scholarly journals Analytical solution of two-phase spherical Stefan problem by heat polynomials and integral error functions

2016 ◽  
Author(s):  
Stanislav N. Kharin ◽  
Merey M. Sarsengeldin ◽  
Hassan Nouri
Keyword(s):  
Analytical Solution ◽  
Stefan Problem ◽  
Two Phase ◽  
Error Functions ◽  
Integral Error

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.

Method of the integral error functions for the solution of the one- and two-phase Stefan problems and its application

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1017-1029 ◽  
Author(s):  
Merey Sarsengeldin ◽  
Stanislav Kharina
Keyword(s):  
A Priori ◽  
Linear Combinations ◽  
Two Phase ◽  
Electrical Arc ◽  
Error Functions ◽  
Stefan Problems ◽  
Finite Sums ◽  
Integral Error ◽  
Faà Di Bruno Formula ◽  
The One

The analytical solutions of the one- and two-phase Stefan problems are found in the form of series containing linear combinations of the integral error functions which satisfy a priori the heat equation. The unknown coefficients are defined from the initial and boundary conditions by the comparison of the like power terms of the series using the Faa di Bruno formula. The convergence of the series for the temperature and for the free boundary is proved. The approximate solution is found using the replacement of series by the corresponding finite sums and the collocation method. The presented test examples confirm a good approximation of such approach. This method is applied for the solution of the Stefan problem describing the dynamics of the interaction of the electrical arc with electrodes and corresponding erosion.

scholarly journals PROBLEM FROM THE THEORY OF BRIDGE EROSION

2018 ◽  
pp. 68-74
Author(s):  
S.N. Kharin ◽  
S.A. Kassabek ◽  
M. Slyamkhan
Keyword(s):  
Exact Solution ◽  
Mathematical Models ◽  
Stefan Problem ◽  
Error Function ◽  
Two Phase ◽  
Bridge Problem ◽  
Radial Heat ◽  
Integral Error

In this paper, we represent the exact solution of a two phase Stefan problem. Radial heat polynomialsand integral error function are used for solving bridge problem. The recurrent expressions for the coefficients of these series are presented. The mathematical models describe the dynamics of contact opening and bridging. Keywords: radial heat polynomials, Stefan problem.

Development of Analytical Solution for a Two-Phase Stefan Problem in Artificial Ground Freezing Using Singular Perturbation Theory

2020 ◽  
Vol 142 (12) ◽  
Author(s):  
Minghan Xu ◽  
Saad Akhtar ◽  
Ahmad F. Zueter ◽  
Victor Auger ◽  
Mahmoud A. Alzoubi ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Singular Perturbation ◽  
Stefan Problem ◽  
Underground Structure ◽  
Computational Cost ◽  
Singular Perturbation Theory ◽  
Singular Perturbation Method ◽  
Two Phase ◽  
Ground Freezing ◽  
Artificial Ground Freezing

Abstract Artificial ground freezing (AGF) has historically been used to stabilize underground structure. Numerical methods generally require high computational power to be applicable in practice. Therefore, it is of interest to develop accurate and reliable analytical frameworks for minimizing computational cost. This paper proposes a singular perturbation solution for a two-phase Stefan problem that describes outward solidification in AGF. Specifically, the singular perturbation method separates two distinct temporal scales to capture the subcooling and freezing stages in the ground. The ground was considered as a porous medium with volume-averaged thermophysical properties. Further, Stefan number was assumed to be small, and effects of a few site-dependent parameters were investigated. The analytical solution was verified by numerical results and found to have similar conclusions yet with much lesser computational cost. Keywords: artificial ground freezing, Stefan-like problems, singular perturbation, porous media, outward solidification.

scholarly journals A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2157
Author(s):  
Marek Błasik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Numerical Method ◽  
Stefan Problem ◽  
Moving Boundary ◽  
Second Phase ◽  
Time Step ◽  
Partial Differential ◽  
Two Phase

In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

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