Fractional-order theory of heat transport in rigid bodies

Abstract

a b s t r a c tThe non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53], Mongioví and Zingales, 2013 [54]) that thermal energy transport is due to two phenomena: (i) A short-range heat flux ruled by a local transport equation; (ii) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law decay of the distance yielding a novel temperature equation in terms of a-order Marchaud fractional-order derivative ð0 6 a 6 1Þ. Thermodynamical consistency of the model is provided in the context of Clausius-Plank inequality. The effects induced by the boundary conditions on the temperature field are investigated for diffusive as well as ballistic local heat flux. Deviations of the temperature field from the linear distributions in the neighborhood of the thermostated zones of small-scale conductors are qualitatively predicted by the used fractional-order heat transport model, as shown by means of molecular dynamics simulations.

The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

“…In particular, in ref. [38][39][40][41] powerlaw distance-decaying functions have been chosen, resulting in a fractional-order heat conduction equation with some specific fractional operators [38][39][40][41]. Results have been found in agreement with non-classical thermodynamic behavior observed, for instance, in micro-and nano-devices [42][43][44].…”

“…(1), ρ(x)u l (x,t) and ρ(x)u nl (x,t) denote the local and long-range contributions to the total internal energy at location x, related to: (a) the local flux energy vector q(x,t); for instance, it can be provided by the Fourier law; (b) a long-range residual contribution that can be expressed using fractional derivative operators [38][39][40][41].…”

“…The proposed non-local model of thermodynamics [38][39][40][41] relies on the assumption that the internal energy density of the body, ρ(x)u(x, t), comprises two contributions so that:…”

“…In the resulting heat conduction equation at a given location, volume integral accounts for the long-range thermal energy transfer between the given location and all surrounding adjacent and non-adjacent locations, including the material-dependent distance-decaying function that governs the long-range heat transfer [38][39][40][41]. It has been demonstrated that any strictly positive distance-decaying function can be selected, for the model to fulfill the second principle of thermodynamics [38][39][40][41]. In particular, in ref.…”

“…Very recently a non-local model of thermal energy transport has been proposed by the authors [38][39][40][41], based on the assumption that two contributions coexist at a considered observation scale: (i) a local thermal energy transfer, described by the Fourier transport law; (ii) a non-local, long-range thermal energy transport, involving couples of adjacent and non-adjacent elementary volumes, and modeled as proportional to the product of the masses of the interacting volumes, and to the relative temperature at their locations, through a material-dependent distance-decaying function. In the resulting heat conduction equation at a given location, volume integral accounts for the long-range thermal energy transfer between the given location and all surrounding adjacent and non-adjacent locations, including the material-dependent distance-decaying function that governs the long-range heat transfer [38][39][40][41].…”