scholarly journals ON THE SINGULAR PERTURBATION SOLUTION OF BOLTZMANN EQUATION (Ⅱ) "INITIAL LAYER" SOLUTION

1985 ◽  
Vol 34 (1) ◽  
pp. 77
Author(s):  
DING E-JIANG ◽  
HUANG ZU-QIA
Keyword(s):  
Boltzmann Equation ◽  
Singular Perturbation ◽  
Perturbation Solution ◽  
Initial Layer ◽  
Layer Solution

Singular Perturbation Solution for a Two-Phase Stefan Problem in Outward Solidification

2019 ◽  
Author(s):  
Minghan Xu ◽  
Saad Akhtar ◽  
Mahmoud A. Alzoubi ◽  
Agus P. Sasmito
Keyword(s):  
Boundary Condition ◽  
Porous Media ◽  
Analytical Solution ◽  
Singular Perturbation ◽  
Stefan Problem ◽  
Moving Boundary ◽  
Computational Cost ◽  
Perturbation Solution ◽  
Two Phase ◽  
Moving Boundary Condition

Abstract Mathematical modeling of phase change process in porous media can help ensure the efficient design and operation of thermal energy storage and pipe freezing. Numerical methods generally require high computational power to be applicable in practice. Therefore, it is of great interest to develop accurate and reliable analytical frameworks. This study proposes a singular perturbation solution for a two-phase Stefan problem that describes outward solidification in a finite annular space. The problem solves cylindrical heat conduction equations for both solid and liquid phases, with consideration of a moving boundary condition. Perturbation method takes the advantages of small Stefan number as the perturbation parameter, which intrinsically occurs in porous media. Furthermore, a boundary-fixing technique is used to remove nonlinearity in the moving boundary condition. Two different time scales are separately expanded and evaluated to facilitate the construction of a composite asymptotic solution. The analytical solution is verified against a general numerical model using enthalpy method and local volume-averaged thermal properties. The results indicate that the temperature profile of both phases can be well modeled by singular perturbation theory. The analytical solution is found to have similar conclusions to the numerical analysis with much lesser computational cost.

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