Thermal conductivity of a cubic lattice of spheres with capillary bridges

There have been many attempts to develop expressions for thermal conductivity (λ) as a function of liquid content (θ) in granular media such as powders, gravel and soil. We present a new approach based upon a cubic array of solid spheres, with liquid capillary bridges surrounding the solid–solid contacts. Numerical results agree with analytical solutions for the endpoints. For small filling angles (low liquid contents, such that menisci do not coalesce) and for a fixed conductivity ratio α ≡ λsolid/λliquid = λliquid/λgas, we find a simple relationship λ (θ) = λ (θ = 0) + aθb, where the prefactor and exponent are simple functions of α. The power-law relationship also holds for compressed spheres, if the dry conductivity λ (θ = 0) is adjusted for the new porosity. While the results are not trivially applicable to real granular media, they provide a quantitative explanation for the observed phenomenon in soils of the rapid rise in conductivity caused by the addition of a small volume of liquid.