Stefan problem as the limiting case of the problem of phase transition in a temperature range

About the authors:

Boris G. Aksenov, Dr. Sci. (Phys.-Math.), Professor, Department of Industrial Thermal Power Engineering, Industrial University of Tyumen; aksenovbg@tyuiu.ru

Svetlana V. Karyakina, Cand. Sci. (Tech.), Associate Professor, Department of Business Informatics and Mathematics, Industrial University of Tyumen; karyakinaswetlan@mail.ru

Abstract:

The article offers a theory which enables to apply the method of estimates to the solution of Stefan problem. The method of estimates suggests differential and integral inequalities for the equations of parabolic or elliptic type describing processes of non-stationary or stationary heat conductivity. does not allow Such inequalities are inapplicable for the Stefan problem because at phase transition boundary the main equation is not defined. Here the Stefan problem is not treated in its classical statement, but as a limiting case of a more general quasi-linear problem of phase transition in a temperature range. It is shown that under certain conditions there exists an exact equality between the solution of a quasi-linear problem and some front problem. This allows using the inequalities for a continuous quasi-linear problem, to estimate the solution of the Stefan problem. The authors formulate the principles which allow to generate approximate solutions of the Stefan problem for various boundary conditions. The theory is applied to the problem based on freeze-thaw cycles of the moist soil. In coarse-dispersed soils the pore moisture freezes (thaws) at a fixed temperature. This process is typically described with the Stefan front problem. In finely dispersed soils the pore moisture is in the bound state, and therefore, the front of phase transition is not formed, while Joulean heat is released (absorbed) in some temperature range. For each type of finely dispersed soil the phase composition of moisture at negative temperatures is described by the so-called curve of the not-frozen moisture. Thus, both compared problems (quasi-linear and front) are immediately applicable to describe the freeze-thaw cycles of the moist soil.

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