THERMAL CONDUCTIVITY OF METALLIC FOAM

THERMAL CONDUCTIVITY OF METALLIC FOAM: SIMULATION ON REAL X-RAY TOMOGRAPHIED POROUS MEDIUM AND PHOTOTHERMAL EXPERIMENTS J. Vicente, F. Topin, J-V. Daurelle, F. Rigollet IUSTI Laboratory - CNRS UMR 6595 - Polytech’ Marseille, Marseille, France

Abstract Metallic foams are highly porous materials which present a complex structure of three-dimensional open cells. The determination of effective transport properties is essential for these widely used new materials. The thermal conductivity of metallic foams is determined with the solid phase skeleton obtained from X-ray tomography images where the energy equation is solved. A transient photo-thermal method is used to identify thermo-physical parameters of the foam. Predicted and measured conductivity values are in good agreement. Specific structure morphology measurements are performed separately. We measure directional tortuosities of the solid matrix and correlate them to the cell shape and orientation. We quantify the dependence of the conductivity with tortuosity.

1. Introduction

Figure 1. 3D rendering: solid matrix and several segmented cells

The high open porosity, the low relative density, the high thermal conductivity of the cell edges and the large accessible surface area per unit volume make this new class of materials attractive for numerous applications. (Ashby, Evans et al. 2000). Moreover, they also facilitate mixing and have excellent mechanical properties. Metallic foams are thus used in the field of compact heat exchangers, reformers, biphasic cooling systems and spreaders (Tadrist, Miscevic et al. 2004; Dukhan, Quinones-Ramos et al. 2005). Foams have also been used in high-power compact batteries and catalytic reactor applications such as fuel cell systems (Catillon, Louis et al. 2004). Due to their novelty, their complex three-dimensional structure and varied manufacturing processes, metal foams are still incompletely characterized.

Thermo-physical and flow properties strongly depend as well as on the local morphology of the pore and on the solid matrix. The local change of the structure can influence the properties (e.g. constriction, strut cross section, surface roughness…). It is highly important to assess these properties to understand thermo-physical phenomena and their designs for typical applications. The literature models of effective thermal properties widely used in low porosity media can no longer be applied to high porosity materials. Most of the modeling approaches are based on arbitrary periodic structures which represent the texture of the foam. (Bastawros, Evans et al. 1998) only characterized the foam from the cell ligament diameter and the relative density. There is a lack of morphological tools to characterize the real microstructures of these foams and very few works use the real geometry (usually obtained from 3D X-ray tomography) to determine physical properties (Vicente, Daurelle et al. 2005; Zeghondy, Iacona et al. 2006). The first part of this paper is devoted to the foam morphology characterization. Independent measurements demonstrate that both poral space and solid matrix present correlated angular

dependencies. The second part deals with conductive heat transfer phenomena in the solid matrix. We compute directional effective conductivities through a network approach from the actual skeleton which derives from X-ray tomography. Thermo-physical parameters of the same foam sample are identified via an unsteady photo-thermal method. The anisotropy of the conductivity is observed and the relationship between microstructure and properties is analyzed. We present several results obtained on a Nickel foam sample.

2. Morphology of the foam The studied sample is a cylinder (40 mm diameter, 13 mm thickness) of reinforced NickelChromium foam (Recemat NCX-1723-13). The scanned zone is only 10 mm thick. The high resolution microtomographic acquisition was performed on the ID19 beam line of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). The FRELON CCD camera, developed at ESRF, allows 2048 x 2048 pixels radiographs. According to the characteristics of the optical setup, a pixel of the CCD camera will correspond to a physical area of the sample with a size ranging from 40x40 to 0.3x0.3 µm2. An initial set-up was selected to provide a pixel size of 29.47 x 29.47 µm2 in order to include the totality of our sample. From these radiographs, a last process called “reconstruction” uses an appropriate algorithm (filtered back projection) to obtain a series of 2D slices which build the 3D images. At a given energy, X-ray absorption depends on several physical parameters, mainly the local density and the atomic number of the crossed material. In a porous material, if the X-ray absorption of the solid phase components slightly differs, the absorption 3Dimage can be transformed into a porosity 3D binary image. We have developed a morphological analysis tool to characterize both pore space and solid matrix. It provides geometrical measurements (e.g. specific area, geometrical tortuosity of the phases, porosity…) and allows the segmentation of objects (cells, struts,..) which gives access to structural properties (Vicente, Daurelle et al. 2005). 2.1. Segmentation of the poral space An optimal threshold based on the density histogram is calculated for each X-ray images and allows the identification of pore and solid voxels. The segmentation of the pore into individualized cells allows the porosimetry and morphometry of the foam. We use a specific watershed method (Dillard, N’Guyen et al. 2003) to individualize and close the open-celled pores. To obtain a statistically representative result, a large initial volume containing 1224 cells is analyzed. Incomplete cells at the boundary are excluded from the analysis. Figure 1 shows a 3D rendering of a foam sample and several segmented cells. This segmentation directly allows the measurement of cell morphological parameters. The volume of each cell is determined by counting the voxels inside the cells. We use the mean volume, deduced from the cell volume distribution (Figure 2), to define the mean cell diameter, dv, which is the diameter of the equivalent volume sphere. We previously measured the aperture diameter da (Vicente, Daurelle et al. 2005) which is the diameter of the maximal included sphere. da is systematically smaller than dv (Table 1) and therefore cells are not spherical. Indeed, the process of the polyurethane foam template is mainly affected by gravity, which favors cell elongation toward the vertical direction. Several forces can also modify the cell shape during the nickel foam manufacturing process. Table 1. Morphological parameters. Specific Area Sp (m2/m3) 1658

Dpore dv (µm) 1840.9

Aperture diameter da (µm) 1532.6

Equivalent Ellipsoid a b c (µm) (µm) (µm) 1051±142 817±92 668±92

Cells elongation 2a/(b+c) b/c 1.42±0.17

1.23±0.14

We determine the equivalent ellipsoid of each individualized cell. Ellipsoid dimensions as well as their orientations are calculated from the 3D inertia matrix of the cell voxels. The eigenvalues of

this matrix give the three main ellipsoid axis lengths, noted a, b, and c (with a > b > c). Their distributions are monomodal and rather narrow (Figure 3). One can also note that the distribution of the a-parameter is the most scattered and that the three distributions are overlapping. As expected, the three axis lengths are different and allow the calculation of the cell elongations (Table 1.). Ellipsoid directions are determined from the eigenvectors associated to the eigenvalues (a, b, c) of the cell inertia matrix. Figure 4 presents the polar orientation distribution of these vectors. Two quantities are reported, elevation on figure 4.a and azimuth on figure 4.b. 400 200

350

volume cells

half axis A lenght half axis B lenght half axis C lenght

Frequency (number)

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Figure 3. Distribution of half axis length.

Figure 2. Segmented cells volume distribution

The elevation distribution shows that a–axes are almost vertical; the mean deviation is approximately 20°. The weak difference between b and c elevations shows that the plane bOc is not completely parallel to the original slice plane (XOY). Figure 6.b presents the azimuth distribution (axis orientation into the plane XOY). As the sample is a cylinder, the horizontal and vertical directions of the sample images are arbitrary. The mean baxis azimuth and the mean c-projection axis are close to 115° and 25° respectively. The a-projection is more scattered than b- and c-projections. This was expected as the cells are oriented vertically. Besides, the azimuthal rotation of a-projection only shows the tilt of bOc plane.

Frequency (number)

0 160 140 120 100 80 60 40 20 0 20 40 60 80 100 120 140 160

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Figure 4. Polar representation of ellipsoid axis orientation distribution: left) Elevation (ϕ), right) Azimuth (θ)

We deduce that all the cells are oriented accordingly inside the porous. We will see how the cell organization has an influence on the solid structure and in particular on the solid tortuosity values.

2.2. Geometrical tortuosity (Adler 1992) proposes a geometrical tortuosity definition for a couple of points contained in the same phase: ⎡ L ( p , p )⎤ τ g ( p1 , p2 ) = ⎢ min 1 2 ⎥ ⎢⎣ p1 − p2 ⎥⎦

2

(1) With Lmin(p1,p2), the length of the shortest path joining p1 to p2. We calculate geodesics in the medium with a technique based on a numerical fast marching implementation. The Fast Marching Method (FMM) (Sethian and Kimmel 1988; Sethian 1999) is mainly used for the construction of geodesics on surfaces and the calculation of the optimal ways to circumvent obstacles (Deschamps and Cohen 2001). These methods rely on a fixed grid (adapted to discrete 3D binary images) and consist in propagating a moving interface at a given velocity into the media. A moving interface Γ(t) can be formulated as the zero level curve of a scalar-valued function ϕ : R3xR Æ R, where: (2) Γ(t ) = {x ∈ ℜ3 : ϕ ( x, t ) = 0} ϕ(x) is the crossing time map, a function which gives the time when the moving front crosses point x. Thus, ϕ -1(0) is the initial position of the front and at any later time t the front position is given by ϕ -1(t). The crossing time map is constructed by solving the equation : r 1 (3) ∇ϕ ( x ) = , with F ( x )front velocity at the point x. F ( x) We use this method to compute efficiently the shortest path between any couple of points in a given phase. A constant front propagation velocity for the FMM calculation gives the true geodesic path. We thus calculate the geometrical tortuosity between two points p1 and p2 belonging to the same phase as follows: 1 (4) τ g ( p1 , p2 ) = ϕ ( p 2 ), with ϕ −1 (0 ) = p1 p1 − p2

To measure the tortuosity toward a given direction (e.g. X-direction), we set the initial position of the front ϕ -1(0) as the solid voxels belonging to the plane (x0,y,z). The front propagates across the sample in the solid phase. We collect the arrival times of this front at the plane (x1,y,z); the directional tortuosity is then given by: τ X (x 0 , x1 ) =

1 1 n ∑ ϕ ( pi ) x1 − x 0 n I +1

(5)

with ϕ −1 (0 ) = {p ( x 0, y , z )∈ solid phase}, and pi ( x1 , yi , zi )∈ solid phase . and n is the number of solid voxels of the plane (x1,y,z)

We measure the geometrical tortuosity of the solid phase between two far-off parallel planes in the sample. This evaluation is made for both orthogonal directions X and Y which correspond to the initial horizontal and vertical image orientation. To quantify the angular dependence of the tortuosity, we measure τX and τY for several image rotations around the Z-axis. Figure 5 shows the angular variation of the tortuosity during a complete revolution of the sample. We observe that tortuosity values are quite different for some orientations (25° modulo 90°) and equal for almost orthogonal orientations (70° modulo 90°). To understand these results, we have reported the tortuosity τX on the azimuth distribution of the cell orientation (Figure 6). Solid tortuosity is maximal along the direction of the c-ellipsoid cells. On the other hand, tortuosity is minimal toward the orthogonal direction corresponding to the bellipsoid axis length. We have proved that the solid path is more tortuous toward the small axis cell direction and correlated the solid phase to the cell shape orientation.

directional geometrical tortuosity

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Figure 5. Horizontal and vertical tortuosity for different angular sample rotations

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Figure 6. tortuosity values (τx). Principal axis b and c azimuth are superimposed.

2.3. Solid matrix skeleton A centerline extraction method allows to model the solid matrix as a network of linear connected segments. A skeleton is a subset of the foreground object which is topologically equivalent to the original image. It has the same number of connected components, holes and cavities as the original image. A skeleton is centrally located within the foreground object and could either be constituted of lines or surfaces depending on the topological nature of the foreground object. A thin line centered skeleton (filar skeleton) is one point wide, except at junction points, where the connectivity requires several points. The use of a filar skeleton in the case of open celled foam will give access to the calculation of the strut network. The statistical analysis of the strut length and orientation, as well as the throat reconstruction could be deduced from the idealized network made of interconnected linear segments. We choose the Distance Ordered Homotopic Thinning (Pudney 1998) (DOHT), which uses an iterative deletion of “simple” points in the increasing distance map order leading to a centered skeleton. The skeleton is thus computed by peeling off the boundary of the object iteratively, layerby-layer. A point is said “simple” if its deletion preserves the object topology (Lohou and Bertrand 2002). If all the simple points are removed the result object is topologically equivalent to the original one, but far simpler. In order to preserve the rotation invariance, we add to this method a directional strategy as proposed by (Pudney 1998).The obtained skeleton (Figure 7) is connected, topologically equivalent to the object, centered and thin. Paragraph 3.2 presents the numerical calculation of the effective conductivity from the calculated skeleton. To study the angular dependence of the effective conductivity, we extract from the complete skeleton of the sample different box-shaped skeletons for different rotations around the z-direction.

3. Thermal conductivity (Decker, Möbbauer et al. 2000) provided a detailed experimental characterization and a numerical modeling of the heat and mass transport properties of highly porous media for solar receivers and porous burners. Several researchers focused on determining the effective thermal conductivity. (Pan, Pickenacker et al. 2002) determined experimentally the effective thermal conductivities of foams made out of aluminum and silicon alloys. (Writz 1997) presented a semi-empirical model for the combined conduction and convection heat transfers in a thin porous wall. (Bastawros 1998) provided experimental measurements and modeling of the thermal and hydraulic aspects of cellular

metals subject to transverse airflow. (Kim, Koo et al. 2001) investigated numerically the anisotropy in permeability and effective thermal conductivity on the performance of an aluminum heat sink.

1.29

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Figure 7. Box-shaped skeleton sample (2x2x0.93cm)

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directional effective thermlal conductivity

Tx Ty Keffx/Kmat Keffy/Kmat

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Figure 8 – Directional thermal conductivity and directional tortuosity. Comparison of angular variations

3.1. Thermal conductivity tensor calculation Several authors have implemented prediction models of effective thermal conductivity of foams in a purely conductive regime. These models are based on the knowledge of thermophysical properties of the material constituting the metallic matrix, of the fluid within and generally of a structure parameter of the foam (for example ratio between the average sizes of struts and nodes) (Calmidi and Mahajan 1999; Boomsma and Poulikakos 2001; Bhattacharya, Calmidi et al. 2002). All these methods do not take into account the real morphology of the foam but assume an idealized periodic pattern. Besides, measurements were achieved with both stationary (Calmidi and Mahajan 1999) and transient methods (Fetoui, Albouchi et al. 2004). The computation of heat transfers over the skeleton of the actual solid matrix allows to determine directional effective conductivities. Different box-shaped skeletons are achieved to study the influence of the sample orientation on the effective conductivity. The orientation refers to the orientation of box faces around z axis. Because of the poor thickness of our foam, we have only studied orientation on the xOy plane. A network included in a box (2cm x 2cm x 0.9cm) is cut into the sample. Non-dimensional effective thermal conductivity K ieff k solid on each direction Ox, Oy, Oz is determined alternatively. We calculate the total heat flux Φi across the network with the nodal temperature deduced from simple one-dimensional conduction heat transfers on each strut. On each node p of the network the energy balance is given by: k Φ pj = − solid S pj (T j − T p ) = 0 ∑ ∑ l pj j∈Nodes j∈Nodes connected to p connected to p (6) Where Φpj , lpj and Spj are respectively the heat flux, the length and the cross section of the strut connecting node p to node j. ksolid is the thermal conductivity of the solid matrix, and Tp, Tj are the nodal temperatures. The cross section of the struts is deduced from the porosity of the foam, the volume of the box and the total length of the network. Eventually, the flux Φi is identified with macroscopic conductive heat fluxes across a homogeneous Φi e eff medium placed in the same conditions and K i = ΔT .

It is supposed, that the fluid phase does not contribute to the effective thermal conductivity and the radiative transfer between solid surfaces is negligible. Considering the foam structure and the sample size, we assume that the tensor keeps a dominant diagonal. To highlight the main directions of the conductivity tensor we rotate a box in the xOy plane. For each orientation of the box we identify the conductivity according to the three directions corresponding to the opposite faces Ox, Oy, Oz. Table 2 : Effective thermal conductivity results for several sample orientations. Box orientation Total network length Box volume Strut section Strut equivalent diameter kx/ksol ky/ksol kz/ksol kx (ksol=30) ky (ksol=30) kz (ksol=30)

(°) (m) (10-6 m3) (10-8 m2) (µm) (%) (%) (%) (Wm-1K-1) (Wm-1K-1) (Wm-1K-1)

-70 7.702 3.32 3.71 217.2 2.21 2.12 2.93 0.66 0.64 0.88

-25 7.682 3.32 3.72 217.5 2.04 2.33 2.80 0.61 0.70 0.84

0 7.685 3.32 3.71 217.5 1.99 2.33 2.79 0.60 0.70 0.84

25 7.684 3.32 3.71 217.4 2.15 2.21 2.80 0.65 0.66 0.84

70 7.695 3.33 3.72 217.6 2.33 2.02 2.84 0.70 0.61 0.85

The total network length varies slightly with the angle but the strut section is quite independent from the box orientation. The equivalent strut diameter is comparable to the mean aperture diameter of the solid matrix (Table 2). The small variations of conductivities with the orientation of the sample confirm the assumption that heat fluxes are mainly orientated according to temperature gradients. Conductivity along Oz axis is greater and varies with orientation. This is an unexpected result as there is no variation of the box orientation around Ox and Oy axes. The thickness of our sample (0.9 cm) corresponds to only four or five cells along z axis. The rotation of the box can, thus, include singularities of the network along z direction. This hypothesis is confirmed by the total strut network length values which are higher for the highest conductivities on Oz axis. Conductivity variations along Ox et Oy are however significant and correlated to tortuosity variations as illustrated on Figure 8 which shows superimposed tortuosity and conductivity in the Ox and Oy directions. The tortuosity calculation is implemented directly on 3D images and thus is not dependent on the skeleton building process. For samples of identical tortuosity their conductivity values are similar whatever the direction of the measurement. Table 2 shows quantitative results in a non dimensional form as well as for the case of an Inconel type solid of thermal conductivity 30 Wm-1K-1 which is a typical value for the material constituting our foam sample. A good agreement is observed with the photo-thermal measurement. 3.2. Thermal conductivity using photo-thermal measurement A rear face photo-thermal experiment set-up was carried out (Figure 9). A radiative source provides a uniform heat flux W (W.m-2) onto the sample (initially isothermal) front face for a known duration. The rear face sample temperature evolution versus time is measured by an infrared camera. A parameter identification method is used to deduce thermo-physical properties of the sample (Rigollet, Fetoui et al. 2005). Two cylindrical samples (diameter 40mm, 13mm and 5 mm thick) of the RCM 1723 foam were tested. The thickest one is the tomographied sample presented above. Each sample is taken between

two soles (black painted aluminium foil (13µm thick + about 20µm thickness paint). The front sole guarantees that the radiative heat flux will not be imposed inside the foam volume but on the sole surface only. For the same reason, the infrared flux is measured on the rear sole. A conductive grease containing copper particles provides a good thermal contact between the foam and the soles. The sample lateral surface is insulated with a fibrous material. Uniform heat flux (halogen lamp)

Outside Black painted aluminium soles

IR Caméra (AGEMA 880 SW)

Measured area

sample

Foam (laterally insulated)

Figure 9 Expermimental set up and rear face infrared image (Temperature value)

The radiative source is a halogen lamp supplying up to 1000W/m². The illumination duration, tc, of the sample (about 30 s) is chosen so that the maximum temperature elevation of the rear face does not exceed 2.5°C. Thermograms are measured during 200s. The infrared camera is a AGEMA 880 SW, working with a liquid nitrogen cooled HgCdTe monodetector (3-5µm sensitivity range). The black body calibration curve of the infrared camera and the black paint emissivity value allow the measurement of the rear face temperature. Finally the temperature is averaged on a disc whose diameter is twice smaller than the sample one. A time averaging of 5 consecutive measurements at 25 Hz is used to obtain 5 Hz measurements statistically independent. The noise standard deviation is equal to 0.025°C. Table 3 : Properties of both samples Thickness 13mm Thickness 5 mm 723.3 ± 0.5% 905.96 ± 0.5%

Density ρ f (kg.m-3)

0.914 ± 1% Porosity ε 5 -3 -1 Thermal Volume Capacity ρ f C pf , (Jm K ) 3.23.10 ± 2%

0.892 ± 1% 4.041 e5 ± 2%

Identified β1 = α f / e f , (s-1)

0.0152 ± 6%

0.107 ± 10%

Identified β 2 = h e f / k f

0.343 ± 2%

0.117 ± 1%

Identified β 3 = W / ρ f C pf e f , (Ks-1)

0.146 ± 6%

0.314 ± 9%

Identified β 4 = ρ s C ps e s / ρ f C pf e f

0.526 ± 11%

0.680 ± 16%

2

-6

Thermal diffusivity α f , (m²s-1)

2.58.10 ± 7%

2.67.10-6 ± 11%

Thermal conductivity k f , (W.m-1K-1)

0.83 ± 9%

1.08 ± 13%

Heat exchange coefficient h, (Wm-2K-1)

22 ± 12%

25 ± 15%

The thermal quadrupoles method is used to model the thermal behavior of the multilayer sample (foam between soles). The model describes the foam, the two soles (purely capacitive layers) and the convecto-radiative exchanges between each sole and its surrounding (supposed identical on each face). The four natural groups of parameters are presented in Table 3. The superposition principle is used into the equation 7.a. The inverse Laplace transform L-1{} is numerically computed with the Stehfest algorithm (Maillet 2000). The foam density ρf is calculated from mass and dimension measurements. Porosity ε is then deduced from the knowledge of density of the solid material constituting the foam ρNiCr=8387kg.m-3

and ε=1-ρf/ρNiCr (neglecting air density). The volume heat capacity of the foam is then given by ρ f C pf = (1 − ε ) ρ NiCr C pNiCr + ερ air C pair where C pNiCr = 446J .kg −1 .K −1 . for t ≤ t c T (t , β ) = Tech (t , β ) and Tech (t , β ) = L−1 {θ ech ( p, β )}

(7a) (7b)

B = β 4σ β 4σ 2 + 2 β 2 + σ + β 22 σ

(7c)

[

A = 2 β 4σ 2 + 2 β 2 ,

β = (β1 , β 2 , β 3 , β 4 )

for t > t c T (t , β ) = Tech (t , β ) - Tech (t − t c , β ) β 1 with θ ech ( p, β ) = 2 3 2 β 1 σ A. cosh(σ ) + B. sinh(σ )

]

σ = p β1

(7d)

The most interesting parameter is β1 because it contains the diffusivity of the foam. The conductivity will be deduced from the knowledge of the volume heat capacity. The non-correlation of the four sensitivity coefficients has been verified with simulated noisy measurements and the parameter identification (Figure 10.b). This latter consists in minimizing the least square criterion between the measurement and the model response with respect to the four parameters. This analysis shows that it is possible to estimate the four parameters simultaneously thanks to the iterative sequential Gauss-Newton method. The convergence is possible despite its very great sensitivity to the initial values of the parameters. a)

2

measurements optimal model

°C

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Figure 10 a) Example of experimental and model thermal responses (13mm thick sample). b) Reduced sensitivities to the four parameters

The influence of the thermal contact resistance between soles and foam has been tested. It appears that the best residuals (which have the same statistical properties as noise measurement) are obtained for the model with nil contact resistance (Figure 10.a). Moreover, the obtained sequential parameter values (estimated at each time step) converge clearly in the second half of the time range (about 100s). This is a good indication of the relevancy of identified parameters (Figure 10). Optimal parameter values are presented in Table 3. We obtain a higher value of β4 (capacitive parameter) than expected. Grease and black paint may have a higher heat capacity than expected or the layer thickness may be underestimated. The obtained thermal conductivity values are coherent between each other and well agree with the calculations presented above (Table 2). Another approach using volume excitation (i.e. without soles) is in progress in order to eliminate the thermal contact resistance and to measure the full conductivity tensor.

K.s-1

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Figure 11 Sequential estimations of the 4 parameters. Red lines delimit 95% confidence intervals

4. Conclusion We developed a morphological analysis tool to quantify the main structural parameters of metallic foams. It provides the functions of geometrical measurements (specific area, pore size distribution...). Cell shapes are measured and specific cell orientations are quantified. An original method based on the numerical fast marching implementation has been developed to measure the geometrical tortuosity of both phases (solid and pore). Anisotropy of the solid matrix is observed. The tortuosity measurements carried out for different directions in the sample show a clear correlation between cell orientation and tortuosity values of the solid matrix. An efficient centerline extraction method gives the skeleton of the solid phase. The identification of nodes, segments, and connectivity of the solid network has been carried out. We have been able to calculate heat fluxes and temperature gradients by solving the energy equation within the network. The effective thermal conductivity tensor is thus identified. It helps understanding heat transfers in the foam and how they are correlated to the local structure. This constitutes an important step to the prediction of the conductivity tensor of the foam from its real structure. However, this work is not yet a fully quantitative prediction method of conductivity tensor values. A systematic study of the impact of morphology on transport properties is in progress with other samples of different textures. On the other hand, a specific experimental set-up is implemented to give accurate values of the thermal conductivity for the other sample directions.

Acknowledgments The authors thanks Recemat Company for providing the samples, the ID19 beam team for the helpful assistance at ESRF synchrotron facility, the ministry of education and research and CNRS who supported this project “Specimousse”.

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