Images of the individual pores extracted from the x-ray microtomography data sets for two of the FRMs are provided in Figure 4. Clearly, the two materials exhibit vastly different pore structures. The gypsum-based FRM C on the left side of Figure 4 contains basically isolated spherical pores, while the pores in the fiber/portland cement-based FRM A on the right side of Figure 4 are much more continuous and anisotropic in nature. The pore sizes and their connectivity would be expected to influence heat and mass transfer through these materials. From the viewpoint of minimizing both the mass transfer of reaction gases and radiative heat transfer through the FRMs, isolated small pores are preferable (see equation 2 for example). Direct evidence for this can be found in the fumed-silica board insulation materials that exhibit effective thermal conductivities between about 0.02 W/ (m·K) and 0.04 W/(m·K), in the temperature range from room temperature to 800 ºC (10, 14). These insulations have been formulated to contain nanometer-sized pores and also contain an opacifier to minimize radiative transfer. It can be observed that their effective thermal conductivities are a factor of five to ten times lower than the values measured/computed for typical FRMs currently on the market (Figures 3, 5, and 6). This in itself suggests that there is considerable room for improvement in the thermal performance of FRMs.
The images in Figure 4 were used as input into the finite difference/conjugate gradient solver to obtain computational estimates of the thermal conductivities of the FRMs as a function of temperature. Results for the gypsum-based FRM C are summarized in Figure 5. One issue for this particular material is what value to use for the thermal conductivity of the solid (powder) material. According to Horai (18), the thermal conductivity of anhydrite (dehydrated gypsum) is about 4 times that of gypsum. Thus, to obtain a set of bounding curves, the computations of thermal conductivity were conducted using first a thermal conductivity based on gypsum over the entire temperature range and then using a thermal conductivity accounting for the gypsum to anhydrite dehydration, assumed to occur at a temperature near 300 ºC. Of course, in the real FRM specimen, the dehydration reactions are occurring as a front that progressively penetrates the specimen and are present over some range of mean specimen temperatures. Thus, while good agreement is observed between computed and measured values for temperatures of 200 ºC and below in Figure 5, for temperatures between 200 ºC and 800 ºC, the measured values are underestimated by the constant gypsum assumption and overestimated by the complete gypsum to anhydrite dehydration assumption, respectively. These results indicate that while microstructural characterization can yield valuable information concerning thermal conductivity and can even be used to compute estimated effective thermal conductivity values, care must be taken to consider the possible complicating factors such as reactions within the material and enhanced mass transfer and radiative heat transfer, as pointed out in reference (4).
Figure 4 False color renderings of the individual pores isolated using the watershed segmentation algorithm in a gypsum-based FRM C (left: 120 by 120 by 120 voxels) and a fiber/portland cement-based FRM A (right: 200 by 200 by 200 voxels), from ref. (4).
Figure 5 Measured and predicted thermal conductivities for a gypsum-based FRM C as a function of temperature, for the case where individual pores were identified using a watershed segmentation algorithm (5), from ref. (4).