Derivation of a homogenized two-temperature model from the heat equation

TL;DR

This paper derives a macroscopic two-temperature model for a two-phase material with spherical inclusions by applying homogenization techniques to the heat equation, capturing heat exchange between phases.

Contribution

It introduces a homogenized coupled PDE system for phase temperatures, extending previous homogenization methods to two-phase heat transfer modeling.

Findings

Derived a coupled PDE system for phase temperatures.

Quantified heat exchange terms via homogenization.

Provided a rigorous mathematical framework for multi-phase heat transfer.

Abstract

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat: Coll\`ege de France Seminar vol. 2. (Paris 1979-1980) Res. Notes in Math. vol. 60, pp. 98-138. Pitman, Boston, London, 1982.]