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Solution of the Boltzmann Equationfor Phonon Transport

  • Denis Lemonnier
Chapter
Part of the Topics in Applied Physics book series (TAP, volume 107)

Abstract

We discuss two popular methods for solving the radiative transfer equation in the field of thermal radiation, which can be used to calculate conduction on nanoscales under certain hypotheses. After a brief summary of the theory leading to a radiative transfer equation for phonons, we present the P 1 method and the discrete ordinate method. The first is based on a global approach to transfer in which the only unknown is the local internal energy. In particular, it uses an approximate treatment of the boundary conditions and for this reason becomes somewhat inaccurate when transfer is dominated by ballistic phonons from the bounding surfaces. The second method takes into account the directional aspect of the transfer and yields better results than the P 1 method, except near the diffusive regime. It solves a transport equation in a discrete set of directions. Integrated quantities such as the internal energy and flux are evaluated using quadrature formulas. Whereas the partial differential equation derived in the P 1 approach can be solved by standard methods, the numerical system associated with the discrete ordinate method is more specific, particularly in cylindrical geometries.

Keywords

65.80.+n 82.53.Mj 81.16.-c 44.10.+i 44.40.+a 82.80.Kq 

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References

  1. N. W. Ashcroft, N. D. Mermin: Solid State Physics (Harcourt College Publishers 1976) Google Scholar
  2. C. Kittel: Introduction to Solid State Physics (John Wiley, New York 1996) zbMATHGoogle Scholar
  3. J. Ziman: Electrons and Phonons. The Theory of Transport Phenomena in Solids (Clarendon Press, Oxford 1960) zbMATHGoogle Scholar
  4. G. P. Srivastava: The Physics of Phonons (Adam Hilger 1990) Google Scholar
  5. A. Majumdar: Microscale heat conduction in dielectric thin films, J. Heat Transf. 115, 7–16 (1993) CrossRefGoogle Scholar
  6. J. H. Jeans: The equation of radiative transfer of energy, Monthly Notices, Roy. Astronom. Soc. 78, 28–36 (1917) ADSGoogle Scholar
  7. R. L. Murray: Nuclear Reactor Physics (Prentice Hall 1957) Google Scholar
  8. M. F. Modest: Radiative Heat Transfer, 2 ed. (Academic Press 2003) Google Scholar
  9. G. Chen: Ballistic-diffusive equations for transient heat conduction from nano to microscales, J. Heat Transf. 124, 320–328 (2002) CrossRefGoogle Scholar
  10. R. E. Marchak: Note on the spherical harmonic method as applied to the Milne problem for a sphere, Phys. Rev. 71, 443–446 (1947) CrossRefADSGoogle Scholar
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery: Numerical Recipes in Fortran 77, 2 ed. (Cambridge University Press, Cambridge 1992) zbMATHGoogle Scholar
  12. S. Chandrasekar: Radiative Transfer (Dover Publications, New York 1960) Google Scholar
  13. B. G. Carlson, K. D. Lathrop: Transport theory. The method of discrete ordinates, in H. Greenspan, C. V. N. Kelber, D. Okrent (Eds.): Computing Methods in Reactor Physics (Gordon and Breach 1968) Google Scholar
  14. W. A. Fiveland: A discrete ordinate method for predicting radiative heat transfer in axisymmetric enclosures, ASME Paper 82-HTD-20 (1982) Google Scholar
  15. W. A. Fiveland: Discrete ordinate solutions of the radiative transport equation for rectangular enclosures, J. Heat Transf. 106, 699–706 (1984) CrossRefGoogle Scholar
  16. W. A. Fiveland: Three-dimensional radiative heat transfer solutions by the discrete ordinate method, J. Thermophys. Heat Transfer 2, 309–316 (1988) CrossRefGoogle Scholar
  17. A. A. Joshi, A. Majumdar: Transient ballistic and diffusive phonon heat transport, Appl. Phys. 74, 31–39 (1993) CrossRefGoogle Scholar
  18. J. D. Chung, M. Kaviany: Effect of phonon pore scattering and pore randomness on effective conductivity of porous silicon, Int. J. Heat Mass Transf. 43, 521–538 (2000) zbMATHCrossRefGoogle Scholar
  19. S. Volz, D. Lemonnier, J. B. Saulnier: Clamped nanowire thermal conductivity based on phonon transport equation, Microscale Thermophys. Eng. 5, 191–207 (2001) CrossRefGoogle Scholar
  20. D. Balsara: Fast and accurate discrete ordinate methods for multidimensional radiative transfer. Part I: Basic methods, J. Quant. Spectrosc. Rad. Transf. 69, 671–707 (2001) CrossRefADSGoogle Scholar
  21. R. Koch, R. Becker: Evaluation of quadrature schemes for the discrete ordinate method, J. Quant. Spectrosc. Rad. Transf. 84, 423–435 (2004) CrossRefADSGoogle Scholar
  22. K. D. Lathrop: Spatial differencing of the transport equation. Positivity versus accuracy, J. Comp. Phys. 4, 475–498 (1969) zbMATHCrossRefADSGoogle Scholar
  23. S. Jendoubi, H. S. Lee, T. K. Kim: Discrete ordinate solutions for radiatively participating media in cylindrical enclosures, J. Thermophys. Heat Transfer 7, 213–219 (1993) ADSCrossRefGoogle Scholar
  24. W. A. Fiveland: The selection of discrete ordinate quadrature sets for anisotropic scattering, ASME HTD 160, 89–96 (1991) Google Scholar

Authors and Affiliations

  • Denis Lemonnier
    • 1
  1. 1.Laboratoire d’Etudes Thermiques (UMR CNRS 6608)ENSMAFuturoscope Chasseneuil Cedex

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