Thermal Conductivity:

A wide variety of experimental techniques exist for measuring the thermal conductivity of materials at elevated temperatures: high temperature guarded hot plate (ASTM C177), heat flow meter apparatus (ASTM C518), laser flash diffusivity methods (ASTM E1461), and transient line/hot wire (ASTM C1113) and plane source methods.^{2, 8-10} Similar to the discussion presented for concrete by Flynn,⁸ these measurements are always complicated by the dynamic nature of the FRM which is undergoing degradation even as its thermal conductivity is being measured.

An alternative to measuring the thermal conductivities of FRMs at high temperatures is to measure the value at room temperature (or perhaps up to 100 ºC) and "predict" the higher temperature values based on some theory for the conductivity of composite (porous) materials. Example theories that are closer to reality than the simplest parallel and series models include those of Russell,¹¹ Frey, ¹² and Bruggeman. ¹³ For example, the theory of Russell estimates the thermal conductivity of the porous material, k, as¹¹:

For a spherical pore of radius r, the radiation contribution to the overall thermal conductivity of the pore is¹⁴:

Knowing the densities of the FRM and the base solid components (by grinding to a powder and measuring in an alcohol solution, for instance), one can calculate the porosity of the FRM. This, along with estimates of the solid's thermal conductivity and the material's typical pore radius, and the tabulated thermal conductivity of air as a function of temperature, ^{3, 15} permits the estimation of the thermal conductivity of the FRM at elevated temperatures. As shown in Figures 4 and 5, application of this theory to both portland cement-based and gypsum-based spray-applied FRMs yields results in good agreement with existing measurements. While the measured values of ρmax and ρmatl were used in the calculations, in each case, the pore radius was a floating parameter that was adjusted to give a reasonable fit to the experimental data. But, in each case, the adjusted value for the pore radius is in agreement with visual optical microscopy observations of the characteristic pore sizes in these materials (Figures 6 to 8). These figures illustrate the potential of applying this approach in lieu of or to minimize the number of complicated and costly high temperature measurements for these materials. The approach also points out the advantage of incorporating smaller pores into the FRM structure, as the insulating performance of materials with larger ones will suffer significantly due to radiation effects at higher temperatures.

Figure 4: Measured thermal conductivities1 and predictions based on theory of Russell/Loeb11, 14 for two similar portland-cement based spray-applied FRMs.

Figure 5: Measured thermal conductivities1 and predictions based on theory of Russell/Loeb11, 14 for a gypsum-based spray-applied FRM.

Figure 7: Optical micrograph for portland-cement based spray-applied FRM-B. Typical pore diameter as indicated by the labeled scale bars in the middle of the two images is on the order of 1.5 mm corresponding to a pore radius of 0.75 mm. The original image is on the left and a contrast-enhanced version that better highlights the porosity is shown on the right.

Successful application of this theory requires a detailed understanding of the dynamic microstructure of the FRM. For example, one widely used spray-applied FRM utilizes expanded polystyrene (EPS) beads as a lightweight aggregate. When these highly porous beads burn out at elevated temperatures, even though the total porosity will not change significantly, a new larger size of characteristic pores will be created within the microstructure, potentially leading to an increase in thermal conductivity. Intumescents will also be a challenging application, as in this case, the pore size and total porosity are both dynamic variables that change dramatically during the fire exposure and charring of the coating.

A more detailed microstructural analysis is possible via the utilization of x-ray microtomography which can capture the three-dimensional microstructure of materials with a voxel dimension on the order of micrometers. ¹⁶ Example two-dimensional images (slices) obtained for both gypsum-based and portland cement-based FRMs using one of the X-ray microtomography units available at the Center for Quantitative Imaging at Pennsylvania State University are provided in Figure 9^*. These digital image-based three-dimensional microstructures can be segmented into solid and pore phases, and finite element and finite difference techniques applied to compute their equivalent thermal conductivity. ¹⁷ For example, a numerical temperature gradient could be placed across the microstructure and the computed heat flow used to determine the thermal conductivity of the composite 3-D microstructure. Thus, this approach is similar to that used in conventional computational thermal analysis, but it is being applied at the microstructure scale instead of the conventional macro (structure) scale.