The Effective Thermal Conductivities of Composites with 2-D Arrays of Circular and Square Cylinders

Abstract

A boundary collocation scheme suitable for calculation of the effective thermal conductivity of 2-dimensional periodic arrays is developed. Investigated 2-dimensional arrays include the square arrays of infinitely long circular and square cylinders and hexagonal arrays of infinitely long circular cylinders. The effective thermal conductivity is studied subject to variation in two system parameters: the inclusion volume fraction, f, and the conductivity ratio of the inclusion to the matrix, k₂/ k₁ (= σ). The parameter domain investigated is complete, ranging from 0 to the close packing in inclusion volume fractions and from 0 to ∞ in conductivity ratios. The effectiveness of different configurations in improving (if k₂/ k₁ > 1) or reducing (if k₂/ k₁ < 1) the effective thermal conductivity of the composite is studied with the three arrays. With enhancing (insulating) inclusions, the square array of infinitely long square cylinders is found to be the most efficient configuration in improving (reducing) the effective thermal conductivity of the composite among the three arrays for f below a first crossing volume fraction, f _c1, around 0.56 For f greater than f_c1, the square array of infinitely long circular cylinders becomes the most effective configuration. Contour plots of the relative effective thermal conductivity, k _eff/ k₁, in the f - log (σ) space are obtained, which can be used to infer values of σ from measurement of k _eff/ k₁ at some given f.