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On small-time similarity-solution behaviour in the solidification shrinkage of binary alloys

Part of: Basic methods in fluid mechanics Phase transformations in solids

Published online by Cambridge University Press:  17 April 2020

M. ASSUNÇÃO Open the ORCID record for M. ASSUNÇÃO [Opens in a new window] ,
M. VYNNYCKY Open the ORCID record for M. VYNNYCKY [Opens in a new window]  and
S.L. MITCHELL
M. ASSUNÇÃO
Affiliation:
Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of São Paulo at São Carlos, PO Box 668, São Carlos, 13560-970SP, Brazil, email: assuncao@icmc.usp.br
M. VYNNYCKY
Affiliation:
Department of Mathematics and Statistics, University of Limerick, LimerickV94 T9PX, Ireland
S.L. MITCHELL
Affiliation:
Department of Mathematics and Statistics, University of Limerick, LimerickV94 T9PX, Ireland
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Abstract

In the one-dimensional solidification of a binary alloy undergoing shrinkage, there is a relative motion between solid and liquid phases in the mushy zone, leading to the possibility of macrosegregation; thus, the problem constitutes an invaluable benchmark for the testing of numerical codes that model these phenomena. Here, we revisit an earlier obtained solution for this problem, that was posed on a semi-infinite spatial domain and valid for the case of low superheat, with a view to extending it to the more general situation of a finite spatial domain, arbitrarily large superheat and both eutectic and non-eutectic solidification. We find that a similarity solution is available for short times which contains a boundary layer on the liquid side of the mush–liquid interface; this solution is believed to constitute the correct initial condition for the subsequent numerical solution of the full non-similar problem, which is deferred to future work.

Keywords

Moving boundary problemsdynamics of phase boundariesdimensional analysis and similarity

MSC classification

Primary: 76M55: Dimensional analysis and similarity
Secondary: 74N20: Dynamics of phase boundaries
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 199 - 225
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

The first author acknowledges the financial support of Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for the researcher grant (grant number 2016/12678-0). The second author acknowledges the financial support of FAPESP for the award of a visiting researcher grant (grant number 2018/07643-8).

References

Alexiades, V. & Solomon, A. D. (1993) Mathematical Modelling of Melting and Freezing Processes, Hemisphere Publishing Corporation, Washington, DC.Google Scholar
Amberg, G. (1991) Computation of macrosegregation in an iron-carbon cast. Int. J. Heat Mass Transfer 34, 217–227.CrossRefGoogle Scholar
Assunção, M. & Vynnycky, M. (2018) Mathematical modelling of macrosegregation in ingot casting. Engenharia Térmica (Therm. Eng.) 17(2), 74–79.Google Scholar
Bennon, W. D. & Incropera, F. P. (1987) A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems – I. Model formulation. Int. J. Heat Mass Transfer 30(10), 2161–2170.CrossRefGoogle Scholar
Brody, H. B. & Flemings, M. C. (1966) Solute redistribution in dendritic solidification. Trans. AIME 236(12), 615–624.Google 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, 876–913.CrossRefGoogle Scholar
Carslaw, H. S. & Jaeger, J. C. (1959) Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford.Google Scholar
Chakraborty, S. & Dutta, P. (2002) An analytical solution for conduction-dominated unidirectional solidification of binary mixtures. Appl. Math. Mod. 26, 545–561.CrossRefGoogle Scholar
Chen, J. H. & Tsai, H. L. (1993) Inverse segregation for a unidirectional solidification of aluminium-copper alloys. Int. J. Heat Mass Transfer 36, 3069–3075.CrossRefGoogle Scholar
Clyne, T. W. & Kurz, W. (1981) Solute redistribution during solidification with rapid solid-state diffusion. Met. Trans. A 12(6), 965–971.CrossRefGoogle Scholar
Crank, J. (1984) Free and Moving Boundary Problems, Clarendon Press, Oxford.Google Scholar
Davis, G. B. & Hill, J. M. (1982) A moving boundary problem for the sphere. IMA J. Appl. Math. 29, 99–111.CrossRefGoogle Scholar
Diao, Q. Z. & Tsai, H. L. (1993) Modelling of solute redistribution in the mushy zone during solidification of aluminium-copper alloys. Metall. Trans. 24A, 963–973.CrossRefGoogle Scholar
Diao, Q. Z. & Tsai, H. L. (1993) The formation of negative- and positive-segregated bands during solidification of aluminum-copper alloys. Int. J. Heat Mass Transfer 36, 4299–4305.CrossRefGoogle Scholar
Du, Q., Eskin, D. G. & Katgerman, L. (2009) Numerical issues in modelling macrosegregation during DC casting of a multi-component aluminium alloy. Int. J. Num. Meth. Heat Fluid Flow 19, 910–930.CrossRefGoogle Scholar
Feltham, D. L. & Garside, J. (2001) Analytical and numerical solutions describing the inward solidification of a binary melt. Chem. Eng. Sci. 56, 2357–2370.CrossRefGoogle Scholar
Flemings, M. C. (1974) Solidification Processing, McGraw-Hill, New York, NY.CrossRefGoogle Scholar
Jakhar, A., Rath, P. & Mahapatra, S. K. (2016) A similarity solution for phase change of binary alloy with shrinkage or expansion. Eng. Sci. Technol - Int. J. - JESTECH 19, 1390–1399.Google Scholar
Jalanti, T., Swierkosz, M., Gremaud, M. & Rappaz, M. (2006) Modelling of macrosegregation in continuous casting of aluminium. In: Ehrke, K. and Schneider, W. (editors), Continuous Casting, WILEY-VCH Verlag GmbH, Weinheim, pp. 191–198.CrossRefGoogle Scholar
Kucera, A. & Hill, J. M. (1986) On inward solidifying cylinders and spheres initially not at their fusion temperature. Int. J. Nonlinear Mech. 21, 73–82.CrossRefGoogle Scholar
McCue, S. W., Bisheng, W. & Hill, J. M. (2008) Classical two-phase Stefan problem for spheres. Proc. R. Soc. A 464, 2055–2076.Google Scholar
Minakawa, S., Samarasekera, I. V. & Weinberg, F. (1985) Inverse segregation. Metall. Trans. 16B, 595–604.CrossRefGoogle Scholar
Mitchell, S. L. & Vynnycky, M. (2009) Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems. Appl. Math. Comp. 215, 1609–1621.CrossRefGoogle Scholar
Mitchell, S. L. & Vynnycky, M. (2014) On the numerical solution of two-phase Stefan problems with heat-flux boundary conditions. J. Comp. Appl. Math. 264, 49–64.CrossRefGoogle Scholar
Mitchell, S. L. & Vynnycky, M. (2016) On the accurate numerical solution of a two-phase Stefan problem with phase formation and depletion. J. Comp. Appl. Math. 300, 259–274.CrossRefGoogle Scholar
Mo, A. (1993) Mathematical modelling of surface segregation in aluminum DC casting caused by exudation. Int. J. Heat Mass Transfer 36, 4335–4340.CrossRefGoogle Scholar
Mullins, W. W. & Sekerka, R. F. (1964) Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35, 444–451.CrossRefGoogle Scholar
Swaminathan, C. R. & Voller, V. R. (1997) Towards a general numerical scheme for solidification systems. Int. J. Heat Mass Transfer 40, 2859–2868.CrossRefGoogle Scholar
Voller, V. R., Brent, A. D. & Prakash, C. (1989). The modelling of heat, mass and solute transport in solidification systems. Int. J. Heat Mass Transfer 32(9), 1719–1731.CrossRefGoogle Scholar
Voller, V. R. & Sundarraj, S. (1995) A model of inverse segregation: the role of microporosity. Int. J. Heat Mass Transfer 38, 1009–1018.CrossRefGoogle Scholar
Voller, V. R. (1997) A similarity solution for the solidification of a multicomponent alloy. Int. J. Heat Mass Transfer 40(12), 2869–2877.CrossRefGoogle Scholar
Voller, V. R. (1998) A numerical scheme for solidification of an alloy. Can. Met. Quart. 37, 169–177.CrossRefGoogle Scholar
Voller, V. R., Mouchmov, A. & Cross, M. (2004) An explicit scheme for coupling temperature and concentration fields in solidification models. Appl. Math. Mod. 28, 79–94.CrossRefGoogle Scholar
Vušanović, I. & Voller, V. R. (2014) Understanding channel segregates in numerical models of alloy solidification: a case of converge first ask questions later. Mater. Sci. Forum 790–791, 74–78.Google Scholar
Vušanović, I. & Voller, V. R. (2016) Best practice for measuring grid convergence in numerical models of alloy solidification. Int. J. Numer. Methods Heat Fluid Flow 26, 427–439.CrossRefGoogle Scholar
Vynnycky, M. & Kimura, S. (2007) An analytical and numerical study of coupled transient natural convection and solidification in a rectangular enclosure. Int. J. Heat Mass Transfer 50, 5204–5214.CrossRefGoogle Scholar
Vynnycky, M. & Kimura, S. (2015) Can natural convection alone explain the Mpemba effect? Int. J. Heat Mass Transfer 80, 243–255.CrossRefGoogle Scholar
Vynnycky, M. & Saleem, S. (2017) On the explicit resolution of the mushy zone in the modelling of the continuous casting of alloys. Appl. Math. Mod. 50, 544–568.CrossRefGoogle Scholar
Vynnycky, M., Saleem, S. & Fredriksson, H. (2018) An asymptotic approach to solidification shrinkage-induced macrosegregation in the continuous casting of binary alloys. Appl. Math. Mod. 54, 605–626.CrossRefGoogle Scholar
Waclawczyk, T. & Schaefer, M. (2018) Verification of a binary fluid solidification model using semi-analytical solution of 1D heat diffusion equation. Chem. Process Eng.-Inzynieria Chemiczna i Procesowa 39, 85–102.Google Scholar
Xiong, M. & Kuznetsov, A. (2001) Comparison between lever and Scheil rules for modeling of microporosity formation during solidification. Flow Turbul. Combust. 67, 305–323.CrossRefGoogle Scholar