Analytical Modelling of the Radiative Properties of Metallic Foams: Contribution of X‐Ray Tomography

Abstract

Solid foams and notably metallic and ceramic foams exhibit thermal, mechanical and exchange properties that make them very interesting in various applications such as burners, electrodes for electric battery, high temperature spatial insulation, heat exchangers, etc. In most of these applications, the knowledge of the thermophysical properties is of primary importance for the dimensioning of structures. However, these properties, and especially the radiative properties, are directly related to the foam microstructure. [1,2] In order to optimize the thermal efficiency of such materials it is thus necessary to understand how the microstructure affects the radiative properties.It is particularly difficult to estimate radiative heat transfer in solid foams due to the complexity of the architecture in these materials. Several methods are used to predict the radiative properties of dispersed media. [3,4] Some of them consider the porous structure equivalent to a random arrangement of particles of given shapes and use the Mie theory or the geometric optics laws. [5][6][7][8] Other methods are based on reflectance and transmittance measurements of the medium and use the inverse method. [9,10] Finally, another approach consists in realising Monte Carlo simulation at the local microscopic scale and taking into account the complex morphology of the porous medium. [11,12] If, in these former studies, the architecture of the porous medium was taken from a model shape, recent studies use the real structure reconstruction using the X-ray tomography technique. [13,14] However, Monte Carlo simulations in tomographed samples require a huge computational effort, and should be restricted to the study of some special aspects such as the detailed effect of the architecture.Among the radiative properties, it is particularly interesting to estimate the extinction coefficient because it is not a local property (as the phase function and the albedo). It depends on the porosity and on the morphological structure of the foam. Moreover it enables to calculate the radiative conductivity k R . If the medium is optically thick and exhibits an isotropic scattering phase function, the Rosseland diffusion approximation is valid. This approach consists in treating the radiative transfer as a diffusion process. Then one may define the radiative conductivity as: [15] k R 16n 2 rT 3 3b R 1where r is the Stefan-Boltzmann constant (= 5.67 × 10 -8 W/m 2 K 4 ) and n is the effective index of refraction of the heterogeneous medium. The Rosseland mean extinction coefficient b R can be computed from its definition:where b à k is the weighted extinction coefficient and (dI b,k )/dT is the derivative of the blackbody spectral intensity I b,k with respect to the temperature T. Note that b à k is corrected in order to consider the well known anisotropy of open-cell foams scattering. [16,6,7] From these observations, the purpose of this study is to present the multiple possibilities offered by the tomography technique for the determination of the radiative prope...

Quantitative X-ray tomography

“…This has mainly been achieved for highly porous materials. [410][411][412][413] The thermal properties of nuclear graphite are also of technological concern and have been predicted directly from CT images. 414 …”

Pore-level engineering of macroporous media for increased performance of solar-driven thermochemical fuel processing

“…9. The models of Hsu and Howell [27] and Loretz et al [28], introduced in Table 2, are fitted with the empirical fitting parameter W equals 2.398 and 1.765, respectively (RMS = 25.0% and 3.8%, respectively). Hsu and Howell [27] modeled the porous foam as a suspension of mono dispersed, independently scattering particles with a diameter equal to the pore diameter d. Despite some consensus of the model with the extinction coefficients for foam like structures [61,27], it is not consistent with the present MC data.…”

“…Two models used to predict the effective extinction coefficient are based on geometrical optics for porous media consisting of a suspension of mono-dispersed and independently scattering particles. Hsu and Howell [27] considered spherical particles whereas Loretz et al [28] investigated multi-faced particles.…”

Materials Selection for Optimal Design of a Porous Radiant Burner for Environmentally Driven Requirements

“…[1][2][3][4][5][6] and their relationship with the constitutive material and architecture. [11][12][13][14][15][16][17] The current numerical model is constituted of 36 coupled equations (27 equations for species transport according to the famous Mech 1 chemistry mechanisms; [18] 3 equations for energy transfer and gas phase velocity (i.e., Eqs. 1, 5, and 6); and 6 RTEs corresponding to 6 Gaussian angular discretization [19] between 0 and p).…”