Modeling of heat conduction via fractional derivatives

Modeling of heat conduction via fractional derivatives The modeling of heat conduction is considered by letting the time derivative, in the Cattaneo–Maxwell equation, be replaced by a derivative of fractional order. The purpose of this new approach is to overcome some drawbacks of the Cattaneo–Maxwell equation, for instance possible fluctuations which violate the non-negativity of the absolute temperature. Consistency with thermodynamics is shown to hold for a suitable free energy potential, that is in fact a functional of the summed history of the heat flux, subject to a suitable restriction on the set of admissible histories. Compatibility with wave propagation at a finite speed is investigated in connection with temperature-rate waves. It follows that though, as expected, this is the case for the Cattaneo–Maxwell equation, the model involving the fractional derivative does not allow the propagation at a finite speed. Nevertheless, this new model provides a good description of wave-like profiles in thermal propagation phenomena, whereas Fourier’s law does not. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Heat and Mass Transfer Springer Journals

Modeling of heat conduction via fractional derivatives

Modeling of heat conduction via fractional derivatives

Heat Mass Transfer (2017) 53:2785–2797 DOI 10.1007/s00231-017-1985-8 ORIGINAL 1 2 3 Mauro Fabrizio · Claudio Giorgi · Angelo Morro Received: 29 October 2015 / Accepted: 20 January 2017 / Published online: 24 March 2017 © Springer-Verlag Berlin Heidelberg 2017 Abstract The modeling of heat conduction is considered κ Heat conductivity per unit of time by letting the time derivative, in the Cattaneo–Maxwell c Heat capacity equation, be replaced by a derivative of fractional order. r External heat supply The purpose of this new approach is to overcome some ε, ε ˆ Internal energy densities drawbacks of the Cattaneo–Maxwell equation, for instance η Entropy density possible fluctuations which violate the non-negativity of ψ ,ψ Helmholtz free energy densities the absolute temperature. Consistency with thermodynam-  Thermal displacement ics is shown to hold for a suitable free energy potential, ξ Internal dissipation rate that is in fact a functional of the summed history of the heat θ (·) Summed history of the absolute flux, subject to a suitable restriction on the set of admis - temperature sible histories. Compatibility with wave propagation at a g (·) Past history of the temperature gradient finite speed is investigated in connection with temperature- g (·) Summed history of the temperature rate waves. It follows that though, as expected, this is the gradient case for the Cattaneo–Maxwell equation, the model involv- ζ (·) Relative history of the heat flux ing the fractional derivative does not allow the propagation q(t, ·) Summed history of the heat flux at a finite speed. Nevertheless, this new model provides a K, β , γ , h, g, j Memory kernels good description of wave-like profiles in thermal propaga - tion phenomena, whereas Fourier’s law does not. 1 Introduction List of symbols x, x Space variables The...
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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Engineering; Engineering Thermodynamics, Heat and Mass Transfer; Industrial Chemistry/Chemical Engineering; Thermodynamics
ISSN
0947-7411
eISSN
1432-1181
D.O.I.
10.1007/s00231-017-1985-8
Publisher site
See Article on Publisher Site

Abstract

The modeling of heat conduction is considered by letting the time derivative, in the Cattaneo–Maxwell equation, be replaced by a derivative of fractional order. The purpose of this new approach is to overcome some drawbacks of the Cattaneo–Maxwell equation, for instance possible fluctuations which violate the non-negativity of the absolute temperature. Consistency with thermodynamics is shown to hold for a suitable free energy potential, that is in fact a functional of the summed history of the heat flux, subject to a suitable restriction on the set of admissible histories. Compatibility with wave propagation at a finite speed is investigated in connection with temperature-rate waves. It follows that though, as expected, this is the case for the Cattaneo–Maxwell equation, the model involving the fractional derivative does not allow the propagation at a finite speed. Nevertheless, this new model provides a good description of wave-like profiles in thermal propagation phenomena, whereas Fourier’s law does not.

Journal

Heat and Mass TransferSpringer Journals

Published: Mar 24, 2017

References

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