Emergence of Non-Fourier Hierarchies

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.

Keywords: Guyer-Krumhansl equation; heat pulse experiments; non-Fourier heat conduction; pseudo-temperature; thermal expansion.

Figure 1

Setup of our heat pulse experiment [40].

Figure 2

Data recorded for basalt rock sample with thickness of $1.86$ mm. The dashed line shows the prediction of Fourier’s law.

Figure 3

Data recorded for basalt rock sample with thickness of $2.75$ mm. The dashed line shows the prediction of Fourier’s law.

Figure 4

Data recorded for basalt rock sample with thickness of $3.84$ mm. The dashed line shows the prediction of Fourier’s law.

Figure 5

Data recorded using the basalt with thickness of $1.86$ mm. The dashed line shows the prediction of GK equation.

Figure 6

Rear-side temperature history; solid line: $a={10}^{-6}\mathrm{m}{}^2/\mathrm{s}$, dashed line: $a=2.5·{10}^{-7}\mathrm{m}{}^2/\mathrm{s}$, $L=2\mathrm{mm}$.

Figure 7

Rear-side temperature histories.