Microstructure Characterization

Cast and fracture surfaces of the materials were viewed under an optical microscope to obtain an estimate of each material's representative pore size. In addition, for each material, cylindrical cores were carefully cut from each specimen (using a utility knife) and placed in 27 mm ID polypropylene tubes for viewing in an x-ray microtomography system. Details of the microtomography study are provided in reference (4). The three-dimensional microtomography data sets were subsampled to extract a representative 200 x 200 x 200 voxel subvolume. The subvolumes were segmented (binarized) using a simple thresholding operation to separate the porosity from the "solids". The volumes of each individual pore were determined after separating the pores using either a watershed segmentation algorithm (5) or a combination of erosions and dilations. The individual pores were then each assigned a temperature-dependent conductivity depending on their radius (volume), k_pore:

(1)

where k_gas = thermal conductivity of air at temperature T, from (6), and k_rad is a radiation term, as proposed by Loeb (7), for spherical pores:

(2)

with r = pore radius (m),

σ = Stefan-Boltzmann constant (5.669x10⁻⁸ W/m²·K⁴),

E = emissivity of solid material (1.0 for a black body), and

T = absolute temperature (K).

Because the "solids" imaged using the microtomography are themselves microporous, they were assigned a thermal conductivity, k_PS, using the theory of Russell for porous solids (8):

(3)

where v = k_pore/k_solid,

p = porosity of the "solid" voxels,

k_solid = thermal conductivity of solid (powder) material, and

k_pore = thermal conductivity of pores in the "solid" voxels = k_gas + k_rad.

For the microporous solid voxels, the pore radius for use in equation 2 was set at an upper bound value of half of the smallest voxel dimension for each three-dimensional data set following the analysis in reference (4). For each material, the (average) porosity of the solid voxels was determined based on the measured bulk density of the material, the measured density of the material ground to a fine powder, and the x-ray microtomography-measured "coarse" porosity. Finally, the digital image representations of the FRM microstructures, with their assigned thermal conductivities for each voxel, were used as input into a finite difference/conjugate gradient program (9) to compute the effective thermal conductivity of each porous material as a function of temperature.