Micro-macro heat conduction model for the prediction of local, transient temperature in composite media

Abstract

Heat flow in heterogeneous media with complex microstructure follows tortuous path and therefore the determination of local temperature distribution is a challenging task. The paper presents a micro-macro heat conduction model for the prediction of this distribution. The proposed solution is based on the prior determination of the macroscopic temperature in a quasi-homogeneous medium characterized by effective properties and then on using the relations between the macroscopic and microscopic temperatures. The Ensemble Averaging Method (EAM) was applied to derive the macroscopic transport equations while the combination of the EAM with the Green's Function Theory (GFT) was used to obtain relations between microscopic and macroscopic temperature fields. The expansion of these relations versus powers of the ratio of the micro to macro dimensions was then used to obtain a two-scale approximation of the mentioned relations. The model is applicable wherever the characteristic dimensions of the microstructure, i.e., distance between inclusions, are smaller than macro-dimension associated with time and spatial variation of macroscopic temperature distribution. The latter can be found from the respective expansion of the macroscopic temperature distribution into the Fourier series. The verification of the proposed model was carried out for 2D transient heat conduction in a periodic composite with the fixed microstructure by comparing direct numerical solution of the energy equation with the results of the approximate micro-macroscopic modelling. The model will be further extended to the prediction of temperature distribution in the case when the thermal contact resistance between constituents, species concentration in heterogeneous media, phase change phenomena and formation of a medium microstructure are present.