scholarly journals An Accurate Approximation of the Two-Phase Stefan Problem with Coefficient Smoothing

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1924
Author(s):  
Vasily Vasil’ev ◽  
Maria Vasilyeva
Keyword(s):  
Phase Change ◽  
Stefan Problem ◽  
Error Function ◽  
Discontinuous Coefficients ◽  
Two Phase ◽  
Spatial Interval ◽  
The One ◽  
Transfer Problems ◽  
The Mathematical Model ◽  
The Given

In this work, we consider the heat transfer problems with phase change. The mathematical model is described through a two-phase Stefan problem and defined in the whole domain that contains frozen and thawed subdomains. For the numerical solution of the problem, we present three schemes based on different smoothing of the sharp phase change interface. We propose the method using smooth coefficient approximation based on the analytical smoothing of discontinuous coefficients through an error function with a given smoothing interval. The second method is based on smoothing in one spatial interval (cell) and provides a minimal length of smoothing calculated automatically for the given values of temperatures on the mesh. The third scheme is a convenient scheme using a linear approximation of the coefficient on the smoothing interval. The results of the numerical computations on a model problem with an exact solution are presented for the one-dimensional formulation. The extension of the method is presented for the solution of the two-dimensional problem with numerical results.

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.

Numerical Solution of the Two-Phase Stefan Problem in the Enthalpy Formulation with Smoothing the Coefficients

2021 ◽  
pp. 4-23
Author(s):  
V.I. Vasilyev ◽  
M.V. Vasilyeva ◽  
S.P. Stepanov ◽  
N.I. Sidnyaev ◽  
O.I. Matveeva ◽  
...  
Keyword(s):  
Numerical Solution ◽  
Stefan Problem ◽  
Large Diameter ◽  
Discontinuous Coefficients ◽  
Numerical Calculations ◽  
Smooth Functions ◽  
Two Phase ◽  
Time Layer ◽  
Dirac Delta ◽  
Selection Of

To simulate heat transfer processes with phase transitions, the classical enthalpy model of Stefan is used, accompanied by phase transformations of the medium with absorption and release of latent heat of a change in the state of aggregation. The paper introduces a solution to the two-phase Stefan problem for a one-dimensional quasilinear second-order parabolic equation with discontinuous coefficients. A method for smearing the Dirac delta function using the smoothing of discontinuous coefficients by smooth functions is proposed. The method is based on the use of the integral of errors and the Gaussian normal distribution with an automated selection of the value of the interval of their smoothing by the desired function (temperature). The discontinuous coefficients are replaced by bounded smooth temperature functions. For the numerical solution, the finite difference method and the finite element method with an automated selection of the smearing and smoothing parameters for the coefficients at each time layer are used. The results of numerical calculations are compared with the solution of Stefan’s two-phase self-similar problem --- with a mathematical model of the formation of the ice cover of the reservoir. Numerical simulation of the thawing effect of installing additional piles on the existing pile field is carried out. The temperature on the day surface of the base of the structure is set with account for the amplitude of air temperature fluctuations, taken from the data of the Yakutsk meteorological station. The study presents the results of numerical calculations for concrete piles installed in the summer in large-diameter drilled wells using cement-sand mortars with a temperature of 25 °С. The distributions of soil temperature are obtained for different points in time

scholarly journals Developing methods for mathematical modeling of a two-phase dielectric-electrolyte microsystem

2021 ◽  
Vol 2057 (1) ◽  
pp. 012119
Author(s):  
E V Gorbacheva ◽  
E N Kalaidin
Keyword(s):  
Electric Fields ◽  
Stokes Equations ◽  
Constant Electric Field ◽  
Lower Surface ◽  
Short Wave ◽  
Two Phase ◽  
One Dimensional ◽  
Alternating Electric Fields ◽  
The One ◽  
The Mathematical Model

Abstract In this paper, we propose a numerical solution to the problem of stability of a two-phase dielectric / electrolyte system under direct and alternating electric fields. The lower wall adjacent to the electrolyte is assumed to be a charged surface, while the upper one is electrically insulated. The charge on the lower surface is supposed to be stationary, and the surface charge on the free interface between liquids is assumed to be mobile. The model is described by a system of Nernst-Planck-Poisson-Stokes equations. The mathematical model is closed by the corresponding boundary conditions. The linear stability of the one-dimensional flow is investigated. At a constant electric field, and the presence of two types of instabilities is found: short-wave and long-wave.

A feedback temperature control in a chemical reactor with high inertia of its cooling system

1984 ◽  
Vol 49 (7) ◽  
pp. 1642-1652 ◽  
Author(s):  
Josef Horák ◽  
František Jiráček ◽  
Libuše Ježová
Keyword(s):  
Mathematical Model ◽  
Temperature Control ◽  
Cooling System ◽  
Batch Reactor ◽  
Time Derivative ◽  
Exothermic Reaction ◽  
Chemical Reactor ◽  
The One ◽  
The Mathematical Model ◽  
The Given

In this work we compare simple algorithms for the one-off feedback temperature control of the reaction mixture in a batch reactor during an exothermic reaction. The aim of the control was to maintain the temperature of the mixture within the given range, and simultaneously, to minimize the number of the regulator switchings. The temperature control of the mixture was being performed at conditions when working states of the reactor in an open regulation loop are unstable and when the response of the cooler to regulation is slow. The following control algorithms were compared: P - regulator, PD - regulator and algorithms based on a prediction mathematical model including its adaptive variant. The results indicate that the algorithms based on the mathematical model are more efficient. However, the precision of the control can be diminished due to error in the time derivative of the temperature of the reaction mixture which forms the input to the prediction model. The adaptive variant of the algorithms was advantageous in cases when it was necessary to make up for significant errors in initial estimates of parameters of the prediction mathematical model.

scholarly journals Identification of the heat transfer coefficient in phase change problems

2010 ◽  
Vol 31 (1) ◽  
pp. 61-78
Author(s):  
Damian Słota
Keyword(s):  
Heat Transfer ◽  
Exact Solution ◽  
Heat Transfer Coefficient ◽  
Phase Change ◽  
Transfer Coefficient ◽  
Solid Phase ◽  
Two Phase ◽  
Additional Information ◽  
The Given ◽  
The Heat Transfer Coefficient

Identification of the heat transfer coefficient in phase change problemsIn this paper, an algorithm will be presented that enables solving the two-phase inverse Stefan problem, where the additional information consists of temperature measurements in selected points of the solid phase. The problem consists in the reconstruction of the function describing the heat transfer coefficient, so that the temperature in the given points of the solid phase would differ as little as possible from the predefined values. The featured examples of calculations show a very good approximation of the exact solution and stability of the algorithm.

Asymptotic analysis of a two-phase Stefan problem in annulus: Application to outward solidification in phase change materials

2021 ◽  
Vol 408 ◽  
pp. 126343
Author(s):  
Minghan Xu ◽  
Saad Akhtar ◽  
Ahmad F. Zueter ◽  
Mahmoud A. Alzoubi ◽  
Laxmi Sushama ◽  
...  
Keyword(s):  
Asymptotic Analysis ◽  
Phase Change ◽  
Stefan Problem ◽  
Phase Change Materials ◽  
Two Phase

Classical solutions of the one-dimensional, two-phase Stefan problem

1975 ◽  
Vol 107 (1) ◽  
pp. 311-341 ◽  
Author(s):  
John R. Cannon ◽  
Daniel B. Henry ◽  
Daniel B. Kotlow
Keyword(s):  
Stefan Problem ◽  
Classical Solutions ◽  
Two Phase ◽  
One Dimensional ◽  
The One

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.

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