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Research Papers: Micro/Nanoscale Heat Transfer

Generalized Ballistic-Diffusive Formulation and Hybrid SN -PN Solution of the Boltzmann Transport Equation for Phonons for Nonequilibrium Heat Conduction

[+] Author and Article Information
Arpit Mittal

Fellow ASME Department of Mechanical and Aerospace Engineering,  The Ohio State University, Columbus, OH 43210mazumder.2@osu.edu

Sandip Mazumder1

Fellow ASME Department of Mechanical and Aerospace Engineering,  The Ohio State University, Columbus, OH 43210mazumder.2@osu.edu

1

Corresponding author.

J. Heat Transfer 133(9), 092402 (Jul 07, 2011) (11 pages) doi:10.1115/1.4003961 History: Received September 11, 2010; Revised April 01, 2011; Published July 07, 2011; Online July 07, 2011

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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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Geometry and boundary conditions for first test case

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Figure 8

Comparison of temperature distribution obtained using two different methods for Knudsen number equal to unity and after t=τ: (a) CADOM BTE and (b) CADOM-P1 BDE

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Figure 9

Comparison of temperature distribution obtained using two different methods for Knudsen number equal to unity and after t=10τ: (a) CADOM BTE and (b) CADOM-P1 BDE

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Figure 6

Description of the second test case: (a) geometry and boundary conditions and (b) computational mesh

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Figure 7

Comparison of temperature distribution obtained using two different methods for Knudsen number equal to unity and after t=0.1τ: (a) CADOM BTE and (b) CADOM-P1 BDE

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Figure 2

Results of the first test case for Knudsen number of 0.01: (a) unsteady temperature profiles along the centerline, (b) steady state heat flux distribution along the bottom wall. The error bars for the Monte Carlo results correspond to ±3σ.

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Figure 3

Results of the first test case for Knudsen number of 1: (a) unsteady temperature profiles along the centerline, (b) steady state heat flux distribution along the bottom wall. The error bars for the Monte Carlo results correspond to ±3σ.

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Figure 4

Results of the first test case for Knudsen number of 100: (a) unsteady temperature profiles along the centerline, (b) steady state heat flux distribution along the bottom wall. The error bars for the Monte Carlo results correspond to ±3σ.

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Figure 5

Comparison of steady state temperature distributions obtained using (a) DOM and (b) CADOM for solution of the BTE for Knudsen number of 100

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