Abstract

The classical constitutive equation for heat conduction, Fourier's law, plays an essential role in the engineering practice and holds only for homogeneous materials. However, most of the materials consist of some kind of heterogeneity, such as porosity, cracks, or different materials are in contact.

One outstanding example is the thermal behaviour of rocks. We report the results of heat pulse (or flash) experiments. This is a standard method in the engineering practice to measure the thermal diffusivity of a material. We observed two effects in these experiments.

Firstly, a size effect emerges, that is, for the same type of rock with different size, different thermal diffusivity is measured. Secondly, we also observed the deviation from Fourier's law in a particular time interval. Thus the modelling requires some extension for the constitutive equation.

The variety of their constituents and the porosity makes it difficult to derive a general constitutive law. Here, in this paper, we briefly present the framework of non-equilibrium thermodynamics in which we are able to derive an appropriate extension for Fourier's law. The resulting model is the so-called Guyer-Krumhansl equation, which is independent of the structure, therefore able to model the thermal behaviour of various samples.

We conclude that the Guyer-Krumhansl equation is an appropriate extension for Fourier's law, in accordance with the previous measurements and evaluations. Furthermore, we observed that the deviation depends on the size of the sample, too. Finally, we communicate the measured thermal diffusivities for each sample, showing a size effect as well.

1 Introduction

From an experimental point of view, the history of non-Fourier heat conduction began with the low-temperature measurements of Peshkov (1944), in which the damped wave-form of heat conduction, the so-called second sound has been measured. Later on, the third way of heat propagation, the ballistic conduction, is observed by McNelly et al. (1970), which is a thermo-mechanical effect, a temperature wave that propagates with the speed of sound. It is characteristic of low-temperature situations, too.

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