Mathematical modelling of solidification and melting: A review

Modelling and Simulation in Materials Science and Engineering

Volumn 4, Issue 4, 1996, Pages 371-396

  Hu, Henry ^a   Argyropoulos, Stavros A ^b  

^a Institute of Magnesium Technology   (Canada)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

LIQUIDS; MELTING; NUMERICAL METHODS; SOLIDIFICATION; THERMAL CONDUCTIVITY;

GRAVITATIONAL ACCELERATION; STEFAN NUMBER;

MATHEMATICAL MODELS;

EID: 0030181722     PISSN: 09650393     EISSN: None     Source Type: Journal    
DOI: 10.1088/0965-0393/4/4/004     Document Type: Review

References (100)

1. Lame G and Clapeyron B P E 1831 Memoire sur la solidification par refroidissment d'un globe solid Ann. Chem. Phys. 47 250-60
2. Stefan J 1889 S B Wien Akad. Mat. Natur. 98 473-84, 965-83
3. Stefan J 1891 Uber die theorie der eisbildung, inbesondere uber die eisbildung im polarmeer Ann. Chem. Phys. 42 269-86
4. Crank J 1984 Free and Moving Boundary Problems (Oxford: Clarendon)
5. Neumannn F 1912 Die Paruellen Differentialgleichungen der Mathematischen Physik vol 2 (Reimann-Weber) p 121
6. Paterson S 1952-53 Propagation of a boundary of fusion Proc. Glasgow Math. Assoc. 1 42-7
7. Goodman T R 1958 The heat-balance integral and its application to problems involving a change of phase Trans. AMSE 80 335-42
8. Pohlhausen K 1921 Zur naherungsweisen integration der differentialgleichunger der laminaren grenzschicht Zeitschrift für angewandte Mathematik und Mechanik 1 252-8
9. Goodman T R and Shea J J 1960 The melting of finite slabs J. Appl. Mech. 27 16-27
10. Noble B 1975 Heat balance methods in melting problems Moving Boundary Problems in Heat Flow and Diffusion ed J R Ockendon and W R Hodgkins (Oxford: Clarendon)
11. Bell G E 1978 A refinement of heat balance integral methods applied to a melting problem Int. J. Heat Mass Transfer 21 1357-61
12. Crank J 1975 The Mathematics of Diffusion (Oxford: Clarendon)
13. Murray W D and Landis F 1959 Numerical and machine solutions of transient heat-conduction problems involving melting or freezing J. Heat Transfer 81 106-12
14. Ciment M and Guenther R B 1974 Numerical solution of a free boundary value problem for parabolic equations Appl. Anal. 4 39-62
15. Ciment M and Sweet R A 1973 Mesh refinements for parabolic equations J. Comput. Phys. 12 513-25
16. Lazaridis A 1970 A numerical solution of the multidimensional solidification (or melting) problem Int. J. Heat Mass Transfer 13 1459-77
17. Basu B and Date A W 1988 Numerical modelling of melting and solidification problems: a review Sadhana 13 169-213
18. Voller V R, Swaminathan C R and Thomas B G 1990 Fixed grid techniques for phase change problems: a review Int. J. Numer. Meth. Eng. 30 875-98
19. Douglas J and Gallie T M 1955 On the numerical integration of a parabolic differential equation subject to a moving boundary condition Duke Math. J. 22 557-71
20. Gupta R S and Kumar D 1980 A modified variable time step method for the one-dimensional Stefan problem Comput. Meth. Appl. Mech. Eng. 23 101-9
21. Goodling J S and Khader M S 1974 Inward solidification with radiation-convection boundary condition J. Heat Transfer 96 114-15
22. Gupta R S and Kumar D 1981 Variable time step methods for one-dimensional Stefan problem with mixed boundary condition Int. J. Mass Heat Transfer 24 251-9
23. Gupta R S and Kumar D 1981 Complete numerical solution of the oxygen diffusion problem involving a moving boundary Comput. Meth. Appl. Mech. Eng. 29 233-9
24. Hansen E B and Hougaard P 1974 On a moving boundary problem from biomechanics J. Inst. Math. Appl. 13 385-98
25. Dahmardah H O and Mayers D F 1983 A Fourier-series solution of the Crank-Gupta equation IMA J. Numer. Analysis 3 81-5
26. Heitz W L and Westwater J W 1970 Extension of the numerical method for melting and freezing problems Int. J. Heat Mass Transfer 13 1371-5
27. Tien L C and Churchill S W 1965 Freezing front motion and heat transfer outside an infinite isothermal cylinder AIChEJ 11 790-3
28. Rathjen K A and Jiji L M 1971 Heat conduction with melting or freezing in a corner J. Heat Transfer 93 101-4
29. Tien L C and Wilkes J O 1970 Axisymmetrical normal freezing with convection above Proc. IV Int. Heat Transfer Conf. ed U Grigull and E Hahne (Amsterdam: Elsevier)
30. Crank J and Gupta R S 1972 A method for solving moving boundary problems in heat flow using cubic splines or polynomials J. Inst. Math. Appl. 10 296-304
31. Gupta R S 1974 Moving grid method without interpolations Comput. Meth. Appl. Mech. Eng. 4 143-52
32. Zerroukat M and Chatwin CR 1994 Computational Moving Boundary Problems (New York: Wiley)
33. Hashemi H T and Sliepcevich C M 1967 A numerical method for solving two-dimensional problems of heat conduction with change of phase Chem. Eng. Prog. Symp. Series 63 34-41
34. Comini G, Del Guidice S, Lewis R W and Zienkiewicz O C 1974 Finite element solution of non-linear heat conduction problems with special reference to phase change Int. J. Numer. Meth. Eng. 8 613-24
35. Poirier D and Salcudean M 1988 On numerical methods used in mathematical modeling of phase change in liquid metals Trans. ASMEJ. Heat Transfer 110 562-70
36. Poirier D 1986 On numerical methods used in mathematical modeling of phase change in liquid metals MASc Thesis Department of Mechanical Engineering, University of Ottawa
37. Dusinberre G M 1945 Numerical methods for transient heat flow Trans ASME 67 703-12
38. Rolph W D and Bathe K J 1982 An efficient algorithm for analysis of nonlinear heat transfer with phase changes Int. J. Numer. Meth. Eng. 18 119-34
39. Argyropoulos S A and Guthrie RIL 1979 The exothermic dissolution of 50 wt% ferro-silicon in molten steel Can. Metall. Q. 18 267-81
40. Argyropoulos S A and Guthrie R I L 1984 The dissolution of titanium in liquid steel Metall. Trans B 15 47-58
41. Argyropoulos S A 1981 Dissolution of high melting point additions in liquid steel PhD Thesis Department of Mining and Metallurgical Engineering, McGill University
42. Sismanis P G and Argyropoulos S A 1988 Modeling of exothermic dissolution Can. Metall. Q. 27 123-33
43. Voller V R and Swaminathan C R 1991 General source-based method for solidification phase change Numer. Heat Transfer 19 175-89
44. Patankar S V 1980 Numerical Heat Transfer and Fluid Flow (New York: Hemisphere)
45. Zienkiewicz O C 1980 The Finite Element Method (New York: McGraw-Hill)
46. Salcudean M and Abdullah Z 1988 On the numerical modelling of heat transfer during solidification process Int. J. Numer. Meth. Eng. 25 445-73
47. Eyres N R, Hartree D R, Ingham J, Jackson R, Sarjant R J and Wagstaff JB 1946 The calculation of variable heat flow in solid Phil. Trans. R. Soc. A 240 1-57
48. Rose M E 1960 A method for calculating solutions of parabolic equations with a free boundary Math. Comput. 14 249-56
49. Shamsunder N and Sparrow E M 1975 Analysis of multidimensional conduction phase change via the enthalpy model J. Heat Transfer 97 333-40
50. Bell G E and Wood A S 1983 On the performance of the enthalpy method in the region of a singularity Int. J. Numer. Meth. Eng. 19 1583-92
51. Carslaw H S and Jaeger J C 1959 Conduction of Heat in Solid (Oxford: Clarendon)
52. Tacke K H 1985 Discretization of the explicit enthalpy method for planar phase change Int. J. Numer. Meth. Eng. 21 543-54
53. Voller V and Cross M 1981 Accurate solutions of moving boundary problems using the enthalpy method Int. J. Heat Mass Transfer 24 545-56
54. Voller V and Cross M 1983 An explicit numerical method to track a moving phase change front Int. J. Heat Mass Transfer 26 147-50
55. Swaminathan C R and Voller V R 1992 A general enthalpy method for modeling solidification processes Metall. Trans. B 23 651-64
56. Alexiades V and Solomon A D 1993 Mathematical Modeling of Melting and Freezing Processes (Washington, DC: Hemisphere)
57. Solomon A D, Morris M D, Martin J and Olszewski M 1986 The development of a simulation code for a latent heat thermal energy storage system in a space station, ORNL-6213
58. Hunter J W and Kuttler J R 1989 The enthalpy method for heat conduction problems with moving boundaries J. Heat Transfer 111 239-42
59. Cao Y, Faghri A and Chang W S 1989 A numerical analysis of Stefan problems for generalized multi-dimensional phase-change structures using the enthalpy transforming model Int. J. Heat Mass Transfer 32 1289-98
60. Zeng X and Faghri A 1993 A temperature-transforming model with a fixed-grid numerical methodology for binary solid-liquid phase change problems Proc. on Heat Transfer In Melting, Solidification, and Crystal Growth (ASME) pp 43-57
61. Sparrow E M, Patankar S V and Ramadhyani S 1977 A analysis of melting in the presence of natural convection in the melt region J. Heat Transfer 99 520-6
62. Ramsey J W and Sparrow E M 1978 Melting and natural convection due to a vertical embedded heater J. Heat Transfer 100 368-70
63. Hale N W Jr and Viskanta R 1978 Photographic observation of the solid-liquid interface motion during melting of a solid heat from an isothermal vertical wall Lett. Heat Mass Transfer 5 329-37
64. Bathelt A G, Viskanta R and Leidenfrost W 1979 An experimental investigation of natural convection in the melted region around a heated horizontal cylinder J. Fluid Mech. 90 227-39
65. Van Buren P D and Viskanta R 1980 Interferometric measurement of heat transfer during melting from a vertical surface Int. J. Heat Mass Transfer 23 568-71
66. Cole G S and Boiling GF 1965 The importance of natural convection in casting Trans. TMS-AIME 233 1568-72
67. Harrison C and Weinberg F 1985 The influence of convection on heat transfer in liquid tin Metall. Trans. B 16 355-7
68. Huang S C 1985 Analytical solution for the buoyancy flow during the melting of a vertical semi-infinite region Int. J. Heat Mass Transfer 28 1231-3
69. Gerald C F and Wheatley P O 1989 Applied Numerical Analysis (New York: Addison-Wesley)
70. Wilkes J O and Churchill S W 1977 The finite difference computation of natural convection in a rectangular enclosure AIChE J. 12 161-6
71. Kublbeck K, Merker G P and Struab J 1980 Advanced numerical computation of two dimensional time dependent free convection in cavities Int. J. Heat Mass Transfer 23 203-17
72. Kee R J and Mckillop AA 1977 A numerical method for predicting natural convection in horizontal cylinders with asymmetric boundary condition Comp. Fluids 5 1-14
73. Ramachandran N and Gupta J P 1982 Thermal and fluid flow effects during solidification in rectangular enclosure Int. J. Heat Mass Transfer 25 187-94
74. Incropera F P and Dewitt D P 1990 Introduction to Heat Transfer (New York: Wiley)
75. Okada M 1984 Analysis of heat transfer during melting from a vertical wall Int. J. Heat Mass Transfer 27 2057-66
76. Ho C J and Chen S 1986 Numerical simulation of melting of ice around a horizontal cylinder Int. J. Heat Mass Transfer 29 1359-69
77. Guenigault R and Poots G 1985 Effects of natural convection on the inward solidification of spheres and cylinders Int. J. Heat Mass Transfer 28 1229-31
78. Vabishchevich P N and Iliev O P 1989 Numerical investigation of heat and mass transfer during the crystallization of metal in a mould Comm. Appl. Numer. Meth. 5 515-26
79. Harlow F H and Welch J E 1965 Numerical calculation of time-dependent viscous incompressible flow Phys. Fluids 8 2182-93
80. Nichols B D, Hirt C W and Hotchkiss R S 1980 SOLA-VOF: a solution algorithm for transient fluid flow with multiple free boundaries Los Alamos Scientific Laboratory Report LA-8355
81. Salcudean M and Guthrie RIL 1973 A three-dimensional representation of fluid flow induced in ladles or holding vessels by the action of liquid metal jet Metall. Trans. 4 1379-88
82. Stoher R and Hwang W S 1983 Modelling of the flow of molten metal having a free surface during entry into molds Proc. Engineering Foundation Conf. - Modelling of Casting and Welding Processes II (The Metallurgical Society of AIME)
83. Hirt C W and Nicols B D 1981 Volume of fluid (VOF) method for the dynamics of free boundaries J. Comp. Phys. 39 23-31
84. Benard C, Gobin D and Zanoli A 1986 Moving boundary problem: heat conduction in the solid phase of a phase-change material during melting driven by natural convection in the liquid Int. J. Heat Mass Transfer 29 1669-81
85. Neilson D G and Incropera F P 1990 Numerical simulation of solidification in a horizontal cylindrical annulus charged with an aqueous salt solution Int. J. Heat Mass Transfer 33 367-80
86. Kim C J and Kaviany M 1992 A numerical method for phase-change problems with convection and diffusion Int. J. Heat Mass Transfer 35 457-67
87. Chabchoub F, Argyropoulos S A and Mostaghimi J 1994 Mathematical modelling and experimental measurements on the horizontal ohno continuous casting process for pure tin Can. Metall. Q. 33 73-88
88. Trovant M and Argyropoulos S A 1996 Mathematical modeling and experimental measurements of shrinkage in a casting of metals Can. Metall. Q. 35 75-84
89. Hu H and Argyropoulos S A 1984 Modelling of Stefan problems in complex configurations involving two different materials using the enthalpy method J. Modell. Sim. Mater. Sci. Eng. 2 53-64
90. Hu H and Argyropoulos S A 1995 Mathematical modeling and experimental measurements of moving boundary problems associated with exothermic heat of mixing Int. J. Heat Mass Transfer 39 1005-21
91. Hu H and Argyropoulos S A 1996 Mathematical simulation and experimental verification of melting resulting from the coupled effect of natural convection and exothermic heat of mixing Metall. Trans. B at press
92. Hu H 1996 Mathematical modeling and experimental measurements of melting of a solid in a liquid associated with exothermic heat of mixing PhD Thesis Department of Metallurgy and Materials Science, University of Toronto
93. Samarskii A A, Vabishchevich P N, Iliev O P and Churbanov A G 1993 Numerical simulation of convection/diffusion phase change problems - a review Int. J. Heat Mass Transfer 36 4095-106
94. Stefanescu D M 1993 Critical review of the second generation of solidification models for castings: macro transport-transformation kinetics codes Modeling of Casting, Welding and Advanced Solidification Process-VI ed T S Piwonka, V Voller and L Katgerman (Warrendale, PA: TMS) pp 3-20
95. Stefanescu D M 1995 Methodologies for modeling of solidification microstructure and their capabilities ISIJ Int. 35 637-50
96. Lu S and Hunt J D 1992 A numerical analysis of dendritic and cellular array growth: the spacing adjustment mechanisms J. Crystal Growth 123 17-34
97. Boettinger W J, Wheeler A A, Murray B T and McFadden G B 1994 Prediction of solute trapping at high solidification rates using a diffuse interface phase-field theory of alloy solidification Mater. Sci. Eng. A 78 217-23
98. Gandin Ch-A, Charbon Ch and Rappaz M 1995 Stochastic modeling of solidification grain structures ISIJ Int. 35 651-7
99. Kurz W, Giovanola B and Trivedi R 1986 Theory of microstructural development during rapid solidification Acta Metall. 34 823-30
100. Estrin L 1994 A deeper look at casting solidification software Modern Casting 20-24 July