Statistical bounds for the effective thermal conductivity of microsphere and fibrous insulation

The present work consists of an analytical investigation of the effective thermal conductivity of microsphere and fi- brous insulation media. Because of the statistical nature of random packing patterns of microspheres and fibers, there ex- ists a range of effective thermal conductivities for any given solid fraction of the microsphere or fibrous insulation medium. An analysis that applies two standard variational principles to some model media simulating the microsphere and fibrous insulation has been carried out to determine the statistical upper and lower bounds of the effective thermal conductivity. The physical models composed of cells with different packing configurations in case of microspheres and special fiber-matrix arrangements in case of fibrous insulation significantly sim- plify the evaluation of the statistical bounds. In general, the pertinent parameters for determining the effective con- ductivity are thermal conductivities of the constituent phases, the volume fraction, and a constant that is a function of the cell geometry. Furthermore, contemplating the empirical re- sults that particular types of packings occur more than others in the microsphere case, a computer simulation imparting this effect into determining the effective bounds has been devised. The present microsphere results are in excellent agreement with the previous analytical work using a regular-cell approach. Nomenclature