Skip to main content

Recent developments in modelling metal flow and solidification

  • Chapter
Modelling the Flow and Solidification of Metals

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.

    Google Scholar 

  2. 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).

    Google Scholar 

  3. 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.

    Google Scholar 

  4. 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.

    Google Scholar 

  5. G.K. Batchelor: Mass transfer from small particles suspended into turbulent fluid. J Fluid Mech 98 (1980) 609–623.

    Article  CAS  Google Scholar 

  6. G.E. Bell: On the performance of the enthalpy method. Int J Heat Mass Transfer 25 (1982) 587–589.

    Article  CAS  Google Scholar 

  7. 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.

    Google Scholar 

  8. 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.

    Article  Google Scholar 

  9. 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.

    Google Scholar 

  10. J.C. Brice: The Growth of Crystals from Liquid. North-Holland, Amsterdam (1973).

    Google Scholar 

  11. 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.

    Google Scholar 

  12. 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.

    Article  CAS  Google Scholar 

  13. 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.

    Article  CAS  Google Scholar 

  14. T.W. Clyne: The use of heat flow modelling to explore solidification phenomena. Metall Trans B 13B (1982) 471–478.

    Article  CAS  Google Scholar 

  15. T.W. Clyne: Numerical treatment of rapid solidification. Metall Trans B 15B (1984) 369–381.

    Article  CAS  Google Scholar 

  16. T.W. Clyne: Heat flow, solidification and energy aspects of DC and strip casting of aluminium alloys. Metals Tech 11 (1984) 350–357.

    CAS  Google Scholar 

  17. 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.

    Google Scholar 

  18. 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.

    Google Scholar 

  19. G.S. Cole: Temperature measurements and fluid flow distributions ahead of solid-liquid interfaces. Trans Metall Soc AIME 239 (1967) 1287–1295.

    CAS  Google Scholar 

  20. G.S. Cole and G.F. Boiling: The importance of natural convection in casting. Trans Metall Soc AIME 233 (1965) 1568–1572.

    CAS  Google Scholar 

  21. A.B. Crowley and J.R. Ockendon: Modelling mushy regions. App Sci Res 44 (1987) 1–7.

    Article  CAS  Google Scholar 

  22. 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.

    Article  Google Scholar 

  23. 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.

    Google Scholar 

  24. 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.

    Google Scholar 

  25. 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.

    Google Scholar 

  26. 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.

    Google Scholar 

  27. L. DeSocio and G. Gualtieri: A hyperbolic Stefan problem. Quart Appl Math 41 (1983) 253–259.

    Google Scholar 

  28. C. Dietsche and U. Müller: Influence of Benard convection on solid-liquid interfaces. J Fluid Mech 161 (1985) 249–268.

    Article  CAS  Google Scholar 

  29. E.R. Evans: Fluidity of molten cast iron. Foundry Trade J 128 (1955) 757–763.

    Google Scholar 

  30. S.C. Flood and J.D. Hunt: A model of a casting. Appl Sci Res 44 (1987) 27–42.

    Article  CAS  Google Scholar 

  31. R.M. Furzeland: A comparative study of numerical methods for moving boundary problems. J Int Maths Applics 26 (1980) 411–429.

    Article  Google Scholar 

  32. 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.

    Google Scholar 

  33. 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.

    Article  CAS  Google Scholar 

  34. 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.

    Article  CAS  Google Scholar 

  35. 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.

    Google Scholar 

  36. R. Guenigault and G. Poots: Effects of natural convection on the inward solidification of cylinders. Int J Heat Mass Transf 28 (1985) 1229–1231.

    Article  Google Scholar 

  37. 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.

    Article  Google Scholar 

  38. C Harrison and E. Weinberg: The influence of convection on heat transfer in liquid tin. Metall Trans B 16B (1985) 355–357.

    Article  CAS  Google Scholar 

  39. 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.

    Article  CAS  Google Scholar 

  40. 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.

    Article  Google Scholar 

  41. K. Ho and R.D. Pehlke: Mechanisms of heat transfer at a metal-mould interface. Trans Am Foundrymen’s Soc 92 (1984) 587–598.

    Google Scholar 

  42. 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.

    CAS  Google Scholar 

  43. 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.

    Article  Google Scholar 

  44. J.CR. Hunt: Turbulence structure in thermal convection and shear-free boundary layers. J Fluid Mech 138 (1984) 161–184.

    Article  Google Scholar 

  45. J.CR. Hunt: Turbulent diffusion from sources in complex flows. Ann Rev Fluid Mech 17 (1985) 447–485.

    Article  Google Scholar 

  46. J.D. Hunt and D.G. McCartney: Numerical finite difference model for steady state array growth. Appl Sci Res 44 (1987) 9–26.

    Article  CAS  Google Scholar 

  47. H.E. Huppert: From multi-branched snowflakes to precious minerals. Nature 323 (1986) 202–203.

    Article  Google Scholar 

  48. H.E. Huppert: The intrusion of fluid mechanics into geology. J Fluid Mech 173 (1986) 557–594.

    Article  Google Scholar 

  49. H.E. Huppert and J.S. Turner: A laboratory model of a replenished magma chamber. Earth Planet Sci Lett 54 (1981) 144–142.

    Article  CAS  Google Scholar 

  50. H.E. Huppert and M.G. Worster: Dynamic solidification of a binary melt. Nature 314 (1985) 703–707.

    Article  CAS  Google Scholar 

  51. I. Imafuku and K. Chijiiwa: A mathematical model for shrinkage cavity prediction in steel castings. Trans Am Foundrymen’s Soc 91 (1985) 527–520.

    Google Scholar 

  52. 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.

    Google Scholar 

  53. 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.

    Article  Google Scholar 

  54. 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.

    Article  Google Scholar 

  55. 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.

    Article  Google Scholar 

  56. 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.

    Google Scholar 

  57. H. Jones: Rapid Solidification of Metals and Alloys. Institution of Metallurgists, London (1982).

    Google Scholar 

  58. 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.

    Article  Google Scholar 

  59. 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.

    Google Scholar 

  60. 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.

    Google Scholar 

  61. B.H. Kear, B.C. Giessen and M. Cohen (eds): Rapidly-solidified Amorphous and Crystalline Alloys. North-Holland, New York (1982).

    Google Scholar 

  62. 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.

    Google Scholar 

  63. W. Kunda and G. Poots: A mathematical model of ladle stirring by bath agitation in steel making. Appli Sci Res 44 (1987) 209–224.

    Article  Google Scholar 

  64. 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.

    Google Scholar 

  65. R.W. Lewis and P.M. Roberts: Finite element simulation of solidification problems. Appl Sci Res 44 (1987) 61–92.

    Article  CAS  Google Scholar 

  66. 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.

    Google Scholar 

  67. T. Matsumiya and M.C. Flemings: Modelling of continuous strip production by rheo-casting. Metall Trans B 12B (1981) 17–31.

    Article  CAS  Google Scholar 

  68. 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.

    Google Scholar 

  69. 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.

    Google Scholar 

  70. K. Morgan: A numerical analysis of freezing and melting with convection. Comp Meth Appl Mech Engng 28 (1981) 275–284.

    Article  Google Scholar 

  71. Y. Nagasaka, J. Ohnaka and T. Fukusako: Effect of heat transfer during pouring on solidification of steel plate castings. Imono 56 (1984) 22–27.

    Google Scholar 

  72. 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.

    Google Scholar 

  73. J. Pullins and M.K. Walther: Simulation for designing metal-casting moulds. Comp Aided Engr (1984) 62–68.

    Google Scholar 

  74. 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.

    Article  CAS  Google Scholar 

  75. 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.

    CAS  Google Scholar 

  76. S. Rogers, L. Katgerman, P.G. Enright and N.A. Darby: Modelling of liquid-liquid metal mixing. Appl Sci Res 44 (1987) 175–196.

    Article  Google Scholar 

  77. F. Rosenberger: Fundamentals of Crystal Growth Vol 1. Springer, Berlin (1979).

    Google Scholar 

  78. R.W. Ruddle: The Solidification of Castings. Institute of Metals Monograph and Reports Series No 7 (2nd ed) (1957).

    Google Scholar 

  79. M. Sadd and J. Didlake: Non-fourier melting of a semi-infinite solid. Trans AIME J Heat Trans 99 (1979) 25–28.

    Article  Google Scholar 

  80. P.R. Sahm, H. Jones and CM. Adam: Science and Technology of the Under cooled Melt. Martinus Nijhoff Publishers, Dordrecht (1986).

    Google Scholar 

  81. 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.

    Article  Google Scholar 

  82. W. Schneider: A local analysis of solidification in horizontal continuous casting. Arch Eisenhüttenwes 54 (1983) 487–490.

    CAS  Google Scholar 

  83. A. Schröder: Fluid flow considerations in the filling of lost moulds with open feeders. Giessereiforschung 37 (1985) 65–79.

    Google Scholar 

  84. 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.

    Article  Google Scholar 

  85. T.J. Smith: Adaptive mesh schemes for computational fluid dynamics. CFD News 3/84 (1984)11–15.

    Google Scholar 

  86. T.J. Smith: A basis for the control of the casting process. GST Report No: PS 1940.1 GST 33/1.00.

    Google Scholar 

  87. 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.

    Article  CAS  Google Scholar 

  88. 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.

    Google Scholar 

  89. 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.

    Article  Google Scholar 

  90. 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.

    Google Scholar 

  91. 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.

    Google Scholar 

  92. 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.

    Google Scholar 

  93. 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.

    Article  CAS  Google Scholar 

  94. J.S. Turner: Buoyancy Effects in Fluids, 2nd Edition. Cambridge University Press, Cambridge (1979).

    Google Scholar 

  95. 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.

    CAS  Google Scholar 

  96. V.R. Voller: Implicit finite-difference solutions of the enthalpy formulation of Stefan problems. I M A J Numer Anal 5 (1985) 201–214.

    Google Scholar 

  97. 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.

    Article  Google Scholar 

  98. 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.

    Article  CAS  Google Scholar 

  99. 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.

    Google Scholar 

  100. 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.

    Google Scholar 

  101. 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).

    Google Scholar 

  102. R.E. White: An enthalpy formulation of the Stefan problem. SIAM J Numer Anal 19 (1982) 1129–1157.

    Article  Google Scholar 

  103. 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.

    Article  Google Scholar 

  104. 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.

    Article  Google Scholar 

  105. M.G. Worster: Solidification of an alloy from a cooled boundary. J Fluid Mech 167 (1986) 481–501.

    Article  CAS  Google Scholar 

  106. 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.

    Article  Google Scholar 

  107. 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.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints 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

Publish with us

Policies and ethics