Emergence of Non-Fourier Hierarchies.

Tamás Fülöp,Tamás Fodor,Ágnes Rieth,Gyula Gróf,Mátyás Szücs,Róbert Kovács,Péter Ván,Ádám Lovas

doi:10.3390/e20110832

Abstract

The non-Fourier heat conduction phenomenon on room temperature is analyzed from various aspects. The first one shows its experimental side, in what form it occurs, and how we treated it. It is demonstrated that the Guyer-Krumhansl equation can be the next appropriate extension of Fourier’s law for room-temperature phenomena in modeling of heterogeneous materials. The second approach provides an interpretation of generalized heat conduction equations using a simple thermo-mechanical background. Here, Fourier heat conduction is coupled to elasticity via thermal expansion, resulting in a particular generalized heat equation for the temperature field. Both aforementioned approaches show the size dependency of non-Fourier heat conduction. Finally, a third approach is presented, called pseudo-temperature modeling. It is shown that non-Fourier temperature history can be produced by mixing different solutions of Fourier’s law. That kind of explanation indicates the interpretation of underlying heat conduction mechanics behind non-Fourier phenomena.

Highlights

The Fourier’s law [1] → q = −k∇ T (1)is one of the most applicable, well-known elementary physical laws in engineering practice
Phenomena that do not fit into these limits, called non-Fourier heat conduction, appear in many different forms
Some of them occur at low temperature such as the so-called second sound and ballistic propagation [2,3,4,5,6,7]

Summary

Introduction

Is one of the most applicable, well-known elementary physical laws in engineering practice. Some of them occur at low temperature such as the so-called second sound and ballistic (thermal expansion induced) propagation [2,3,4,5,6,7] These phenomena have been experimentally measured several times [8,9,10,11] and many generalized heat equations exist to simulate them [12,13,14,15,16,17,18,19,20]. In most of the room-temperature measurements, the existence of Maxwell-Cattaneo-Vernotte (MCV) type behavior attempted to be proved [23,24] It is this MCV equation that is used to model the aforementioned second sound, the dissipative wave propagation form of heat [3,25,26]. The approach of pseudo-temperature is presented to provide one concrete possible interpretation for non-Fourier heat conduction

Size Dependence

Pseudo-Temperature Approach

Findings

Outlook and Summary