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 fibrous insulation media. Due to the statistical nature of random packing patterns of microspheres and fibers, there exists a range of effective thermal conductivities for any given solid fraction of the microsphere or fibrous insulation medium. An analysis which 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 simplify the evaluation of the statistical bounds. In general, the pertinent parameters for determining the effective conductivity are thermal conductivities of the constituent phases, the volume fraction, and a constant which is a function of the cell geometry.