Workshop on mathematical modelling of energy and mass transfer processes, and applications (original) (raw)
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An explicit solution for a two-phase Stefan problem with a similarity exponential heat sources
MAT Serie A
A two-phase Stefan problem with heat source terms in both liquid and solid phases for a semi-in¯nite phase-change material is considered. The internal heat source functions are given by g j (x; t) = (¡1) j+1 ½l t exp ³ ¡(x 2a j p t + d j) 2´(j = 1 solid phase; j = 2 liquid phase), ½ is the mass density, l is the fusion latent heat by unit of mass; a 2 j is the di®usion coe±cient, x is spatial variable, t is the temporal variable and d j 2 R. A similarity solution is obtained for any data when a temperature boundary condition is imposed at the¯xed face x = 0; when a°ux condition of the type ¡q 0 = p t (q 0 > 0) is imposed on x = 0 then there exists a similarity solution if and only if a restriction on q 0 is satis¯ed.
2019
In this chapter we consider different approximations for the one-dimensional one-phase Stefan problem corresponding to the fusion process of a semi-infinite material with a temperature boundary condition at the fixed face and non-linear temperature-dependent thermal conductivity. The knowledge of the exact solution of this problem, allows to compare it directly with the approximate solutions obtained by applying the heat balance integral method, an alternative form to it and the refined balance integral method, assuming a quadratic temperature profile in space. In all cases, the analysis is carried out in a dimensionless way by the Stefan number (Ste) parameter.
2009
We study a one-phase Stefan problem for a semi-infinite material with temperaturedependent thermal conductivity and a convective term with a constant temperature boundary condition or a heat flux boundary condition of the type −q0/ √ t (q0 > 0) at the fixed face x = 0. We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for t ≥ t0 > 0 with t0 an arbitrary positive time. We improve the results given in Rogers-Broadbridge, ZAMP, 39 (1988), 122-129 and Natale-Tarzia, Int. J. Eng. Sci., 41 (2003), 1685-1698 obtaining explicit solutions through the unique solution of a Cauchy problem with the time as a parameter and we also give an algorithm in order to compute the explicit solutions.
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.
Waves in Random and Complex Media, 2022
In this paper a one-phase Stefan problem with size-dependent thermal conductivity is analysed. Approximate solutions to the problem are found via perturbation and numerical methods, and compared to the Neumann solution for the equivalent Stefan problem with constant conductivity. We find that the size-dependant thermal conductivity, relevant in the context of solidification at the nanoscale, slows down the solidification process. A small time asymptotic analysis reveals that the position of the solidification front in this regime behaves linearly with time, in contrast to the Neumann solution characterized by a square root of time proportionality. This has an important physical consequence, namely the speed of the front predicted by size-dependant conductivity model is finite while the Neumann solution predicts an infinite and, thus, unrealistic speed as t → 0.
MAT Serie A
We study a one-phase Stefan problem for a semi-infinite material with temperaturedependent thermal conductivity and a convective term with a constant temperature boundary condition or a heat flux boundary condition of the type −q 0 / √ t (q 0 > 0) at the fixed face x = 0. We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for t ≥ t 0 > 0 with t 0 an arbitrary positive time. We improve the results given in
MAT Serie A
We study a one-phase Stefan problem for a semi-in¯nite material with temperaturedependent thermal conductivity with a constant temperature or a heat°ux condition of the type ¡q 0 = p t (q 0 > 0) at the¯xed face x = 0. We obtain in both cases su±cient conditions for data in order to have a parametric representation of the solution of the similarity type for t¸t 0 > 0 with t 0 an arbitrary positive time. These explicit solutions are obtained through the unique solution of an integral equation with the time as a parameter.
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.