Numerical Simulation of Stefan’s One-Dimensional Multi-Front Problems

Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy


Release:

2017, Vol. 3. №3

Title: 
Numerical Simulation of Stefan’s One-Dimensional Multi-Front Problems


For citation: Aksenov B. G., Karyakin Yu. E. 2017. “Numerical Simulation of Stefan’s One-Dimensional Multi-Front Problems”. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 3, no 3, pp. 8-16. DOI: 10.21684/2411-7978-2017-3-3-8-16

About the authors:

Boris G. Aksenov, Dr. Sci. (Phys.-Math.), Professor, Department of Industrial Thermal Power Engineering, Industrial University of Tyumen; aksenov_bg@me.com

Yuri E. Karyakin, Cand. Sci. (Tech.), Associate Professor, Department of Information Systems, University of Tyumen; y.e.karyakin@utmn.ru

Abstract:

Heat transfer with the phase transition is traditionally described with the Stefan problem which consists of a system of parabolic differential equations with usual boundary conditions and an extra condition at the phase transition front. It is formally possible to use one equation of heat transfer type but then a delta-function appears in one of the coefficients which corresponds to the latent heat of phase transition emission at the temperature of phase transition. The widely used method of “continuous calculation” involves the substitution of delta-function by a delta-shaped function. It transforms the Stefan problem into a boundary problem for non-linear equation of heat transfer. Yet, in this approach the result of calculations is the temperature field, and identifying the position of the phase transition front is difficult. Meanwhile, the point of calculations is often the estimation of the dynamic of phase transition. That is why there have been developed many methods of the Stefan problem solution to find the front coordinates. A general disadvantage of these methods is the fact that they cannot be used when there are several fronts appearing, disappearing, changing directions, and joining. Following the dynamic of each front is difficult in this case.

In this paper we examine the Stefan problem as the problem of moist soil freezing and melting. A method of solving the Stefan problem which identifies the front as a zero isotherm is developed in this article. This method excludes the necessity to follow the evolution of each front. The Stefan problem is considered as the limiting case of the more general problem of phase transition in some temperature range. A number of standard transformations and Green’s function help to present the Stefan problem as an integral equation. Approximate solution is given by a recurrence formula with an example of numerical simulation. The results correspond with the results of the “continuous calculation” method. However, here we have not the temperature field, but the evolution of the front in time. The numeric simulation shows that the method presented here is convenient for multi-front Stefan problems solution. It should be noted that estimators in form of a system of functions, which majorize the required solution alternately above and below (if we need such estimators), may be obtained only for monotonous Stefan problems. For non-monotonous problems, it is not possible. This issue requires additional investigating.

References:

  1. Aksenov B. G., Danielyan Yu. S. 1985. “Priblizhennoe reshenie nelineynykh nemonotonnykh zadach teorii fil'tratsii” [An Approximate Solution of Nonlinear Nonmonotonous Problems in Filtration Theory]. Izvestiya AN SSSR. Mekhanika Zhidkosti i Gaza, no 4.
  2. Aksenov B. G., Danielyan Yu. S., Rubtsov N. A. 1984. “Priblizhenie resheniya zadachi radiatsionno-konduktivnogo teploobmena” [Approximation of the Solution of the Radiation-Conductive Heat Transfer Problem]. Izvestiya SO AN SSSR. Seriya tekhnicheskie nauki, vol. 4, no 1.
  3. Belyaev N. M., Ryadno A. A. 1982. Metody teorii teploprovodnosti [Methods of the Theory of Heat Conductivity], vol. 2. Moscow: Vysshaya shkola.
  4. Danielyan Yu. S., Aksenov B. G. 1985. “Postroenie otsenok resheniy nekotorykh nemonotonnykh zadach nelineynogo teploobmena” [Construction of Estimates for the Solutions of Certain Nonmonotonous Problems of Nonlinear Heat Transfer]. Izvestiya AN SSSR. TVT, vol. 23, no 5.
  5. Dostovalov B. N., Kudryavtsev V. A. 1967. Obshchee merzlotovedenie [General Permafrost Studies]. Moscow: Izd-vo MGU, 
  6. Lyubov B. A. 1975. Teoriya kristallizatsii v bol'shikh ob"emakh [Theory of Crystallization in Large Volumes]. Moscow: Nauka.
  7. Nikitenko N. N. 1971. Issledovanie nestatsionarnykh protsessov teplo- i massoobmena metodom setok [Investigation of Non-Stationary Processes of Heat and Mass Transfer by the Grid Method]. Kiev: Nauk, dumka.
  8. Tikhonov A. N. 1937. “Ob ostyvanii tel pri lucheispuskanii, sleduyushchem zakonu Stefana-Bol'tsmana” [Cooling of Bodies with Radiation, the Following Stefan-Boltzmann’s Law]. Izvestiya AN SSSR. Seriya geografii i geofiziki, no 3.
  9. Tikhonov A. N., Samarskiy A. A. 1966. Uravneniya matematicheskoy fiziki [Equations of Mathematical Physics]. Moscow: Nauka.
  10. Feldman G. M. 1977. Prognoz temperaturnogo rezhima gruntov i razvitiya kriogennykh protsessov [Forecast of the Temperature Regime of Soils and the Development of Cryogenic Processes]. Novosibirsk: Nauka.