Is the Hamilton-Crosser model of any relevance? – A contribution dedicated to the 150th anniversary of Maxwell's model and the 100th anniversary of Fricke's model

Abstract

The Hamilton-Crosser model (HC model) is designed to describe the thermal conductivity of suspensions and nanofluids with anisometric particles. It is one of the most widely cited models in the nanofluid community. In this paper we show, however, that the HC model (1962) is essentially scientific folklore. It is an incorrect reformulation of a correct model that has been derived almost a century ago, Fricke's model (1924). The latter is a generalization to ellipsoids, in particular spheroids, of Maxwell's model (1873) for spheres. Fricke's model is an effective medium approximation that provides admissible (though not necessarily realistic) predictions of the thermal conductivity of carbon nanotube or graphene nanoplatelet nanofluids, while the HC model is an inadmissible variant of the same model, because it cannot reproduce the correct predictions (Fricke's model) and under certain circumstances it even violates a basic law of physics, the upper Hashin-Shtrikman bound (1962). Thus, it cannot be used for any isotropic system. Therefore, in contrast to common belief, any comparison of experimental data with the HC model predictions is irrelevant and all attempts to “improve” or “renovate” the HC model should be abandoned.