An explicit solution for a two-phase Stefan problem with a similarity exponential heat sources (original) (raw)
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Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients
Nonlinear Analysis: Real World Applications
In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component is considered. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modelled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the solution is provided by using the fixed point Banach theorem.
Latin American applied research Pesquisa aplicada latino americana = Investigación aplicada latinoamericana
We consider one-dimensional two-phase Stefan problems for a finite substance with different boundary conditions at the fixed faces. The goal of this paper is to determine the behavior of the free boundary and the temperature when the thermal coefficients of the material change. We obtain properties of monotony with respect to the latent heat, the common mass density, the specific heat of each phase and the thermal conductivity of the liquid phase. We show that the solution is not monotone with respect to the thermal conductivity of solid phase, in some cases, by computing a numerical solution through a finite difference scheme. The results obtained are important in technological applications as the climate of buildings, the storage of energy in satellites and clothes and the transport of biological substances and telecommunications.
Behavior of the solution of a Stefan problem by changing thermal coefficients of the substance
Applied Mathematics and Computation, 2007
We consider a one-dimensional one-phase Stefan problem for a semi-infinite substance. We suppose that there is a transient heat flux at the fixed face and the thermal coefficients are constant. The goal of this paper is to determine the behavior of the free boundary and the temperature by changing the thermal coefficients. We use the maximum principle in order to obtain properties of monotony with respect to the latent heat of fusion, the specific heat and the mass density. We compute approximate solutions through the quasi-stationary, the Goodman's heat-balance integral and the Biot's variational methods and a numerical solution through a finite difference scheme. We show that the solution is not monotone with respect to the thermal conductivity. The results obtained are important in technological applications.
Analytical solutions to the Stefan problem with internal heat generation
A differential equation modeling the Stefan problem with heat generation is derived. The analytical solutions compare very well with the computational results. The system reaches steady-state faster for larger Stefan numbers. The interface location is proportional to the inverse square root of the heat generation. a b s t r a c t A first-order, ordinary differential equation modeling the Stefan problem (solid–liquid phase change) with internal heat generation in a plane wall is derived and the solutions are compared to the results of a computational fluid dynamics analysis. The internal heat generation term makes the governing equations non-homogeneous so the principle of superposition is used to separate the transient from steady-state portions of the heat equation, which are then solved separately. There is excellent agreement between the solutions to the differential equation and the CFD results for the movement of both the solidification and melting fronts. The solid and liquid temperature profiles show a distinct difference in slope along the interface early in the phase change process. As time increases, the changes in slope decrease and the temperature profiles become parabolic. The system reaches steady-state faster for larger Stefan numbers and inversely, the time to steady-state increases as the Stefan number decreases.
Computational and Applied Mathematics
A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson, and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia (Comput Appl Math 9:201-211, 1990).