Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

Christenson, J. G.; Austin, R. A.; Phillips, R. J.

doi:10.1063/1.5022182

Title: Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

Journal Article · Fri May 04 00:00:00 EDT 2018 · Journal of Applied Physics

DOI:https://doi.org/10.1063/1.5022182· OSTI ID:1558868

Christenson, J. G. ^[1]; Austin, R. A. ^[2]; Phillips, R. J. ^[1]

Univ. of California, Davis, CA (United States)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

In this study, the phonon Boltzmann transport equation is used to analyze model problems in one and two spatial dimensions, under transient and steady-state conditions. New, explicit solutions are obtained by using the P₁ and P₃ approximations, based on expansions in spherical harmonics, and are compared with solutions from the discrete ordinates method. For steady-state energy transfer, it is shown that analytic expressions derived using the P₁ and P₃ approximations agree quantitatively with the discrete ordinates method, in some cases for large Knudsen numbers, and always for Knudsen numbers less than unity. However, for time-dependent energy transfer, the P_N solutions differ qualitatively from converged solutions obtained by the discrete ordinates method. Although they correctly capture the wave-like behavior of energy transfer at short times, the P₁ and P₃ approximations rely on one or two wave velocities, respectively, yielding abrupt, step-changes in temperature profiles that are absent when the angular dependence of the phonon velocities is captured more completely. It is shown that, with the gray approximation, the P₁ approximation is formally equivalent to the so-called “hyperbolic heat equation.” In conclusion, these results support the use of the P_N approximation to find solutions to the phonon Boltzmann transport equation for steady-state conditions. Such solutions can be useful in the design and analysis of devices that involve heat transfer at nanometer length scales, where continuum-scale approaches become inaccurate.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1558868

Alternate ID(s):: OSTI ID: 1436015

Report Number(s):: LLNL-JRNL-740667; 894791; TRN: US2000259

Journal Information:: Journal of Applied Physics, Vol. 123, Issue 17; ISSN 0021-8979

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 3 works

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