C between both models. The pore size and porosity significantly affect the heat transfer through the metal foam. The comparison between numerical simulations and experimental results regarding the temperature fields shows a good agreement allowing the validation of the developed threedimensional model based on the finite element method.
© EDP Sciences, 2018
1 Introduction
Thermal management systems using a latent heat storage issued from phase change materials (PCMs) are widely employed in various usedpower systems such as a microelectronics cooling, a solar energy storage, mobile phones, laptops and other opencell foam solutions [1]. Latent heat storage by PCM is a promising way because it has high energy storage density and could store and release the latent heat at a narrow temperature range [2,3].
In order to improve the thermal response of PCM, numerous applications are studied by using experimental and numerical methods, such as adding metal fins [4], spreading particles with high conductivity in PCM [5], modifying the shape of container [6–10] and using porous metal foams as skeleton [11,12]. The last, consisting of a porous metal structures gathering open cell foams and PCM, represent one of interesting thermal energy storage (TES) system.
As the research continues, mechanical, thermal and acoustical properties of metal foam have been studied. In the thermodynamic filed, authors consider that the opencell metal foam is a promising material for heat exchanger [8]. This is due to its high internal surface area and porous structure that could mainly contribute to the heat transfer process. Comparing with lattice metal fin, the metal foam owns similar surface area density and higher mechanical strength properties [9]. Monno et al. [10] analyzed cost efficient methods to produce aluminum opencell metal foams intended for energy engineering applications in the field of catalytic supports. The performed numerical and experimental studies exhibit the fact that the overall heat transfer performances, comparing to the traditional packed bed of catalyst pellets, are very interesting, with almost 400 W m^{−2} K^{−1} of global heat transfer coefficient. With regard to the applications of opencell foams as catalyst supports in catalytic fixed bed reactors, Bianchi et al. [11] performed numerical simulations in order to assess the effective thermal conductivity of periodic open cellular structures (POCS) of ideal cell geometry. Several configurations were suggested and used for the 3D finite volume analysis. It was shown that the effective thermal conductivity of POCS depends significantly on the distribution of the solid along the strut length.
The heat transfer of the metal foams/PCM composites could be affected by either: porosity, pore size, metal matrix, input power, convective coefficient, etc. Several studies have been done regarding the effect of such metal foam parameters on the thermal conduction and convection in composites. Lafdi et al. [12] examined experimentally the effect of pore size and porosity in phase change process. They found that metal foam with lower porosity or smaller pore sizes enhances heat conduction and higher porosity foam could be beneficial for the natural convection. Sundarram et al. [13] established a numerical threedimensional finite element model (FEM) for thermal calculation of metal foam. It was concluded that smaller pore could help dissipate the heat more rapidly from the heat source. Zhao et al. [14] compared the both melting and solidification process of metal foams/PCM composites and pure PCM. The results revealed that the addition of metal foam could increases the overall heat transfer by 3–10 times. Liu et al. [15] analyzed numerically the heat transfer enhancement of metal foams in a shellandcube type of TES system. It was found that heat transfer could be enhanced by 7 times with the addition of metal foams.
All above studies indicate that the melting rate of PCM is enhanced by the addition of metal foams and conclude that the thermal behavior of PCM embedded in metal foam is better than the pure PCM. Accordingly, metal foam requires deepgoing research in order to obtain optimized structure.
Recently, thermal performance of gradient metal foam attracts the interest of many researchers. Yang et al. [16] proposed the metal foam with linearly changed porosity, and the numerical results demonstrated that this structure could enhance the phase change process in PCM. Tao et al. [17] used Lattice Boltzmann method to investigate the effect of the nonuniform porosity on the heat transfer performance. Yang et al. [18] studied the metallic foam with graded morphologies by experimental method, and it is found that this structure could improve the solidification of the water saturated in it. Tao et al. [17] used Lattice Boltzmann method to investigate the effect of the nonuniform porosity on the heat transfer performance.
In multidisciplinary optimization, thermal optimization is widely used. The heat convection can significantly influence optimal design configurations [19,20] obtained with structural optimization. Methodologies based on the conventional density approach have been successfully proposed by several researchers [21,22]. Moreover, when such approaches are applied to heat transfer, complex geometry can disrupt the analysis. Thus, the density approach like solid isotropic material with penalization (SIMP) method has to be managed with trade constraints.
This paper investigates experimentally and numerically the thermal behavior of the optimized opencell foam for cooling applications at the unsteady state. It highlights the advantages of such optimized model against the lattice model. The paper is organized as follows: in second section, the methodologies and tools used are described through the topological optimization methodology applied in the opencell metal foams to suggest the best geometry which takes into account the heat transfer parameters. This section includes the rapid tooling presentation by binder jetting technology used to manufacture the plaster mold of optimized model and lattice model. It also contains the description of the aluminum casting of opencell foam. The third section presents respectively experimental investigation and numerical FE modeling of the heat transfer and melting process in both metal foam models. Then, the results are discussed. The conclusion is presented in the final section to summarize this research and its outcomes.
2 Methodologies and tools used
2.1 Topological optimization
Topology optimization problem can be defined as the determination of the best allocation or distribution of material in a given design space [23]. There are numerous methods for solving a topological optimization problem: derivative based and level set method, topological gradient method, homogenization method, evolutionary structural optimization, nongradient methods etc. More details on all these methods can be found in [24].
The study is particularly interesting in homogenization method [25] and more precisely SIMP method. Actually, SIMP method has been applied successfully in several domains to obtain innovative structural design and was implemented in most topological optimization commercial software. The aim of the homogenization method is to transform a shape optimization problem into a problem of material density optimization (Shape Optimization by the Homogenization Method). As described in [26], the homogenization method requires a large number of variables, making the implementation computationally expensive.
The aim of the topological optimization is to find the best distribution of material by determining the subdomain ω of a reference domain filled with the material. To solve the topological optimization problem, we usually minimize the objective function f within constraints to define
For instance, the objective function may be represented by: weight, volume, strain energy etc., design variables by dimensions (thickness etc.), type of material etc. and constraints by displacement, mass, frequencies etc.
2.2 Rapid tooling by binder jetting
In the context of structured aluminum foam achievement, the additive manufacturing (AM) technologies allow different possibilities to manufacture complex shape parts of suitable aluminum foam. Such technologies are most commonly used for modeling, prototyping, tooling through an exclusive machine or 3D printer [27]. In order to reduce the time and cost of molds manufacturing, AM technologies are used to develop and manufacture systems of rapid tooling. They are not only broadly used for manufacturing shortterm prototypes but also involved for smallscale series production and tooling applications (Rapid Tooling) [28]. Regardless of the casting method, the foundry industry has as its central process the use of a physical pattern to produce molds into which to cast metal. Although this is true for both the engineering design and production cycles, it is mainly the design earlier stage that will benefit from AM patterns.
The use of Rapid Tooling technologies in the creation of casting patterns allows to quickly manufacture a metal part without using a permanent tooling. Laser technologies applied on metal powder bed (SLS, BEAM and DMLS − Direct Metal Laser Sintering…) are expensive manufacturing processes and the anteriority of project guides the work towards aluminum casting by vacuum. Nevertheless and based on [29], a metal powder product through an AM laser technology keeps a granular structure which is not desirable in this thermal behavior study of metal foam. Through a rapid tooling, this project avoids such granular structure from molten metal powder. Thus, the 3DP plaster technology opens a research path and shows the advantages to realize rapid tooling intended for aluminum foam by casting according to the scale and the complex geometry studied.
2.2.1 3DP technology
In this study, the process uses the ProJet® 460 Plus (3D Systems, formerly ZCorporation) based on a binder jetting technology on powder to produce some plaster molds for casting. A powder hopper provides the manufacturing tray in hydraulic powder by spreading through a roller. The powder consists of Calcium Sulfate Hemihydrate (plaster or gypsum) which is used in making casts, molds, and sculpture. In order to integrate the available technology, the approach defines a manufacturing process to produce the foam structure that corresponds to the cavity shape of the plaster mold.
The optimized model issued from the topological optimization based on the heat exchanges and the lattice model has been chosen according to the geometry of tetrakaidecahedracal model which was proposed by [30,31]. The tetrakaidecahredon is a polyhedron that packs to fill space. Such shape approximates the structural features observed through experiment, nearly satisfies the minimum surface energy condition, and is often used for modeling lowdensity foams.
2.2.2 Rapid tooling process
The process is based on a 3D model in CAD software and then exported as STL format file. This last one is readable by a specific software own to the AM technology which cuts the 3D model in slices to get a new file containing the information for each layer. The specific software generates the hold to maintain the complex geometries automatically with sometimes the possibility to choose some parameters.
After that, the powder mold is produced by binder jetting on powder bed which requires finally a fine cleaning in order to extract the nonhardened powder inside the mold. The final mold diameter is of 80 mm with a height of 44 mm. Final mold in plaster is used as a lost tool through a casting process in order to manufacture the aluminum foam.
2.2.3 Plaster mold preparation
The manufacturing process is achieved by following three steps (i) pretreatment, (ii) casting and (iii) post treatment. Once the plaster mold is manufactured, the pretreatment method should be performed to obtain a plaster mold more suitable for next steps. The negative vacuum casting method is used to fulfill the molten aluminum into the plaster mold. Finally, the post treatment is carried out to remove all plaster remained in the aluminum foam.
The viscosity of molten aluminum increases if the mold temperature is too low regarding the temperature of melting aluminum. Thus, a high viscosity is not convenient to allow molten aluminum filling all cavities of plaster mold. Hence, a high temperature preheating of plaster molds should be used before casting process.
According to the chemical properties of plaster and the results of several pretests, the pretreatment methods with two steps are proposed to improve the quality of the samples. First, the plaster mold is preheated four hours under 200 °C. Then, it is heated again 20 min before casting under 400 °C in order to avoid the liquidity reduction of the melting aluminum and the negative pressure casting is applied.
2.3 Casting of opencell foam
The schematic diagram of infiltration casting method is presented in Figure 1a. The process follows different steps of injection, infiltration, and solidification. Firstly, the plaster mold should be preheated until 400 °C to keep the viscosity of the molten metal during the infiltration process. After the injection of molten metal into the mold, the pressure should be imposed immediately to make the molten metal fills into the spaces and cavities.
The process of infiltration casting consists of three steps: injection, infiltration and solidification. In the injection process, the molten aluminum (at 700 °C) is poured into the mold. The negative pressure is applied to make the aluminum infiltrate through the cavities of plaster mold. The solidification process consists of putting the aluminum/plaster composite in the water to cold it and separate the plaster from the aluminum foam (Fig. 1b).
The aluminum rollers obtained after the infiltrations are then cut (Fig. 2) in order to use them in the experimental setup as described in the next section.
Fig. 1
Schematic diagram of the opencell foam casting. (a) Infiltration casting method. (b) Plaster mold by 3D printing and manufacturing process using the infiltration casting. 
Fig. 2
Aluminum foams; (a) optimized model; (b) lattice model. 
3 Experimental setup and numerical modeling
3.1 Experimental protocol
In order to study the thermal behavior of the manufactured metal foams, PCM is always incorporated in the open foam metal structure. In our case, paraffin wax is used as PCM and the opencell foam obtained is used as metal foam. The paraffin employed is R5658 (Merck Millipore®). The aluminum foam sample (AS7G) was immersed into the melted liquid paraffin. After cooling, a sample was obtained by cutting away the extra paraffin wax.
The test apparatus is designed to observe the melting process and measure the temperature deviation of the composite. As illustrated in Figure 3, the sample of the composite is placed in a rectangular container, a PMMA (Polymethyle Methacrylate) box, with inside dimensions of 57 × 23 × 44 mm. This allow to observe the interface evolution during the melting process. The bottom of the container is the aluminum plate powered by an electric heating film, which is applied as the heat source. The constant heat flux density of 10 000 (W m^{−2}) is realized by DC power. Such boundary condition simulates, for instance, a heat losses issued from electronic devices, PV panels, etc. Top and vertical surfaces are uninsulated and subjected to ambient air convection. In fact, a Newmann boundary condition is applied on the bottom surface while the top and vertical sides are subjected to a Fourier boundary conditions.
Fig. 3 Experimental setup. (a) Experimental system. (b) Schematic representation. 
Three T thermocouples T1, T2 and T3, with an incertitude of ±0.1 °C, are placed at the center (i.e. Y = D/2 and X = L/2) inside the sample at different distances from the heated surface (d1, d2 and d3) according to the zdirection and providing three local temperature measurements which are recorded each second thanks to the data acquisition unit (Fig. 3b). Furthermore, an infrared thermal imaging camera (VarioCAM®) was used to record the temperature filed of the observation surface according to the XZ plane.
3.2 Numerical modeling
3.2.1 Numerical resolution by finite element method of the thermomechanical problem
Consider both conduction and convection in the porous metal with PCM contained in a threedimensional rectangular sample. As explained above, Newmann boundary condition is applied on the bottom surface while the top and vertical sides are subjected to Fourier boundary conditions. All the boundaries are hydrodynamically impermeable. The melting process of the PCM in the porous metal is considered by taking into account the natural convection during the liquid phase. The liquid fraction of the PCM is assumed to be Newtonian fluid and satisfy the Boussinesq approximation. The thermosphysical properties of the considered material are assumed constant in the given temperature's range of variation [20 °C; 80 °C].
The initial and boundary thermomechanical problem (IBVP) is defined in the spacetime domain Ω × [t_{0}, t_{f}] by the following balance equations: (1) where the sub surfaces and forming the boundary Γ_{t} fulfi ll at each time t the following classical relationships and , is the body forces vector, is the force field imposed on and is the force field imposed on . Appropriate initial conditions should be added. The second rank tensor σ_{ij} is the Cauchy stress tensor defined by the thermoelastic constitutive equation: (2)where E_{ij} is the strain second rank tensor, μ_{e}(T) and λ_{e}(T) are the classical thermal dependent Lame's constants with _{E} and v are the Young modulus and Poisson coefficient respectively, is the compressibility or bulk modulus, ξ(T) is the thermal expansion coefficient and T_{0} is the reference temperature assimilated to the outdoor temperature.
Thermal balance equation deduced from the first law of thermodynamics which is combined with Fourier theory is given by: (3) where ρ is the material density, c_{v} is the specific heat coefficient, λ is the heat conduction coefficient, the sub surfaces and forming the boundary Γ_{t} fulfill at each time t the following classical relationships and , is the temperature field imposed on and is the heat flux vector field imposed on . This last heat flux field can include convection and radiation heat parts. Appropriate initial conditions should be added.
The space domain will be discretized using the displacement based Galerkin FEM the reader is referred to the general books dedicated to the FEM for nonlinear problems [32–36]. The discretization by FEM of the week forms of the variation thermal and mechanical problems associated to equations (1) and (3) leads to the following functional for each element (e) in its current (deformed and heated) configuration with volume Ω and boundary Γ_{t}: (4)
In equation (4) the same interpolation functions [N^{e}] have been used to approximate both temperature and displacement fields on each element (e). The matrix [B^{e}] is the matrix of interpolation for the strain tensor and is the one concerning the derivatives of the temperature field. By using the classical assembly procedure, a highly nonlinear algebraic system is obtained. This system is linearized thanks to the iterative Newton–Raphson leading to: (5) in which the components of the tangent stiffness matrix are: (6)with the global functional F and G are obtained by assembling the elementary functionals F^{e} and G^{e} defined by equation (4).
The thermophysical and geometrical parameters of the used materials as well as the boundary conditions are given in Table 1. Because of the influence of the natural air convection surrounding the sample, the convective boundary condition was applied on the four sides as well as the top surface while the fixed heat flow boundary condition was imposed on the heated bottom surface. The average convective heat coefficient may be calculated using free convection empirical correlations for average Nusselt number [32]. Hence, for the laminar flow, is calculated by: (7) where Ra_{L} is the Rayleigh number and is generally of the form: (8)with g (m s^{−2}) is the gravity coefficient, β (K^{−1}) is the air thermal expansion coefficient, T_{s} (°C) is the surface temperature, T_{0} (°C) is the air temperature, v (m^{2} s^{−1}) is the air kinematic viscosity, α (m^{2} s^{−1}) is the thermal diffusivity, and l (m) is the characteristic length defined as: (9)where A (m^{2}) and P (m) are the surface area and perimeter respectively.
(W m^{−2} K^{−1}) is, then, calculated by: (10) where λ_{0} (W m^{−1} K^{−1}) is the air thermal conductivity.
3.2.2 Optimization and control phases
Two phases of modeling were carried out, the optimization phase to obtain new shapes; and the control phase to check a more complex thermal calculation on the proposed shape especially for comparative analysis with experimental measurements. Concerning the optimization phase, the developed model (in SIMP method) assumes that the systems are at the steadystate. When thermal compliance is minimized, temperature at grids where power is applied is minimized. Since it is a smooth convex function, optimization converges much quicker than minimizing the maximum temperature of the entire structure [33].
The aim of the optimization is to find a shape that cools down as much as possible by conduction with respect to a fixed volume constraint (volume of the design area not exceeding 50% of the initial volume). Thus, the thermal compliance is minimized and a model consisting of several predefined patterns and constrain these patterns with a link is performed (see Fig. 4a).
Fig. 4 Optimization process (a) the pattern design model. (b) the control phase model. 
Concerning the control phase, the shape obtained with optimization has been integrated in a global model in relationship with experimental results (see Fig. 4b). Boolean operation between meshes has been used to obtain the model (with Hypermesh© Software [34]). The simulations has been done with the MSC Marc© solver [35,36].
4 Results and discussion
The Figure 5 shows the transient local temperature of the metal foam measured by the embedded thermocouples at different heights from the lower heated surface for both lattice and optimized models. It is shown that the local temperatures rise similarly for such metal foam models until the PCM starts to melt at ∼55 °C whatever the thermocouples location. In the Figure 5, the curves are smooth at the beginning while the temperature fluctuation occurs after the paraffin reaching 55 °C, when PCM starts to melt, thus, it is considered that the effect of the convective heat transfer on the enhancement of the melting process is more pronounced. After that, local temperatures, for both lattice and optimized models, start to diverge reaching significant gap before reaching the steady state.
The melting process is timedelayed depending on the measurement location. For instance, at Y = D/2 and X = L/2, the PCM starts to melt at 27 min; 30 min and 36.5 min respectively for d =10 mm; 20 mm and 30 mm. After PCM completes melting (i.e. ∼66 °C), the temperature continues to rise before reaching the steady state due to the metal heat conduction. The maximum temperature differences are 7 °C; 10 °C and 9 °C respectively for d = 10 mm; 20 mm and 30 mm.
The optimized model which maintains a lower local temperature is more convenient and reliable than the lattice one because of that cooling ability. Hence, for instance, in the microelectronic cooling application, reducing the operating temperature of the integrated circuit chips by 10 °C could reduce the failure rate by half [13].
In order to visualize and analyze the transient thermal behavior of both metal foams at Y = 0, infrared thermal and melting process maps are shown in Figures 6 and 7 from t = 25 min to t = 40 min after melting starting process of PCM and before reaching steady state.
As it can be seen in Figure 6, close to the heated surface according to the zdirection, the temperature inside the lattice model at t = 25 min and 30 min is higher than the optimized model. At the end of the melting process, with no impact of the latent heat (i.e. at t > 30 min), the temperature increases hugely and homogenously in the lattice model. This fact is due to the high sensible heat induced by the high metal thermal conductivity and, as the pore size is relatively high (15 mm), by the convective heat transfer in molten PCM.
However, Figure 7 illustrates the fact that for the optimized model the temperature is less homogenous induced by the effect of the convective heat transfer in cells which is lower than the conductive one when the pore size is relatively small. Furthermore, because of the lower porosity (48.5%) the enhanced fraction of metal increases the conductive heat transfer preserving pores from overheating.
In order to validate the developed threedimensional FEM model, the simulated transient temperature field for the optimized model is compared in Figure 8 with the infrared thermal image before reaching the steady state. One can observe the good accordance between numerical simulation and experimental temperature field of the observation surface (Y = 0) according to the XZ plane. Note that a symmetry effect is observed according to X direction because of the identical convective boundary conditions applied on the sides X = 0 and L. At the steady state (not represented here), the predicted local temperature, at the centre, reaches 77.5 °C at d1 = 10 mm; 73.3 °C at d2 = 20 mm and 67.8 °C at d3 = 30 mm.
Fig. 5
Comparison of local measured temperatures of the PCM infiltrated metal foam at different heights from the heat source for lattice (pore size = 15 mm, porosity = 52%) and optimized (pore size = 12 mm, porosity = 48.5%) metal foam model. (a) d_{1} = 10 mm; (b) d_{2} = 20 mm and (c) d_{3} = 30 mm. 
Fig. 6
Temperature distribution obtained by the infrared thermal camera of the open metal foams at Y = 0. (a) Lattice model; (b) optimized model. 
Fig. 7
Melting process visualization of the open metal foams. (a) Lattice model; (b) optimized model. 
Fig. 8
Temperature distribution for the optimized metal foam at Y = 0. (a) Numerical simulation (scale from left); (b) experimental image (scale from right). 
5 Conclusion
This study focuses on the thermal analysis of a lattice model and an optimized model of opencell metal foams manufactured through an aluminum casting process based on a plaster mold produces by 3D Printing. The study of thermal behavior on the two metal foam structures was conducted based on several measurements, as well as numerical simulations based on the topological optimization. An experimental system was set up in order to measure the local temperature using thermocouples embedded in the metal foam at different levels. Furthermore, the infrared thermal camera was used to provide pictures to visualize the temperature distribution. In parallel, a threedimensional finite element model was developed to simulate and analyze the thermal behavior of such metal foam model.
According to the experimental results, for the given boundary conditions, it was observed that the optimized metal foam presented less and nonhomogenous local temperature than the lattice model with the gap of about 10 °C between both models preserving, for instance, the efficiency of overheated microelectronic systems or photovoltaic panels. The pore size and porosity affect hugely heat transfer through the metal foam. It was proved that conductive heat transfer is more important than the convective one in a lower porosity and lower pore size model.
The comparison between numerical simulations and experimental results regarding the temperature distribution and melting process was qualitatively showed a good agreement allowing the validation of the developed threedimensional model. It was observed the symmetry in terms of temperature distribution and melting process according to the X and Y directions due to the identical convective boundary conditions applied in each side of the sample. Thus, the effect of lateral aspect ratio on thermal behavior in metal foam is not significant.
For future applications, the optimized opencell foam can be helpful to preserve the efficiency and to enhance the lifetime of overheated microelectronic systems or photovoltaic panels.
Nomenclature
e: thickness of the PMMA box [m]
g: gravity coefficient [m s^{−2}]
β: air thermal expansion coefficient [K^{−1}]
T_{s}: surface temperature [°C]
v: kinematic viscosity [m^{2} s^{−1}]
α: thermal diffusivity [m^{2} s^{−1}]
A: surface area of the sample [m^{2}]
P: perimeter of the sample [m]
: average heat transfer coefficient [W m−2 K−1]
λ_{0}: air thermal conductivity, [W m^{−1} K^{−1}]
μ_{e}(T): thermal dependent Lame's constant
λ_{e}(T): thermal dependent Lame's constant
K(T): thermal dependent compressibility modulus
ξ(T): thermal expansion coefficient of the sample [K^{−1}]
ρ: material density [kg m^{−3}]
C_{v}: constant volume specific heat [J Kg^{−1} K^{−1}]
κ: thermal conductivity of the sample [W m^{−1} K^{−1}]
Author contribution statement
The work presented in the paper ap180060 was achieved thanks to the contribution of all aforementioned authors. They have made substantial contributions in the following aspects: topological optimization; modeling and simulation; mathematical formulation; rapid tooling and prototyping (AM); measurements, interpretation of data; drafting of manuscript; and critical revision. Below a comprehensive detail is given:

topological optimization: Dr. Nicolas Gardan;

modeling and simulations: Dr. Nicolas Gardan;

mathematical formulation: Dr. Houssem Badreddine, Dr. Abdelatif Merabtine;

rapid tooling and prototyping (AM): Dr. Julien Gardan;

measurements: Dr. Abdelatif Merabtine, Dr. Chuan Zhang, Dr. Feng Zhu;

analysis and interpretation of data: Dr. Abdelatif Merabtine, Dr. Julien Gardan, Dr. Houssem Badreddine, Dr. Xiao Lu Gong;

drafting of manuscript: All;

critical revision: All.
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Cite this article as: Abdelatif Merabtine, Nicolas Gardan, Julien Gardan, Houssem Badreddine, Chuan Zhang, Feng Zhu, XiaoLu Gong, Experimental and numerical thermal analysis of opencell metal foams developed through a topological optimization and 3D printing process, Eur. Phys. J. Appl. Phys. 83, 10904 (2018)
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All Figures
Fig. 1
Schematic diagram of the opencell foam casting. (a) Infiltration casting method. (b) Plaster mold by 3D printing and manufacturing process using the infiltration casting. 

In the text 
Fig. 2
Aluminum foams; (a) optimized model; (b) lattice model. 

In the text 
Fig. 3 Experimental setup. (a) Experimental system. (b) Schematic representation. 

In the text 
Fig. 4 Optimization process (a) the pattern design model. (b) the control phase model. 

In the text 
Fig. 5
Comparison of local measured temperatures of the PCM infiltrated metal foam at different heights from the heat source for lattice (pore size = 15 mm, porosity = 52%) and optimized (pore size = 12 mm, porosity = 48.5%) metal foam model. (a) d_{1} = 10 mm; (b) d_{2} = 20 mm and (c) d_{3} = 30 mm. 

In the text 
Fig. 6
Temperature distribution obtained by the infrared thermal camera of the open metal foams at Y = 0. (a) Lattice model; (b) optimized model. 

In the text 
Fig. 7
Melting process visualization of the open metal foams. (a) Lattice model; (b) optimized model. 

In the text 
Fig. 8
Temperature distribution for the optimized metal foam at Y = 0. (a) Numerical simulation (scale from left); (b) experimental image (scale from right). 

In the text 
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