High resolution schemes for phonon heat transfer and ideal quantum gas dynamics
Date Issued
2007
Date
2007
Author(s)
Hsieh, Tse-Yang
DOI
zh-TW
Abstract
It is universally acknowledged that substance is composed of molecules. In macroscopic scale, substance can be considered as continuum, and the transport phenomena can be described by macroscopic governing equations, e.g. Fourier law, Euler equation, Navier-Stokes equation, etc. The size effect can no longer be neglected as the characteristic length of the flow filed is comparable with the molecule mean free path. We need to consider motions and interactions of the individual molecules.
In non-equilibrium problems, we investigate phonon heat transfer in micro-scale semiconductor materials. The macroscopic governing equations are no longer valid as the length of the material is comparable with the phonon mean free path. The special heat transport phenomena must be described with the behavior of individual phonons. In micro-scale problems, the governing equation for the phonon distribution function is the phonon Boltzmann equation. The nonlinear integral-differential Boltzmann equation is very difficult to solve in general. Under the relaxation time approximation, Boltzmann equation can be simplified to BGK (Bhatnagar-Gross-Krook) equation. With the discrete ordinate method, the original space, velocity and time continuous equation can be transformed to a set of equations which are continuous in physical space and time only and point-wise in velocity space. The set of equations are hyperbolic partial differential equations with source terms, so they can be solved by utilizing the high resolution conservation law scheme. In the transient thin film problem, we apply the modified discrete ordinate method to eliminate ray effects and cooperate with the high resolution scheme to eliminate false scattering. To reduce the computation time and separate the memory requirement, we adopt the parallel computing strategy. The heat conduction properties of thin film, thin film type, wire type and particle type superlattice are investigated in this thesis.
In quasi-equilibrium problems, we analyze the gases that obey quantum statistics from the molecule’s point of view. The governing equation is quantum Euler equation when the viscosity effect is not considered. We develop a kinetic scheme that suitable for describing ideal quantum gas dynamics. Kinetic schemes utilize the property that macroscopic governing equations can be derived from taking moments to Boltzmann equation and calculate the numerical flux from the distribution function. The ideal quantum gases obey Bose-Einstein or Fermi-Dirac statistics and the flux can be derived from the known distribution function. In this sense, we develop the kinetic schemes that suitable for ideal quantum gas and investigate the influence of the quantum effect. In this thesis we discuss shock tube problems and shock diffraction by wedges and shock diffraction by cylinders for the ideal quantum gases.
Subjects
聲子
波茲曼方程
理想量子氣體
氣體運動算則
Phonon
Boltzmann equation
Ideal quantum gas
Gas kinetic scheme
Type
thesis
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