Guyer-Krumhansl-type heat conduction at room temperature

TL;DR

This paper demonstrates that heat conduction at room temperature in various materials deviates from Fourier's law and is better described by the Guyer-Krumhansl equation, confirming the relevance of non-equilibrium thermodynamics.

Contribution

It provides experimental evidence supporting the applicability of the Guyer-Krumhansl model to room temperature heat conduction in macroscopic samples.

Findings

Temperature evolution deviates from Fourier's law

Guyer-Krumhansl equation accurately models heat conduction

Supports non-equilibrium thermodynamics for thermomechanical processes

Abstract

Results of heat pulse experiments in various artificial and natural materials are reported in the paper. The experiments are performed at room temperature with macroscopic samples. It is shown that temperature evolution does not follow the Fourier's law but well explained by the Guyer-Krumhansl equation. The observations confirm the ability of non-equilibrium thermodynamics to formulate universal constitutive relations for thermomechanical processes.

Tables

Table 1. Table 1: Dimensionless parameters

Sample	$\hat{\alpha }\times {10}^3$	$\hat{\tau }$	b
Capacitor	1.40	95.4	2.23
Limestone	1.124	99.1	2.17
Metal foam 1	0.912	40.2	3.04
Leucocratic	1.56	132.0	1.77

Table 2. Table 2: Dimensional parameters. α F subscript 𝛼 𝐹 \alpha_{F} is the thermal diffusivity from Fourier equation (dashed line), α G ​ K subscript 𝛼 𝐺 𝐾 \alpha_{GK} is the thermal diffusivity with the Guyer-Krumhansl equation (solid line)

Sample	$L\left[mm\right]$	${10}^6{\alpha }_F\left[\frac{m^2}{s}\right]$	${10}^6{\alpha }_{GK}\left[\frac{m^2}{s}\right]$	$\tau \left[s\right]$	l[mm]
Capacitor	3.9	3.45	2.13	0.954	2.79
Limestone	1.7	0.45	2.950	0.991	0.80
Metal foam 1	5.1	3.04	2.373	0.402	1.70
Leucocratic	1.75	7.14	4.77	1.32	1.06

Table 3. Table 3: Material parameters of the direct calculation

	Density	Heat capacity	Conductivity	Diffusivity
	[$kg/m^2$]	[$J/kgK$]	[$W/mK$]	[$m^2/s$]
Aluminium	2707	905	237	9.61$\times {10}^{-5}$
Polystyrol	1040	1350	0.15	${10}^{-7}$

Equations

$$\tau \dot{\mathbf{q}}+\mathbf{q}+{\lambda }_F\nabla T-{\beta }^{\prime }\Delta \mathbf{q}-{\beta }^{\prime \prime }\nabla \cdot \nabla \mathbf{q}=0.$$

$$\rho c\frac{\partial T}{\partial t}+\frac{\partial q}{\partial x}=0.$$

$$\tau \frac{\partial q}{\partial t}+q+{\lambda }_F\frac{\partial T}{\partial x}-l^2\frac{{\partial }^{2}q}{\partial x^2}=0.$$

$$\tau \frac{\partial }{\partial t}\left(\frac{\partial T}{\partial t}-b\alpha \frac{{\partial }^{2}T}{\partial x^2}\right)+\left(\frac{\partial T}{\partial t}-\alpha \frac{{\partial }^{2}T}{\partial x^2}\right)=0.$$

$$\hat{t}=\frac{t}{t_p},\hat{x}=\frac{x}{L},\hat{q}=\frac{q}{q_{\max }},\hat{T}=s\frac{T-T_0}{T_{\mathrm{ref}}-T_0},$$

$$q_{\max }=\frac{1}{t_p}{\int }_0^{t_p}{q}_0(t)dt,T_{\mathrm{end}}=T_0+\frac{q_{\max }{t}_p}{\rho cL},s=\frac{T_{\mathrm{ref}}-T_0}{T_{\mathrm{end}}-T_0},$$

$$\hat{\tau }=\frac{\tau }{t_p};\hat{\alpha }=\alpha \frac{t_p}{L^2};\hat{l}=\frac{l}{L};\hat{b}=\frac{{\hat{l}}^2}{\hat{\tau }\hat{\alpha }}=b;\hat{B}=B\frac{t_p}{\rho cL}.$$

$$s\frac{\partial \hat{T}}{\partial \hat{t}}+\frac{\partial \hat{q}}{\partial \hat{x}}$$

$$\hat{\tau }\frac{\partial \hat{q}}{\partial \hat{t}}+\hat{q}+s\hat{\alpha }\frac{\partial \hat{T}}{\partial \hat{x}}-b\hat{\tau }\hat{\alpha }\frac{{\partial }^2\hat{q}}{\partial {\hat{x}}^2}$$

Full text

heat pulse experiment; flash method; Guyer-Krumhansl equation; over-diffusive regime

Deviation from Fourier’s law at low temperatures was predicted by Tisza and Landau and then observed first by Peshkov in liquid Helium Tisz+ . The wave like propagation of heat, the second sound, is later explained by a hyperbolic extension of Fourier’s law called Maxwell-Cattaneo-Vernotte (MCV) equation Cat+ . The next important experimental step, the detection of second sound and ballistic propagation in solids AckGuy+ required an extension of the theory and a generalization lead to the Guyer-Krumhansl equation Cal+ , which is intensively studied also nowadays, first of all in case of nanomaterials SelEta+ ; Zhu+ .

The Guyer-Krumhansl (GK) equation is as follows:

[TABLE]

Here $\mathbf{q}$ is the heat flux, the current density of the internal energy in a rest frame of the continuum, $\lambda_{F}$ is the Fourier heat conduction coefficient, $\tau$ is a relaxation time, while $\beta^{\prime}$ and $\beta^{\prime\prime}$ are Guyer-Krumhansl coefficients in isotropic media. In rarefied gases these are related to the relaxation times of the Callaway collision integral Cal+ . The overdot denotes the time derivative, $\nabla$ is the space derivative, and the coefficients are considered as constant. The first three terms of (1) form the Maxwell–Cattaneo–Vernotte equation, the second and the third ones are Fourier’s law.

The conceptual foundation of looking non-Fourier phenomena in macroscopic samples at room temperature is originated in the phenomenological theories: phonons mean free paths and relaxation times are not relevant concepts in sand or frozen meat at room temperature heat conduction. Therefore, among the many phenomenological ideas leading to and explaining the origin of MCV equation (see e.g. JosPre ; Cim+ ) non-equilibrium thermodynamics plays a distinguished role. With a consistent phenomenology the obtained constitutive relations are independent of the microscopic details, like Fourier’s law in local equilibrium GroMaz+ . Their validity is based only on the second law of thermodynamics and in this sense they are universal. Therefore, after the development of Extended Thermodynamics Gya+ ; MulRug98b there were several predictions of similar phenomena with various assumptions and conditions JosPre , including heterogeneous materials at room temperature. The first promising positive experimental results in granular and biological media were reported about Maxwell-Cattaneo-Vernotte type heat conduction Kam+ . However, these measurements were not confirmed HerBec+ .

In the framework of Extended Thermodynamics, Guyer-Krumhansl equation can be obtained by a modification of the entropy flux in all approaches, usually together with other assumptions Rug+ . A minimal functional deviation of the entropy flux from the local equilibrium – introducing Nyíri multipliers Nyi91a1 – leads to the GK equation without any further ado. It is also straightforward to obtain most of the viable suggested generalisations of Fourier’s law and MCV equation in a uniform framework VanFul12a , which is also compatible with kinetic theory KovVan15a .

Motivated by the universality of non-equilibrium thermodynamics, most recently an experimental-theoretical study has been performed in order to identify suitable qualitative signatures of detecting non-Fourier heat conduction beyond the MCV equation in heterogeneous, macroscopic samples at room temperature CzeEta+ . The first indication of the effect was measured on artificial samples, with alternating layers of good and bad heat conductors parallel to the heat flux BotEta16a . Here we report experimental results of heat pulse measurements on various artificial and natural materials with GK-type heat conduction.

In our simple experimental device, a flash lamp generates the heat pulse at the front end of the sample, and the temperature is measured by a pin-thermocouple (K-type) at the rear end. The thermocouple and the detector part are insulated from the heat pulse and from the electromagnetic noises. The heat pulse is measured directly at the front end by a photovoltaic cell, triggering data acquisition. A typical pulse shape is triangular and is $10$ ms long.

A sketch of the experimental setup is shown on Figure 1. The thickness of the studied specimens is smaller at about a magnitude than their diameter and the front face is homogeneously heated by the heat pulse, therefore, heat conduction can be considered one-dimensional.

At the front side a thin black coating is applied to ensure uniform boundary conditions, as well as to eliminate the transparency of the sample. At the rear side silver painting is used, therefore, the thermocouple measures an effective temperature. The effect of the coating is negligible according to control measurements. The experimental device was calibrated by using several samples with known heat diffusivity.

In a one dimensional setting the balance of internal energy for rigid heat conductors is

[TABLE]

Here $c$ is the specific heat, $T$ is the temperature, and the specific internal energy is $e_{sp}=cT$. $\rho$ is the density, and $q$ is the heat flux in the direction of the heat propagation. The one dimensional version of (1) is:

[TABLE]

Here $k$ is the Fourier heat conduction coefficient, $\tau$ is the relaxation time and $l^{2}=\beta^{\prime}+\beta^{\prime\prime}$ is a nonnegative material parameter of the GK equation, and is expressed with the help of a characteristic length scale $l$.

Remarkable and instructive to recognise a hierarchical structure in the above system of equations, (2)–(3), when expressed for the temperature BerEta+ . It is best seen by eliminating the heat flux and rearranging the system as follows:

[TABLE]

where $\alpha=\frac{k}{\rho c}$ is the thermal diffusivity and $b=\frac{l^{2}}{\tau\alpha}$ is the coefficient characterising the deviation from Fourier heat conduction. One can observe that solutions of the Fourier equation are solutions of (4), whenever $b=1$, that is $\alpha=\frac{l^{2}}{\tau}$. If $b<1$ then the solutions of (2)–(3) show wavelike characteristics; if $b>1$ then the solutions are over-diffusive TanAra00a ; KovVan15a .

We are looking for solutions of (2)–(3) with heat pulse boundary condition and surface heat exchange at the front side. Therefore,

q(x=0,t)=\left\{\begin{array}[]{cl}q_{\rm max}\left(1-\cos\left(2\pi\frac{t}{t_{p}}\right)\right)-B(T(0,t)-T_{0})&\textrm{if }\ 0<t\leq t_{p},\\ -B(T(0,t)-T_{0})&\textrm{if }\ t>t_{p}.\end{array}\right.

Here $t_{p}$ is the pulse length, $T_{0}$ is the ambient temperature and $B$ is the heat exchange coefficient. The particular shape of the pulse does not influence the effect as long as the length of the pulse is much shorter than the characteristic time scale of the experiment. The backside boundary is considered adiabatic, $q(L,t)=0$, because of the insulating cover. Initially, the temperature distribution is uniform and the heat flux is zero along the sample, that is, $T(x,0)=T_{0}$ and $q(x,0)=0$, this is provided by the measurement protocol, with 30-60 minutes temperature equilibration periods between the measurements.

A suitable dimensionless form of the variables are:

[TABLE]

where $q_{0}(t)=q(x=0,t)+B(T(0,t)-T_{0})$ is the boundary condition without cooling. $T_{\rm ref}$ is a reference temperature, e.g. the measured maximum temperature, which is different from the limiting adiabatic temperature, $T_{\rm end}$, because of the cooling. The dimensionless parameters are, consequently,

[TABLE]

The nondimensional form of the equations is

[TABLE]

The corresponding boundary and initial conditions are:

\hat{q}(0,\hat{t})=\left\{\begin{array}[]{cl}1-\cos\left(2\pi\cdot\hat{t}\right)-s\hat{B}\hat{T}(0,t)&\textrm{if }0<\hat{t}\leq 1,\\ -s\hat{B}\hat{T}(0,t)&\textrm{if }\hat{t}>1.\end{array}\right.

$\hat{q}(1,\hat{t})=0$, $\hat{T}(\hat{x},0)=0$ and $\hat{q}(\hat{x},0)=0$. Here the parameters $\tau$, $\alpha$ and $b$ characterise the material, the temperature scale, $s$, and the heat exchange coefficient, $B$, are not.

Among the investigated samples four had been chosen, where the deviation from Fourier heat conduction was the most apparent. These were a capacitor, the same as in BotEta16a , a crystalline limestone sample from Villány, southern Hungary, a leucocratic rock sample and a metal foam sample (see Figure 2.). The corresponding backside temperature is shown in Figures 3. and 4. The system of equations (7)–(8) are solved and the parameters are fitted to the data with the built-in nonlinear regression algorithm of Mathematica 10.0, using the solution of the system of partial differential equations as an input function. On the figures the nondimensional temperature is scaled approximately to the adiabatic limiting temperature ($0.9T_{max}$), the data is slightly smoothed with 3-point running average. The solid lines show the best GK fit and the dashed line is the best Fourier fit with the correct asymptotics.

The dimensionless parameters are given in Table 1., and the related physical parameters in Table 2.

All measurements indicate an overdiffusive, $b>1$, Guyer–Krumhansl-type heat conduction.

In case of the capacitor sample the effect of regular heterogeneity can be calculated directly. The geometry of the calculation is shown in Figure 5. It is assumed, that the cylindrical symmetry is not important and also an aluminium layer is applied at the upper (rear) side instead of the silver coating. The original capacitor is not symmetric, the aluminium and polystyrol layers are rolled and continuous, therefore, a two-dimensional representation seems to be proper approximation.

The applied material parameters are given in Table 3.

The Fourier equation ((4) with $\tau=0$) is solved numerically with a finite volume algorithm developed to deal with material heterogeneity BerEta++ . The time step was chosen $\Delta t=0.01t_{p}=10^{-4}s$ and the spatial dimensions were $185\times 40$ space steps, 5 and 15 steps for the $5\mu m$ aluminium and for the $15\mu m$ polystyrol, respectively. The calculations are performed for $10s=100000$ time steps in two cases. In the first case a pure aluminium sample was assumed and the comparison with the data of Both et al. BotEta16a is shown on the left side of Figure 6. The measured and simulated dimensionless backside temperature is shown as a function of time by the thin and thick solid lines, respectively. With the geometry of Figure 5. the material parameters of Table 3., the backside temperature is shown on the right side of Figure 6.

One can see that for pure aluminium, after an initial coincidence, the slope of the computed curve becomes very different from the experimental values. For the heterogeneous case the simulated curve is closer to the experimental one, but the difference still exist.

We have shown deviation from Fourier’s law in room temperature experiments. These were correctly modelled by the GK equation, known to be valid in case of low temperature solids. Our experimental samples exhibit heterogeneity. However, in case of the simplest, regular heterogeneity a direct simulation does not emulate the effect. These observations together indicate the robust universality of the GK equation in accordance with the theoretical expectations VanFul12a ; KovVan15a .

A practical use of our findings is the possibility to improve technological data of transient and stationary heat conduction in heterogeneous structures. This can simplify and guide microstructure based detailed analyses and computations.

The work was supported by the grants NKFIH K104260, K116197 and K116375. Thanks to László Kovács and the Kőmerő Kft. for the preparation of the rock sample. Thanks to Tamás Bárczy and the Admatis Kft. for providing the metal foam sample.