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Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 16–27 www.elsevier.com/locate/jqsrt

Metallic foams: Radiative properties/comparison between different models M. Loretza,, R. Coquarda, D. Baillisa, E. Maireb a

CETHIL, UMR5008, CNRS, INSA-Lyon, Universite´ Lyon1, F-69621 Villeurbanne, France b GEMPPM, UMR 5510, CNRS, INSA-Lyon, F-69621 Villeurbanne, France Received 19 October 2006; received in revised form 16 May 2007; accepted 25 May 2007

Abstract The aim of this study is to determine the radiative properties, which are the extinction coefficient, the scattering albedo and the scattering phase function, of highly porous open-cell aluminium foam, using more-or-less simple predictive models, and to compare all these models. The radiative properties are predicted using geometric optics laws to model the interaction of radiation with the particles forming the foam. Moreover, the particles forming the foam are large compared with the considered wavelength and are supposed to be sufficiently distant from each other to scatter radiation independently. Thus, the radiative characteristics of the foam can be determined by adding the contributions of each particle. A particular attention is paid on microstructure analysis and modelling. We considered different kinds of cell shapes and struts cross-section, using microscopic and tomographic analysis. Furthermore, a new phase function modelling is presented. Finally, we compare the results of each method with the radiative properties obtained from experimental measurements of directional and hemispherical transmittances and hemispherical reflectance. r 2007 Elsevier Ltd. All rights reserved. Keywords: Metallic foams; Radiative properties; Tomography; Independent scattering; Polyhedral cells

1. Introduction A wide variety of engineering systems involve semi-transparent media, like porous materials and notably foams [1]. They play a crucial role in the heat transfer phenomenon and thus the modelling of radiative transfer is of primary importance for the prediction and optimization of the capabilities and performances of such materials. Radiative heat transfer prediction in porous media requires the determination of their radiative properties, which are the extinction coefficient b, the scattering albedo o and the scattering phase function F, appearing in the radiative transfer equation. This paper deals with the determination of the radiative properties of a highly porous open-cell aluminium foam sample from different methods, which are compared between themselves. The radiative properties of foam depend on microscopic structure and optical properties of the solid matrix. There are two main groups Corresponding author. Fax: +33 0 4 72 43 88 11.

E-mail address: [email protected] (M. Loretz). 0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2007.05.007

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Nomenclature Latin symbols a bmoy D ps G Lv Nv Rh Td Th V1 V2

strut length (m) average strut thickness (m) average cell diameter (m) parameter of specularity average area of a particle projection on a plan perpendicular to the incident ray propagation direction (m2) total length of struts per unit volume (m2) total number of particles per unit volume (m3) normal–hemispherical reflectance normal–normal transmittance normal–hemispherical transmittance strut volume (m3) strut juncture volume (m3)

Greek symbols b F o r l e j y

extinction coefficient (m1) scattering phase function scattering albedo reflectivity of the solid matrix wavelength (mm) porosity azimuthal angle (rad) polar angle (rad)

of methods used for determination of the radiative properties of porous media: analytical modeling and experimental approaches. In order to understand the complex physical radiative phenomena occurring in the foam, the first group of methods is useful. Radiative properties can be obtained by adding up the effects of all particles constituting the foam. Indeed, for a highly porous foam, dependent scattering can be neglected. Previous works which already deal with the modelling of foam structure have pointed out quasi-regular polyhedral geometry of the pores. Indeed, a unit cell closely resembles a pentagon dodecahedron. Thus, Glicksman and Torpey [2] considered polyurethane foams as a set of randomly oriented blackbody struts with constant triangular thickness (Fig. 1), assuming an absorption efficiency factor of one. Microscopic analysis of foam cross-sections permitted them to conclude that they occupied two-thirds of the area of an equilateral triangle formed at the vertices. The resulting extinction coefficient is a function of the cell diameter D and the foam porosity e. In addition, Kuhn et al. [3] used infinitely long randomly oriented cylinders to describe struts of polyurethane and expanded polystyrene foams. The triangular cross-sections were converted into circular ones with the same geometric mean cross-section (Fig. 2). Mie theory is used to predict the radiative

Fig. 1. Perspective view and real cross-section of a strut.

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Fig. 2. Perspective view and cross-section.

Fig. 3. Decomposition of a strut into three parts and representation of a volume of intersection.

properties. Moreover, Baillis et al. [4] proposed a model for prediction of the radiative properties of open-cell carbon foams based on a combination of diffraction and geometric optic laws. It was applied to two different types of particles: struts with varying thickness and strut junctures (Fig. 3). Finally, Placido et al. [5] have developed a geometrical cell model to predict the radiative and conductive properties of different types of insulation foam with different morphological structures such as expanded polystyrene, extruded polystyrene and polyurethane foams. Mie theory is used as in Kuhn et al.’s works. The effective thermal conductivities of foams are compared with experimental data. Until now, radiative heat transfer in metallic foams was not studied. Thus, in this work, we are interested in predicting the radiative properties of open-cell aluminium foam using these different methods. We model the structure of the foam in a more or less precise way by studying it using microscopic and tomographic analysis, and we look for the more adequate model. Firstly, we focus on the structural analysis of the foam, using microscopic and tomographic analysis. Indeed, the properties of interest have a direct relation with the morphology of the cellular material [6]. Then, we calculate the radiative properties using the hypothesis of independent scattering, usually used for highly porous foams. This hypothesis assumes that the interaction of one particle in the foam with the radiation field is not influenced by the presence of others particles. The dimensions of the particles composing the foam are much greater than the wavelength l of the thermal radiation studied, which range between 5 and 50 mm for a radiative heat transfer calculation at 300 K. As a result, the interaction between radiation and matter can be treated using the geometric optic laws. Finally, the influence of diffraction on the radiative heat transfer is neglected. Indeed, according to Brewster [7], the diffraction phase function for particles with a large size parameter is predominantly oriented in the forward direction and diffracted rays are very close to transmitted rays. We then assume, here, that diffraction can be treated as transmission. Finally, we determine the extinction coefficient, the scattering albedo and the scattering phase function parameter, thanks to an identification method using infrared spectrometer transmittances and reflectances measurements. This method is based on the partial diffuse and partial specular reflection behaviour of particles, whereas until now diffuse reflection was assumed by Baillis et al. [8] for open-cell carbon foams. This permits us to compare the results of the different methods with these experimental results.

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2. Structural analysis The investigated material is commercialized by the ERG company. It is obtained by liquid infiltration of a ceramic preform by molten aluminium. The structure is very homogeneous in terms of strut thickness and cell size. The dimensions of the sample are 12 mm  12 mm  9 mm for a porosity of 93%. The number of cells in the thickness of the sample is approximately 5 and its surface includes a great number of cells. Thus, the aluminium foam sample can be regarded as a homogeneous medium. 2.1. Microscopic analysis Aluminium foam has a cellular structure, whose shape can be deduced from microscopic analysis (Fig. 4). According to these pictures, cells present not only a pentagonal shape, as it is usually assumed, but also a tetracaedecaedric shape. Tetracaedecaedric cell is made up of fourteen faces including six square faces and eight hexagonal faces. In addition, stronger enlargement photographs show that the network of interconnected edges forming the cells is composed of nonconstant curved triangular struts section as observed by Baillis et al. [4] for open-cell carbon foams (Fig. 5). The averaged geometrical parameters required for the determination of the properties for each model were obtained for a single sample using numerous microscopic analyses of several struts (Table 1). In order to take into account the average projected surface, the parameters of Table 1 are obtained using the following formulas: ( a¯ S ¼ aS p4 ; a¯ P ¼ aP p4 ; (1) b¯ min ¼ bmin 3 ; b¯ P ¼ bP 3 ; b¯ max ¼ bmax 3 ; p

p

p

where a¯ S , a¯ P , b¯ min , b¯ P and b¯ max are the average lengths projected on all the orientations. 2.2. Tomographic analysis In order to have more information such as cell diameter, it seems necessary to consider the real structure of the foam by using X-ray tomography. Fig. 6 shows a three-dimensional (3D) image of the aluminium foam, with a tomography resolution of 60 mm. The reconstruction method is a filtered back-projection algorithm widely used in X-ray tomography [9] reconstruction and especially implemented for parallel beam geometry reconstruction. This image confirms that the struts have a nonconstant section of triangular form.

Fig. 4. Photographs of aluminium foam cuts with the SEM. a0 is the projection of the total length of a strut (length between two strut junctures).

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Fig. 5. Photographs of aluminium foam cuts with the SEM.

Table 1 Geometrical parameters of the aluminium foam in mm with standard deviation from average values (SEM) a

aS

ap

bmin

bp

bmax

b

1.6470.08

1.170.08

0.2970.08

0.5270.03

0.0870,03

0.7070.03

0.5870.03

b is the average thickness of a strut calculated over all its length.

Fig. 6. 3D Image of the Aluminium foam.

In the studied images, the threshold is straightforward because the attenuation contrast is very good between aluminium and air and also because the resolution is adapted to the size of the features to image. As a consequence, the threshold has a weak influence on the geometrical parameters determined. A granulometric analysis on the 3D image provides the size distributions of the struts’ thickness bmoy and of the cell diameter D. Indeed, granulometry [10] consists in making successive openings with increasingly large octahedral or spherical structuring elements, which permit to progressively remove the particles having the size of the structuring element. The difference between the image before and after an opening corresponds to the total volume of the disappeared particles. It is then possible to obtain the size distributions of the cells’ equivalent diameter but also, by working on the opposite image, the size distributions of the struts thickness. A monomodale distribution is found, for the struts and for the cells. Octahedral or spherical structuring

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Table 2 Geometrical parameters of the aluminium foam in mm with standard deviation from average values (granulometry from the tomographic images) D

bmoy

4.970.4

0.5470.2

elements give almost the same results. Geometrical parameters obtained for the sample of the aluminium foam are shown in Table 2. 3. Morphology modelling and prediction of the radiative properties for foam As a consequence of structural analysis, two approaches are considered: the first assumes that aluminium foam is made up of pentagon dodecahedron cells, as usually assumed for highly porous foams, and the second considers that it is made up of tetracaedecaedric cells. In addition, we have considered two types of input data: geometrical parameters of the aluminium foam obtained using microscopic analysis, and those obtained with tomographic analysis, in order to study their influence on the result. Finally, we calculate the extinction coefficient for each model and using each type of input data, assuming independent scattering and using geometric optics laws. The corresponding extinction coefficients formulas as well as the porosity calculations for each model are summarized in Table 3. A model will be valid if it allows simultaneously to approach the experimental results of the extinction coefficient and to obtain a porosity close to the real one. For example, the extinction coefficient obtained by the Glicksman and Torpey model [2]: rffiffiffiffiffiffiffiffiffiffiffi 1 b ¼ 4:09 (2) D2 is here calculated from struts dimensions. Indeed, ¯1 b ¼ NvG

(3)

with Nv ¼

1:305 a3

(4)

if the cells are pentagon dodecahedrons as it is assumed in this model, GT1 3ab . ¼ 4 4

(5)

1:305 3b 3 ¼ Lv b a2 4 4

(6)

¯1 ¼ G So, b¼ with Lv ¼

1:305 . a2

(7)

Finally, 3 b b ¼ 1:305 2 , 4 a

(8)

where a is calculated from a0 , which is the measurement of the projection of the total length of a strut (Fig. 4).

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Table 3 Results of all the studied models Cellular medium

Struts Strut SEM only junctures b

SEM+granulometric analysis Formulas

1e

n

No

3 b 1:305 2 4 a

}

No

3 1:305 ðaS bmin þ 2ap bmin þ 2ap bp Þ 3 4 a

J

No

1 2

&

Yes

pffiffiffi   3 2 1:305 3 b b þ 2a b þ 2a b Þ þ ða S min p min p p max 8 a3 4

ðV 1 þðV 2 =2ÞÞ (12) 1:305 a3

Tetracaedecahedron n

No

3 b 1:061 2 4 a

b (13) 0:306 a2

}

No

3 1:061 ðaS bmin þ 2aP bmin þ 2ap bp Þ 3 4 a

(14) 0:306 bmax a2

J

No

1 2

&

Yes

pffiffiffi   3 2 1:061 3 bmax ðaS bmin þ 2aP bmin þ 2ap bp Þ þ 3 8 a 4

Pentagon dodecahedron

rffiffiffiffiffiffiffiffiffiffiffi p 1:305 pffiffiffib 2: 3 a2

rffiffiffiffiffiffiffiffiffiffiffi p 1:061 pffiffiffib 2: 3 a2

2

b (9) 0:377 a2

2

(10) 0:377 bmax a2

Formulas in function of D ¼ 2.57a (instead of a) in Eqs. (9)–(12) bmoy (instead of b) in Eqs. (9) and (11), both determined by granulometric analysis

2

b (11) 0:377 a2

2

2

Formulas in function of D ¼ 2.995a (instead of a) in Eqs. (13)–(16), bmoy (instead of b) in Eqs. (13) and (15), both determined by granulometric analysis

2

b (15) 0:306 a2

ðV 1 þðV 2 =2ÞÞ (16) 1:061 a3

n constant triangular section struts. } nonconstant triangular section struts. J circular section struts. & struts+strut junctures.

4. Experimental radiative properties Experimental results are based on identification procedure using spectral transmittances and reflectances measurements. Since the foam is metallic, the particles are assumed opaque; the scattering is thus limited to the reflection. As a consequence, the scattering albedo ol is equal to the reflectivity rl. In the work of Baillis et al. on carbon foam [8], diffuse reflection was assumed. In the current work, we assume that the foam presents an intermediate reflecting behaviour comprised between diffuse and specular reflection. Thus, the phase function is represented by a combination of diffuse and specular scattering phase functions of opaque particles [7], pondered by a parameter. It is called parameter of specularity and denoted psl. The zero value of psl corresponds to the diffuse reflection, whereas when psl ¼ 1 reflection is specular. The determination of the scattering albedo and the phase function requires only the knowledge of rl and psl. The reflectivity values found in the literature present a great dispersion and do not correspond necessarily to the wavelengths considered in this study. As a consequence, an identification method using spectrometric measurements was developed.

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4.1. Spectrometric measurements A spectrometer of manufacture Biorad, model FTS 60A, associated with a gold-coated integrating sphere, is used in the spectral range 1.5–25 mm. It allows the measurements of normal–normal transmittances (Td), normal–hemispherical transmittances (Th) and normal–hemispherical reflectances (Rh). Six spectrometric measurements are made by repositioning the sample, in order to obtain average normal–normal transmittance, average normal–hemispherical transmittance and average normal–hemispherical reflectance of the aluminium foam with standard deviations. 4.2. Identification procedure The procedure is iterative and requires the calculation of theoretical transmittances and reflectance of aluminium foam. Indeed, by assimilating the aluminium foam to a semi-transparent absorbing and scattering medium, with radiative properties bl, ol ( ¼ rl) and F (psl), it is possible to determine theoretical transmittances and reflectance of the sample by solving the RTE inside the foam. An iterative process is then used in order to determine the values of bl, ol (or rl), and psl minimizing the difference between theoretical transmittances and reflectance of the sample and the measured experimental values Tdlm, Thlm and Rhlm. The iterative process assumes that theoretical transmittances and reflectance strongly depend on one of the parameters to identify as follows:



normal–normal transmittance Tdl strongly depends on the extinction coefficient bl, but varies very slightly with ol and psl,  the sum Thl þ Rhl represents the proportion of incident energy leaving the foam without being absorbed, and thus depends primarily on ol,  the ratio Rhl =Thl translates the relationship between reflected and transmitted energies and varies primarily with psl. The theoretical calculation of transmittances and reflectance is carried out by solving the one-dimensional Cartesian Radiative Transfer Equation by the Discrete Ordinates Method, using a uniform quadrature constituted of 90 directions between 0 and p. The iterative procedure, enabling to determine simultaneously bl, rl and psl, which minimize the difference between the theoretical and experimental values, unrolls as follows. In the first step, theoretical transmittances and reflectance Tdl, Thl and Rhl are calculated starting from values of bl, rl and psl arbitrarily chosen. Then, the difference between theoretical and experimental values of Tdl, Thl þ Rhl and Rhl =Thl are used to determine the increments Dbl, Dol and Dpsl needed for the calculation of bl, ol and psl at the next iteration. Finally, the procedure goes back to the beginning of the first step with the new values of bl, rl and psl. The procedure is repeated until the relative differences between theoretical and experimental values of Tdl, Thl þ Rhl and Rhl =Thl are all lower than a tolerance, typically 0.001. The calculation of the phase function, which depends on psl, is required for the calculation of theoretical transmittances and reflectance. To avoid recomputing the phase function at each iteration, and thus to minimize the computation time, a series of calculations was carried out as a preliminary, in order to know all the possible phase functions for psl varying from 0 to 1. 4.3. Calculation of the phase function The phase function is computed as follows. In the first step, ps ¼ 0 is taken. Then, in a second step, a counter of the reflected rays, compt, is set to 0. The values of ps and of a random number D ranging between 0 and 1 are compared. If D 4 ps, then the reflection is diffuse and the direction of the reflected ray is randomly pffiffiffiffiffiffi chosen using new random numbers Di (i ¼ y, j): yr ¼ cos1 ð Dy Þ and jr ¼ 2pDj : 8 > < dxrefl ¼ sin yr sin jr ; dyrefl ¼ sin yr cos jr ; (17) > : dz ¼ cos y : refl r

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Fig. 7. Illustration of the specular reflection.

Else, the reflection is specular and the direction of the reflected ray is symmetrical with that of the incident ray, compared with the normal on the surface (Fig. 7): 8 > < dxrefl ¼ dxinc ; dyrefl ¼ dyinc ; (18) > : dz ¼ dz : refl

inc

The determination of the direction of the reflected ray permits to calculate the angle y0 between the incident and scattered directions: y0 ¼ 2 sin1 ðdist=2Þ

(19)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with dist ¼ ½ðdxinc  dxrefl Þ2 þ ðdyinc  dyrefl Þ2 þ ðdzinc  dzrefl Þ2  and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2i þ dy2i þ dz2i ¼ 1;

i ¼ inc; refl.

The counter S(y0 ) of the phase function, where y0 is the angle between the incident and scattered directions, is then incremented. Then, the procedure goes back to the beginning of the second step with a new random number D and compt ¼ compt+1, until the counter of the reflected rays reaches a very large value fixed in advance, typically 1,000,000 in this case. In practice, at each reflection, the angle between the incident direction and the direction of reflection is stored. The average angular distribution of the reflected rays S(y0 ) is then discretized for y0 0 ¼ 01, y0 1 ¼ 11, yy0 180 ¼ 1801, and all the rays reflected in a direction forming an angle y0 ranging between y0 i0.51 and y0 i+0.51 with the direction of incidence are brought together in S(y0 i). Then, the phase function can be determined from this angular distribution. Indeed:

 

S(y0 ) represents the probability for a ray arriving on the solid matter of foam to be reflected in a direction forming an angle ranging between y0 –0.51 and y0 +0.51 with its direction of incidence, and ð1=4pÞ Fðy0 ÞDO0 represents the probability for the incident ray to be scattered in the solid angle DO0 centred on the direction, which forms an angle y0 with respect to the direction of incidence. So we have: ð1=4pÞFðy0 ÞDO0 ¼ Sðy0 Þ and Fðy0i Þ ¼ ð4pSðy0i Þ=ðDOðy0i ÞÞ with 0pip180. 0 0 0 Moreover, DOðyi Þ ¼ 2p sinðyi ÞDyi .

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Fig. 8. Examples of scattering phase functions.

And thus Fðy0i Þ ¼ 2

Sðy0i Þ sinðy0i ÞDy0i

with Dy0i ¼ 1 .

(20)

Finally, the phase function is normalized in order to satisfy the equation: 180 1 X ðFðy0i ÞÞ  2p sinðy0i ÞDy0i ¼ 1. 4p i¼0

(21)

Lastly, the procedure goes back to the beginning of the first step with a new value of ps, in order to determine a new phase function, until all the phase functions for ps varying from 0 to 1 with a step of 0.05 are obtained. Fig. 8 presents examples of scattering phase functions. Curves are plotted for all the 51 to be more understandable. The limiting cases can be verified: for ps ¼ 0, the phase function is related to reflection from a large opaque diffusely reflecting sphere; for ps ¼ 1 the isotropic phase function is obtained. 5. Results and discussion 5.1. Variation of the identified parameters with the wavelength The results of the identification are represented in Figs. 9–11 for the sample. We can note the following:

  

The extinction coefficient is almost independent on the wavelength. This confirms that geometrical optic laws can be applied to determine the radiative characteristics of the foam. The reflectivity is raised enough, 90% on average, which is logical, taking into account the strong reflective ability of aluminium matter. Finally, the parameter of specularity is close to 0 for most of wavelengths. It tends to confirm that the aluminium foam has a scattering behaviour close to that of an opaque sphere with diffuse reflection. This assumption was used by Baillis in the case of carbon foams with open pores.

This extinction coefficient identified using experimental measurements is the reference to compare the different models of prediction. 5.2. Theoretical and experimental extinction coefficient The extinction coefficients as well as the checking of porosity for each model (Table 3) are represented in Fig. 12. The uncertainties on calculations are about 1% for the extinction coefficient and 0.2% for the porosity. They are calculated from the standard deviations obtained for the geometrical parameters.

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Fig. 9. Identified extinction coefficient with the wavelength.

Fig. 10. Identified reflectivity with the wavelength.

Fig. 11. Identified parameter of specularity with the wavelength.

We can note that tetracaedecaedric cells give better results than dodecahedral cells. The approach using tetracaedecaedric cells is the only one that gives practically the same results when measurements are made with the SEM alone (a and b), or with the SEM and by granulometric analysis (D and bmoy). In fact, there is a distribution of polyhedral shapes in foams, and the results obtained tend to show that tetracaedecaedric cells on an average better approximate the bulk. Moreover, it means the importance of the choice of the model structure on radiative properties. The model comparing foam to a medium made up of tetracaedecahedron, formed by nonconstant triangular section struts and strut junctures, is the most satisfactory. Thus, the choice of the model and the precision of the foam microstructural entry parameters play an important role in the determination of the extinction coefficient. It can be noted that the most commonly used extinction coefficient, obtained by the Glicksman and Torpey formula (Eq. (2)), which neglects the strut junctures and depends, contrary to the formulas presented in Table 3, on the porosity, gives b ¼ 220:21  0:45 m1 . This represents an error of about 15% compared with the experimental extinction coefficient. 6. Conclusions If the radiative properties of carbon, expanded polystyrene, extruded polystyrene and polyurethane foams have already been studied, it is not the case for aluminium foams. Thus, the current study underlines the importance of radiative scattering in these foams. Indeed, the model of Glicksman and Torpey, neglecting scattering, is less suitable for the studied material, because of the high reflectivity of aluminium. In addition,

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Fig. 12. (a) Extinction coefficient (m1) function of porosity—pentagon dodecahedron cells. (b) Extinction coefficient (m1) function of porosity—tetracaedecahedron cells.

pentagon dodecahedrons are usually used to model the complex morphology of the foam. This study highlights tetracaedecahedron cells, and shows that struts, strut junctures and cell geometry are recommended to be considered. A new method of phase function determination, combining specular and diffuse phase functions, has been developed, in order to take into account the real scattering behaviour of the foam. Thus, this study shows that the aluminium foam has a scattering behaviour close to that of an opaque sphere with diffuse reflection. Moreover, the proposed experimental method is promising to determine the radiative properties of other materials like ceramic foams, and the determination of radiative properties dependent on temperature will be conducted. References [1] Weaire D, Hutzler S. The physics of foams. Oxford: Oxford University Press; 1999. [2] Glicksman LR, Torpey M. A study of radiative heat transfer through foam insulation. Report prepared by the Massachusetts Institute of Technology under subcontract no. 19X-09099C, Cambridge, 1988. [3] Kuhn J, Ebert HP, Arduini-Schuster MC, Buttner D, Fricke J. Thermal transport in polystyrene and polyurethane foam insulations. Int J Heat Mass Transfer 1992;35(7):1795–801. [4] Baillis D, Raynaud M, Sacadura JF. Determination of spectral radiative properties of open cell foam. Model validation. J Thermophys Heat Transfer 2000;14(2):137–43. [5] Placido E, Arduini-Schuster MC, Kuhn J. Thermal properties predictive model for insulating foams. Infrared Phys Technol 2004;46(3):219–31. [6] Gibson LJ, Ashby MF. Cellular solids: structure and properties. 2nd ed. Cambridge, UK: Cambridge University Press; 1997. [7] Brewster MQ. Thermal radiative transfer and properties. New York: Wiley; 1992. [8] Baillis D, Arduini-Schuster M, Sacadura JF. Identification of spectral radiative properties of polyurethane foam from hemispherical and bi-directional transmittance and reflectance measurements. JQSRT 2002;73(2–5):297–306. [9] Baruchel J, Buffie`re JY, Maire E, Merle P, Peix G. X-ray tomography in material science. HERMES Science Publications; 2000. [10] Maire E, Colombo P, Adrien J, Babout L, Biasetto L. Characterization of the morphology of cellular ceramics by 3D image processing of X-ray tomography. J Eur Ceram Soc 2007;27(4):1973–81.