Influence of Scattering Phase Function on Estimated Thermal Properties of Al₂O₃ Ceramic Foams

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 Al₂O₃ ceramic foam.

Appendix

The transient temperature responses in the medium can be obtained by solving nonlinear energy equation in combination with radiative transfer equation. The following assumptions are introduced for this research: 1) a plane-parallel, open-cellular porous plate is installed horizontally and bounded by two opaque diffuse solid plates; 2) the porous plate is heated from above and cooled from below; 3) heat transfer across the plate is due to conduction and radiation; 4) the porous plate is homogeneous and capable of emitting, absorbing and anisotropically scattering thermal radiation; and 5) there exists unique equivalent value for each thermal parameter in the entire temperature range of interest to substitute the real thermal property for simulation of global equivalent thermal behavior of material. Under these assumptions, the energy equation can be written as follows [46, 47]:

$$ \rho C\left( T \right)\frac{{\partial T\left( {x,t} \right)}}{\partial t} = - \frac{{\partial q_{c} \left( {x,t} \right)}}{\partial x} - \frac{{\partial q_{r} \left( {x,t} \right)}}{\partial x} $$

(4)

Mathematically, the initial and thermal boundary conditions are thus the following:

$$ T\left( {x,0} \right) = T_{0} $$

(5)

$$ T\left( {0,t} \right) = T_{hot} \left( t \right) $$

(6)

$$ T\left( {L,t} \right) = T_{cold} \left( t \right) $$

(7)

The conductive heat flux q_c is defined by:

$$ q_{c} \left( {x,t} \right) = - \tilde{\lambda }_{two - phase} \frac{{\partial T\left( {x,t} \right)}}{\partial x} $$

(8)

Given the radiation intensity at each point of the medium, in each direction and at each time, the radiative heat flux q_r is then defined by:

$$ q_{r} \left( {x,t} \right) = 2\pi \mathop \int \limits_{ - 1}^{1} I\left( {x,\mu ,t} \right)\mu d\mu $$

(9)

The radiation intensity field in an absorbing-emitting-scattering gray medium is governed by the radiative transfer equation (RTE). For a 1D radiation heat transfer, the RTE can be expressed by [46]

$$ {\upmu}\frac{{\partial I\left({x,\mu,t}\right)}}{\partial x}=-\tilde{\beta}I\left({x,\mu,t} \right) + \tilde{\beta }\left( {1 - \tilde{\omega }} \right)I_{b} \left( T \right) + \frac{{\tilde{\beta }\tilde{\omega }}}{2}\mathop \int \limits_{ - 1}^{1} {{\tilde{\Phi }}}\left( {\mu^{\prime } ,\mu } \right)I\left( {x,\mu^{\prime } ,t} \right)d\mu^{\prime } $$

(10)

In Eq. 10, the terms on the right-hand side describe, respectively, the extinction phenomena, the internal emission and the intensity of the scattering in the μ direction.

When the surface bounding of the medium is gray and emits and reflects diffusely, then the radiative boundary conditions for Eq. 10 are given by

$$ I\left( {0,\mu ,t} \right) = \varepsilon_{1} I_{b} \left( T \right) + 2\left( {1 - \varepsilon_{1} } \right)\mathop \int \limits_{ - 1}^{0} I\left( {0,\mu^{\prime } ,t} \right)\left| {\mu^{\prime } } \right|d\mu^{\prime } \quad 0 < {\upmu } \le 1 $$

(11a)

$$ I\left( {L,\mu ,t} \right) = \varepsilon_{2} I_{b} \left( T \right) + 2\left( {1 - \varepsilon_{2} } \right)\mathop \int \limits_{0}^{1} I\left( {L,\mu^{\prime } ,t} \right)\mu^{\prime } d\mu^{\prime } \quad - 1 \le {\upmu }<0 $$

(11b)