A generalized form of the ballistic-diffusive equations (BDEs) for approximate solution of the Boltzmann Transport equation (BTE) for phonons is formulated. The formulation presented here is new and general in the sense that, unlike previously published formulations of the BDE, it does not require a priori knowledge of the specific heat capacity of the material. Furthermore, it does not introduce artifacts such as media and ballistic temperatures. As a consequence, the boundary conditions have clear physical meaning. In formulating the BDE, the phonon intensity is split into two components: ballistic and diffusive. The ballistic component is traditionally determined using a viewfactor formulation, while the diffusive component is solved by invoking spherical harmonics expansions. Use of the viewfactor approach for the ballistic component is prohibitive for complex large-scale geometries. Instead, in this work, the ballistic equation is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM) and the control angle discrete ordinates method (CADOM). Results of each method for solving the BDE are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two different two-dimensional transient heat conduction problems at various Knudsen numbers. It is found that standalone CADOM (for BTE) and hybrid CADOM-P1 (for BDE) yield the best accuracy. The hybrid CADOM-P1 is found to be the best method in terms of computational efficiency.
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e-mail: mazumder.2@osu.edu
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September 2011
This article was originally published in
Journal of Heat Transfer
Research Papers
Generalized Ballistic-Diffusive Formulation and Hybrid SN-PN Solution of the Boltzmann Transport Equation for Phonons for Nonequilibrium Heat Conduction
Arpit Mittal,
Arpit Mittal
Fellow ASME Department of Mechanical and Aerospace Engineering,
The Ohio State University
, Columbus
, OH 43210
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Sandip Mazumder
Sandip Mazumder
Fellow ASME Department of Mechanical and Aerospace Engineering,
e-mail: mazumder.2@osu.edu
The Ohio State University
, Columbus
, OH 43210
Search for other works by this author on:
Arpit Mittal
Fellow ASME Department of Mechanical and Aerospace Engineering,
The Ohio State University
, Columbus
, OH 43210
Sandip Mazumder
Fellow ASME Department of Mechanical and Aerospace Engineering,
The Ohio State University
, Columbus
, OH 43210e-mail: mazumder.2@osu.edu
J. Heat Transfer. Sep 2011, 133(9): 092402 (11 pages)
Published Online: July 7, 2011
Article history
Received:
September 11, 2010
Revised:
April 1, 2011
Online:
July 7, 2011
Published:
July 7, 2011
Citation
Mittal, A., and Mazumder, S. (July 7, 2011). "Generalized Ballistic-Diffusive Formulation and Hybrid SN-PN Solution of the Boltzmann Transport Equation for Phonons for Nonequilibrium Heat Conduction." ASME. J. Heat Transfer. September 2011; 133(9): 092402. https://doi.org/10.1115/1.4003961
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