Thermal or electrical conduction through a granular material
George Keith Batchelor, R. W. O'Brien
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Source: Crossref
Published: Jul 13, 1977
DOI: 10.1098/rspa.1977.0100
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Abstract The material under investigation consists of particles of relatively large conductivity embedded or immersed in a matrix, the volume fraction of the particles being so high that they are in, or nearly in, contact. The particles are arranged randomly, and the material is statistically homogeneous. A general formula gives the effective conductivity of the material in terms of the average thermal (or electrical) dipole strength of a particle. The thermal flux across the surface of a particle is concentrated in areas near points of contact with another particle, and the dipole strength is approximately equal to a weighted sum of the fluxes across the areas near contact points. It is thus necessary to calculate the flux between two adjoining particles at different temperatures, and we do this by solving numerically an integral equation for the distribution of temperature over the (locally spherical) surface of one of the particles near the contact point. The flux between the two particles is found to be proportional to loge ah when a2 2h/a≫ 1 and to logea when a 2h/a≪ 1, where h is the minimum gap between the particle surfaces, a~1 the mean of their local curvatures, and a the ratio of the conductivities of the particles and the matrix. In the case of two particles pressed together to form a circular flat spot of radius p, the flux occurs almost wholly in the particle material, and is proportional to p when ap/a≫ 1. Explicit approximate results are obtained for the effective conductivity of the granular material in the case of uniform spherical particles. For a close-packed bed of particles making point contact the effective conductivity is found to be 4.0k logea where k is the matrix conductivity. This asymptotic relation (applicable when a ≫ 1) is seen to be consistent with the available measurements of the conductivity of packed beds of spheres. Values of the effective conductivity for packed beds of particles of different shape are not expected to be greatly different.
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