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

Abstract

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-statemore » 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.« less

Authors:

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

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+ Show Author Affiliations

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

Publication Date:

Fri May 04 00:00:00 EDT 2018

Research Org.:

Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:

1558868

Alternate Identifier(s):

OSTI ID: 1436015

Report Number(s):

LLNL-JRNL-740667
Journal ID: ISSN 0021-8979; 894791; TRN: US2000259

Grant/Contract Number:  

AC52-07NA27344

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Applied Physics

Additional Journal Information:

Journal Volume: 123; Journal Issue: 17; Journal ID: ISSN 0021-8979

Publisher:

American Institute of Physics (AIP)

Country of Publication:

United States

Language:

English

Subject:

36 MATERIALS SCIENCE; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS