# Influence of Scattering Phase Function on Estimated Thermal Properties of Al_{2}O_{3} Ceramic Foams

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## Abstract

Open ceramic foams are usually constituted of three-dimensional networks with randomly interconnected solid struts and fluid within pores. The heat transport within this material can be understood as the coupling of conduction, convection as well as radiation. The cooperative control of different heat transfer mechanisms is critical to successful design and optimization of components working at high temperatures. An answer to this problem usually requires good knowledge and understanding of the corresponding thermal properties in a wide range of temperatures. In the present paper, an inverse identification method was developed to determine coupled thermal properties from transient thermal measurements at temperatures up to 900 K for full description of conduction/radiation heat transports of foam media with absorbing, emitting, and anisotropic scattering effects. The influence of postulated phase function on the identified equivalent extinction coefficient, scattering albedo, anisotropic scattering factor, and two-phase thermal conductivity was discussed for better understanding of thermal behavior within ceramic foams. The estimated thermal properties under each postulated phase function of the sample at transient temperature profiles were used to calculate equivalent thermal conductivities, which were then compared with the measured results at more than 1000 K. The accordance between them indicated that linear anisotropic scattering phase function demonstrates superiority in description of radiation behavior within Al_{2}O_{3} ceramic foam.

## Keywords

Anisotropic scattering Ceramic foams Inverse method Phase function Thermal properties## List of symbols

*C*Specific heat of the sample (J·kg

^{−1}·K^{−1})*g*Anisotropic scattering factor of phase function

- \( \tilde{g} \)
Equivalent scattering factor of phase function

*I*Total radiation intensity (W·m

^{−2})*I*_{b}Total blackbody radiation intensity (W·m

^{−2})- k
_{rad} Radiative thermal conductivity (W·m

^{−1}·K^{−1})*L*Thickness of ceramic foam sample (m)

*q*_{c}Conductive heat flux (W·m

^{−2})*q*_{r}Radiant heat flux (W·m

^{−2})*t*Time (s)

*T*Temperature (K)

*T*_{0}Initial temperature (K)

*T*_{cold}Cold surface temperature (K)

*T*_{hot}Hot surface temperature (K)

*x*Spatial coordinate through the sample thickness (m)

## Greek symbols

*β*Extinction coefficient (m

^{−1})*β*^{*}Weighted extinction coefficient (m

^{−1})- \( \tilde{\beta } \)
Equivalent extinction coefficient (m

^{−1})- \( \tilde{\varvec{\beta }}^{\varvec{ * }} \)
Weighted equivalent extinction coefficient (m

^{−1})*ε*_{1}Emissity of the upper bounding surface

*ε*_{2}Emissity of the lower bounding surface

*θ*Polar angle rad

- Θ
Scattering angle rad

*λ*_{two- phase}Two-phase thermal conductivity (W·m

^{−1}·K^{−1})- \( \tilde{\lambda }_{{two{ - }phase}} \)
Equivalent two-phase thermal conductivity (W·m

^{−1}·K^{−1})*μ*Cosine of the angle between

*x*-axis and direction of radiation propagation*μ*′Cosine of the angle between

*x*-axis and another direction of radiation*ρ*Density (kg·m

^{−3})*σ*Stefan–Boltzmann constant (W·m

^{−2}·K^{−4})*σ*_{a}Absorption coefficient (m

^{−1})*σ*_{s}Scattering coefficient (m

^{−1})- \( \sigma_{s}^{*} \)
Weighted scattering coefficient (m

^{−1})- Φ
Scattering phase function

- \( {\tilde{\varPhi}} \)
Equivalent scattering phase function

*ω*Scattering albedo

- \( \omega^{*} \)
Weighted scattering albedo

- \( \tilde{\omega } \)
Equivalent scattering albedo

- \( \tilde{\varvec{\omega }}^{\varvec{*}} \)
Weighted equivalent scattering albedo

## Notes

### Acknowledgements

This work was supported by Pre-research Key Laboratory Foundation of General Armament Department of China (Grant No. JZ20180035).

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