Thermal wave oscillations and thermal relaxation time determination in a hyperbolic heat transport model
Introduction
Heat transport has been traditionally studied based on Fourier law, which is supported by an impressive quantity of useful and successful results showing very good agreement with experimental data for most of the analyzed experimental conditions [1], [2].
However, it is well known that Fourier heat diffusion law predicts an infinite velocity for heat propagation, in such a way that a temperature change in any part of the material results in an instantaneous perturbation at each point of the sample [3], [6]. The origin of this fundamental problem is due to the fact that Fourier law establishes that, when a temperature gradient at time t is imposed, the heat flux starts instantaneously at the same time t. Considering that heat transport is due to microscopic motion and collisions of electrons and phonons, it can be inferred that the Fourier condition on the velocity of heat transport cannot be sustained [3], [4], [5], [6], [7], [8].
One of the most simple and accepted approaches that surmounts the limitation of Fourier law [3], [4], was suggested by Cattaneo [9] and independently by Vernotte [10]. It consists of modifying the heat flux equation, incorporating the finite propagation speed of heat. The one-dimensional form of Cattaneo–Vernotte equation is,where t is the time, x is the spatial coordinate, J [W/m2] is the heat flux, T (K) is the absolute temperature, k [W/m K] is the thermal conductivity and τ(s) is the thermal relaxation time, which represents the time necessary for the initiation of the heat flux after a temperature gradient has been imposed. Eq. (1) establishes that the heat flux does not start instantaneously, but rather grows gradually, depending on the thermal relaxation time, after the application of the temperature gradient [3].
From Eq. (1), expanding the heat flux in Taylor series around τ = 0, and approximating at first order in τ,
On the other hand, energy conservation equation is given by [1]where ρ [kg/m3] is the density, c [J/kg K] is the specific heat of the medium and the source term S [W/m3] represents the rate per unit volume at which the heat flux is generated. Combining Eqs. (2), (3), the hyperbolic Cattaneo–Vernotte heat conduction equation is obtained [3], [9], [10], [11]:where α [m2/s] is the thermal diffusivity of the material [12]. On the left hand side of this equation, the second order time derivative term indicates that heat propagates as a wave with a characteristic speed and the first order time derivative corresponds to a diffusive process, which damps spatially the heat wave. Eq. (4) reduces to the parabolic heat conduction equation (based on Fourier law) for τ = 0 or in steady-state conditions [3], [13].
Thermal relaxation time is associated with the average communication time among the collisions of electrons and phonons [3], and it has been theoretically estimated for metals, superconductors and semiconductors to be of the order of microseconds (10−6 s) to picoseconds (10−12 s) [11], [14], [15]. These small values of the thermal relaxation time indicate that its effects will not be significant if the physical time scales are of the order of microseconds or larger. In these situations Fourier law provides an adequate approach. However, in modern applications such as in analysis and processing of materials using ultrashort laser pulses and high speed electronic devices, the finite value of the thermal relaxation time should be considered [13], [14], [15], [16], [17], [18].
The measurement of the thermal relaxation time and the subsequent hyperbolic effects in heat transport have been elusive problems [15], [19]. In fact there are research groups indicating that hyperbolic effects can be easily observed, and that they are associated with thermal relaxation times of the orders of seconds in materials with non-homogeneous structure [16], [17], [18]. In contrast, other authors have criticized their experimental methodologies and have suggested alternative experimental arrangements; obtaining results that do not seem to be affected by hyperbolic effects [20], [21]. Therefore, in this last case, if Cattaneo equation is considered adequate, it would indicate that the observed results would be associated with very short thermal relaxation times [20]. More careful experiments, using a modulated heat source, for similar materials, have shown that in order to determine the thermal relaxation time, it is important to measure simultaneously the thermal diffusivity, also using a hyperbolic approach [18]. In this context, Roetzel et al. [18] have found hyperbolic effects, associated with thermal relaxation times of at least one order of magnitude smaller when compared with the results of Kaminski [16] and Mitra et al. [17]. In the experiments reported by Roetzel et al., the methodology is based on a phase lag method, however given that this is a feature that is also present in the parabolic approach, it is not easy to accept or reject the possibility of a hyperbolic behavior. It has also been shown that the thermal diffusivity and thermal relaxation time can be determined from the thermal profiles; however the suggested methodology makes difficult the accurate determination of these parameters, due to the fact that a small change in the thermal diffusivity can result in a large change in the thermal relaxation time [18].
It is of the main importance to develop a methodology to establish when without a doubt hyperbolic effects are being observed and how from these results, a measurement of the thermal relaxation time can be obtained. For steady-state boundary conditions, a comprehensive discussion of both the hyperbolic effects and the methods for the measurement of the thermal relaxation time have already been presented in the recent book by Wang, Zhou and Wei [19] and in the articles of Mengi and Turhan [22], and Tan and Yang [23]. Otherwise, for modulated heat sources, it could be expected that the advantages found in photothermal science in the analysis of thermal depth profiles using Fourier equation can also arise when the same kind of heat sources is used with hyperbolic Cattaneo–Vernotte equation [11].
In this paper, heat transport governed by Cattaneo–Vernotte equation in a system formed by a finite layer in perfect thermal contact [24] with a semi-infinite layer of a different material is analyzed when the first layer is excited with a periodic source. It is shown that, when the thermal relaxation time of the finite layer is near to its thermalization time, it is possible to find a frequency range, at which the temperature at both sides of the finite layer shows an evident hyperbolic behavior, depending on the boundary conditions. In such conditions, oscillations of the spatial component of the surface temperature as a function of the modulation frequency are obtained.
When the thermal effusivities of the layers are quite different or when the thermal relaxation times of both layers are similar, it is shown that analytical expressions for the values of the maxima and minima as well as the frequencies at which they occur, can be obtained. From these results, the thermal relaxation time can be determined. It is also shown that depending on the ratio of thermal effusivities of the layers and the boundary conditions, the thermal properties of the finite layer can be determined simultaneously.
Section snippets
Formulation of the problem and solutions
Let us consider the configuration shown in Fig. 1, in which the system is excited at the surface x = 0, with a modulated heat source at frequency f, of the form [11], [25]:where ω = 2πf, Re(ξ) is the real part of ξ and F [W/m3] is the spatial distribution of deposited energy over the sample per unit volume and unit time. The temperature at any point of the sample is given by:where Tamb corresponds to the ambient temperature, Tdc(x)
Analysis and discussions
The analysis of the thermal profiles is performed at x = l for the Dirichlet boundary condition and at x = 0 and x = l for the Neumann boundary condition.
Conclusions
In this work the heat transport governed by the Cattaneo–Vernotte hyperbolic equation in a system formed by a finite layer in thermal contact with a semi-infinite layer when a periodic heat source is applied to the first one is considered. It has been shown that remarkable oscillations of the amplitude and phase of the spatial component of the surface temperature are obtained in the high frequency regime for Dirichlet and Neumann boundary conditions. The observation, in an experiment of these
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