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Dynamical Governing Equations of Non-Fourier Heat Conduction

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Part of the Springer Theses book series (Springer Theses)

Abstract

Thermal energy has its corresponding equivalent mass according to Einstein’s mass–energy equivalence, which is termed as thermomass. The thermomass theory established the continuous governing equation for the non-Fourier heat conduction. The mass balance equation of thermomass gives the energy conservation equation while the momentum balance equation of thermomass gives the general heat conduction law. The microscopic foundation of the general heat conduction law based on the thermomass theory is investigated. The derivation based on the phonon Boltzmann equation shows that the second order expansion of phonon distribution function leads to the spatial inertia (or convective) term in the general heat conduction law, which makes the difference from the previous phonon hydrodynamic model. Limiting to the first order expansion will give the Cattaneo-Vernotte model, while the zeroth order expansion gives the classical Fourier’s law. Comparison with other derivations of phonon Boltzmann equation such as the Chapman–Enskog expansion and eigenvalue analysis is presented.

Keywords

Heat Flux Heat Wave Drift Velocity Rest Mass Momentum Balance Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Moller C (1972) The theory of relativity. Clarendon Press, OxfordGoogle Scholar
  2. 2.
    Misner CM (1973) Gravitation. WH Freeman and Company, San FranciscoGoogle Scholar
  3. 3.
    Schroder UE (1990) Special relativity. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  4. 4.
    Rindler W (1982) Introduction to special relativity. Clarendon Press, OxfordzbMATHGoogle Scholar
  5. 5.
    Einstein A, Infield L (1938) The evolution of physics: the growth of ideas from early concepts to relativity and quanta. Simon and Schuster, New YorkGoogle Scholar
  6. 6.
    Taylor EF, Wheeler JA (1966) Spacetime physics. W.H. Freeman and Company, New YorkGoogle Scholar
  7. 7.
    Kittel C (1996) Introduction to solid state physics, 7th edn. Wiley, New YorkzbMATHGoogle Scholar
  8. 8.
    Reissland JA (1973) The physics of phonons. Wiley, LondonGoogle Scholar
  9. 9.
    Guo ZY, Cao BY, Zhu HY et al (2007) State equation of phonon gas and conservation equations for phonon gas motion. Acta Phys Sin 56(6):3306–3312Google Scholar
  10. 10.
    Guo ZY, Cao BY (2008) A general heat conduction law based on the concept of motion of thermal mass. Acta Phys Sin 57(7):4273–4281Google Scholar
  11. 11.
    Cao BY, Guo ZY (2007) Equation of motion of a phonon gas and non-Fourier heat conduction. J Appl Phys 102(5):053503ADSCrossRefGoogle Scholar
  12. 12.
    Tzou DY, Guo ZY (2010) Nonlocal behavior in thermal lagging. Int J Therm Sci 49(7):1133–1137CrossRefGoogle Scholar
  13. 13.
    Guo ZY, Hou QW (2010) Thermal wave based on the thermomass model. J Heat Transfer 132(7):072403CrossRefGoogle Scholar
  14. 14.
    Wang M, Guo ZY (2010) Understanding of temperature and size dependences of effective thermal conductivity of nanotubes. Phys Lett A 374(42):4312–4315ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang M, Yang N, Guo ZY (2011) Non-Fourier heat conductions in nanomaterials. J Appl Phys 110(6):064310ADSCrossRefGoogle Scholar
  16. 16.
    Wang M, Shan X, Yang N (2012) Understanding length dependences of effective thermal conductivity of nanowires. Phys Lett A 376(46):3514–3517ADSCrossRefGoogle Scholar
  17. 17.
    Wang HD, Cao BY, Guo ZY (2010) Heat flow choking in carbon nanotubes. Int J Heat Mass Transf 53(9):1796–1800CrossRefzbMATHGoogle Scholar
  18. 18.
    Christov CI, Jordan PM (2005) Heat conduction paradox involving second-sound propagation in moving media. Phys Rev Lett 94:154301ADSCrossRefGoogle Scholar
  19. 19.
    Müller I, Ruggeri T (1993) Extended thermodynamics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  20. 20.
    Guyer RA, Krumhansl JA (1966) Solution of the linearized phonon Boltzmann equation. Phys Rev 148(2):766–778ADSCrossRefGoogle Scholar
  21. 21.
    Guyer RA, Krumhansl JA (1966) Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals. Phys Rev 148(2):778ADSCrossRefGoogle Scholar
  22. 22.
    Sussmann JA, Thellung A (1963) Thermal conductivity of perfect dielectric crystals in the absence of umklapp processes. Proc Phys Soc 81(6):1122ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Hardy RJ (1970) Phonon Boltzmann equation and second sound in solids. Phys Rev B 2(4):1193ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Hardy RJ, Albers DL (1974) Hydrodynamic approximation to the phonon Boltzmann equation. Phys Rev B 10(8):3546ADSCrossRefGoogle Scholar
  25. 25.
    Gurevich VL (1986) Transport in phonon systems. North-Holland, AmsterdamGoogle Scholar
  26. 26.
    Enz CP (1968) One-particle densities, thermal propagation, and second sound in dielectric crystals. Ann Phys 46(1):114–173ADSCrossRefGoogle Scholar
  27. 27.
    Sahasrabudhe GG, Lambade SD (1999) Temperature dependence of the collective phonon relaxation time and acoustic damping in Ge and Si. J Phys Chem Solids 60(6):773–785ADSCrossRefGoogle Scholar
  28. 28.
    Banach Z, Larecki W (2008) Chapman-Enskog method for a phonon gas with finite heat flux. J Phys A Math Theor 41(37):375502CrossRefzbMATHGoogle Scholar
  29. 29.
    Jiaung WS, Ho JR (2008) Lattice-Boltzmann modeling of phonon hydrodynamics. Phys Rev E 77(6):066710ADSCrossRefGoogle Scholar
  30. 30.
    Krumhansl JA (1965) Thermal conductivity of insulating crystals in the presence of normal processes. Proc Phys Soc 85(5):921ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Key Laboratory for Thermal Science and Power Engineering of Ministry of EducationDepartment of Engineering Mechanics, Tsinghua UniversityBeijingChina

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