Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat
10 February 2021
Last update: 09/02/21 15:14
Stefan problems are a special type of partial differential boundary value problems, introduced around 1890 by the Slovenian physicist Josef Stefan [1]. Originally the problem is motivated by the modelling of ice formations in the arctic oceans, which is a challenging mathematical problem due to the boundaries of the ice changing continuously by melting and freezing. Problems like these where boundaries change over time are referred to free boundary value problems and the challenging point, in this case, is to combine the time evolution of the heat distribution with the changing boundaries of the ice.
In thermodynamics, the transitions of substances between states of matter are called phase-transitions and a one-phase transition problem is a problem where two possible states of matter are considered, for example, a liquid and a solid part of the system. The heat that results in phase-transitions without resulting in a change of temperature is called latent heat. This heat is used to overcome atomic or molecular bounds enabling the substance to change its state of matter.
One of the latest articles in the European Journal of Applied Mathematics that is strongly connected to this field is the article “Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat” by Julieta Bollati and Domingo A. Tarzia from the FCE-Universidad Austral.
In their work, the authors consider different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. As Stephan-problems belong to the class of non-linear problems, exact solutions are only available in a few cases making it necessary to solve the equation numerically.
Bollati and Tarzia consider two problems that differ in their boundary condition and approximate the solutions by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). The HBIM method “consists in the transformation of the heat equation into an ordinary differential equation in time, assuming a quadratic profile in space for the temperature” [2] and the RIM methods suggests solving an approximate problem that replaces the heat equation. Hence, the paper aims at comparing these solutions and evaluating their accuracy. For this, the authors regard special cases where the exact solution is known and where they can compare the exact solutions with the approximated numerical solutions. Moreover, the authors provide convergence results for the approximated solutions.
Bollati and Tarzia summarize their work by the statement that their numerical simulations show that “in the majority of cases, the modified integral method is the most reliable of accuracy” [2] and the least accurate method is the classical HBIM “not only to the high percentage error committed but also because” they “could not obtain a result that assures uniqueness of the approximate solution”.
References:
[1] Josef Stefan. Uber die Theorie der Eisbildung. Monatshefte Mat. Phys., 1:1–6, 1890. https://doi.org/10.1002/andp.18912780206
[2] Julieta Bollati and Domingo A. Tarzia. Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat – European Journal of Applied Mathematics. https://doi.org/10.1017/S0956792520000170
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