Abstract
The dual-phase lagging (DPL) model has been considered as one of the most promising theoretical approaches to generalize the classical Fourier law for heat conduction involving short time and space scales. Its applicability, potential, equivalences, and possible drawbacks have been discussed in the current literature. In this study, the implications of solving the exact DPL model of heat conduction in a three-dimensional bounded domain solution are explored. Based on the principle of causality, it is shown that the temperature gradient must be always the cause and the heat flux must be the effect in the process of heat transfer under the dual-phase model. This fact establishes explicitly that the single- and DPL models with different physical origins are mathematically equivalent. In addition, taking into account the properties of the Lambert W function and by requiring that the temperature remains stable, in such a way that it does not go to infinity when the time increases, it is shown that the DPL model in its exact form cannot provide a general description of the heat conduction phenomena.
Highlights
Nanoscale heat transfer involves a highly complex process, as has been witnessed in the last years in which remarkable novel phenomena related to very short time and spatial scales, such as enhancement of thermal conductivity in nanofluids, granular materials, thin layers, and composite systems among others, have been reported [1,2,3,4,5]
When analyzing two-phase systems, one of the simplest heat conduction models that considers the microstructure is known as the two-equation model [9,10], which has been developed writing the Fourier law of heat conduction [11] for each phase and performing a volume averaging procedure [9]
According to the principle of causality, it has been shown that the temperature gradient must precede the heat flux
Summary
Nanoscale heat transfer involves a highly complex process, as has been witnessed in the last years in which remarkable novel phenomena related to very short time and spatial scales, such as enhancement of thermal conductivity in nanofluids, granular materials, thin layers, and composite systems among others, have been reported [1,2,3,4,5]. For any kind of homogeneous boundary conditions, its solutions go to infinity in the long time domain This explosive characteristic of the temperature predicted by Equation 2 indicates that the DPL model, in its exact form, cannot be considered as a valid model of heat conduction. By expanding both sides of Equation 1 in a Taylor series and considering a first-order approximation in the phase lags, this author found that tT = 100 tq for metals This discrepancy with the causality principle indicates that the predictions of the DPL model in its approximate and exact forms may be remarkably different. This fact reveals that the small-phase lags can have great effects, as it has been shown in the theory of delayed differential equations [23].
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