Abstract
This paper discusses the modelling of a number of metallurgical systems involving flow and solidification. The discussion is neither definitive nor critical, but is an attempt to provide information on important processes not covered elsewhere in this volume. The subjects covered include rapid solidification, semi-solid casting, natural convection and alternative formulations for solidification modelling. In addition, some new results for hyperbolic heat transfer and heat losses during mould filling are presented. Throughout the discussion, the practical application of the various models is stressed and wherever possible use is made of industrial examples. Finally, a substantial, although by no means exhaustive bibliography is provided to indicate where further details of the work may be found.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M.R. Albert and K. O’Neill: Transient two-dimensional phase change with convection using deforming finite elements. In: R.W. Lewis, K. Morgan, J.A. Johnson and W.R. Smith (eds) Computational Techniques in Heat Transfer, Volume 1. Pineridge Press, Swansea (1985) pp 229–243.
A.A. Amsden and F.H. Harlow: The SMAC method, a numerical technique for calculating incompressible fluid flows. Los Alamos Scientific Laboratory Technical Report No: LA-4370 (1970).
T.R. Auton, J.C.R. Hunt, K.J. Sene and N.H. Thomas: Distribution of a disperse phase in vertical or near-vertical turbulent shear flows. In: Extended Abstracts of the International Symposium on Two-Phase Annular and Dispersed Flows, ETF, Pisa (1984) 209–214.
V. Baskaran, J. Ghias and W.R. Wilcox: Modelling the influence of convection on eutectic microstructures. In: Modelling of Casting and Welding Processes II. The Metallurgical Society of AIME, Warrendale, Pennsylvania (1984) 115–118.
G.K. Batchelor: Mass transfer from small particles suspended into turbulent fluid. J Fluid Mech 98 (1980) 609–623.
G.E. Bell: On the performance of the enthalpy method. Int J Heat Mass Transfer 25 (1982) 587–589.
G.E. Bell: Inverse formulations as a method for solving certain melting and freezing problems. In: R.W. Lewis and K. Morgan (eds) Numerical Methods in Heat Transfer Volume III. John Wiley and Sons, Chichester (1985) pp 59–78.
G.E. Bell and A.S. Wood: On the performance of the enthalpy method in the region of a singularity. Int J Numer Meth Engng 19 (1983) 1583–1592.
W.J. Boettinger and S.R. Correll: Microstructure formation in rapidly solidified alloys. In: P.R. Sahm, H. Jones and CM. Adam (eds) Science and Technology of the Under-cooled Melt. Martinus Nijhoff Publishers, Dordrecht (1986) pp 87–108.
J.C. Brice: The Growth of Crystals from Liquid. North-Holland, Amsterdam (1973).
R.W. Brown, C.J. Chang and P.M. Adornato: Finite element analysis of directional solidification of dilute and concentrated binary alloys. In: Modelling of Casting and Welding Processes II. The Metallurgical Society of AIME, Warrendale, Pennsylvania (1984) pp 95–114.
F.M. Chiesa and R.I.L. Gutherie: An experimental study of natural convection and wall effect in liquid metals contained in vertical cylinders. Metall Trans 2 (1971) 2833–2838.
F.M. Chiesa and R.I.L. Gutherie: Natural convective heat transfer rates during the solidification and melting of metals and alloy systems. Trans ASME J Heat Transf 96 (1974) 377–384.
T.W. Clyne: The use of heat flow modelling to explore solidification phenomena. Metall Trans B 13B (1982) 471–478.
T.W. Clyne: Numerical treatment of rapid solidification. Metall Trans B 15B (1984) 369–381.
T.W. Clyne: Heat flow, solidification and energy aspects of DC and strip casting of aluminium alloys. Metals Tech 11 (1984) 350–357.
T.W. Clyne, R.A. Ricks and P.J. Goodhew: The production of rapidly-solidified aluminium powder by ultrasonic gas atomisation. Part I: heat and fluid flow. Int J Rapid Solid 1 (1984–85) 59–80.
T.W. Clyne, R.A. Ricks and P.J. Goodhew. The production of rapidly-solidified aluminium powder by ultrasonic gas atomisation. Part II: solidification structure Int J Rapid Solid 1 (1984–85) 85–101.
G.S. Cole: Temperature measurements and fluid flow distributions ahead of solid-liquid interfaces. Trans Metall Soc AIME 239 (1967) 1287–1295.
G.S. Cole and G.F. Boiling: The importance of natural convection in casting. Trans Metall Soc AIME 233 (1965) 1568–1572.
A.B. Crowley and J.R. Ockendon: Modelling mushy regions. App Sci Res 44 (1987) 1–7.
L.A. Crivelli and S.R. Idelsohn: A temperature based finite element solution for phase-change problems. Int J Num Meth Engng 23 (1986) 99–119.
P.V. Desai, J.T. Berry and C. Kim: Computer simulation of forced and natural convection during filling of a casting. Trans Am Foundrymen’s Soc 92 (1984) 519–528.
P.V. Desai and C. Kim: On convection in liquid metal moulds. In: R.W. Lewis, K. Morgan and B.A. Schrefler (eds) Numerical Methods in Thermal Problems, Volume II, Pineridge Press, Swansea (1981) pp 119–129.
P.V. Desai and C. Kim: Heat losses in runner channels. In: Modelling Casting and Welding Processes II. The Metallurgical Society of AI ME, Warrendale, Pennsylvania (1984) pp 59–66.
P.V. Desai and F. Rastegar: Convection in mould cavities. In: Modelling of Casting and Welding Processes. The Metallurgical Society of AIME, Warrendale, Pennsylvania (1981) pp 351–360.
L. DeSocio and G. Gualtieri: A hyperbolic Stefan problem. Quart Appl Math 41 (1983) 253–259.
C. Dietsche and U. Müller: Influence of Benard convection on solid-liquid interfaces. J Fluid Mech 161 (1985) 249–268.
E.R. Evans: Fluidity of molten cast iron. Foundry Trade J 128 (1955) 757–763.
S.C. Flood and J.D. Hunt: A model of a casting. Appl Sci Res 44 (1987) 27–42.
R.M. Furzeland: A comparative study of numerical methods for moving boundary problems. J Int Maths Applics 26 (1980) 411–429.
D.K. Gartling: Finite element analysis of convective heat transfer problems with change of phase. In: K. Morgan, C. Taylor and C.A. Brebbia (eds) Computer Methods in Fluids. Pentech Press, Plymouth (1980) pp 257–269.
C. Gau and R. Viskanta: Melting and solidification of a metal system in a rectangular cavity. Int J Heat Mass Transf 27 (1984) 113–123.
G. Gau and R. Viskanta: Effect of natural convection on solidification from above and melting from below of a pure metal. Int J Heat Mass Transf 28 (1985) 573–587.
N.J. Grant: Engineering properties and applications of rapidly solidified materials. In: P.R. Sahm, J. Jones and CM. Adam (eds). Science and Technology of the Undercooled Melt. Martinus Nijhoff Publishers, Dordrecht (1986) pp 210–228.
R. Guenigault and G. Poots: Effects of natural convection on the inward solidification of cylinders. Int J Heat Mass Transf 28 (1985) 1229–1231.
N.W. Hale and R. Viskanta: Solid-liquid phase-change heat transfer and interface motion in materials cooled from above or below. Int J Heat Mass Transf 23 (1980) 283–357.
C Harrison and E. Weinberg: The influence of convection on heat transfer in liquid tin. Metall Trans B 16B (1985) 355–357.
W.L. Heitz and J.W. Westwater: Extension of the numerical method for melting and freezing problems. Int J Heat Mass Transf 13 (1970) 1371–1375.
R.N. Hills, D.E. Loper and P.H. Roberts: A thermodynamically consistent model of a mushy zone. Q J Mech Appl Math 36 (1983) 505–539.
K. Ho and R.D. Pehlke: Mechanisms of heat transfer at a metal-mould interface. Trans Am Foundrymen’s Soc 92 (1984) 587–598.
CP. Hong, T. Umeda and Y. Kimura: Solidification simulation of shaped castings by the boundary element method and prediction of shrinkage cavity. Imono 56 (1984) 758–764.
S.C. Huang: Analytical solution for the buoyancy flow during the melting of a vertical semi-infinite region. Int J Heat Mass Transf 28 (1985) 1231–1233.
J.CR. Hunt: Turbulence structure in thermal convection and shear-free boundary layers. J Fluid Mech 138 (1984) 161–184.
J.CR. Hunt: Turbulent diffusion from sources in complex flows. Ann Rev Fluid Mech 17 (1985) 447–485.
J.D. Hunt and D.G. McCartney: Numerical finite difference model for steady state array growth. Appl Sci Res 44 (1987) 9–26.
H.E. Huppert: From multi-branched snowflakes to precious minerals. Nature 323 (1986) 202–203.
H.E. Huppert: The intrusion of fluid mechanics into geology. J Fluid Mech 173 (1986) 557–594.
H.E. Huppert and J.S. Turner: A laboratory model of a replenished magma chamber. Earth Planet Sci Lett 54 (1981) 144–142.
H.E. Huppert and M.G. Worster: Dynamic solidification of a binary melt. Nature 314 (1985) 703–707.
I. Imafuku and K. Chijiiwa: A mathematical model for shrinkage cavity prediction in steel castings. Trans Am Foundrymen’s Soc 91 (1985) 527–520.
J. Isaac, G.P. Reddy and G.K. Sharma: Variations of heat transfer coefficients during solidification of castings in metal moulds. Brit Foundrymen 78 (1985) 465–468.
S.T. Johansen, F. Boysen and W.H. Ayers: Mathematical modelling of bubble driven flows in metallurgical processes. Appl Sci Res 44 (1987) 197–207.
P.B. Johns: A simple explicit and unconditionally stable numerical routine for the solution of the diffusion equation. Int J Num Meth Engng 11 (1979) 1307–1328.
P.B. Johns and G. Butler: The consistency and accuracy of the TLM method for diffusion and its relationship to existing methods. Int J Num Meth Engng 19 (1983) 1549–1554.
E.W. Jones, W.M. Sleigelmann and G.P. Wachtell: Heat transfer from molten metals to sand mould runners. Trans Am Foundrymen’s Soc 71 (1963) 817–825.
H. Jones: Rapid Solidification of Metals and Alloys. Institution of Metallurgists, London (1982).
E.G. Josberger and S. Martin: A laboratory and theoretical study of the boundary layer adjacent to a vertical melting ice wall in salt water. J Fluid Mech 11 (1981) 439–473.
L. Katgerman: Computer simulation of cellular growth during rapid solidification of binary aluminium alloys. In: Modelling of Casting and Welding Processes II. The Metallurgical Society of AIME, Warrendale, Pennsylvania (1984) pp 135–143.
L. Kategerman: Effect of process conditions during melt spinning on solidification morphology of aluminium alloys. In: S. Steeb and H. Warliment (eds). Rapidly Quenched Metals. Elsevier Science Publishers BV, The Hague (1985) pp 819–822.
B.H. Kear, B.C. Giessen and M. Cohen (eds): Rapidly-solidified Amorphous and Crystalline Alloys. North-Holland, New York (1982).
M. Kölling and U. Grigull: A mathematical model to calculate the fluidity of pure metals. In: D.B. Spalding and N.H. Afgan (eds). Heat and Mass Transfer in Metallurgical Systems. McGraw-Hill, Washington (1981) pp 329–340.
W. Kunda and G. Poots: A mathematical model of ladle stirring by bath agitation in steel making. Appli Sci Res 44 (1987) 209–224.
R.W. Lewis, K. Morgan and P.M. Roberts: Determination of thermal stresses in solidification problems. In: J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (eds). Numerical Analysis of Forming Processes. John Wiley and Sons Limited, Chichester (1984) pp 405–431.
R.W. Lewis and P.M. Roberts: Finite element simulation of solidification problems. Appl Sci Res 44 (1987) 61–92.
M. Matsuda and M. Ohmi: Heat transfer analysis and fluidity of flowing metal in a cylindrical mould cavity. AFS Int Cast Metals J 6 (1981) 18–27.
T. Matsumiya and M.C. Flemings: Modelling of continuous strip production by rheo-casting. Metall Trans B 12B (1981) 17–31.
G. Menges, V. Masberg, B. Gesenhues and C. Berry: Numerical simulation of three-dimensional non-Newtonian flow in thermoplastics extrusion dies with finite element methods. In: J.FT. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (eds). Numerical Analysis of Forming Processes. John Wiley and Sons, Chichester (1984) pp 307–350.
J.L. Meyer and F. Durand: Analysis of the transient effects of convection during solidification with or without electromagnetic stirring. In: Modelling of Casting and Welding Processes II. The Metallurgical Society of AIME, Warrendale, Pennsylvania (1984) 179–197.
K. Morgan: A numerical analysis of freezing and melting with convection. Comp Meth Appl Mech Engng 28 (1981) 275–284.
Y. Nagasaka, J. Ohnaka and T. Fukusako: Effect of heat transfer during pouring on solidification of steel plate castings. Imono 56 (1984) 22–27.
J.F.T. Pittman and A. Nakazawa: Finite element analysis of polymer processing operations. In: J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (eds). Numerical Analysis of Forming Processes. John Wiley and Sons, Chichester (1984) pp 165–218.
J. Pullins and M.K. Walther: Simulation for designing metal-casting moulds. Comp Aided Engr (1984) 62–68.
N. Ramachandran, J.P. Gupta and Y. Jahuria: Thermal and fluid flow effects during solidification in a rectangular enclosure. Int J Heat Mass Transf 25 (1982) 187–194.
T. Robertson, P. Moore and R.J. Hawkins: Computational flow model as aid to solution of fluid flow problems in the steel industry. Ironmaking and Steelmaking 13 (1986) 195–203.
S. Rogers, L. Katgerman, P.G. Enright and N.A. Darby: Modelling of liquid-liquid metal mixing. Appl Sci Res 44 (1987) 175–196.
F. Rosenberger: Fundamentals of Crystal Growth Vol 1. Springer, Berlin (1979).
R.W. Ruddle: The Solidification of Castings. Institute of Metals Monograph and Reports Series No 7 (2nd ed) (1957).
M. Sadd and J. Didlake: Non-fourier melting of a semi-infinite solid. Trans AIME J Heat Trans 99 (1979) 25–28.
P.R. Sahm, H. Jones and CM. Adam: Science and Technology of the Under cooled Melt. Martinus Nijhoff Publishers, Dordrecht (1986).
B.L. Sawford and J.C.R. Hunt: Effects of turbulence structure, molecular diffusion and source size on scalar fluctuations in homogeneous turbulence. J Fluid Mech 165 (1986) 373–400.
W. Schneider: A local analysis of solidification in horizontal continuous casting. Arch Eisenhüttenwes 54 (1983) 487–490.
A. Schröder: Fluid flow considerations in the filling of lost moulds with open feeders. Giessereiforschung 37 (1985) 65–79.
H.J. Schulze, P.M. Beckett, J.A. Howarth and G. Poots: Analytical and numerical solutions to two-dimensional moving interface problems with applications to the solidification of killed steel ingots. Proc R Soc Lond A 385 (1983) 313–343.
T.J. Smith: Adaptive mesh schemes for computational fluid dynamics. CFD News 3/84 (1984)11–15.
T.J. Smith: A basis for the control of the casting process. GST Report No: PS 1940.1 GST 33/1.00.
T.J. Smith, A.F.A. Hoadley and D.M. Scott: On the sensitivity of numerical simulations of solidification to the physical properties of the melt and the mould. Appl Sci Res 44 (1987) 93–110.
T.J. Smith, A.F.A. Hoadley and D.M. Scott: The incorporation of natural convection effects in solidification simulation. In: R.W. Lewis and K. Morgan (eds) Numerical Methods in Thermal Problems, Vol V, Pineridge Press, Swansea (1987) in the press.
T.J. Smith and D.B. Welbourn: The integration of geometric modelling with finite element analysis for the computer-aided design of castings. Appl Sci Res 44 (1987) 139–160.
A.D. Solomon, V. Alexiades, D.G. Wilson and J. Drake: On the formulation of hyperbolic Stefan problems. Quart Appl Math XLIII (1985) 295–304.
N.D. Steward: Semi-solid casting (abstract). In: T.J. Smith (ed), Mixing, Stirring and Solidification in Metallurgical Processes. University of Cambridge, Department of Engineering (1985) p 7.
R.A. Stoehr and W.S. Hwang: Modelling the flow of molten metal having a free surface during entry into moulds. In: Modelling of Casting and Welding Processes II. The Metallurgical Society of AIME, , Pennsylvania (1984) pp 47–58.
J. Szekely and A.S. Jassal: An experimental and analytical study of the solidification of a binary dendritic system. Metall Trans B 9B (1978) 389–398.
J.S. Turner: Buoyancy Effects in Fluids, 2nd Edition. Cambridge University Press, Cambridge (1979).
J.S. Turner, H.E. Huppert and R.S.J. Sparks: Komatiites II: experimental and theoretical investigations of post-emplacement cooling and crystallization. J Petrol 27 (1986) 397–437.
V.R. Voller: Implicit finite-difference solutions of the enthalpy formulation of Stefan problems. I M A J Numer Anal 5 (1985) 201–214.
V.R. Voller and M. Cross: Accurate solutions of moving boundary problems using the enthalpy method. Int J Heat Mass Transf 24 (1981) 545–556.
V.R. Voller and M. Cross: An explicit method to track a moving phase change front. Int J Heat Mass Transf 26 (1983) 147–150.
V.R. Voller and M. Cross: Applications of control volume enthalpy methods in the solution of Stefan problems. In: R.W. Lewis, K. Morgan, J.A. Johnson and W.R. Smith. Computational Techniques in Heat Transfer, Vol 1, Pineridge Press, Swansea (1985) pp 245–275.
V.R. Voller, N. Markatos and M. Cross: Techniques for accounting for the moving interface in convection-diffusion phase change. In: R.W. Lewis and K. Morgan (eds). Numerical Methods in Thermal Problems, Vol IV, Pineridge Press, Swansea (1985) pp 595–609.
J.E. Welsh, F.H. Harlow, J.P. Shannon and B.J. Bally: The MAC method, a computing technique for solving viscous, incompressible, transcient fluid flow problems involving free surfaces. Techn Rept LA-3425 Los-Alamos Scientific Laboratory (1966).
R.E. White: An enthalpy formulation of the Stefan problem. SIAM J Numer Anal 19 (1982) 1129–1157.
K. Wollhover, Ch. Körber, M.W. Scheiwe and U. Hartmann: Unidirectional freezing of binary aqueous solutions: an analysis of transient diffusion of heat and mass. Int J Heat Mass Transf 28 (1985) 761–770.
K. Wollhover, M.W. Scheine, U. Hartmann and Ch. Körber: On morphological stability of planar phase boundaries during unidirectional transient solidification of binary aqueous solutions. Int J Heat Mass Transf 28 (1985) 897–902.
M.G. Worster: Solidification of an alloy from a cooled boundary. J Fluid Mech 167 (1986) 481–501.
J. Yoo and B. Rubinsky: A finite element method for the study of solidification processes in the presence of natural convection. Int J Numer Meth Engng 23 (1986) 1785–1805.
O.C. Zienkiewicz: Flow formulation for numerical solution of forming processes. In: J.F.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (eds). Numerical Analysis of Forming Processes. John Wiley and Sons, Chichester (1984) pp 1–44.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Martinus Nijhoff Publisher, Dordrecht
About this chapter
Cite this chapter
Smith, T.J., Hoadley, A.F.A. (1987). Recent developments in modelling metal flow and solidification. In: Smith, T.J. (eds) Modelling the Flow and Solidification of Metals. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3617-1_17
Download citation
DOI: https://doi.org/10.1007/978-94-009-3617-1_17
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8119-1
Online ISBN: 978-94-009-3617-1
eBook Packages: Springer Book Archive