Problem of the Moving Boundary in Continuous Casting Solved by The Analytic-Numerical Method

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Abstract

Mathematical modeling of thermal processes combined with the reversible phase transitions of type: solid phase - liquid phase leads to formulation of the parabolic or elliptic moving boundary problem. Solution of such defined problem requires, most often, to use some sophisticated numerical techniques and far advanced mathematical tools. The paper presents an analytic-numerical method, especially attractive from the engineer’s point of view, applied for finding the approximate solutions of the selected class of problems which can be reduced to the one-phase solidification problem of a plate with the unknown a priori, varying in time boundary of the region in which the solution is sought. Proposed method is based on the known formalism of initial expansion of a sought function, describing the field of temperature, into the power series, some coefficients of which are determined with the aid of boundary conditions, and on the approximation of a function defining the freezing front location with the broken line, parameters of which are determined numerically. The method represents a combination of the analytical and numerical techniques and seems to be an effective and relatively easy in using tool for solving problems of considered kind.

[1] Samojlowič. I.A. (1977). Formirowanie slitka, Mietallurgia. Moskwa (in russian).

[2] Piasecka Belkhayat. A. (2008). Numerical modeling of solidification process using interval boundary element method. Archives of Foundry Engineering. 8(4), 171-176.

[3] Mochnacki, B. & Suchy, J.S. (1995). Numerical methods incomputations of foundry processes, Kraków: PFTA.

[4] Majchrzak, E., Mochnacki, B., Dziewoński, M. & Jasiński, M. (2008). Identification of boundary heat flux on the continuous casting surface. Archives of FoundryEngineering. 8(4), 105-110.

[5] Mendakiewicz, J., Piasecka, Belkhayat, A & Szopa, R. (2000). Modeling of the Stefan Problem Using the BEM. Solidification of Metals and Alloys. 2(44), 223-228.

[6] Mochnacki, B. & Pawlak, E. (2007). Identification of boundary condition on the contact surface of continuous casting mould. Archives of Foundry Engineering. 7(4), 202-206.

[7] Majchrzak, E. & Witek, H. (1994). Analysis of complex casting solidification using combined boundary element. Solidification of Metals and Alloys. 19, 193-201.

[8] Hetmaniok, E., Słota, D., Wituła, R. & Zielonka, A. (2011). Comparison of the Adomian decomposition method and the variational iteration method in solving the moving boundary problem. Computers and Mathematics with Applications. 61, 1931-1934. DOI:10.1016/j.camwa.2010.07.050.

[9] Słota, D. (2011). Solving Inverse Problems of Solidificationwith the Use of Genetic Algorithm. Gliwice, Silesian University of Technology Press (in Polish).

[10] Hetmaniok, E., Słota, D., Zielonka, A. & Wituła, R. (2012). Comparison of ABC and ACO Algorithms Applied for Solving the Inverse Heat Conduction Problem. LectureNotes in Computer Science. 7269, 249-257. DOI: 10.1007/978-3-642-29353-5_29.

[11] Ockendon, J.R. & Hodgkins, W.R. (1975). MovingBoundary Problems in Heat Flow and Diffusion, Oxford: Clarendon Press.

[12] Wilson, D.G., Solomon, A.D. & Boggs, P.T. (1978). MovingBoundary Problems, New York: Academic Press.

[13] Crank, J. (1984). Free and Moving Boundary Problems, Oxford : Clarendon Press.

[14] Hetmaniok, E. & Pleszczyński, M. (2011). Analitycal method of determining the freezing front location. ZeszytyNaukowe Politechniki Śląskiej Matematyka Stosowana. 1, 121-136.

[15] Grzymkowski, R., Hetmaniok, E. & Pleszczyński, M. (2011). Analytic-numerical method of determining the freezing front location. Archives of Foundry Engineering. 11(3), 75-80.

[16] Hetmaniok, E. & Pleszczyński, M. (2012). Application of an analytic-numerical method in solving the problem with a moving boundary. Zeszyty Naukowe Politechniki ŚląskiejMatematyka Stosowana. 1, (in press).

Archives of Foundry Engineering

The Journal of Polish Academy of Sciences

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CiteScore 2016: 0.42

SCImago Journal Rank (SJR) 2016: 0.192
Source Normalized Impact per Paper (SNIP) 2016: 0.316

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