Science and Engineering of Casting Solidification Second Edition

Transcription

1 Science and Engineering of Casting Solidification Second Edition

2 Doru Michael Stefanescu Science and Engineering of Casting Solidification Second Edition

3 Doru Michael Stefanescu Department of Materials Science and Engineering The Ohio State University Columbus, OH USA ISBN e-isbn Library of Congress Control Number: Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

4 To my teachers: my mother my father Prof. Laurentie Sofroni Prof. Suzana Gâdea Prof. Carl Loper Max, my grandson my first and dearest my first metallurgy teacher my cast iron teacher my doctorate advisor my American teacher for recasting my understanding of this world

5 PREFACE "The book of nature is written in mathematical language" Galileo We come to know about the world in two distinctive ways: by direct perception and by application of rational reasoning which, in its highest form, is mathematical thinking. The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. In other words, we only know what we can describe through mathematical models. Casting of metals has evolved first as witchcraft, to gradually become an art, then a technology, and only recently a science. Many of the processes used in metal casting are still empirical in nature, but many others are deep-rooted in mathematics. In whatever form, casting of metals is an activity fundamental to the very existence of our world, as we know it today. Foundry reports indicate that solidification modeling is not only a costeffective investment but also a major technical asset. It helps foundries move into markets with more complex and technically demanding work. The ability to predict internal soundness allows foundries to improve quality and deliveries, and provides the information required to make key manufacturing decisions based on accurate cost estimates before pattern construction even begins. The acceptance of computational modeling of solidification by the industry is a direct result of the gigantic strides made by solidification science in the last two decades. Yet, solidification science is of paramount importance not only in understanding macro- and microscopic changes during the solidification of castings, but is also the basis of many new processes and materials such as semi-solid casting, laser melting, powder atomization, metal matrix composites, bulk metallic glasses. This book is the second attempt by the author to synthesize the information that can be used for engineering calculations pertinent to computational modeling of casting solidification. It includes additional material on the fundamentals of rapid solidification and bulk metallic glasses. This book is based on the author s more than forty years experience of teaching, research and industrial practice of solidification science as applied to casting processes. It is an attempt to describe solidification theory through the complex mathematical apparatus that includes partial differential equations and

6 viii Preface numerical analysis, required for a fundamental treatment of the problem. The mathematics is however restricted to the elements essential to attain a working knowledge in the field. This is in line with the main goal of the book, which is to educate the reader in the fast moving area of computational modeling of solidification processes. While the book is not intended to be a monograph, for the sake of completion, a special effort has been made to introduce the reader to the latest developments in solidification theory, even when they have no engineering applications at this time. In this respect this is a unique attempt to integrate the newest information in a text book format. The text is designed to be self-contained. The author s teaching experience demonstrates that some of the students interested in solidification science are not fully proficient in partial differential equations (PDE) and/or numerical analysis. Accordingly, elements of PDE and numerical analysis required to obtain a working knowledge of computational solidification modeling have been introduced in the text, while attempting to avoid the interruption of the fluency of the subject. Numerous modeling and calculation examples using the Excel spreadsheet as an engineering tool are provided. The book is addressed to graduate students and seniors interested in solidification science, as well as to industrial researchers that work in the field of solidification in general and casting modeling in particular. The book is divided in 15 major chapters. After introducing the length scale of solidification analysis in the first chapter, the reader is exposed to the basic concepts of driving force for solidification, undercooling, local equilibrium, and interface non-equilibrium from the thermodynamic perspective (Chapter 2). The following three chapters present a detailed analysis of the governing transport equations and their application at the macro-scale level to predict such features of interest in casting solidification as segregation, shrinkage cavity, solidification time and velocity, and temperature gradients. Numerical approximation methods with an emphasis on finite difference approximations are presented in Chapter 6 together with numerous examples of solidification modeling through analytical and numerical methods solved on the Excel spreadsheet. In this chapter, the reader is also introduced to the applications of macro modeling of solidification in today s casting technology. Chapters 7 through 11 extend the transport equations to the study of microscale phenomena and the formation of casting microstructure. Nucleation is discussed from the engineering standpoint that is emphasizing possible methodologies for quantification in solidification analysis of castings. A detailed analysis of existing models for dendritic, eutectic, peritectic and monotectic growth is provided. Again, the emphasis is on the use of this knowledge to build computational solidification models. To achieve this goal, each section of this chapter includes a comprehensive discussion of the applicability and limitations of transferring the information available from steady state analysis to continuous cooling solidification. Chapter 12 extends the concepts introduced earlier to the evolution of microstructure during rapid solidification. Rapidly solidified crystalline alloys and metallic glasses are briefly discussed. The solidification behavior in the presence of a third phase (gaseous or solid impurities) is covered in the 13 th chapter.

7 Preface ix Chapter 14 is dedicated to the fast moving field of numerical modeling of solidification at the micro-scale. Deterministic and cellular automaton models are covered in detail, while phase field modeling is briefly summarized. The analysis of nucleation and growth at the atomic scale level, required for a complete understanding of solidification and the associated phenomena is presented in chapter 15. Since the current level of understanding does not permit the use of this information directly in computational modeling of solidification, the emphasis is on the physics rather than on engineering.

8 CONTENTS 1 Length-scale in solidification analysis 1 References 4 2 Equilibrium and non-equilibrium during solidification Equilibrium The undercooling requirement Curvature undercooling Thermal undercooling Constitutional undercooling Pressure undercooling Kinetic undercooling Departure from equilibrium Local interface equilibrium Interface non-equilibrium Applications 23 References 23 3 Macro-scale phenomena - general equations Relevant Transport Equations Introduction to diffusive transport Flux laws The differential equation for macroscopic heat transport 30 References 31 4 Macro-mass transport Solute diffusion controlled segregation Equilibrium solidification No diffusion in solid, complete diffusion in liquid (the Gulliver-Scheil model) No diffusion in solid, limited diffusion in liquid Limited diffusion in solid, complete diffusion in liquid Limited diffusion in solid and liquid Partial mixing in liquid, no diffusion in solid Zone melting Fluid dynamics during mold filling Fluidity of molten metals Capillary flow Gating systems for castings 51

9 xii Contents 4.3 Fluid dynamics during solidification Shrinkage flow Natural convection Surface tension driven (Marangoni) convection Flow through the mushy zone Macrosegregation Fluid flow controlled segregation Fluid flow /solute diffusion controlled segregation Fluid dynamics during casting solidification - macroshrinkage formation Metal shrinkage and feeding Shrinkage defects Applications 69 References 74 5 Macro-energy transport Governing equation for energy transport Boundary conditions Analytical solutions for steady-state solidification of castings Analytical solutions for non-steady-state solidification of castings Resistance in the mold Resistance at the mold/solid interface The heat transfer coefficient Resistance in the solid Applications 93 References 96 6 Numerical Macro-modeling of solidification Problem formulation The Enthalpy Method The Specific Heat Method The Temperature Recovery Method Discretization of governing equations The Finite Difference Method - Explicit formulation The Finite Difference Method - implicit formulation The Finite Difference Method - general implicit and explicit formulation Control-volume formulation Solution of the discretized equations Macrosegregation modeling Macroshrinkage modeling Thermal models Thermal/volume calculation models Thermal/fluid flow models Applications of macro-modeling of solidification Applications 121 References Micro-scale phenomena and interface dynamics Nucleation 128

10 Contents xiii Heterogeneous nucleation models Dynamic nucleation models Micro-solute redistribution in alloys and microsegregation Interface stability Thermal instability Solutal instability Thermal, solutal, and surface energy driven morphological instability Influence of convection on interface stability Applications 154 References Cellular and dendritic growth Morphology of primary phases Analytical tip velocity models Solute diffusion controlled growth (isothermal growth) of the dendrite tip Thermal diffusion controlled growth Solutal, thermal, and capillary controlled growth Interface anisotropy and the dendrite tip selection parameter Effect of fluid flow on dendrite tip velocity Multicomponent alloys Dendritic array models Dendritic arm spacing and coarsening Primary spacing Secondary arm spacing The columnar-to-equiaxed transition Applications 188 References Eutectic solidification Classification of eutectics Cooperative eutectics Models for regular eutectic growth Models for irregular eutectic growth The unified eutectic growth model Divorced eutectics Interface stability of eutectics Equiaxed eutectic grain growth Solidification of cast iron Nucleation and growth of austenite dendrites Crystallization of graphite from the liquid Eutectic solidification The gray-to-white structural transition Solidification of aluminum-silicon alloys Nucleation and growth of primary aluminum dendrites Eutectic solidification Applications 240

11 xiv Contents References Peritectic solidification Classification of peritectics Peritectic microstructures and phase selection Mechanism of peritectic solidification The rate of the peritectic reaction The rate of the peritectic transformation Growth of banded (layered) peritectic structure Applications 261 References Monotectic solidification Classification of monotectics Mechanism of monotectic solidification 266 References Microstructures obtained through rapid solidification Rapidly solidified crystalline alloys Metallic glasses 276 References Solidification in the presence of a third phase Interaction of solid inclusions with the solid/liquid interface Particle interaction with a planar interface Material properties models Kinetic models Mechanism of engulfment (planar S/L interface) Particle interaction with a cellular/dendritic interface Shrinkage porosity The physics of shrinkage porosity formation Analytical models including nucleation and growth of gas pores Analysis of shrinkage porosity models and defect prevention 312 References Numerical micro-modeling of solidification Deterministic models Problem formulation Coupling of MT and TK codes Models for dendritic microstructures Microporosity models Stochastic models Monte-Carlo models Cellular automaton models Phase field models 355 References Atomic scale phenomena - Nucelation and growth Nucleation Steady-state nucleation - homogeneous nucleation Steady-state nucleation - Heterogeneous Nucleation Time-dependent (transient) nucleation 373

12 Contents xv 15.2 Growth Kinetics Types of interfaces Continuous growth Lateral growth Applications 379 References 382 Appendix A 383 Appendix B 385 Appendix C 391

13 NOMENCLATURE C, C o alloy composition p probability * CS interface composition in the solid q diffusion flux C L * interface composition in the liquid r radius (m) D species diffusivity (m 2 s -1 ) t time (s) E internal energy (J mole -1 or J m -3 ) v volume (m 3 ) F G Helmholtz free energy (J mole -1 or J m -3 ) Gibbs free energy (J mole -1 or J m -3 ) gradient v a atomic volume (m 3 atom -1 ) v m molar volume (m 3 mole -1 ) C o H enthalpy (J mole -1, J m -3, J kg -1 ) G v concentration difference between liquid and solid at the solidus temperature change in volumetric free energy (J m -3 ) I intensity of nucleation (m -3 ) H change in volumetric enthalpy (J m -3 ) J mass flux H f K curvature (m -1 ) S f permeability of porous medium (m 2 ) T latent heat of fusion (J mol -1, J kg -1, J m -3 ) entropy of fusion (J mol -1 K -1 or J m -3 K -1 ) undercooling (K) equilibrium constant (Sievert s law) T c constitutional undercooling (K) P pressure (Pa) T k kinetic undercooling (K) Péclet number T o liquidus-solidus interval (K)

14 xviii Nomenclature Q volumetric flow rate (m 3 s) T r curvature undercooling (K) R gas constant (J mol -1 K -1 ) Γ general diffusion coefficient T temperature (K or C) Gibbs-Thomson coefficient (m K) T L liquidus temperature (K) Φ phase quantity T S solidus temperature (K) α thermal diffusivity (m 2 s -1 ) S entropy (J mol -1 K -1 or J m -3 K -1 ) dimensionless back-diffusion coefficient V velocity (m s -1 ) β T thermal expansion coefficient (K -1 ) V o speed of sound (m s -1 ) β c solutal expansion coefficient (wt% -1 ) c specific heat (J m -3 K -1 ) γ surface energy (J m -2 ) f mass fraction of phase δ boundary layer, disregistry g volume fraction of phase ν kinematic viscosity(m 2 s) g, g gravitational acceleration (m s -2 ) vibration frequency h heat transfer coefficient (J m -2 K -1 s - 1 ) ρ density (kg m -3 ) k solute partition coefficient λ interphase spacing (m) thermal conductivity (W m -1 K -1 ) µ growth constant k B Boltzman constant chemical potential (J mole -1 ) l length (m) dynamic viscosity (N m -2 s) m slope of the liquidus line (K wt% -1 ) θ contact angle mass (kg) τ momentum flux n number of atoms (moles) superscripts subscripts het heterogeneous cr critical hom homogeneous e equilibrium m molar eut eutectic r property related to radius of curvature f fusion * interface g glass het hom heterogeneous homogeneous

15 Nomenclature xix subscripts subscripts E equivalent, eutectic i component, interface G gas k kinetic L liquid met metastable P particle, pressure n atoms per unit volume S solid r property related to radius of curvature T thermal s surface, stability c constitutional, solutal st stable v property related to volume