## Science and Engineering of Casting Solidification, Second EditionScience and Engineering of Casting Solidification, Second Edition covers the essentials of solidification science of metals and alloys at macro- and micro-length scales at cooling rates specific to commercial castings and rapid solidification processing. The mathematical fundamentals necessary to build a working knowledge in the field, specifically partial differential equations and numerical analysis, are introduced. Each topic begins with the description of the underlying physics, followed by the mathematics required to build analytical and numerical models. Wherever possible, a detailed description of the architecture of the numerical model is provided, followed by examples of models built on the Excel spreadsheet. Features of this new edition include:Expanded sections on peritectic solidification and shrinkage porosity mechanisms and modeling,A new chapter addressing rapid solidification and bulk metallic glasses,Additional solved problems,Revised and simplified derivations of several models.Science and Engineering of Casting Solidification, Second Edition will prove useful to senior undergraduate and graduate students, as well as to industrial researchers that work in the field of solidification in general and casting modeling in particular. The detailed coverage of casting defects will also make it useful to industrial practitioners of metal casting. |

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### Contents

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23 Curvature undercooling | 9 |

24 Thermal undercooling | 11 |

25 Constitutional undercooling | 12 |

26 Pressure undercooling | 15 |

822 Thermal diffusion controlled growth | 163 |

823 Solutal thermal and capillary controlled growth | 164 |

824 Interface anisotropy and the dendrite tip selection parameter | 171 |

825 Effect of fluid flow on dendrite tip velocity | 172 |

826 Multicomponent alloys | 174 |

83 Dendritic array models | 175 |

84 Dendritic arm spacing and coarsening | 177 |

842 Secondary arm spacing | 179 |

28 Departure from equilibrium | 17 |

281 Local interface equilibrium | 19 |

282 Interface nonequilibrium | 20 |

29 Applications | 23 |

Macroscale Phenomenageneral equations | 25 |

32 Introduction to diffusive transport | 29 |

322 The differential equation for macroscopic heat transport | 30 |

References | 31 |

Macromass transport | 32 |

411 Equilibrium solidification | 36 |

412 No diffusion in solid complete diffusion in liquid the GulliverScheil model | 38 |

413 No diffusion in solid limited diffusion in liquid | 39 |

414 Limited diffusion in solid complete diffusion in liquid | 41 |

415 Limited diffusion in solid and liquid | 44 |

417 Zone melting | 47 |

42 Fluid dynamics during mold filling | 49 |

423 Gating systems for castings | 51 |

43 Fluid dynamics during solidification | 54 |

431 Shrinkage flow | 55 |

433 Surface tension driven Marangoni convection | 58 |

44 Macrosegregation | 60 |

441 Fluid flow controlled segregation | 61 |

442 Fluid flow solute diffusion controlled segregation | 62 |

45 Fluid dynamics during casting solidification macroshrinkage formation | 64 |

451 Metal shrinkage and feeding | 65 |

452 Shrinkage defects | 68 |

46 Applications | 69 |

References | 74 |

Macroenergy transport | 75 |

51 Governing equation for energy transport | 76 |

52 Boundary conditions | 77 |

Analytical solutions for steadystate solidification of castings | 79 |

54 Analytical solutions for nonsteadystate solidification of castings | 81 |

541 Resistance in the mold | 84 |

542 Resistance at the moldsolid interface | 86 |

543 The heat transfer coefficient | 89 |

544 Resistance in the solid | 91 |

55 Applications | 93 |

References | 96 |

Numerical Macromodeling of Solidification | 97 |

611 The Enthalpy Method | 98 |

612 The Specific Heat Method | 99 |

62 Discretization of governing equations | 100 |

622 The Finite Difference Method implicit formulation | 105 |

624 Controlvolume formulation | 106 |

63 Solution of the discretized equations | 107 |

65 Macroshrinkage modeling | 111 |

651 Thermal models | 112 |

652 Thermalvolume calculation models | 114 |

653 Thermalfluid flow models | 115 |

66 Applications of macromodeling of solidification | 118 |

67 Applications | 121 |

References | 125 |

Microscale Phenomena and interface dynamics | 127 |

71 Nucleation | 128 |

711 Heterogeneous nucleation models | 131 |

712 Dynamic nucleation models | 135 |

73 Interface stability | 142 |

731 Thermal instability | 143 |

732 Solutal instability | 144 |

733 Thermal solutal and surface energy driven morphological instability | 148 |

73 4 Influence of convection on interface stability | 153 |

74 Applications | 154 |

References | 155 |

Cellular and dendritic growth | 157 |

82 Analytical tip velocity models | 160 |

85 The columartoequiaxed transition | 183 |

86 Applications | 188 |

References | 193 |

Eutectic solidification | 195 |

92 Cooperative Eutectics | 197 |

921 Models for regular eutectic growth | 199 |

922 Models for irregular eutectic growth | 205 |

923 The unified eutectic growth model | 207 |

93 Divorced eutectics | 211 |

94 Interface stability of eutectics | 214 |

95 Equiaxed eutectic grain growth | 218 |

96 Solidification of Cast Iron | 219 |

962 Crystallization of graphite from the liquid | 222 |

963 Eutectic Solidification | 226 |

964 The graytowhite structural transition | 231 |

97 Solidification of aluminumsilicon alloys | 233 |

98 Applications | 240 |

References | 244 |

Peritectic solidification | 247 |

102 Peritectics microstructures and phase selection | 249 |

103 Mechanism of peritectic solidification | 254 |

1031 The rate of the peritectic reaction | 255 |

1032 The rate of the peritectic transformation | 257 |

1033 Growth of banded layered peritectic structure | 259 |

104 Applications | 261 |

References | 262 |

Monotectic solidification | 265 |

111 Classification of monotectics | 266 |

References | 270 |

Microstructures obtained through rapid solidification | 271 |

121 Rapidly solidified crystalline alloys | 272 |

122 Metallic glasses | 276 |

References | 280 |

Solidification in the presence of a third phase | 282 |

1311 Particle interaction with a planar interface | 285 |

1312 Material properties models | 287 |

1313 Kinetic models | 288 |

1314 Mechanism of engulfment planar SL interface | 300 |

1315 Particle interaction with a cellulardendritic interface | 301 |

132 Shrinkage porosity | 303 |

1322 Analytical models including nucleation and growth of gas pores | 310 |

1323 Analysis of shrinkage porosity models and defect prevention | 312 |

References | 313 |

Numerical micromodeling of solidification | 317 |

141 Deterministic models | 318 |

1412 Coupling of MT and TK codes | 322 |

1413 Models for dendritic microsctructures | 323 |

1414 Microporosity models | 333 |

142 Stochastic models | 341 |

1421 MonteCarlo models | 342 |

1422 Cellular automaton models | 346 |

143 Phase field models | 355 |

References | 358 |

Atomic scale phenomena Nucelation and growth | 361 |

1511 Steadystate nucleation homogeneous nucleation | 362 |

1512 Steadystate nucleation Heterogeneous Nucleation | 368 |

1513 Timedependent transient nucleation | 373 |

152 Growth Kinetics | 374 |

1522 Continuous growth | 377 |

1523 Lateral growth | 378 |

153 Applications | 379 |

382 | |

Appendix A | 383 |

Appendix B | 385 |

Appendix C | 391 |

### Common terms and phrases

Al-Si alloys Application assumed atoms austenite Beckermann boundary conditions calculated cast iron cell columnar composition computational constant constitutional undercooling convection cooling curve cooling rate criterion crystal curvature D.M. Stefanescu decreases dendrite dendrite tip dendritic growth density derived diffusion directional solidification engulfment equiaxed grains equilibrium eutectic evolution experimental fluid flow flux formation fraction of solid fraction solid free energy function governing equation graphite growth velocity increases kinetics Kurz lamellar lever rule liquid macrosegregation Mater melt metal microstructure mold morphology mushy zone Nastac nucleation nuclei obtained occurs particle partition coefficient peritectic phase diagram planar pore porosity predict pressure radius S/L interface Scheil equation segregation shown in Figure shrinkage simulation solid fraction solidifica Solidification Processes solidification velocity solute solved Stefanescu D.M. structure surface tion Trans Trivedi typical undercooling viscosity volume element Warrendale γ γ ρ ρ