Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T17:43:19.199Z Has data issue: false hasContentIssue false
  • English
  • Français

Extended Stefan problem for the solidification of binary alloys in a sphere

Part of: Thermodynamics and heat transfer Miscellaneous topics - Partial differential equations Asymptotic theory

Published online by Cambridge University Press:  04 May 2020

FERRAN BROSA PLANELLA ,
COLIN P. PLEASE  and
ROBERT A. VAN GORDER Open the ORCID record for ROBERT A. VAN GORDER [Opens in a new window]
FERRAN BROSA PLANELLA
Affiliation:
1WMG, University of Warwick, Gibbet Hill Road, CoventryCV4 7AL, UK
COLIN P. PLEASE
Affiliation:
2Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
ROBERT A. VAN GORDER
Affiliation:
3Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin9054, New Zealand email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the extended Stefan problem which includes constitutional supercooling for the solidification of a binary alloy in a finite spherical domain. We perform an asymptotic analysis in the limits of large Lewis number and small Stefan number which allows us to identify a number of spatio-temporal regimes signifying distinct behaviours in the solidification process, resulting in an intricate boundary layer structure. Our results generalise those present in the literature by considering all time regimes for the Stefan problem while also accounting for impurities and constitutional supercooling. These results also generalise recent work on the extended Stefan problem for finite planar domains to spherical domains, and we shall highlight key differences in the asymptotic solutions and the underlying boundary layer structure which result from this change in geometry. We compare our asymptotic solutions with both numerical simulations and real experimental data arising from the casting of molten metallurgical grade silicon through the water granulation process, with our analysis highlighting the role played by supercooling in the solidification of binary alloys appearing in such applications.

Keywords

Two-phase Stefan problemsolidification of binary alloysmatched asymptotic analysis

MSC classification

Primary: 80A22: Stefan problems, phase changes, etc. 35R37: Moving boundary problems
Secondary: 34E10: Perturbations, asymptotics 80A20: Heat and mass transfer, heat flow
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 242 - 279
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allen, D. N. D. G. & Severn, R. T. (1952) The application of relaxation methods to the solution of non-elliptic partial differential equations: II. The solidification of liquids. Q. J. Mech. Appl. Math. 5(4), 447–454.CrossRefGoogle Scholar
Benham, G. P., Hildal, K., Please, C. P. & Van Gorder, R. A. (2016) Solidification of silicon in a one-dimensional slab and a two-dimensional wedge. Int. J. Heat Mass Transfer 98, 530–540.CrossRefGoogle Scholar
Brosa Planella, F., Please, C. P. & Van Gorder, R. A. (2018) Instability in the self-similar motion of a planar solidification front. IMA J. Appl. Math. 83(1), 106–130.CrossRefGoogle Scholar
Brosa Planella, F., Please, C. P. & Van Gorder, R. A. (2019) Extended Stefan problem for solidification of binary alloys in a finite planar domain. SIAM J. Appl. Math. 79(3), 876–913.CrossRefGoogle Scholar
Chadam, J., Howison, S. & Ortoleva, P. (1987) Existence and stability for spherical crystals growing in a supersaturated solution. IMA J. Appl. Math. 39(1), 1–15.CrossRefGoogle Scholar
Chen, S., Merriman, B., Osher, S. & Smereka, P. (1997) A simple level set method for solving Stefan problems. J. Comput. Phys. 135(1), 8–29.CrossRefGoogle Scholar
Crank, J. & Gupta, R. S. (1975) Isotherm migration method in two dimensions. Int. J. Heat Mass Transfer 18(9), 1101–1107.CrossRefGoogle Scholar
Crowley, A. B. (1978) Numerical solution of Stefan problems. Int. J. Heat Mass Transfer 21(2), 215–219.CrossRefGoogle Scholar
Davis, G. B. and Hill, J. M. (1982) A moving boundary problem for the sphere. IMA J. Appl. Math. 29(1), 99–111.CrossRefGoogle Scholar
Eftekhari, A. A. & Schüller, K. (2018) FVTool: a finite volume toolbox for Matlab. https://github.com/simulkade/FVTool.Google Scholar
Elliott, C. M. & Ockendon, J. R. (1982) Weak and Variational Methods for Moving Boundary Problems, Vol. 59, Pitman Publishing, Boston.Google Scholar
Fasano, A., Primicerio, M., Howison, S. D. & Ockendon, J. R. (1989) On the singularities of one-dimensional stefan problems with supercooling. In: Rodrigues, J. F. (editor) Mathematical Models for Phase Change Problems. International Series of Numerical Mathematics, Vol 88, Birkhäuser Basel, Springer, pp. 215–226.CrossRefGoogle Scholar
Fasano, A., Primicerio, M., Howison, S. D. & Ockendon, J. R. (1990) Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension. Q. Appl. Math. 48(1), 153–168.CrossRefGoogle Scholar
Fasano, A., Primicerio, M. & Lacey, A. A. (1981) New results on some classical parabolic free-boundary problems. Q. Appl. Math. 38(4), 439–460.CrossRefGoogle Scholar
Feltham, D. L. & Garside, J. (2001) Analytical and numerical solutions describing the inward solidification of a binary melt. Chem. Eng. Sci. 56(7), 2357–2370.CrossRefGoogle Scholar
Gupta, S. C. (1990) Numerical and analytical solutions of one-dimensional freezing of dilute binary alloys with coupled heat and mass transfer. Int. J. Heat Mass Transfer, 33(4), 593–602.CrossRefGoogle Scholar
Herrero, M. A. & Velázquez, J. J. L. (1996) Singularity formation in the one-dimensional supercooled Stefan problem. Eur.J.Appl. Math. 7(2), 119–150.CrossRefGoogle Scholar
Hill, J. M. & Kucera, A. (1983a) Freezing a saturated liquid inside a sphere. Int. J. Heat Mass Transfer 26(11), 1631–1637.CrossRefGoogle Scholar
Hill, J. M. & Kucera, A. (1983b) The time to complete reaction or solidification of a sphere. Chem. Eng. Sci. 38(8), 1360–1362.CrossRefGoogle Scholar
Howison, S. (1988) Similarity solutions to the Stefan problem and the binary alloy problem. IMA J. Appl. Math. 40(3), 147–161.CrossRefGoogle Scholar
Jaafar, M. A., Rousse, D. R., Gibout, S. & Bédécarrats, J.-P. (2017) A review of dendritic growth during solidification: mathematical modeling and numerical simulations. Renewable Sustainable Energy Rev. 74, 1064–1079.CrossRefGoogle Scholar
Jiji, L. M. & Weinbaum, S. (1978) Perturbation solutions for melting or freezing in annular regions initially not at the fusion temperature. Int. J. Heat Mass Transfer 21(5), 581–592.CrossRefGoogle Scholar
King, J. R., Riley, D. S. & Wallman, A. M. (1999) Two–dimensional solidification in a corner. Proc. R. Soc. London A Math. Phys. Eng. Sci. 455(1989), 3449–3470.CrossRefGoogle Scholar
Kucera, A. & Hill, J. M. (1986) On inward solidifying cylinders and spheres initially not at their fusion temperature. Int. J. Non-linear Mech. 21(1), 73–82.CrossRefGoogle Scholar
Lamé, G. & Clapeyron, B. P. E. (1831) Mémoire sur la solidification par refroidissement d’un globe liquide. Annales Chimie Physique 47, 250–256.Google Scholar
Lazaridis, A. (1970) A numerical solution of the multidimensional solidification (or melting) problem. Int. J. Heat Mass Transfer 13(9), 1459–1477.CrossRefGoogle Scholar
Liu, F. & McElwain, D. L. S. (1997) A computationally efficient solution technique for moving-boundary problems in finite media. IMA J. Appl. Math. 59(1), 71–84.CrossRefGoogle Scholar
McCue, S. W., King, J. R. & Riley, D. S. (2003) Extinction behaviour for two–dimensional inward-solidification problems. Proc. R. Soc. London A Math. Phys. Eng. Sci. 459(2032), 977–999.Google Scholar
McCue, S. W., King, J. R. & Riley, D. S. (2005) The extinction problem for three-dimensional inward solidification. J. Eng. Math. 52(4), 389–409.CrossRefGoogle Scholar
McCue, S. W., Wu, B. & Hill, J. M. (2008) Classical two-phase Stefan problem for spheres. Proc. R. Soc. London A Math. Phys. Eng. Sci. 464(2096), 2055–2076.Google Scholar
Nelson, L., Brooks, P., Bonazza, R., Corradini, M., Hildal, K. & Bergstrom, T. (2005) Steam explosions of single drops of molten silicon-rich alloys. In: Proceedings of the Ninth International Ferroalloys Congress (INFACON 9), pp. 338–351.Google Scholar
Nygaard, L., Brekken, H., Lie, H., Magnussen, T. E. & Sveine, A. (1995) Water granulation of ferrosilicon and silicon metal. In: INFACON 7, Trondheim, Norway, pp. 665–671.Google Scholar
Pedroso, R. I. & Domoto, G. A. (1973) Perturbation solutions for spherical solidification of saturated liquids. J. Heat Transfer 95(1), 42–46.CrossRefGoogle Scholar
Riley, D. S., Smith, F. T. & Poots, G. (1974) The inward solidification of spheres and circular cylinders. Int. J. Heat Mass Transfer 17(12), 1507–1516.CrossRefGoogle Scholar
Rubinstein, L. I. (1971) The Stefan Problem, Translations of Mathematical Monographs, Vol. 27, American Mathematical Society, Providence, Rhode Island.Google Scholar
Selim, M. S. & Seagrave, R. C. (1973) Solution of moving-boundary transport problems in finite media by integral transforms. II. Problems with a cylindrical or spherical moving boundary. Ind. Eng. Chem. Fundam. 12(1), 9–13.Google Scholar
Soward, A. M. (1980) A unified approach to Stefan’s problem for spheres and cylinders. Proc. R. Soc. London A Math. Phys. Eng. Sci. 373(1752), 131–147.Google Scholar
Stefan, J. (1890) Über die Theorie der Eisbildung. Monatshefte für Mathematik 1(1), 1–6.CrossRefGoogle Scholar
Stewartson, K. & Waechter, R. T. (1976) On Stefan’s problem for spheres. Proc. R. Soc. London Ser. A Math. Phys. Sci. 348(1655), 415–426.Google Scholar
Tao, L. C. (1967) Generalized numerical solutions of freezing a saturated liquid in cylinders and spheres. AIChE J. 13(1), 165–169.CrossRefGoogle Scholar
Theillard, M., Gibou, F. & Pollock, T. (2015) A sharp computational method for the simulation of the solidification of binary alloys. J. Sci. Comput. 63(2), 330–354.CrossRefGoogle Scholar
Van Dyke, M. (1975) Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, California.Google Scholar
Voller, V. & Cross, M. (1981) Accurate solutions of moving boundary problems using the enthalpy method. Int. J. Heat Mass Transfer 24(3), 545–556.CrossRefGoogle Scholar
Wallman, A. M., King, J. R. & Riley, D. S. (1997) Asymptotic and numerical solutions for the two-dimensional solidification of a liquid half-space. Proc. R. Soc. London A Math. Phys. Eng. Sci. 453(1962), 1397–1410.Google Scholar
Wang, G.-X., Prasad, V. & Matthys, E. F. (1997) An interface-tracking numerical method for rapid planar solidification of binary alloys with application to microsegregation. Mater. Sci. Eng. A, 225(1–2), 47–58.CrossRefGoogle Scholar
Wheeler, A. A., Boettinger, W. J. & McFadden, G. B. (1993a) Phase-field model of solute trapping during solidification. Phys.Rev. E, 47(3), 1893.CrossRefGoogle ScholarPubMed
Wheeler, A. A., Murray, B. T. & Schaefer, R. J. (1993b) Computation of dendrites using a phase field model. Phys. D Nonlinear Phenom. 66(1–2), 243–262.CrossRefGoogle Scholar
Yang, J., Zhao, C.-Y. & Hutchins, D. (2012) Modelling the effect of binary phase composition on inward solidification of a particle. Int. J. Heat Mass Transfer 55(23), 6766–6774.CrossRefGoogle Scholar