Error Analysis and Equations for the Thermal Conductivity of Composites

Abstract

The overall effective thermal conductivity (K) of a composite is a function of the thermal conductivities (K₁) and volume fractions (ϕ_i) of each of the constituents. Probable errors (ΔK. and Δϕ_i) for each of the constituents lead to a probable error (ΔK) in the overall thermal conductivity. For realistic problems, the exact function K(K_i, ϕ_i) is not known since it depends on the way the constituents are distributed. For this reason equations which describe the bounds on the thermal conductivity, for given K_i and ϕ_i, are very useful. Using standard calculational techniques, we plot K, K + ΔK, and K − ΔK versus volume fraction of one constituent for several cases of two-phase systems. These include the series and parallel cases as “absolute bounds” and the Maxwell equations which are the bounds for systems which are isotropic and homogeneous. We find, for common values of K₁/K₂, that the bounding equations overlap when errors are considered. This implies three things: (1) A measurement of thermal conductivity of a composite, when errors are taken into account, may not allow discrimination between cases. (2) For many cases of experimental interest a rough calculation is quite sufficient. (3) The bounds on the thermal conductivity are really not unconditional when errors are taken into account. We also consider two-phase composites where cylinders are embedded in a matrix. Complications in the calculation of the thermal conductivity include the effect of radiation, the effect of moisture and convection, and thermal resistance in the interface between phases.