Computational Procedures

The three-dimensional FRM microstructure data sets were processed using the following computational procedures:

1) segmentation into binary images- utilizing the commercially available Image Pro software package, each two-dimensional FRM slice image was segmented into a binary image of "pores" and "solids" by manually choosing a greylevel threshold; the chosen threshold was held constant for all of the two-dimensional slices comprising each FRM material but varied between the three different materials, due to inherent contrast variations.

2) extraction of a subvolume- a 200 by 200 by 200 voxel subvolume was extracted from each three-dimensional data set; in each case, the subvolume was chosen away from the sample edges and in an attempt to select as representative a volume of the overall material as possible.

3) isolation of pores and quantification of pore volumes- each binary subvolume data set was further analyzed to identify individual pores and determine their volumes (by voxel counting); computer programs were written in the C programming language to perform pore separations utilizing such common image processing algorithms as erosion/dilation and watershed segmentation (Russ and Russ, 1988). 4) prediction of thermal conductivity- the processed subvolumes were used as input into a finite difference program (Garboczi, 1998) to estimate the thermal conductivity of the composite FRMs based on an electrical analogy; a voltage (temperature) gradient was placed across the microstructure and the resulting currents (heat flows) were computed for each microstructure element (node).

To utilize the finite difference program, it is necessary to assign thermal conductivities to the "pore" and "solid" components of the underlying microstructure. Knowing the pore volume of each individual pore, equivalent radii were calculated assuming a spherical pore geometry. Then, the thermal conductivity of a pore of radius r at material temperature T was given by:

where:
k_gas = thermal conductivity of air at temperature T, and according to (Loeb, 1954),

(2)

with

r = pore radius (m),
σ = Stefan-Boltzmann constant (5.669x10⁻⁸ W/m²·K⁴),
E = emissivity of solid material (1.0 for black bodies), and
T = absolute temperature (K).

In this way, each different size pore in the three-dimensional microstructure was assigned a different thermal conductivity value.

When considering the “correct” thermal conductivity to assign to the "solids", several complications arise. First, while the microtomography clearly indicates the coarser pores (50 µm to 100 µm and greater in diameter) present in the microstructure, the remaining solid phases are themselves porous. While it may have been possible to estimate the local "micro-porosity" based on the greylevel intensity at each solid voxel in the three-dimensional microstructure, instead, a single "fine" porosity value within all solid voxels was estimated based on the measured densities of the original FRMs, the measured densities obtained by grinding them to a fine powder (hopefully removing all internal porosity), and the microtomography-measured coarse porosity volume fraction for each material. Then, the theory of (Russell, 1935) was used to estimate the thermal conductivity of the porous solid component of the microstructures, k_ps, as:

(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 calculations of k_rad for equation (3), an upper bound pore diameter equal to the smallest of the voxel dimensions was chosen for each material (e.g., 0.0273 mm or 0.0586 mm). The thermal conductivities of the solid powders, k_solid, were chosen as 0.8 W/m·K for the gypsum-based FRM and 0.4 W/m·K for the fiber/cement-based FRMs, to provide predictions in agreement with the room temperature measured values of thermal conductivity (Anter Laboratories, 2004). In applying the theory of Russell, the density (porosity) of the porous material was adjusted to account for its measured mass loss as a function of temperature. The gypsum-based FRM loses about 25 % of its mass upon heating to 1000 ºC, while the fiber/cement-based FRMs lose between 10 % and 15 % (Anter Laboratories, 2004).

For the gypsum-based FRM, a further complication is that the thermal conductivity of anhydrite (dehydrated gypsum) is nearly four times that of gypsum (Horai, 1971). Based on the measured mass loss of the gypsum-based FRM and utilizing the Hashin-Shtrikman upper and lowers bounds for thermal conductivity (Hashin and Shtrikman, 1962), it was estimated that upon complete conversion of the gypsum to anhydrite, the thermal conductivity of the solid FRM powder should increase to be on the order of 1.66 W/m·K. Thus, for temperatures above the nominal dehydration temperature for gypsum of about 300 ºC, the finite difference computations were performed with k_ps values based on both gypsum and anhydrite for comparison to experimental data.