Abstract
In this work, we derive different systems of mesoscopic moment equations for the heat-conduction problem and analyze the basic features that they must hold. We discuss two- and three-equation systems, showing that the resulting mesoscopic equation from two-equation systems is of the telegraphist’s type and complies with the Cattaneo equation in the Extended Irreversible Thermodynamics Framework. The solution of the proposed systems is analyzed, and it is shown that it accounts for two modes: a slow diffusive mode, and a fast advective mode. This latter additional mode makes them suitable for heat transfer phenomena on fast time-scales, such as high-frequency pulses and heat transfer in small-scale devices. We finally show that, if proper initial conditions are provided, the advective mode disappears, and the solution of the system tends asymptotically to the transient solution of the classical parabolic heat-conduction equation.
Highlights
Over the last decades, considerable efforts have been spent on the goal to extend the phenomenological concept of irreversible thermodynamics into the region beyond the classical hydrodynamic description [1]
We focus on kinetic approaches, which have been extensively used to obtain new heat-conduction equations derived from the Boltzmann equation [26,27,28,29] and to investigate regimes in which multiple scales and sub-continuum effects are important [30,31,32]
We show that, considering two-equation systems, the resulting mesoscopic equation for temperature complies with the Cattaneo equation in the Extended Irreversible Thermodynamics Framework
Summary
Considerable efforts have been spent on the goal to extend the phenomenological concept of irreversible thermodynamics into the region beyond the classical hydrodynamic description [1]. Contemporary technology strives for high speed and miniaturization; transport equations should be able to cope with the related phenomena Another reason, perhaps even more important, is the unphysical behavior of the classical parabolic partial differential equations, which imply that perturbations propagate with an infinite speed. The proposed models provide an extended description that is able to account for heat transfer phenomena occurring at fast time-scales, which are not recovered by Fourier’s description. Practical applications for these models, under proper considerations, span over nano-technologies and.
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