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On Entropic Framework Based on Standard and Fractional Phonon Boltzmann Transport Equations
1Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
This study explores entropy in non-Fourier heat conduction using standard and fractional phonon Boltzmann transport equations (BTEs). Results show fractional BTEs introduce memory effects via fractional-order operators in entropy flux and production.
Area of Science:
- Thermodynamics
- Heat Transfer
- Statistical Mechanics
Background:
- Generalized entropy expressions in non-Fourier heat conduction are gaining research interest.
- Existing frameworks like classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT) provide a basis for understanding heat transport.
- The need for more comprehensive models that capture complex thermal behaviors, including memory effects, is evident.
Purpose of the Study:
- To investigate entropic functionals (density, flux, production rate) within standard and fractional phonon Boltzmann transport equations (BTEs).
- To derive macroscopic approximations for these entropic concepts.
- To compare the entropic behavior predicted by standard BTEs with that of fractional BTEs, particularly those related to the generalized Cattaneo equation (GCE).
Main Methods:
- Utilized standard and fractional phonon Boltzmann transport equations (BTEs).
- Employed the relaxation time approximation and power series expansion to derive macroscopic approximations.
- Analyzed entropic functionals including entropy density, entropy flux, and entropy production rate.
Main Results:
- For standard BTEs, the derived entropic frameworks align with classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT), assuming an effective thermal conductivity.
- For fractional BTEs (GCE class), entropy flux and entropy production rate deviate from CIT and EIT forms.
- Fractional BTEs result in entropy flux and production rate containing fractional-order operators, signifying inherent memory effects.
Conclusions:
- The study provides generalized entropic expressions for non-Fourier heat conduction.
- Fractional BTEs offer a more nuanced description of heat transport by incorporating memory effects through fractional-order operators.
- This work extends the understanding of thermodynamic principles in complex heat conduction scenarios.
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