Modeling and Simulation of Microstructure Evolution in Solidifying Alloys (Mathematics and Its Applications)

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MODELING AND SIMULATION OF MICROSTRUCTURE EVOLUTION IN SOLIDIFYING ALLOYS

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MODELING AND SIMULATION OF MICROSTRUCTURE EVOLUTION IN SOLIDIFYING ALLOYS

by

Laurentiu Nastac

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

CD-ROM available only in print edition eBook ISBN: 1-4020-7832-3 Print ISBN: 1-4020-7831-5

©2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2004 Kluwer Academic Publishers Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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Dedication

To my lovely wife, Mihaela To my fantastic kids, Gabriel and Michael To others...

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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Contents

1. Length-Scales and Generations of Modeling Methodologies for Predicting the Evolution of the Solidification Structure 1.1 Introduction 1.2 Length Scales in Modeling of the Solidification Structures 1.3 Generations of Modeling Techniques, Modeling Capabilities and Limitations

1 1 1 2

2. Deterministic Macro-Modeling: Transport of Energy, Momentum, Species, Mass, and Hydrodynamics During the Solidification Processes 5 2.1 Introduction 5 2.2 A Macroscopic Model for Calculating Energy, Momentum, Mass and Species Transport 6 11 2.3 References Appendix: Methods for Coupling HT-SK Models 13 A2.1 Introduction 13 13 A2.2 HT-SK Model 13 A2.2.1 HT Model A2.2.2 Solidification Kinetics Model 15 15 A2.2.3 Stability Criterion A2.2.4 Coupling of HT-SK Models 16

3. Deterministic Micro-Modeling: Mathematical Models for Evolution of Dendritic and Eutectic Phases 23 3.1 Introduction

viii

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys 3.2 A Microscopic Model for Predicting the Evolution of the Fraction of Solid 3.3 Theoretical Analysis 3.3.1 Comparison between Hemispherical and Parabolic Growth 3.3.2 Comparison between Calculated and Experimental Growth Velocities of Dendrite Tip for Succinitrile 3.4 References

4. Stochastic/Mesoscopic Modeling of Solidification Structure 4.1 Introduction 4.2 Mesoscale Model for Dendritic Growth 4.3 Solution Methodology 4.4 Algorithm 4.5 Results and Discussion 4.6 References 5. Solute Transport Effects on Macrosegregation and Solidification Structure 5.1 Analytical Modeling of Solute Redistribution during Unidirectional Solidification 5.1.1 Introduction 5.1.2 Mathematical Formulation and Analytical Solution 5.1.3 Model Validation 5.1.4 Size of Initial Transient Region 5.1.5 Solid/Liquid Interface Instability of Dilute Binary Alloys 5.2 Numerical Modeling of Segregation 5.3 References 6. Micro-Solute Transport Effects on Microstructure and Microsegregation 6.1 Introduction 6.2 Dendrite Coherency and Grain Size Evolution 6.3 Deterministic Modeling of Microsegregation 6.3.1 Introduction 6.3.2 Models based on the “Closed System” Assumption 6.3.3 An analytical Model for Estimation of Microsegregation in Open and Expanding Domains 6.3.4 Partition Coefficient Evaluation 6.3.5 Predictions of Microsegregation in commercial alloys 6.3.6 Microsegregation Index (MSI) 6.4 Deterministic Modeling of Secondary Phases

23 26 26 26 28 29 29 32 34 40 41 51

53 53 53 54 58 60 61 65 72

75 75 75 78 78 79 82 91 95 97 99

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys 6.5 References 7. Probabilistic (Monte Carlo) Modeling of Solidification Structure 7.1 Fourth Generation of Solidification Modeling 7.2 Results and Discussion 7.3 References Appendix: Monte Carlo Program 8. Modeling and Simulation of Solidification Structure in Shaped and Centrifugal Castings 8.1 Shaped Castings 8.1.1 Prediction of Grain Structure and of Columnar-toEquiaxed Transition in Steel Castings 8.1.2 Gray-to-White Transition in Cast Iron 8.1.3 Modeling of Solidification Structure in Al-based Alloy Castings 8.1.4 Modeling of Solidification Structure in RS5 Alloys 8.2 Centrifugal Castings 8.2.1 Introduction 8.2.2 Model Description 8.2.3 Results and Discussion 8.2.4 Summary of the Parametric Studies 8.3 References 9. Modeling and Simulation of Ingot Solidification Structure in Primary and Secondary Remelt Processes 9.1 Introduction 9.2 Description of a Modeling Approach for Simulation of Remelt Ingots 9.2.1 A Deterministic Macroscopic Model for Calculation of Mass and Energy Transport in Cast Ingots 9.2.2 A Stochastic Mesoscopic Model for Simulation of Structure Evolution in Solidifying Ingots 9.2.3 Computational Aspects for Modeling of Remelt Ingots 9.2.4 Primary and Secondary Dendrite Arm Spacings in Commercial alloys (Deterministic Modeling) 9.2.5 A Stochastic Model for Modeling Secondary Phases During Solidification of Alloy 718 Ingots 9.3 Simulation Results for Some Commercial Applications 9.3.1 Modeling Parameters 9.3.2 Global Comparison of VAR, ESR, and PAM Processes

ix

107 109 109 111 112 113

115 115 115 117 128 128 136 136 138 143 145 148

151 151 153 153 156 164 165 166 170 170 173

x

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

9.3.3 VAR Process Modeling 9.3.4 ESR Process Modeling 9.3.5 PAM Process Modeling 9.3.6 Process Optimization 9.3.7 Alloy Systems and Solidification Maps 9.3.8 Prediction of Primary and Secondary Dendrite Arm Spacings 9.3.9 Stochastic Modeling of Secondary Phases 9.3.10 Experimental Technique for Composition Measurements and for Estimating the Secondary Arm Spacing in Ti-17 Alloy 9.4 References Appendix: Model Analysis A9.1 Sensitivity Analysis A9.1.1 Melt Rate Effects A9.1.2 Mesh Resolution - Columnar Growth A9.1.3 Mesh Resolution - Equiaxed Growth A9.1.4 Time Step A9.1.5 Impingement Factor - Columnar Growth A9.1.6 Impingement Factor - Equiaxed Growth A9.1.7 Nucleation - Columnar Growth A9.1.8 Nucleation - Equiaxed Growth A9.1.9 Summary A9.2 Grain Growth Analysis A9.2.1 Growth Parameter m - Columnar A9.2.2 Growth Parameter n - Columnar A9.2.3 Time Step A9.2.4 Growth Parameters for High Melt Rate A9.2.5 Growth Parameters for Base Melt Rate A9.2.6 Summary A9.3 CET Analysis A9.3.1 Melt Rate A9.3.2 Thermal Gradient A9.3.3 Nucleation - Columnar A9.3.4 Nucleation - Equiaxed A9.3.5 Time Step A9.3.6 Comparison with Experiments A9.3.7 Summary

175 177 181 183 184 188 192 203 207 211 211 211 212 212 212 218 219 219 222 222 226 226 227 227 230 232 233 234 235 235 237 238 238 238 245

10. Practical Techniques with Simulation Examples for Controlling the 247 Solidification Structure 247 10.1 Introduction 248 10.2 Electromagnetic Stirring

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys 10.2.1 Mathematical Formulation 10.2.2 Solution Methodology and MHD Model Validation 10.2.3 Results and Discussion for PAM-Processed Ti-6-4 Ingots (Experiments and 3D Computed Results with and without EMS) 10.3 Micro-Chilling in Steel Castings 10.3.1 Use of Steel Powder (Micro-Chills) for Efficient Superheat Removal 10.4 Ultrasonic Vibration 10.4.1 Ultrasonic Vibration Effects in Fluids 10.4.2 Modeling of Ultrasonic Vibration in Fluids 10.5 Modeling of Electromagnetic Separation of Phases to Produce In-Situ Composites 10.6 References

xi

249 251 252 253 259 259 262 262 264 265 267

Subject Index

269

About the Author

281

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Preface

The purpose of this monograph was to cover both the fundamentals and the state-of-the-art techniques used in mathematical modeling and computer simulation of the microstructure evolution during the solidification processes. Selected simulation results published in the last decade on the evolution of the solidification structures (both macro- and microstructures) will be presented throughout the book. The book is intended for graduate students and seniors interested in the science and engineering of solidification technology. It can also be used as a reference book for engineers in industry as well as researchers in academia and research institutes. A considerable research effort was done during the last decade in this area. Although a significant number of technical papers on solidification structure modeling were published in technical journals and conference proceedings in the last decade, just few book chapters have attempted to provide a systematic introduction to the modeling and simulation of solidification structure evolution. This book describes in detail some of the most commonly recognized state-of-the-art techniques in this field. The aim of the present book is to describe in a clear mathematical language the physics of the solidification structure evolution of cast alloys. The concepts and methodologies presented here for the net-shaped casting and the ingot remelt

xiv

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

processes can be applied, with some modifications, to model other solidification processes such as welding and deposition processes. Modeling and simulation of solidification structure evolution requires complex multi-scale computational areas, from computational fluid dynamics macroscopic modeling through mesoscopic to microscopic modeling, as well as strategies to link various length-scales emerged in modeling of microstructural evolution. Another aim of the book is to provide simulation examples of the solidification structure modeling in some crucial commercial casting technologies as well as to provide practical techniques for controlling the structure formation during the solidification processes. The simulation codes (PC versions) for 2D (includes fluid flow computations) and 3D mesoscale dendrite growth models (for 2D and 3D simple casting geometries) written in Visual Fortran 90 are provided in a CD attached to the book. The reader can use the dendrite growth codes to compute and visualize on the computer screen the evolving dendritic morphologies for his choices of material and process parameters. Finally, the author would like to thank his friends and colleagues that made writing this book possible, particularly, Prof. Doru Stefanescu for being my mentor and Ph.D advisor and Dr. Roxana Ruxanda for her valuable comments on this book. Laurentiu Nastac

1

LENGTH-SCALES AND GENERATIONS OF MODELING METHODOLOGIES FOR PREDICTING THE EVOLUTION OF THE SOLIDIFICATION STRUCTURE

1.1 INTRODUCTION Process modeling has become a viable tool to optimize the casting and solidification processes and is currently being applied to the remelt processes. Solidification structure is usually generated as part of a casting, welding or remelt process. Very often the solidification structure is the final structure of the component or it is the main contributor to the mechanical behavior of the final component. Accordingly, the mechanical properties of the manufactured parts, which are a direct outcome of the solidification structure, can be tailored through a controlled solidification process.

1.2 LENGTH-SCALES IN MODELING OF SOLIDIFICATION STRUCTURES A comprehensive modeling approach to simulate solidification phenomena in these processes should include computations for macroscopic mass, heat transfer, fluid flow, electromagnetic, and species transport to

2

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

provide temperature, velocity, and concentration fields as outputs. From the macroscopic temperature field the pool profile and pool size, as well as the shape and size of the mushy region, can be determined. In addition, from the concentration field, macrosegregation-related defects can be obtained. The approach should also include microscopic computations to predict the evolution of microstructure in solidifying ingots. The micromodel should compute the grain size and columnar-to-equiaxed transition (CET), as well as microsegregation. Lastly, the approach should also include computations at the dendrite tip length scale (mesoscopic scale) for prediction of dendritic morphology and microsegregation patterns. Figure 1-1 shows the length scales for this modeling approach.

1.3 GENERATIONS OF MODELING TECHNIQUES, MODELING CAPABILITIES AND LIMITATIONS To simulate the microstructure evolution during the solidification processes several modeling techniques were developed in the last 50 years. Their general classification is presented in Table 1-1.

Chapter 1. Length Scales and Generations of Modeling Methodologies

3

The first generation of modeling was started during the middle of this century, simultaneously with computer developments. It is a pure deterministic approach and it can be used to qualitatively predict the final solidification structure. The scale is 1 mm-1 m (macroscale). The second generation of modeling includes the calculation of solid fraction evolution at the micro-scale based on the solidification kinetics approach. This is a totally deterministic approach. The scale is (mesoscale). It cannot accurately predict the formation of the solidification structure. The second generation of computer models (so called macro transport-solidification kinetics codes) is commonly used to: (1) evaluate micro- and macro-segregation, (2) predict microstructural features in terms of amount and size only, (3) predict location and amount of shrinkage and gas pores, and (4) estimate of material properties. The third generation of modeling involves the calculation of structure evolution (including the calculation of the fraction of solid evolution) at the micro-scale based on probabilistic approaches. The transport of mass and energy is calculated at the macro-level with deterministic models. Because of geometrical complexity, the dendrite tip and interdendritic microsegregation is probabilistically calculated. The overall approach is stochastic. The scale is (microscale). The third generation of computer models is composed of two main components: (1) a deterministic macroscopic approach for macroscopic calculations and (2) a stochastic microscopic approach. This generation of modeling is applied to calculate both the formation of the solidification structure (amount, size, and morphology) and the fraction of solid evolution. The fourth generation of modeling involves the calculation of structure evolution at the micro-scale based on the probabilistic/stochastic approaches. The transport of mass and energy is calculated at the micro-level with a “direct” Monte Carlo approach. The overall approach is probabilistic. The scale is (microscale). The significance of the probabilistic approaches is that the simulated structures can be directly compared with the actual structures from experiments at two different scales: grain characteristics can be visualized at the macro-scale, while the amount, size, and distribution of secondary phases can be viewed at the micro-scale. A description of modeling techniques and their predictive capabilities is the aim of the present book. A summary of predictive capabilities is presented in Figure 1-2. Also, some recent developments and results obtained with the third and fourth generations of modeling techniques are summarized in this book.

4

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Another goal of this book is to demonstrate that probabilistic approaches are comprehensive and more accurate than deterministic approaches in predicting the solidification characteristics of various processed alloys. The major problem of deterministic models is shown in Figure 1-3, where it can be seen that the geometry of the micro-volume element in deterministic models is incapable of describing adequately the physics involved in nucleation and growth of microstructures. On the contrary, the macrovolume element in the stochastic/probabilistic models is usually divided in cubic micro-meshes with a typical micro-mesh size of thus, the mathematical models involved at this scale can correctly be solved and then linked with the macro-models. At present, the probabilistic models are mature enough to be used effectively by the manufacturing industry for process development as well as parametric design and optimization studies on microstructure management and control.

2

DETERMINISTIC MACRO-MODELING: TRANSPORT OF ENERGY, MOMENTUM, SPECIES, MASS, AND HYDRODYNAMICS DURING THE SOLIDIFICATION PROCESSES

2.1 INTRODUCTION

Understanding the precise role of each flow mechanism and their effects on heat, mass, and solute transport involves solving the conservation equations for mass, momentum, energy, and species in the geometry of interest. This is the first step required to predict the phase evolution and the associated phenomena in solidifying alloys such as segregation and shrinkage. The magnitude of thermosolutal convection of each particular system can be estimated by using the thermal and solutal Rayleigh numbers:

6

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where the subscripts T and S denote solutal and thermal quantities, respectively, g is the acceleration due to gravity, is the characteristic length scale, v is the kinematic viscosity, is the thermal diffusivity, is the coefficient of thermal expansion, is the coefficient of solutal expansion, is a characteristic temperature difference, and is a characteristic concentration difference. The Rayleigh number compares buoyancy forces to viscous forces. For small Rayleigh numbers, viscous forces should dominate over buoyancy forces, and the conduction regime is maintained with small and stable disturbances in the fluid. Negligible thermosolutal convection effects on macrosegregation should be encountered in this regime. Above a critical value for the Rayleigh number, buoyancy forces become important, disturbances grow, and a convection regime is established. Here, the effects of thermosolutal convection on macrosegregation become significant. For a binary alloy-system, the critical Rayleigh number is about 2000. Below this value, insignificant macrosegregation occurs [3]. In this chapter, a comprehensive macroscopic model for calculating energy, momentum, mass, species transport in solidifying alloys, is described.

2.2 A MACROSCOPIC MODEL FOR CALCULATING ENERGY, MOMENTUM, MASS, AND SPECIES TRANSPORT Using the assumptions of Newtonian, incompressible, and laminar flow, the governing equations for macroscopic transport in cylindrical coordinates (2-D axisymmetric geometry) are as follows [3]: (1) Conservation of mass:

(2) Conservation of momentum:

Chapter 2. Deterministic Macro-Modeling

where u and w are the velocity components, P is the pressure, average density of the control volume, is the viscosity, and friction drag sources in r and z directions, respectively, and buoyancy term given by the Bousinesq approximation as [4-8]:

7

is the are the is the

where is the average liquid concentration, is the reference temperature (equal to the liquidus temperature of the alloy, and is the reference concentration (initial melt concentration). The friction drag sources and are related to the rheology of the system. For equiaxed solidification, these sources can be calculated as follows:

and

where M denotes either eutectic (e) or dendritic (D) solidification, and and are the solid velocities. For eutectic solidification, the drag coefficients are:

where and

is the shape factor for nonspherical particles, is the liquid fraction, is the volume-surface mean diameter. The Equations (2-6) are

termed Ergun’s equations [7, 9, 10] and are appropriate for systems in which the liquid phase flows through the porous mush in the two-phase solid-liquid region. For dendritic solidification, the coefficient in Eq. (2-5) is much smaller than and it is neglected. In this case, the coefficient (here in Eq.

8

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

(2-5) is typically calculated at the level of secondary dendrite arm spacing as [1,2, 11]:

where

and Here, is the fraction of solid, is the solid/liquid (S/L) interface velocity of spherical instabilities, is one half of the average spacing between instabilities (at time t), the initial instability radius at time

is the instability radius at time t,

is

is a stability constant of the order of

is the intrinsic volume average liquid concentration,

is

the undercooling, is the liquid diffusivity, is the Gibbs-Thomson coefficient, is the liquidus slope, and k is the partition ratio. The Equations (2-1)-(2-8) represent the continuum conservation equations for phase change and convection and constitute the relative motion model [12, 13]. For systems in which there is no relative motion between liquid and solid, e.g. dendritic systems after the coherency, a viscosity function based on the solid fraction is used. In this case, the drag source terms defined in Eq. (2-5) are omitted. It will be shown later that the interphase transport terms in the energy and species equations (last two terms in Eq. (2-12) and Sc = 0 in Eq. (2-15)) are also dropped. This is the no-relative-motion model [12, 14]. For multiphase (eutectic, equiaxed dendritic, and columnar dendritic) solidification, both models should be used to describe the pressure drop in the mushy zone. Therefore, in such cases, a hybrid model as that described by Oldenburg and Spera [12] becomes appropriate. The hybrid model [12] uses permeability and viscosity functions to represent the complex behavior of the flow within the dilute mush (at very small solid fraction). At coherency-fraction of solid [1, 2, 11], the mush begins to behave more like a solid than a liquid. Coherency-fraction of solid is dependent on the morphology of the solid phase and can vary from about 0.05 (star dendrite) to maximum 0.8 (globulitic crystals) [11]. To ensure a smooth transition between the dilute and concentrated mush, two switching functions are used as follows [12]:

Chapter 2. Deterministic Macro-Modeling

where

9

is the coherency-fraction of solid. Thus, before coherency, the

interphase transport terms (last two terms in Eq. (2-12) and Sc = 0 in Eq. (215)) are turned off and the viscosity is related to the solid fraction through the following relation [12]:

After coherency, the interphase transport terms are turned on, and the permeability is a function of solid fraction through Ergun’s equations, i.e.,

and An alternative approach for the description of the rheology of the mushy zone was developed by Beckermann and Viskanta [14]. They used a twophase model that required the specification of both effective liquid and solid viscosities over the entire solidification interval. They made the following assumptions: (1) for very small solid fraction, i.e., the effective viscosities are described by:

and

and (2) for flow

through a rigid solid structure, such as packing of equiaxed crystals: and The transition in the macroscopic solid viscosity was obtained by using Krieger’s model (referred to in [14]) for the mixture viscosity of concentrated suspensions, i.e.:

and then solving for (3) The energy transport equation:

with

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

10

and

where T is the temperature, K is the thermal conductivity, is the density, is the specific heat, is the source term associated with the change of equiaxed and columnar phases, which describes the rates of latent heat evolution during the liquid/solid transformation, L is the latent heat of fusion,

and

are the solid fractions of the equiaxed, columnar,

and eutectic phases, respectively, is the liquid fraction, u and w are the superficial velocity components, and and are the solid velocities of the equiaxed and eutectic phases. The governing equations are written in terms of superficial velocities defined as:

Note that, for columnar solidification or after the occurrence of coherency in equiaxed solidification, the solid velocities are null. The continuum thermophysical properties (K, and are weighted by the solid fraction as follows:

(4) The species transport equation: Neglecting solid diffusion at the macro-scale level, the conservation of species can be written as follows:

with

Chapter 2. Deterministic Macro-Modeling

where

is the liquid diffusivity,

11

is the average liquid concentration,

is the average solid concentration, and is the average concentration within the elemental volume and is defined as:

where and are the concentration profiles in the solid and liquid, respectively, and can be obtained through the microsegregation model described in Ref. [11, 15]. The above governing equations are in the conservative form as recommended by Patenkar’s [16] for the numerical solution of heat, mass, and fluid flow problems. To solve these equations, it is required to know the competitive evolution of the solid fractions encountered in the present solidification system, i.e., eutectic, dendritic equiaxed, and dendritic columnar structures. The solidification-kinetics models for the calculation of the evolution of these solid fractions are presented in Chapter 3.

2.3 REFERENCES 1. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 4061-4074, 1996. 2. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 4075-4083, 1996. 3. L. Nastac, Numerical Heat Transfer, Part A, Vol. 35, No. 2, 1999, pp. 173-189. 4. D. R. Poirier, and J. C. Heinrich, in the Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes-VI, T. S. Piwonka (ed.), TMS, pp. 227-234, 1993. 5. B. Gebhart, Buoyancy-Induced Flows and Transport, Hemisphere Pub. Corp., 1988. 6. J. Szekely, Fluid Flow Phenomena in Metals Processing, Academic Press, 1979. 7. J. Szekely, J. W. Evans, and J. K. Brimacombe, The Mathematical and Physical Modeling of Primary Metals Processing Operations, John Wiley & Sons, 1988. Edition, Oxford Science 8. D. J. Tritton, Physical Fluid Dynamics, Publications, 1988.

12

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

9. G. H. Geiger and D. R. Poirier, Transport Phenomena in Metallurgy, Addison-Wesley Publishing Company, 1973. 10. R. W. Fahien, Fundamentals of Transport Phenomena, McGraw-Hill Book Company, 1983. 11. L. Nastac, Simulation of Microstructure Evolution during Solidification Processes, Ph.D. Thesis, The University of Alabama, Tuscaloosa, 1995. 12. C. M. Oldenburg and F. J. Spera, Numerical Heat Transfer, vol. 21B, pp. 217-229, 1992. 13. W. D. Bennon and F. P. Incropera, International Journal of Heat and Mass Transfer, vol. 30, no. 10, pp. 2161-2170, 1987. 14. C. Beckermann and R. Viskanta, Applied Mechanical Reviews, vol. 46, no. 1, pp. 1-27, 1993. 15. L. Nastac and D. M. Stefanescu, Met Trans, vol. 24A, pp. 2107-2118, 1993. 16. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, 1980. 17. M. V. K. Chari and S. J. Salon, Numerical Methods in Electromagnetism (Academic Press, USA, 2000). 18. P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Edition (Cambridge University Press, UK, 1996). 19. K. A. Hoffmann, S. T. L. Chiang, M. S. Siddiqui, M. Papadakis, Fundamental Equations of Fluid Mechanics (Engineering Education System, USA, 1996). 20. Fluent 6.0 User’s Guide Supplement, Magnetohydrodynamics Analysis, Fluent Inc., March 2002, Lebanon, NH, USA. 21. D. M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Met. Trans., Vol. 21A, pp. 997-1005, 1990. 22. L. Nastac and D. M. Stefanescu, Micro / Macro Scale Phenomena in Solidification, HTD-Vol. 218/AMD- Vol. 139, ASME, Aneheim, CA, pp. 27-34, 1992. 23. D. M. Stefanescu and C. Kanetkar, in the proceedings of the State of the Art of Computer Simulation of Casting and Solidification Processes, H.Fredriksson editor, Les Edition de Physique, Courtaboeuf, France, pp. 255-266, 1986. 24. M. Rappaz M. and P. Thevoz, Acta Metall., vol.35, pp. 1487-1497, 1987. 25. M. Rappaz, International Materials Reviews, Vol.34, No.3, pp 93- 123, 1989. 26. Ph. Thevoz Ph., J. L. Desbiolles, and M. Rappaz, Met Trans, Vol. 20A, pp. 311- 322, 1989.

Chapter 2. Deterministic Macro-Modeling

13

APPENDIX: METHODS OF COUPLING HT-SK MODELS A2.1 Introduction

The present section is a discussion of possible schemes for coupling the macroscopic heat transfer (HT model) with the microscopic solidification kinetics (SK model). The main problem in coupling SK and HT codes consists of incorporating the latent heat evolved during solidification, which is calculated through the SK model, in the HT model. The advantage of the HT-SK codes is that, unlike other traditional methods, the solidification path described by the evolution of solid fraction is not imposed a priori. This is of major importance since physically, the solidification path is the result of solidification conditions, and therefore should not be imposed. For example, the solidus temperature is not a material constant and depends upon the solidification path. Unfortunately, HT-SK codes cannot be validated against analytical solutions. The only alternative is to validate them against experimental data. The local evolution of the thermal field in castings can be represented through cooling curves. From these curves information on undercooling, recalescence and local solidification time can be extracted through Computer Aided-Cooling Curve Analysis (CA-CCA) [21]. Such validation was already performed for various types of cast irons and aluminum alloys. For instance, for the case of eutectic lamellar graphite iron, at normal cooling rates, some intergranular compounds, especially carbides and ternary eutectics can be sometimes produced at the end of solidification. Only the HT-SK models allow prediction of such events. Traditionally, the end of solidification is predicted through CA-CCA by determining the time of occurrence of the second minimum on the first derivative curve. This is incorrect [2], and a 5-10% error may occur in the evaluation of the solidification time and of the solidus temperature. A2.2 HT-SK Model A2.2.1 HT Model

A 2D cylindrical coordinates (axisymmetric geometries) heat transfer code used in this work is based on the Control Volume Method. The energy equation can be written as:

14

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where T is the temperature, t is the time, L is the latent heat of fusion,

is

the solid fraction, is the rate of latent heat evolved, and K, and are the thermal conductivity, density, and specific heat as functions of temperature, respectively. In the case of the Specific Heat Method (SHM) one has to include the evolution of latent heat in a modified specific heat term, as follows:

Solving these equations requires knowledge of initial temperature and boundary conditions. Details can be found in [22]. For higher values of the Biot criterion in the mold, i.e. for higher cooling rates, sometimes is a higher risk to omit the latent heat evolved during solidification. Therefore, the Temperature Recovery Method (TRM), schematically shown in Fig. 2-1, can be applied in the modeling of solidification in order to correctly recover the latent heat, especially for the higher cooling rates existing at the metal-mold interface.

In this model, for the first time step, the TRM was used. The first temperature under the equilibrium temperature, TE, which is is calculated from Eq. (2-17) without including the latent heat. This is accomplished by writing an integral energy balance for an elemental volume:

Chapter 2. Deterministic Macro-Modeling

15

From this equation the corrected temperature, is calculated as a function of TE, and Then, the temperature in the volume element is reset as A2.2.2 Solidification Kinetics Model

The heat source term

The term

in Eq. (1) can be calculated as:

can be calculated with appropriate nucleation and growth

laws as follows:

where N is the number of grains per unit volume (grain density), R is the grain radius and V is the growth velocity of the grain. It can be expressed as:

where the growth constant, can be either calculated or determined from experiments. The interface temperature, can be calculated from the overall energy balance within a grain of spherical shape [23]. Eq. (2-21) is valid under the same assumptions as proposed in [21]. It has the advantage that it accounts for the grain impingement, being weighted by the factor Then, solidification kinetics described by Eqs. (220)–(2-22) are coupled to the heat transfer equation described by Eq. (2-17). A2.2.3 Stability Criterion

Stability requirements are always associated with the explicit numerical analysis. The general stability criterion for cylindrical coordinates is derived

16

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

in Appendix A in [22]. When the Control Volume Method (CVM) with the heat source term in the implicit form is used, i.e. Specific Heat Method (SHM) or Enthalpy Method (EM), the required stability criterion is given by Eq. (A12) in [2]. When the heat source term is used explicitly, i.e. Latent Heat Method (LHM) or Micro-Enthalpy Method (MEM), the temperature derivative term is introduced as follows:

Inserting Eqs. (10) and (11) in (12) and then differentiating, the following equation is obtained:

Finally, the stability criterion is obtained by introducing Eq. (2-24) in Eq. (A12) from Appendix A in [22]:

where the terms L, R, U, and D are described in Appendix A in [22]. This stability criterion can be used not only with CV-FDM, but also with FEM forward difference or explicit Euler scheme (conditionally stable). A2.2.4 Coupling of HT-SK Models

The coupling between the macroscopic heat flow and the microscopic growth kinetics can be achieved through various schemes. Basically, the two different coupling schemes which are illustrated in Fig. 2-2 can be used. The most straightforward methods are those described in Fig. 2-2a. The Latent Heat Method has been used to calculate the solidification of various cast irons and Al alloys [21, 23]. The Enthalpy Method can be used in a similar manner. Formulating Eq. (1) with FDM or FEM, the variation of fraction of solid, between t and (time increment at macro level) at all nodes is computed based on the microscopic model of solidification, Eq. (2-21). The variation or can be derived explicitly or implicitly from macroscopic heat flow equations, while the variation is given explicitly at each node, at time t, in order to compute the new temperature field

Chapter 2. Deterministic Macro-Modeling

17

The source term Eq. (2-20), in the heat flow equation is coupling directly the heat transfer and solidification kinetics models. The disadvantage of this coupling scheme is that the time step increment necessary to solve the heat flow equation is limited by the microscopic phenomena. It has to be much smaller than the recalescence period in order to properly describe the microscopic solidification and to obtain a better accuracy in the prediction of local solidification times.

Eq. (2-25) can be successfully used, not only to satisfy the required stability condition for the explicit schemes, but also to correctly describe the microstructural features. Unfortunately, this condition is more severe then the classic stability criterion for explicit schemes (see Eq. (A12) in [22]). The Micro-Latent Heat Method (MLHM), and the Micro-Enthalpy Method, illustrated in Fig. 2-2b are based on the LHM and EM, respectively. The MEM has been first introduced in [24]. The MLHM has been presented for the first time in [22]. The essence of the MLHM and of the MEM is to

18

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

compute heat flow and microstructural evolution at two different scales. At the macro-level the heat flow is computed through one of the standard equations, without including the source term, A large time step, obtained from the general stability criterion, is used. Thus, the macro-enthalpy change, or the macro-temperature chance, can be obtained. Each of the macro-time steps is then divided into a number of small time steps at the micro-level, Again, the stability criterion, Eq. (2-25), is used. Once the variations of enthalpy, or temperature, at all cells are known, the solidification path can be independently computed. Assuming that the heat removal is made at a constant rate during the macro-time steps the enthalpy change, or temperature change, at the microlevel, are obtained after including the source term i.e. nucleation and growth kinetics. Finally, the new micro-temperature, and microstructural features can be obtained. The main advantage of this two time-step procedure is saving of computational time (CPU). Although the basic mechanisms of nucleation and growth are taken into consideration through solidification kinetics, the computing time is only 10% longer than that for standard heat flow calculation [25, 26]. Both MLM and MEM can be further modified as shown in Fig. 2-3. The macroscopic part remains unchanged. The microscopic part is changed. Maintaining the same assumption that the heat extraction per unit volume is constant during one macro-time step, SK calculation can be accomplished by simply integrating the microscopic models into the macroscopic heat flow calculation. The macro-variations or are not divided by the number of micro-time steps. Since the time step given by the classic stability criterion is small, it is not necessary to divide the heat extraction rate into other small values in order to maintain or to improve the convergence of the solution. The Specific Heat Method assumes an equivalent specific heat, Eq. (2), which requires precludes the possibility that

insuring that

This requirement

takes both negative and positive values,

required to predict recalescence. As shown in Fig. 2-4 some minor differences exist when using different coupling methods. MLHM and LHM are very close. Typically, HT-SK models produce larger maximum undercoolings than measured experimentally. Consequently, since LHM shows a smaller maximum undercooling than the other two methods, it can be considered more accurate. Also, since LHM does not include the

Chapter 2. Deterministic Macro-Modeling

19

additional assumption on the heat extraction, it is mathematically more accurate.

An overall comparison between the predictions of different parameters for the three basic coupling methods and the experiment is shown in Table 21, where TER is the temperature of eutectic recalescence and is the maximum undercooling. It is evident that all three methods give very close results. Nevertheless, as shown in Table 2-2, the CPU-time for the LHM scheme is about five times larger than that for MLHM or MEM. The difference in CPU-time between the MMLHM or the MLHM and the standard heat transfer method without SK is only 3 to 6% for the explicit FDM formulation. The MLHM, MMLHM, MEM, or MMEM can be used, in order to save CPU-time, not only with explicit FDM or FEM formulation, but also with implicit formulation. The bouncing effect phenomenon, often

20

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

reported by other investigators [26] in HT-SK calculation was basically nonexistent when the correct stability criterion was used.

Summary. (i) The Latent Heat Method is the most accurate method that can be used in HT-SK codes. The drawback of this method is a much longer computational time. (ii) Assuming a constant heat transfer throughout the micro-solidification path, the Micro-Latent Heat Method and MicroEnthalpy Method can be used with enough accuracy instead of the Latent Heat Method or Enthalpy Method. For example, in the case of eutectic gray iron, for cooling rates between 0.5°C/s and 10°C/s a maximum error of 0.3% was calculated in the prediction of recalescence temperature and solidus temperature, with respect to the LHM. A max. 0.5% error in the prediction of recalescence rate, solidification time, and temperature of eutectic undercooling was also obtained. On the other hand, these methods decrease five times the CPU time. (iii) The Specific Heat Method, especially in its explicit form, should not be used to calculate the evolution of latent heat during phase change in conjunction with HT-SK codes. The main disadvantage is that it cannot predict recalescence during eutectic solidification and, consequently, it results in higher error in predicting microstructural features. (iv) A stability criterion for explicit schemes within the phase transformation region, which allows prediction of recalescence, was proposed. This criterion precluded the occurrence of bouncing effects.

Chapter 2. Deterministic Macro-Modeling

21

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3

DETERMINISTIC MICRO-MODELING: MATHEMATICAL MODELS FOR EVOLUTION OF DENDRITIC AND EUTECTIC PHASES

3.1 INTRODUCTION The most common alloys used in practice, such as aluminum alloys, magnesium alloys, superalloys, steel alloys, and titanium alloys usually solidify with a dendritic structure. Below is a detailed description of a comprehensive deterministic model that can be used to predict dendritic growth kinetics in such alloys.

3.2 A MICROSCOPIC MODEL FOR PREDICTING THE EVOLUTION OF THE FRACTION OF SOLID The coupling between macro-transport and solidification kinetics is accomplished through the fraction of solid evolution that is described at the microscopic scale [1, 2]. This must be done for dendritic columnar, dendritic equiaxed, and eutectic equiaxed growth. For dendritic growth, the model

24

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

developed by Nastac and Stefanescu [1, 3] is used. For dendritic growth (both columnar and equiaxed), the evolution of the fraction of solid is expressed as [1, 3]:

where

is the liquid fraction,

is the solid fraction of the phase p, p

denotes a dendritic columnar, a dendritic equiaxed, or an eutectic phase, the geometrical factor is 3 for dendritic equiaxed growth and 2 for dendritic columnar growth, is the average growth velocity of the envelope (see Eqs. (3-2)–(3-5) in the following paragraphs), is the equivalent dendrite envelope, is the shape factor of the dendrite envelope, is the average growth velocity of the solid instability,

is the radius of the solid

instability, and is the shape factor of the instability. With the present model, three grain growth morphologies can be simulated: equiaxed dendritic, columnar dendritic, and eutectic. Nucleation and growth competition of the three grain morphologies controls the distribution and amount of phases. For the case of columnar dendritic solidification, the growth kinetics of the dendrite tip, is calculated with [3, 5]:

where is the liquid diffusivity, is the Gibbs-Thomson coefficient, k is the partition ratio, is the solidification interval, is the S/L interface undercooling, d is the mesh size, and is the instantaneous cooling rate. For equiaxed dendritic solidification, the model developed by Nastac and Stefanescu [1, 2, 3, 6, 7] is applied. Thus, the growth velocity of the tip is described by:

Chapter 3. Deterministic Micro-Modeling

where m is the liquidus slope, liquid thermal conductivity,

25

is the density, L is the latent heat,

is the

is the liquid interface concentration, and

The melt undercooling for the system under consideration can be calculated based on the following definition/assumption:

where

is the equilibrium liquidus temperature,

volume-averaged liquid concentration,

and

is the intrinsic

is the bulk temperature

defined as the average temperature in the volume element.

and

are calculated with the microsegregation model described in Ref. [4, 8].

The growth of equiaxed eutectic grains is calculated by [8-11]:

Note that the Eq. (3-1) is valid until impingement of the growing grains occurs. The first term in Eq. (3-1) involves calculations at the dendrite length scale, while the second term in Eq. (3-1) includes calculations at the instability length scale and describes both formation and coarsening of instabilities [1, 3]. The position of the equivalent dendrite envelope, is calculated until where the final grain radius is:

Here, and are the volumetric and surface grain density, respectively, and x and z are the coordinates of the microelement within the macrosystem. Note that the solution of the envelope growth velocity in Eq. (3-1) requires solute calculations at the micro-scale level for both interface and intrinsic volume average liquid concentrations. They are obtained by a complete analytical solution of the diffusion field in both liquid and solid phases. The microsegregation model was developed by Nastac and Stefanescu and is described in detail in Ref. [4].

26

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

3.3 THEORETICAL ANALYSIS 3.3.1 Comparison Between Hemispherical and Parabolic Growth

Although there is a significant difference between the hemispherical approximation and the parabolic solution in terms of Péclet (Pe) number [13], small discrepancies are observed when growth velocities vs. either tip radius or solutal Péclet number are plotted. The results are presented in Fig. 3-1. The range of cooling rates used in this analysis is between 0.1 to (that corresponds to the variation of the Péclet number from 0.001 to 1.0 and of the tip radius from 0.2 to Usually, the range of growth velocities encountered in castings is between 0.1 to This corresponds to a variation of Péclet number from 0.001 to 0.50 and of tip radius from 1 to The maximum solutal Pe number used in the present model is 0.5. For this range of Pe numbers the error is max. 5%. Because the Ivantsov’s solution of the tip velocity gives tremendous difficulties in multi-scale coupling and in solving the interface liquid concentration in parabolic coordinates, the hemispherical approximation for the tip was adapted in the present model. 3.3.2 Comparison Between Calculated and Experimental Growth Velocities of Dendrite Tip for Succinonitrile

In many contemporary casting solidification models, dendrite kinetics is calculated assuming a parabolic dendrite tip, while the diffusion field is calculated using spherical coordinates. This is not consistent. A more correct approach would be to use a hemispherical dendrite tip in conjunction with spherical coordinates for diffusion calculation. However, there is an ongoing discussion on the relative merits of the parabolic tip over the hemispherical tip. To verify calculation accuracy or lack of it when using a hemispherical tip, a classic experiment performed on succinonitrile was selected [10]. This particular experiment was conducted isothermally, in a large bath (infinite domain). Since Eqs. (3-3) and where R is the tip radius, describe the non-isothermal solidification into a closed system, two changes were made in these equations. First, the intrinsic volume average concentration of the liquid phase was assumed to be equal to consistent with the infinite domain. Second, the interface liquid concentration was obtained from the hemispherical approximation [1]. The thermophysical parameters of succinonitrile used in calculations are listed in [12, 13, 14]. The growth velocity for succinonitrile-0.07 mole % impurity (assumed to be acetone) calculated with the present modified model is compared with experimental values in Fig. 3-2. The model compares favorably with the

Chapter 3. Deterministic Micro-Modeling

27

experimental data, in particular in the region of moderate velocities, which are typical for castings. The discrepancy shown in Fig. 3-2 at low undercooling may probably be diminished by including thermal convection calculations [15].

28

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

3.4 REFERENCES 1. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 40614074, 1996. 2. L. Nastac and D. M. Stefanescu, Met Trans, vol. 27A, pp. 40754083, 1996. 3. L. Nastac, Simulation of Microstructure Evolution during Solidification Processes, Ph.D. Thesis, The University of Alabama, Tuscaloosa, 1995. 4. L. Nastac and D. M. Stefanescu, Met Trans, vol. 24A, pp. 21072118, 1993. 5. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, vol. 5, no. 4, pp. 391-420, 1997. 6. L. Nastac and D. M. Stefanescu, AFS Trans, Vol. 104, pp. 425-434, 1996. 7. L. Nastac and D. M. Stefanescu, in the Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes-VI, T. S. Piwonka (ed.), TMS, pp. 209-218, 1993. 8. L. Nastac, Mathematical Modeling of Equiaxed Dendritic Solidification-Second Generation of Computer Models, MS Thesis, The University of Alabama, Tuscaloosa, 1993. 9. D. M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Met Trans, vol. 21A, pp. 997-1005, 1990. 10. M. Rappaz, International Materials Reviews, vol. 34, no. 3, pp. 93123, 1989. 11. L. Nastac and D. M. Stefanescu, Micro/Macro Scale phenomena in Solidification, ASME, HTD-Vol. 218/AMD-Vol. 139, pp. 27-34, 1992. 12. J. Lipton, M. E. Glicksman, and W. Kurz, Met. Trans., Vol. 18A, 341-345, 1987. 13. W. Kurz and D. J. Fisher, Fundamentals of Solidification , 2nd ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1986 14. M. E. Glicksman, R. J. Schaefer, and J. D. Ayers, Met Trans, Vol. 7A, 1747-1759, 1976. 15. R. Ananth and W. N. Gill: J. of Crystal Growth, Vol. 108, pp. 17389, 1991.

4

STOCHASTIC/MESOSCOPIC MODELING OF SOLIDIFICATION STRUCTURE The stochastic/mesoscopic modeling is the most powerful and practical approach for simulating the evolution of microstructure during the solidification of castings. The description of a comprehensive stochastic mesoscopic model and typical results showing the capabilities of this model are presented in this chapter.

4.1 INTRODUCTION Dendritic growth is perhaps the most observed phenomenon in solidification of cast alloys. It is also the most studied phenomenon in solidification science. Nevertheless, because of its complexity, it is not yet well understood and further research would be required. In the last two decades, experimental techniques were developed to study dendritic solidification. The soundest experiments for studying the evolution of dendrite morphology in transparent materials were performed by Glicksman and his collaborators [2, 3] (see Fig. 4-1). Several analytical models starting with LGT’s model for equiaxed dendritic growth [4] (see complete list of references in [5]) were developed to calculate various microstructural features of solidifying materials including solidification interface morphology transitions (e.g., stable-to-unstable solid/liquid interface or planar-to-cellular-to-dendritic, columnar-to-equiaxed, etc.) and dendrite morphology parameters (e.g., dendrite tip radius, dendrite tip velocity, primary dendrite arm spacing, and secondary dendrite arm spacing). Also, numerical models were developed and used for simulation of dendritic growth in material processing. They include models based upon deterministic techniques such as “phase field method” or spline mathematics

30

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

approaches for simulation of dendrite morphology and microsegregation patterns, solidification-kinetics deterministic models for calculating the evolution of fraction of solid and dendritic grain size, and probabilistic/stochastic approaches such as Monte Carlo and Cellular Automaton (CA) techniques for simulating the evolution of dendritic morphologies [1, 6–18] (see also the list of references in [1, 6–18]. Other valuable tracking methods of the S/L interface, particularly deterministic approaches that can be used for modeling of dendritic growth are described in details in [19, 20]. The significance of stochastic/probabilistic approaches is that the evolution of simulated microstructures can be directly visualized and compared with the actual microstructures from experiments at two different scales: dendrite grain characteristics such as grain size and location and size of the columnar-to-equiaxed transition can be visualized at the micro-scale, while dendrite morphology (including dendrite tip, various dendrite arm spacings, microsegregation patterns) can be viewed at the mesoscale [9]. Stochastic approaches are mostly used because of their capabilities in simulating (a) the heterogeneous nucleation of grains which is continuous and of probabilistic nature; (b) the crystallographic effects, that is, the growth anisotropy (grain selection/preferential growth) and the probabilistic nature of grain extension; and (c) the nucleation and growth competition of various phases and morphologies (columnar and equiaxed grains, solidification defects, etc.).

Chapter 4. Stochastic/Mesoscopic Modeling

31

Phase Field Method is a popular approach that uses an implicit front tracking method to track a “diffuse” L/S interface. Although the curvature and anisotropy are directly included in the model, the method is computationally intensive and mesh dependent. Also, it would be difficult to account for all physical phenomena by using this method. The entropy form S and the evolution equation for were postulated in [22] as:

where s is the thermodynamic entropy density, is a function of the interface thickness is the phase field variable, e is the internal energy density, is the mesh size, and c is the concentration of the solute B in the solvent A. Examples of simulation results for Cu-Ni alloys are presented in Figure 4-2. Interesting features such as solute redistribution and growth competition can be seen in Fig. 4-2.

Level Set Method was developed for pure substances by Chen et al. [25]. This method uses an implicit Eulerian approach to track the sharp L/S interface. It requires curvature and anisotropy computations and it still computationally intensive but mesh independent. The equation of motion is [25]:

32

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where F is the speed function and is the level set function. At any time t, the front is at zero level set of The surface tension effects are shown in Figs. 4-3a and 4-3b for a hypothetical pure substance [25]. The initial seed was an irregular pentagon. The role of anisotropy can be revealed when comparing Figs. 4-3b (isotropic surface tension) and 4-3c (anisotropic surface tension). It can be seen from Fig. 4-3 that the surface tension has indeed a strong stabilizing effect of the L/S interface.

4.2 MESOSCALE MODEL FOR DENDRITIC GROWTH The mathematical representation of the dendritic solidification process of a binary alloy is considered in a restricted 2-D domain as shown in Fig. 4-4. Here, is the interface normal vector, is the mean curvature of the interface, and the curve represents the solid/liquid (S/L) interface which evolves in time and has to be found as part of the solution. The solidification of binary alloys is governed by the evolution of the temperature (T (x,y,t)) and concentration (C (x,y,t)) fields that have to satisfy several boundary conditions at the moving S/L interface as well as the imposed initial and boundary conditions on the computational domain. The equations that describe the physics of the solidification process are presented in the following paragraphs [27, 28, 29].

Chapter 4. Stochastic/Mesoscopic Modeling

Temperature (T) in

33

(heat transfer equation):

where t is time, is the density, K is the thermal conductivity, is the specific heat, L is the latent heat of solidification, is the liquid fraction, is the solid fraction, is the convective velocity in y direction, and x and y are the domain coordinates.

Concentration (C) in In the liquid phase

where

and

respectively.

(solute diffusion equation):

are the interdiffusion coefficients in the liquid and solid, in Eqs. (4-3) and (4-4) is used for studying

convection effects on dendritic growth and segregation.

34

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Local equilibrium at the S/L interface on interface):

(here, “*” means at

Solute conservation at the S/L interface:

where is the normal velocity of the interface and n denotes the normal to the S/L interface that is pointing into the liquid (see Fig. 4-4). The interface temperature (T * ) is defined as (assuming local equilibrium with both phases):

where

is the equilibrium liquidus temperature of the alloy,

is the

liquidus slope of the phase diagram, is the mean curvature of the S/L interface, is the Gibbs-Thomson coefficient, and is a coefficient used to account for growth anisotropy where is the growth angle (i.e., the angle between the normal and the x-axis) and is the preferential crystallographic orientation angle. In Eq. (4-8), second term in the right side is the constitutional undercooling and the last term in the right side is the curvature undercooling (that reduces the total undercooling at the dendrite tip, that is, has a stabilizing effect on the S/L interface). The interface temperature is also affected by the kinetic undercooling. The kinetic undercooling is not accounted for in this model since its effect becomes significant only at very high solidification velocities (i.e., in the rapid solidification regime).

4.3 SOLUTION METHODOLOGY The solidification process is governed by Eqs. (4-3) to (4-8) and a stochastic model for nucleation and growth. The numerical procedures for calculating the nucleation and growth, temperature and concentration fields as well as the growth velocity of the S/L interface during dendritic solidification are described in details below.

Chapter 4. Stochastic/Mesoscopic Modeling

35

A. Stochastic Model for Nucleation and Growth. The structure of the stochastic model is similar to that described in Ref. [1]. It consists of a regular network of cells that resembles the geometry of interest. The model is characterized by (a) geometry of the cell; (b) state of the cell; (c) neighborhood configuration; and (d) several transition rules that determine the state of the cell. In this work, the geometry of the cell is a square. Each cell has three possible states: “liquid”, “interface”, or “solid”. The selected neighborhood configuration is based on the cubic von Neumann’s definition of neighborhood, that is the first order configuration and it contains the first four nearest neighbors. Solidification behavior depends to a great extend on the transition rules. In this model, the change of state of the cells from “liquid” to “interface” to “solid” is initiated either by nucleation or by growth of the dendrites. At the beginning of the simulation, all cells are liquid, and their state index is set to zero. As nucleation proceeds, some cells become “interface” cells, and their index is changed to an integer larger than zero, n. The cells in contact with the mold wall are identified with a different reference index, m. The index is transferred from the parent cell to adjacent cells, as they become “solid” cells through growth. The integer accounts for the preferential growth of cubic crystals in the direction (for 2dimensions). For graphical representation, each integer has a color associated with it, and each cell is a pixel on the computer screen. Both crystallographic orientation and random location of the new dendrites are chosen randomly among 256 orientation classes that are the first 256 colors used for graphical representation. In 2-D calculations, the probability, that a newly nucleated dendrite has a principal growth direction in the range is given by

where takes into account the four-fold symmetry of the cubic crystal (see integral in Eq. (4-9)). The number of equiaxed and columnar dendrites, and that can nucleate in the volume of the liquid and at the surfaces (boundaries) of the mold (computational domain) during one time step, is calculated by using the nucleation site distributions, and respectively. Thus, assuming no solid movement in the liquid, nucleation rate is given by [1]:

where N is the number of nuclei, is the bulk undercooling, and is a nucleation parameter. The nucleation probabilities for “liquid” cells located

36

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

in the bulk of the liquid and at the surface of the mold nucleate during each time step are given by

where

and

and

to

are the number of cells in the bulk and at the mold

surface, respectively. During the time step calculation both the mold-surface cells and bulk cells are scanned and a random number, rand, is generated for each of them The nucleation of a “liquid” cell will occur only if or It is assumed that a nucleus formed at a particular location will grow based on the growth velocity of the S/L interface given by Eq. (4-7) and the neighborhood configuration rule previously described. As nucleation succeeds, a nucleated “liquid” cell will become an active “interface” cell (e.g., n or m > 0), and will grow until its solid fraction becomes one. Thereafter, the “interface” cell would capture the neighboring cells if a randomly generated number, rand, is smaller than the capture probability, defined as follows:

where takes values from to At the capturing time, the state index of the “interface” cell is transferred to the captured neighboring cells that will become “interface” cells. Then, the “interface” cell becomes a “solid” cell. Further, the same procedure is used until all “liquid” or “interface”cells become “solid” cells. B. Calculation of the Temperature Field. An implicit finite difference scheme (based on the successive over-relaxation method) is used to calculate the temperature of each cell as described by Eq. (4-3). This scheme does not impose any restriction on the time step used in calculations. An initial temperature higher than the equilibrium melting temperature of the alloy was assigned for all cells and convective boundary conditions (through the use of surface heat transfer coefficients) were used for surface cells. C. Calculation of the Time Step. Time step used in calculations is given by

Chapter 4. Stochastic/Mesoscopic Modeling

37

where a is the mesh size (uniform and constant for both x and y directions) and is the maximum growth velocity obtained by scanning the growth velocities of all “interface” cells during each time-step (see Eq. (414)). Equation (4-13) satisfies the conditions for both the explicit tracking scheme of the moving S/L interface and for explicitly calculating the concentration fields in both the solid and liquid phases. D. Calculation of the Concentration Fields in the Liquid and Solid Phases. An explicit finite difference scheme is used for calculating the concentration fields in the liquid and solid phases. This scheme is stable for the time-step condition shown in Eq. (4-13). Zero-flux boundary conditions were used for cells located at the surface of the geometry. The solution algorithm includes the “interface” cells by multiplying the concentration in the liquid by the liquid fraction and the concentration in the solid by the solid fraction of the particular interface cell. Also, during each time-step calculation and for each “interface” cell, the previous values of the liquid and solid concentrations are updated to the current values of the interface liquid and solid concentrations calculated with Eqs. (4-6), (4-7), and (4-14) to (4-17). E. Calculation of the Growth Velocity of the S/L Interface. For “interface” cells, the values of the interface velocities in the x and y directions are obtained from Eqs. (4-6) and (4-7). For the finite difference form is as follows (similar for

F. Calculation of the Fraction of Solid Evolution. Knowing the velocity components in both x and y directions, the solid fraction increment is calculated with [6, 7]:

Then, the solid fraction and growth velocity Eq. (4-15) as:

are computed based on

38

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where superscripts “n” and “o” denote new and old solid fraction values, respectively. G. Calculation of the Interface Liquid Concentration. from Eq. (4-8) as

The procedures for calculating

and

is calculated

are described below.

H. Calculation of the Interface Curvature. The average interface curvature for a cell with the solid fraction is calculated with the following expression:

where N is the number of neighboring cells. In the present calculations, N = 24, that includes all the first and second order neighboring cells. Equation (4-18), a modification of the method proposed in Ref. [21], is a simple counting-cell technique that approximates the mean geometrical curvature (and not the local geometrical curvature). The values of the curvatures calculated with Eq. (4-18) vary from a maximum of 1/a to zero for convex surfaces and from zero to a minimum of –1/a for concave surfaces. The curvature can also be computed with [26-29]:

where the unit normal

is derived from a normal vector

which is the gradient of From Eq. (4-19), the curvature in nonconservative form can be rewritten as [25]:

Chapter 4. Stochastic/Mesoscopic Modeling

where the partial derivatives of

39

are computed using centered finite

difference approximations. is computed from the first 8 neighboring cells only at gridpoints adjacent to the S/L interface. For

3-D

mesoscale

modeling,

the

3-D

curvature

in

nonconservative form can be calculated as:

where

is computed form the first 26 neighboring cells only at

gridpoints adjacent to the S/L interface. I. Calculation of the Anisotropy. The anisotropy of the surface tension (see Eq. (4-8)) is calculated from [6, 7]:

where is calculated with Eq. (4-9) and accounts for the degree of anisotropy. For four-fold symmetry, [22]. J. Reduction in the Mesh Anisotropy. Because of mesh anisotropy dendrites will grow aligned with the axis of the mesh or at 45 degrees independently of the initial crystallographic orientation, which can be called anisotropy in growth direction. Furthermore, dendrites aligned with the axis would have narrower tips than predicted by classic theories, which can be termed anisotropy in growth kinetics. To reduce the mesh anisotropy, Sanchez and Stefanescu [32] proposed the following formulation:

40

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

4.4 ALGORITHM The flowchart for the stochastic model is presented in Fig. 4-5, where T is the new temperature distribution, is the equilibrium liquidus temperature of the alloy, C and the new concentration distribution, is the current time step used in calculations, and IC and BC stand for initial and boundary conditions. The simulation software was written in Visual Fortran 90. The output of the model consists of screen plotting at any chosen time of C, T, or color indexes of all cells. Also, the final values of C, T, and color indexes of all cells are saved at the end of computations on the computer disk.

Chapter 4. Stochastic/Mesoscopic Modeling

41

4.5 RESULTS AND DISCUSSION Thermophysical properties of the alloys used in simulations are presented in Table 4-1. The grid size of the domain is unless otherwise specified. This grid size is fine enough to approximately resolve the dendrite tip radius that, in the present solidification conditions, is typically larger than Newton cooling boundary conditions were applied at the boundaries of the computational domain for multidirectional solidification simulations. For directional solidification simulations, Newton cooling boundary condition was applied only to the bottom of the computational domain. The surface heat transfer coefficient and convective velocity used in the present simulations is

and

respectively, unless

otherwise specified. Zero-flux solute boundary conditions were applied at the boundaries of the computational domain (i.e., a closed system was assumed). The initial melt temperature was assumed to be the liquidus temperature of the alloy under consideration (i.e., no superheat), unless otherwise specified. Also, an initial concentration equal to was assumed everywhere on the computational domain.

First, the effects of constitutional undercooling on the solute redistribution and on dendrite morphology were analyzed. A constitutional undercooling parameter, defined as is used here. Thus, using the material properties in Table 4-1, we can calculate the parameter A as follows: for Al-7%Si, A = 280, for Pb-10%Sn, A = 52, for IN718-5%Nb, A = 57, and for Fe-0.6%C, A = 93. A comparison of simulated equiaxed dendritic morphologies for the above alloy systems (shown at approximately similar total solid fractions) is presented in Fig. 4-6 for a domain, where the legend indicates the solute concentration levels.

42

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Chapter 4. Stochastic/Mesoscopic Modeling

43

Coarsening of secondary dendrite arms and growth of tertiary dendrite arms (or branching of secondary arms) can be observed in Figs 4-6c and 4-6f for Pb-10%Sn and IN718-5%Nb alloys, respectively. The secondary and tertiary dendrite arm spacings of IN718-5%Nb alloy are a bit smaller than those of Pb-10%Sn alloy. Note also the evolution of C segregation in the liquid phase for the Fe-0.6%C alloy in Figures 4-6g and 4-6h. Overall, a small effect of the constitutional undercooling parameter (A) on the morphology of equiaxed dendrites was observed. Nevertheless, it was observed from time-evolution of these systems that alloys with a smaller A (e.g., Pb-10%Sn and IN718-5%Nb) have faster dendritic growth than alloys with larger A (e.g., Al-7%Si and Fe-0.6%C). The morphological evolution of a single columnar dendrite in IN7185%Nb alloy is presented in Fig. 4-7. Solidification starts first with dendritic growth until the tip of the dendrite reaches the top of the simulated domain. Then, coarsening and some branching of secondary dendrite arms take place concomitantly with the dendritic growth process. The effects of the variation (one order of magnitude variation) in the (i.e., surface tension) on the evolution of Nb solute redistribution patterns and dendritic morphologies in IN718-5%Nb alloy system are shown in Fig. 4-8. Insignificant branching (i.e., growth of secondary/tertiary dendrite arms) takes place for the case of higher In Fig. 4-9, the competition between nucleation and growth of multiple columnar dendrites assuming unidirectional solidification of IN718-5%Nb alloy is presented. The strong growth competition from the sample bottom (10 dendrites) to 1/3 of the sample height (5 dendrites) to the sample top (2 dendrites) can be observed.

44

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Simulated microstructure (dendritic morphologies and CET formation) and Nb segregation patterns in multidirectional solidification of IN718-5 wt.% Nb alloy are shown in Fig. 4-10. Here, the competition of equiaxed and columnar morphologies controls the formation of CET. Last, the simulated microstructure (columnar cellular/equiaxed dendritic morphologies and CET formation) in unidirectional solidification of IN718-5 wt.% Nb alloy is presented in Fig. 4-11. To study the effect of the heat extraction rate (i.e., increase in the G/V ratio, where G is the thermal gradient in the mushy region and V is the growth velocity) a one order of magnitude increase in the value of h was used (i.e., In this case, columnar cellular growth occurs as opposed with previous cases where columnar dendritic growth was observed to take place. Note also from Fig. 4-11 the sharp CET that takes place at approximately one half of the sample height where the thermal gradient in the mushy region is about 3000 K/m. The CET is sharp because of faster growth of equiaxed dendrites that have nucleated in the undercooled liquid ahead of the columnar front. A comparison of the curvature models described by Eqs. (4-18) and (419) is provided in Fig. 4-12 for a Pb-10%Sn alloy. The dimensional curvatures, plotted in Fig. 4-12 as a function of the total solid fraction in the computational domain, were computed as the average of all cell curvatures in the solid phase and then normalized by cell size. Although large

Chapter 4. Stochastic/Mesoscopic Modeling

45

differences exist between the two curvatures at the onset of solidification, both curvatures converge to the same value thereafter.

46

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Figure 4-13 shows the evolution of a simulated equiaxed dendrite and of Nb concentration in IN718 alloy. The curvature model described by Eqs. (419) and (4-20) was used to compute the curvature undercooling. Note the evolution of Nb segregation in the liquid phase for the IN718-5 wt.% Nb alloy in Figs. 4-13a to 4-14c. Simulated microstructures (columnar/equiaxed morphologies and CET formation) and Sn segregation patterns during the unidirectional solidification of Pb-10 wt.% Sn alloy cast in microgravity conditions [30] are presented in Fig. 4-14. Note from Fig. 4-14 the sharp CET that takes place at approximately 2/3 of the sample height where the thermal gradient in the mushy region is less than about 3000 K/m. The CET is sharp because of faster growth of equiaxed dendrites that have nucleated in the undercooled liquid ahead of the columnar front. The influence of convection on dendritic growth and segregation patterns in IN718-5 wt.% Nb is shown in Fig. 4-15a. A convective velocity, mm/s, was used in computations and only the left-side wall was cooled off Note from Fig. 4-15a that the columnar dendrites grow opposing the convective flow direction. This is because convective flow changed the temperature and concentration fields in the liquid around each growing dendrite. Thus, solute is depleted at the S/L interface (i.e., dendrite tip) and since smaller solute content gives higher growth velocity of the S/L

Chapter 4. Stochastic/Mesoscopic Modeling

47

interface, the dendritic growth will be preferentially in the upstream direction. Interestingly, the deflection of the dendrite growth direction is also controlled by the strong competitive growth of the primary arms and secondary arms which are growing preferrentially in the upstream direction, as shown in Fig. 4-15a. This phenomenon is described in details in [31]. Also, SCN dendrites (see experiments in [3] and Fig. 4-1b) not growing parallel to gravity are influenced by the asymmetric flow field. It was also demonstrated in [2] that the fluid flow direction with respect to the dendrite growth direction affects both the morphology and stability of dendritic crystals. Figure 4-15b shows the effect of growth morphologies (columnar vs. equiaxed) on Nb segregation during unidirectional solidification of IN718 alloy. As expected, segregation of Nb is reduced during equiaxed growth, at least during the initial transient. In practice, alloy innoculation for enhancing equiaxed solidification and reducing equiaxed grain size can be addressed to decrease alloy segregation in castings. The legends in Figs. 4-6 to 4-10 and 4-13 to 4-15 display solute compositions in both the liquid and solid phases using the following nondimensional quantities: and

Thus, the solute concentration in the liquid

phase, can be displayed simultaneously with the solute concentration in the solid phase, The legend in Fig. 4-15 represents either Sn composition or dendrite color indexes. The legend in Figs. 4-11, 4-14, and 4-15 shows 256 color indexes (as 16 classes, where each class contains 16 colors) that are used to display dendritic morphologies.

48

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Figure 4-16 presents a simulation of equiaxed solidification of Al4wt.%Cu alloy with the improved model [32] that accounts for the reduction in the anisotropy of the mesh in the crystallographic orientation and in growth kinetics. The simulation is performed in a square domain, with three nucleated grains with different crystallographic orientations. Note that each dendrite can grow in its preferred crystallographic growth direction. Figure 4-17 shows the cooling curve recorded at the center of the domain and the evolution of the total solid fraction of the entire simulation domain presented in Fig. 4-16. Typical features for equiaxed solidification such as undercooling, recalescence, and final temperature drop at the end of solidification are correctly predicted by the model.

Chapter 4. Stochastic/Mesoscopic Modeling

49

50

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The computer memory and CPU-time requirements in the present stochastic calculations are discussed briefly below. A RAM memory size of 120 bytes/cell is needed in the present stochastic calculations. For example, to simulate the solidification microstructure and segregation patterns of the geometry shown in Fig. 4-11 (10mm x 20mm), a grid resolution of 500x1000 mesh size) was used. The RAM and CPU-time required to obtain the simulated results presented in Fig. 4-11 are 60 Mbytes and about 6 CPU hours on a PC-150 MHz (about 30 CPU minutes on a PC2GHz), respectively. The CPU-time for the computations is very competitive when compared, for example, with computations based on the phase-field approach. To simulate a single equiaxed dendrite with the phase-field

Chapter 4. Stochastic/Mesoscopic Modeling

51

approach, 7500 CPU seconds on the Cray Y-MP4E/232 was required [22]. The CPU-time of the current computations is reasonably close to classical stochastic CA models for simulating dendritic/cellular grains [9, 24].

4.6 REFERENCES 1. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, Institute of Physics Publishing, Vol. 5, No. 4, pp. 391-420, 1997. 2. M. E. Glicksman, E. Winsa, R. C. Hahn, T. A. Lograsso, S. H. Tirmizi, and M. E. Selleck, Met Trans, Vol. 19A, pp. 1945-1953, 1987. 3. M. A. Chopra, M. E. Glicksman, and N. B. Singh, Met Trans, Vol. 19A, pp. 3087-3096, 1988. 4. J. Lipton, M. E. Glicksman, and W. Kurz, Met Trans, Vol. 18A, pp. 341345, 1987. 5. W. Kurz and D. J. Fisher, Fundamentals of Solidification, 3rd ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1989. 6. U. Dilthey, V. Pavlik, and T. Reichel, Mathematical Modeling of Weld Phenomena 3, Eds. H. Cerjak and H. k. D. H. Bhadeshia, The Institute of Materials, pp. 85-105, 1997. 7. U. Dilthey and V. Pavlik, Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes-VIII, Eds. B. G. Thomas and C. Beckermann, pp. 589-596, 1998. 8. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 27A, pp. 4061-4074 and pp.4075-4084; 1996, Met Trans, Vol. 28A, pp. 1582-87; 1997, AFS Trans pp. 425-34, 1996. 9. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, J. of Metals, TMS, pp. 3035, March 1998. 10. J. A. Spittle and S. G. R. Brown, Ada Metall. Vol. 37, No. 7, 1989, pp. 1803-10 and J. Materials Science, Vol. 30, pp. 3989-3994, 1995. 11. H. W. Hesselbarth and I. R. Goebel, Acta Metall., Vol. 39, No. 9, pp. 2135-43, 1991. 12. M. P. Anderson, D. J. Srolovitz, G. S. Crest, and P. S. Sahni, Acta Metall., Vol. 32, No. 5, pp. 783-91, 1984. 13. G. S. Crest, D. J. Srolovitz, and M. P. Anderson, Acta Metall., Vol. 33, No. 3, pp. 509-20, 1985. 14. M. Rappaz and Gh. A. Gandin, 1993 Acta Metall., Vol. 41, No. 2, 345-60, 1993. 15. N. H. Packard, Proceedings of the First International Symposium for Science on Form, University of Tsukuba, Japan, November 26-30, 1985, Eds. S. Ishizaka, Y. Kato, R. Takaki, and J. Toriwaki, KTK Scientific Publishers, 1987.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

16. H. Pang and D. M. Stefanescu, Solidification Science and Processing, Eds. I. Ohnaka and D. M. Stefanescu, TMS, 1996. 17. D. M. Stefanescu and H. Pang, Canadian Metallurgical Quarterly, Vol. 37, No. 3, pp. 229-240, 1998. 18. I. Steinbach, F. Pezzolla, B. Nestler, R. Prieler, G. J. Schmitz, and J. L. L. Rezende, Physica D, Vol. 94, pp. 135-147, 1996. 19. W. Shyy, H. S. Udaykumar, M. M. Rao, and R. W. Smith, Computational Fluid Dynamics with Moving Boundaries, Taylor & Francis, 1996. 20. H. S. Udaykumar and W. Shyy, Int. J. Heat and Mass Transfer, Vol. 38, No. 11, pp. 2057-2073, 1995. 21. R. Sasikumar and R. Sreenivisan, Acta Metall., Vol. 42, No. 7, pp. 23812386, 1994. 22. J. A. Warren and W. J. Boettinger, Acta Metall., Vol. 43, No. 2, pp. 689703, 1995. 23. L. Nastac, Numerical Heat Transfer, Part A, Vol. 35, No. 2, pp. 173-189, 1999. 24. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, Proceedings of the International Symposium on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, Eds. A. Mitchell and P. Aubertin, pp. 145-165; 1997, Proceedings of the Fourth International Special Emphasis Symposium on “Superalloy 718, 625, 706, and Derivatives”, Ed. E. A. Loria, pp. 55-66, 1997. 25. S. Chen, B. Merriman, S. Osher, and P. Smereka, J. Comp. Physics, Vol. 135, pp. 8-29, 1997. 26. D. B. Kothe, R. C. Mjolsness, and M. D. Torrey, RIPPLE: A Computer Program for Incompressible Flows with free surfaces, Los Alamos National Lab., LA-10612-MS, Los Alamos, NM, 1991. 27. L. Nastac, Acta Materialia, Vol. 47, No. 17, pp. 4253-4262, 1999. 28. L. Nastac, Pacific Rim International Conference on Modeling of Casting and Solidification Processes (MCSP-4), Ed. C. P. Hong, Yonsei University, Seoul, Korea, September 5-8, 1999. 29. L. Nastac, Proceedings of the “Modelling of Casting, Welding, and Advanced Solidification Processes IX”, Engineering Foundation, Aachen, Germany, August 20-25, 2000. 30. L. Nastac and S. Sen, Influence of Gravitational Acceleration on the Segregation and Solidification Structure of Dendritic Alloys, NASA proposal, NRA-98-HEDS-05, 1999. 31.] K. Murakami, T. Fujiyama, A. Koike, and T. Okamoto, Acta Metall., Vol. 31, pp. 1425-1432, 1983. 32. L. Beltran-Sanchez and D. M. Stefanescu, Proceedings of the “Modelling of Casting, Welding, and Advanced Solidification Processes X”, TMS, Destin, FL, USA, pp. 75-82, May 25-30, 2003.

5

SOLUTE TRANSPORT EFFECTS ON MACROSEGREGATION AND SOLIDIFICATION STRUCTURE Solute transport during solidifying alloys is by diffusion and convection. Analytical and numerical modeling of macrosegregation and its effects on the solidification structure are discussed in this chapter.

5.1 ANALYTICAL MODELING OF SOLUTE REDISTRIBUTION DURING UNIDIRECTIONAL SOLIDIFICATION 5.1.1 Introduction

The importance of investigating solute redistribution during the dilute alloy solidification is broadly discussed in the literature [1-12]. One of the most important applications of this investigation would be the mathematical modeling of the equiaxed and columnar solidification [3-5, 7, 8]. This includes the instability of the solid/liquid interface. There are several other mass and heat transfer processes that involve the calculation of solute redistribution during directional solidification. For instance, this is the case of the power-down process used for making turbine blades, high temperature gradient liquid metal cooling furnace [6], various laboratory and industrial directional solidification processes, continuously cast processes which include the remelting processes [7, 8], Czochralski crystal growth technique

54

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

and floating zone techniques used for producing ingot diameters less than 10 mm.

5.1.2 Mathematical Formulation and Analytical Solution

The equation that describes the transfer of solute in the liquid region during the unidirectional solidification with an axially moving boundary is:

Here, t is time, z is the axial coordinate in the moving coordinate system, is the liquid concentration, is the diffusion coefficient of solute in the liquid phase, and W is the solid/liquid interface velocity in the z-direction. Equation (5-1) is solved in a semi-infinite domain together with the following boundary and initial conditions:

where

is the initial liquid concentration, k is the equilibrium distribution

coefficient, and and are the interface liquid and solid concentrations, respectively. Note that the flux balance at the moving boundary (Eq. 5-2) is used to correctly describe the time-evolution of the interface solid (or liquid) concentration. Equations (5-1) to (5-5) represent the mathematical formulation of the transient unidirectional solutal transport in the liquid phase with an axially moving boundary. To be in line with the quasi-steady state theory, this formulation also assumes no convection (fluid flow), constant moving frame velocity, no diffusion in the solid phase (e.g., no back-diffusion) and local equilibrium at the solid/liquid interface (Eq. 5-5). The assumption related to

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

55

the absence of convection (or dominant diffusive solutal transport) is particularly valid for either process involving a horizontal moving frame velocity or microgravity experiments. The solutions for the liquid concentration profile, and interface solid concentration,

calculated at z = 0, are as follows [12]:

Here, erf and erfc are the error and complementary error functions, respectively. Rearranging terms and introducing the coordinate z = Wt in Eq. (5-7), where z represents the distance measure from the beginning of the sample, we found the same solution as that obtained by Smith, Tiller, and Rutter (Eq. (10) in [13]). The liquid concentration gradient at the solid/liquid interface is calculated as

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

56

where is the time dependent concentration gradient at the solid/liquid interface. The transient and steady-state solutal boundary layer thicknesses, and are calculated with:

A “local boundary layer” can also be defined as:

where

is dimensionless time.

A comparison between

and

in Fig. 5-1 for k = 0.1. Small differences exist between

is shown and

at

any and k. However, at any and k, the ratio between the equivalent and local boundary layers for both the transient state and the steady state equals 2. Which boundary layer has a greater fundamental significance? Perhaps, based upon their definitions (e.g., Eq. (5-10) and (5-11)), the “local boundary layer” should be more correct than the “equivalent boundary layer” for studying local phenomena, such as interactions among diffusion fields of various dendrite tips or time evolution of various dendrite arm spacings.

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

57

The variation with time of the interface solid concentration is plotted in Fig 5-2 for different values of k. It can be seen that large differences exists between the transient and the quasi-steady solutions. The unsteady species transport tends to quasi-steady state with time, that is, for and The error between the unsteady state and quasisteady state calculations also decreases with time. The use of the conventional quasi-steady state solution for provides good accuracy (less than 5 % errors) (see Fig. 5-3). For instance, for k = 0.1,

at at and at The transient interface concentration gradient varies from at

to

at

The variation with time of the dimensionless interface liquid concentration gradient is plotted in Fig. 5-3 for different values of k. The absolute values of the interface liquid concentration gradients, can be found by multiplying the values of in Fig. 5-3 by The variation of the interface liquid concentration gradient during the unsteady state is significant, in particular for small k (k < 0.2). For instance, for k = 0.1, the interface liquid concentration gradient decreases with time (from until almost one order of magnitude. Also, the interface liquid concentration

58

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

gradient is positive for k > 1 and negative for k < 1, with and It is noteworthy to mention that for k > 1, the absolute value of the interface liquid concentration gradient decreases with time whereas for k < 1, it increases with time. At steady state, the higher k, the closer to one is

5.1.3

Model Validation

The validation of the analytical model described by Eq. (5-7) is performed against two well-performed experiments [20]: (1) Kagawa and Okamoto (14) used a floating zone melting technique to measure the redistribution of Si during the directional austenite-graphite eutectic solidification of Fe-C-0.56 wt.% Si alloys; and (2) Favier et al. [15] measured the redistribution of Bi during the directional solidification of Sn0.5 at.% Bi alloys by using the Mephisto-USMP1 microgravity experiments. The comparisons between the calculated and experimental results are presented in Figs. 5-4 and 5-5. The thermophysical parameters used in calculations are also shown in these figures. Si segregates negatively (k > 1) during the directional austenite-graphite eutectic solidification of Fe-C-0.56 wt.% Si alloys (see Fig. 5-4), while on the contrary Bi redistributes positively (k < 1) during the directional solidification of Sn-0.5 at.% Bi alloys (see Fig. 5-5). Both the and k are key parameters in modeling the solute redistribution during the directional solidification. They have opposite effects on the solute redistribution in the

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

initial transient state, that is, the higher

the smaller

59

and the higher k

the higher Therefore, results using at least two sets of values for k are presented in Figs. 5-4 and 5-5.

and

The calculated results compare closely with these well-performed experiments implying that: (i) the assumptions used in developing the mathematical formulation were correctly chosen, in particular the assumption related to the dominant diffusive solutal transport (i.e., noconvection) in the liquid phase; and (ii) the analytical solution presented in Eqs. (5-6) and (5-7) is accurate.

60

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

5.1.4 Size of the Initial Transient Region

The initial transition effects during the directional solidification in the absence of convection and solid diffusion occurs over a length that can approximately be calculated as:

where

is the length of the initial transient region and

is the time

to reach quasi-steady state. Equation (5-12) shows that both k and W have a destabilizing effect on the solid/liquid interface, that is, can be increased by decreasing both the value of k (related to the alloy type) and W. For example, for the case of Al-1.0 wt.% Cu, [3, 19] and [3], changes from Taking to 105 mm, when W varies from to From Eq. (5-12), the length of the initial transient region is inverse proportional with k, and therefore, for a

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

61

higher k a smaller time is required for reaching steady state. Furthermore, transposing the present theory to the unconstrained spherical growth theory (applicable for equiaxed solidification) it can be shown that the equiaxed grains would solidify during the initial transient region and the steady state growth region would rarely be attained. This is because the equiaxed grain size is usually less than and, as shown above, can reach In both the calculated and experimental results shown in Fig. 5-4 for FeC-0.56 wt.% Si alloy, steady-state is reached at approximately 20 mm from the base, while for Sn-0.5 at.% Bi alloy (Fig. 5-5), the initial unsteady state length is approximately 5 mm. Similar values for the length of the initial unsteady state are obtained with Eq. (5-12) for both cases presented in Figs. 5-4 and 5-5. 5.1.5 Solid/Liquid Interface Instability of Dilute Binary Alloys

The concept of constitutional undercooling (CS) [21] can be used to estimate the growth conditions where stability or instability can be expected for the solid/liquid interface of a dilute binary alloy during the unidirectional solidification at constrained velocity W. The CS criterion simply states that the presence of constitutional supercooling would correspond to morphological instability and its absence to morphological stability [2]. The appropriate equation that described mathematically the demarcation between the presence and the absence of constitutional supercooling is

where is the thermal gradient (see Refs. [1, 3, 22] for its definition) at the solid/liquid interface and is described by

Assuming that the initial unsteady interface concentration gradient calculated with Eq. (5-8), can replace in Eq. (5-13) the quasi-steady state gradient calculated with Eq. (5-14), the CS criterion can be extended to include the initial unsteady growth state. It was shown in [11]

62

that

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

can vary from

at

to

at

Further, a critical velocity from stable to unstable growth conditions, can be defined based on Eq. (5-13) as [12]:

Equation (5-15) represents an extended CS stability criterion of a planar solid/liquid interface during the solidification of dilute binary alloys in the absence of convection and solid diffusion. It shows that, during the initial stage of growth and for selected solidification conditions, an alloy that would solidify with an unstable solid/liquid interface based on the steady state CS criterion (of Chalmers et al. [21]), it would probably solidify with a planar interface based on the present calculations. A CS stability diagram based on Eq. (5-15) is presented in Fig. 5-6 for Al-1.0 wt.% Cu alloy. Experimental data obtained under microgravity conditions [23] and terrestrial conditions [23, 24] are also shown in Fig. 5-6. The experimental values were normalized to 1.0 wt.% Cu. The solid/liquid interface becomes more unstable under microgravity solidification conditions than under terrestrial conditions as no chemical mixing occurs in the liquid bulk because of reduced convection. In Fig. 5-7 the effect of the equilibrium partition coefficient, k, on interface instability is plotted. As expected, at small growth rates as encountered under normal solidification conditions, the increase of W would destabilize the solid/liquid interface. Also, a value of k close to one would provide a more stable interface. This is also shown by Eqs. (5-8), (5-12), (513), and (5-15). The effects of convection on interface instability during the solidification of alloys can be expressed through the variation of both k and The variation of the effective partition coefficient, with the growth rate was developed by Burton et al. [25] based on the classical boundary layer theory. Therefore, can be expressed as [25]: An effective liquid diffusion coefficient, was calculated by Saques and Horsthemke [26] for spatially periodic

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

hydrodynamic flows. Their main result is

63

where d is the

diameter of the tube and is the average flow velocity. Their result clearly demonstrates that the contribution from the fluid flow can be significant for liquid alloys, where is typically of the order of Note that CS stability criterion discussed above was developed based on thermodynamic considerations and a more consistent kinetic analysis of the instability phenomenon is considered by the theory of morphological stability [1-3]. The morphological stability condition of an interface in binary dilute alloys was first derived by Mullins and Sekerka [1]. They showed that for the quasi-steady state growth:

where

is the wavelength,

is the Gibbs-Thomson coefficient,

and f is the magnitude of the maximum value of a frequency-dependent stability function as defined by Eq. 21 in [1]. In fact, f includes the curvature effects on the modified constitutional supercooling criterion (CS).

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

64

A dimensionless parameter similar to [3], p. 198, as: S = f ( A * , k ) , where

is defined in It can be seen from

Eq. (5-16) that f [3], or S as shown in Ref. [3], has a stabilizing effect on growth (see also [4]). As also discussed in Refs. [1-3], for practical purposes, since (e.g., S > 0.9 ), it is probably safe to take f = 0 in Eq. (5-16) that is, Eq. (5-16) will be similar to Eq. (5-13). A thorough discussion on tests of the onset of interface instability is presented by Sekerka [2]. It is noteworthy to mention that Hecht and Kerr [27] have investigated the stable/unstable interface transition in Bi-Sn alloys and have found, contrary to previous findings, that the interface was more stable than predicted by either steady-state stability analysis or steady state CS. An important outcome of the stability analysis is the theoretical calculation of the dendrite tip radius at the limit of morphological stability. The dendrite tip radius, R, can be approximated as [2, 3]: where interface, and

is the wavelength of instability of the solid/liquid is a stability constant of the order of

By using

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

65

instead of a more accurate evolution of the dendrite tip radius would be predicted. Again, during the initial transient state, the magnitude of the dendrite tip radius is higher than that predicted by the steady-state theory, denoting a more stable solid/liquid interface. Warren and Langer [28] have attempted a transient analysis to predict dendritic spacings during the directional solidification. Their calculation for matches the experiments much closer than obtained from the steadystate Mullins and Sekerka linear stability analysis (see Fig. 4 in [28]), suggesting again a more stable solid/liquid interface in the unsteady state. Summary: The analytical model presented in this chapter can be successfully used to calculate the solute redistribution during the initial unsteady unidirectional solidification of dilute binary alloys. It was shown that the results obtained with the analytical model presented in Eq. (5-5 to 5-8) compare very well with experimental results for Fe-C-0.56 wt.% Si alloys, directionally solidified by using a floating zone melting technique, and with Sn-0.5 at.% Bi alloys, directionally solidified by using Mephisto-USMP1 microgravity experiments. One of the most important applications of this analytical model is the possibility of obtaining accurate values of some thermophysical properties of dilute binary alloys in the absence of convection, such as the diffusion coefficient of solute in the liquid phase and the equilibrium distribution coefficient, if the solute concentration profile in the solid is known from directional solidification experiments. An extended CS stability criterion of a planar solid/liquid interface during the solidification of dilute binary alloys in the absence of convection and solid diffusion has been derived (e.g., Eq. 5-15). It shows that, during the initial stage of growth and for selected solidification conditions, an alloy that would solidify with an unstable solid/liquid interface based on the steady state CS criterion, would probably solidify with a planar interface based on the present calculations. This may have important implications in solidification processing of binary alloy systems. This theory can further be used to study the morphological transition from the initial unsteady (unperturbed) growth to the dendritic (perturbed) growth during the dendritic solidification.

5.2 NUMERICAL MODELING OF SEGREGATION The numerical model presented in chapter 2 has been extensively validated against experimental data for cast iron [30], vacuum arc remelted (VAR) and electroslag remelted (ESR) alloy 718 ingots [31], Pb-26.5 wt. %

66

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Sn and Sn-16 wt.% Pb VAR ingots alloys [31-33], Sn-15 wt.% Pb and Al4.4 wt.% Cu ESR ingots [31, 34], and Pb-10 wt.% Sn unidirectional solidified ingots [31, 35]. Data required to perform the present simulations are shown in Tables 5-1 to 5-3. The sample geometry and boundary conditions are presented in Fig. 5-8. The sample dimensions are similar to those used in the existent NASA’s Isothermal Casting Furnace. Numerical calculations have been performed for thermal and solutal Rayleigh numbers corresponding to gravitational acceleration of 1 g, 0.1 g, 0.05 g, and 0.01 g. A gravity acceleration of 0.01 g is a typical value attainable in the NASA’s KC-135 aircraft. Some numerical results are presented in Fig. 5-9 and 5-10 for Pb-10 wt.% Sn alloy. Simulated Sn concentration (defined as the average total Sn concentration within the elemental volume, i.e., in Eq. 2-15) contours are shown in Fig. 5-9 for a normal gravity environment (1 g).

Chapter 5. Solute Transport Effects on Macrosegregation and Solidification Structure

67

For a low gravity environment (below 0.05 g), the macrosegregation tendency for Sn is very low. In this case, Sn concentration varies between 9.97 to 10.05 wt.% Sn. However, for a normal gravity environment (1 g), macrosegregation of Sn is large (see Fig. 5-9), that is, Sn concentration changes from 6.2 to 19.189 wt.% Sn (eutectic composition). Freckles, fingers (which are caused by plumes), and channels enriched in Sn as well as isolated pockets poor in Sn are seen in Fig. 5-9. Here fingers are defined as

68

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

continuous extensions of the boundaries, while pockets are discrete configurations. Some of the pockets originate from plumes, which break away from the boundaries (see the pockets beneath the plumes in Fig. 5-9). Freckles are long channels formed at the center and edges of the sample. Macrosegregation of Sn in freckles is high. Two large symmetrical Benard cells are formed for this geometry where the ratio between the height and width of the ingot sample is 2.

The studied case is a typical mode of convection which can develop when the liquid is compositionally buoyant and light but statically and thermally stable, i.e. when both and and for sufficiently low thermosolutal Rayleigh numbers. See also [37, 38] for the modes of convection that may develop during the upward solidification (cooling a liquid alloy from below). In this case, the net thermosolutal Rayleigh number (Ra in Eq. 1-1) is approximately for a gravitational acceleration of 1

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g and for a gravitational acceleration of 0.01 g. The selected characteristic length scale was the average mushy zone thickness that is about For the present simulation conditions, no finger-type convection develops for a gravity level below 0.05 g The effect of thermosolutal convection on the local liquid thermal gradients and liquidus isochrones are illustrated in Fig. 5-10a and 5-10b for both the normal gravity (1 g) and low gravity (0.01 g) environments. The liquid/mush interface (represented in Fig. 5-10a and 5-10b by liquidus isochrones) and local liquid thermal gradients are perturbed under 1g condition. No alteration of these solidification parameters is seen in Fig. 510b for a gravity acceleration of 0.01 g.

For a gravity level of 0.01 g, negligible convection takes places in this system. As insignificant back-diffusion occurs in the present case, we expect to see large longitudinal Sn segregation. However, for a gravity level of 0.01 g, negligible longitudinal Sn segregation exists. The rationalism of this phenomenon is explained in the following paragraphs. The initial transition effects during the directional solidification in the absence of convection and solid diffusion occurs over a length that can be calculated with the analytical

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

model developed by Nastac [11, 12, 20] as (i.e., Eq. 5-12). Here, can be increased by decreasing k and W. For Pb-10 wt. % Sn alloy, k = 0.31 and Using an average which is based on the geometry and process parameters shown in Fig. 5-8, equals Above this value, no macrosegregation occurs. The present numerical results for a gravity level below 0.01 g confirm this phenomenon. Note that the variation of Sn concentration during the initial transient cannot be captured by the present numerical model. This is because the length of the initial transient is considerably smaller than the mesh size In Fig. 5-10c, simulated solidification macrostructure is presented for a gravity level of 1 g. A similar appearance for macrostructure was obtained for low gravity environment (0.01 g). The stochastic model used for simulating these macrostructures is presented in details in [7, 8, 29, 39]. It uses the temperature history results obtained with the present numerical model. The resolution used in the stochastic calculations was of 1000x2000 with a uniform mesh size of The surface grain density used in calculations was of The grain size in Fig. 5-10c varies from approximately (measured at 2 mm from the sample bottom) to (measured at 5 mm from the sample top). Overall, the calculated grain size for a gravity level of 0.01 g was about 15 % larger than that obtained under normal gravity conditions (1 g). In the case presented in Fig. 5-10c, the solidification macrostructure is columnar dendritic. The columnar to equiaxed transition (CET) does not take place for the current solidification conditions, i.e., the temperature gradients are larger than and the average solid/liquid interface is greater than (see Fig. 5-10a). In this system, CET occurs for temperature gradients smaller than when the average solid/liquid interface velocity is less than (see [40] for the experimental results in this alloy system). Also, the columnar grain structure shown in Fig. 5-10c is, unlike the microstructure, relatively insensitive to fluid flow and local solutal gradients. The growth direction of the columnar grains is mostly controlled by the direction of the thermal gradients normal to the S/L interface and the grain size (e.g., nucleation and grain growth kinetics) is determined by the cooling rates in the mushy region and the grain selection mechanism (see also [7, 8, 39]). Summary: The magnitude of thermosolutal convection and therefore, macrosegregation intensity are directly related to the gravity level. In Pb-10 wt.% Sn alloy, the mode of convection is solutally unstable and thermally stable. For the geometry and process parameters presented in Fig. 5-8 and Table 5-1 to 5-3, the critical threshold value for gravity level is 0.01 g. This corresponds to a critical Rayleigh number of about Below this value, insignificant macrosegregation occurs during the unidirectional

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solidification of Pb-10 wt.% Sn alloy. This observation is in line with other numerical results [41]. For 1 g and sufficiently low thermosolutal Rayleigh number (approximately freckles, fingers (which are caused by plumes), and channels enriched in Sn as well as isolated pockets poor in Sn develop. Similar appearance for solidification macrostructures simulated in both normal and low gravity conditions was observed. However, the grain size for a gravity level of 0.01 g was overall about 15 % larger than that simulated under normal gravity conditions (1 g). The experimental hardware that can be used for the experimental validation of this work is the existent NASA’s Isothermal Casting Furnace (ICF) that has a temperature range of 373 K to 1623 K, and quenching rate capabilities of 1 to 50 K/s (the sample diameter is 10 mm). Typical ICF experiments are conducted under a low gravity environment (0.01 g for 25 s) during parabolic flights on board a NASA KC-135 aircraft (see [42, 43] for the experimental procedure). Lastly, an example of numerical calculation of macrosegregation in ESR alloy 718 ingots is presented in Fig. 5.11 [31]. The numerical model formulation [32] is similar with that in chapter 2. However, it does not include solidification-kinetics. As shown in Fig. 5-11, the numerical results are in line with the experimental measurements.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

REFERENCES

1. W. W. Mullins and R. F. Sekerka, J. Applied Physics 35, No. 2, p. 444, 1964. 2. R. F. Sekerka, Crystal Growth, ed. P. Hartman, North-Holland Publ. Co., p. 403, 1973. 3. W. Kurz and D. J. Fisher, Fundamentals of Solidification, Trans Tech Publications, 1986. 4. L. Nastac and D. M. Stefanescu, Met Trans 27A, p. 4061 and p. 4075, 1996. 5. L. Nastac and D. M. Stefanescu, Modelling Simul. Mater. Sci. Eng. 5, p. 391, 1997. 6. M. McLean, Directionally Solidified Materials For High Temperature Service, The Metals Society, p. 118, 1983. 7. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, Proceedings of the International Symposium on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, AVS, eds. A. Mitchell and P. Aubertin, p. 145, 1997. 8. L. Nastac, S. Sundarraj, and K. O. Yu, Proceedings of the Fourth International Symposium on Superalloys 718, 625, 706 and Derivatives, TMS, ed. E. Loria, p. 55, 1997. 9. A. V. Catalina and D. M. Stefanescu, Met Trans 27A, p. 4205, 1996. 10. L. Nastac and D. M. Stefanescu, Met Trans 24A, p. 2107, 1993. 11. L. Nastac, International Communications in Heat and Mass Transfer 5, No 3, p. 407, 1998. 12. L. Nastac, J. of Crystal Growth, Vol. 193, No. 1-2, pp. 271-284, 1998. 13. V. G. Smith, W. A. Tiller, and J. W. Rutter, Can. J. Physics 33, p. 723, 1955. 14. A. Kagawa and T. Okamoto, Metal Science, p. 519, 1980. 15. J. J. Favier, P. Lehmann, J. P. Garandet, B. Drevet, and F. Herbillon, Acta Mater. 44, No. 12, p. 4899, 1996. 16. D. R. Poirier and G. H. Geiger, Transport Phenomena in Materials Processing, TMS, p. 450, 1994. 17. J. D. Verhoeven and E. D. Gibson, Met Trans 2, p. 3021, 1971. 18. J. D. Verhoeven, E. D. Gibson, and R. I. Griffith, Met Trans 6B, p. 475, 1975. 19. J. L. Murray, Binary Alloy Phase Diagrams 1, ed. T. B. Massalski, ASM Intern., p. 141, 1990. 20. L. Nastac, Scripta Materialia, Vol. 39, No. 7, pp. 985-989, 1998. 21. J. W. Rutter and B. Chalmers, Can. J. Physics 31, p. 15, 1953. 22. W. A. Tiller, K. A. Jackson, J. W. Rutter, and B. Chalmers, Acta Metall. 1, p. 428, 1953.

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23. J. J. Favier, J. Berthier, Ph. Arragon, Y. Malmejac, V. T. Khryapov, and I. V. Barmin, Acta Astronautica 9, No. 4, p. 255, 1982. 24. M. H. Burden and J. D. Hunt, J. Crystal Growth 22, p. 99, 1974. 25. J. A. Burton, R. C. Primm, and W. P. Slichter, J. Chem. Phys. 21, p. 1987, 1953. 26. F. Sagues and W. Horsthemke, Physical Review A 34, No.5, p. 4136, 1986. 27. M. V. Hecht and H. W. Kerr, J. Crystal Growth, 7, p. 136, 1970. 28. J. Warren and J. S. Langer, Physical Review E 47, No. 4, p. 2702, 1993. 29. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, vol. 5, no. 4, pp. 391-420, 1997. 30. L. Nastac, D. M. Stefanescu, and L. Chuzhoy, Proceedings of the Modeling of Casting, Welding and Advanced Solidification ProcessesVII, M. Cross, M. and J. Campbell (eds.), TMS, pp. 533-540, 1995. 31. J. Chou, L. Nastac, S. Sundarraj, Y. Pang, and Kuang-O Yu, Experimental Evaluation and Computer Model Verification of Secondary Remelt Ingot Structures, Aeromat '98, Tysons Corner, VA, June 1998. 32. S. Sundarraj, L. Nastac, Y. Pang, and K. O. Yu, Proceedings of the Modeling of Casting, Welding and Advanced Solidification ProcessesVIII, C. Beckermann and B. G. Thomas (eds.), TMS, pp. 297-304, 1998. 33. S. D. Ridder, S. Kou, and R. Mehrabian, Met Trans, vol. 12B, pp. 435447, 1981. 34. S. Kou, D. R. Poirier, and M. C. Flemings, Electric Furnace Proceedings, pp. 221-228, 1977. 35. J. R. Sarazin and A. Hellawell, Met Trans, vol. 19A, pp. 1861-1871, 1988. 36. S. D. Felicelli, J. C. Heinrich, and D. R. Poirier, Met Trans, vol. 22B, 847-859, 1991. 37. A. W. Woods, J. of Fluid Mechanics, vol. 239, pp. 429-448, 1992. 38. H. E. Huppert, J. of Fluid Mechanics, vol. 212, pp. 209-240, 1990. 39. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, J. of Metals, TMS, pp. 30-35, March 1998. 40. R. B. Mahapatra and F. Weinberg, Met Trans, Vol. 18B, pp. 425-432, 1987. 41. J. C. Heinrich, S. Felicelli, P. Nandapurkar, and D. R. Poirier, Met Trans, vol. 20B, pp. 883-891, 1989. Pacific 42. J. L. Torres, D. M. Stefanescu, S. Sen, and B. K. Dhindaw, Rim International conference on Modeling of Casting and Solidification Processes, Beijing, pp. 190-197, 1996. 43. L. Nastac, S. Sundarraj, S. Sen, Thermosolutal Effects on Columnar-toEquiaxed Transition during Solidification of Castings, NASA Proposal (NRA-96-HEDS-02), 1997.

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44. L. Nastac, Numerical Heat Transfer, Part A, Vol. 35, No. 2, pp. 173-189, 1999.

6

MICRO-SOLUTE TRANSPORT EFFECTS ON MICROSTRUCTURE AND MICRO SEGREGATION

6.1 INTRODUCTION Microstructure is crucial in controlling the mechanical behavior of the final component. In this chapter, computer models and predictions of important microscopic features, such as dendrite coherency, grain size, microsegregation, and evolution of secondary phases, are presented.

6.2 DENDRITE COHERENCY AND GRAIN SIZE EVOLUTION Dendrite coherency occurs when dendrite tips of adjacent grains come into contact, i.e. when It is one of the most important parameters used to establish the rheology of a particular system. Since it is dependent on the evolution of the dendrite envelope, all factors that augment the evolution of envelope fraction, such as topology and movement of envelope, should reduce the dendrite coherency. The purpose of the following analysis is to evaluate the influence of some process and material parameters on the onset of dendrite coherency. The alloy selected for this analysis is a Fe-0.6 wt.% C. The data used in calculation are given in Table 6-1. These are typical data found for example in [1].

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The evolution of envelope fraction and internal fraction of solid during solidification of the Fe-0.6% C alloy for a cooling rate of 4 °C/s was calculated using the model for the growth of the star dendrite, described in [2]. The results are presented in Figs. 6-1 and 6-2. From Fig. 6-1a it can be seen that coherency is calculated to occur for an envelope fraction of one. For this particular cooling rate coherency occurs at 0.55 fraction solid, when the internal fraction of solid becomes equal to the fraction of solid. Figure 6-1b shows the dendrite coherency during solidification of the same alloy as a function of cooling rate. It is seen that, for the range of cooling rates selected for this analysis, the onset of coherency moves to higher fraction solid as the cooling rate decreases. To evaluate the influence of the diffusion model on the onset of coherency, the interface liquid concentration was calculated using Scheil, equilibrium, and microsegregation model described in [3]. Then, the model has been used to obtain data on the onset of coherency and on the solidus temperature, for these three different assumptions on micro-diffusion calculations. The results presented in Table 6-2 indicate that equilibrium calculation results in higher coherency, while Scheil predicts lower coherency. The model, as expected, predicts an intermediate coherency because it accounts for back diffusion.

In Fig. 6-2, the complex influence of the cooling rate on equiaxed dendritic growth is presented. The cooling rate was calculated immediately above the liquidus temperature. In Fig. 6-2a it is seen that, as the cooling rate

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77

increases, recalescence increases first, and then disappears. The cooling rate also affects grain size and the onset of coherency, as shown in Fig. 6-2b. The model has been incorporated into a commercial macro transport code for modeling of casting solidification (PROCAST). Computation details and experimental validation performed on Inconel 718 castings are described in [2]. Some of the results for IN718 are presented in Fig. 6-3.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

6.3 DETERMINISTIC MODELING OF MICROSEGREGATION 6.3.1. Introduction Assessment of microsegregation occurring in solidifying alloys is important, since it influences mechanical properties. This is especially true for cast crystalline materials. Also, a comprehensive theoretical treatment of dendritic growth requires an accurate evaluation of the solutal field

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(microsegregation) during solidification. Models for calculation of microsegregation differ by the problem they tackle as well as by the approach. Most models are restricted to the case of planar (plate) solidification (Fig.6-4a), or columnar solidification (Fig.6-4b), with the volume element over which the calculation is performed being selected between the primary or the secondary dendrite arms. One dimensional (1D) Cartesian or cylindrical coordinates solution has been proposed. When equiaxed solidification is considered (Fig.6-4c), a 3D problem (or at least 1D spherical coordinates) must be considered.

The majority of the current models are based on the “closed system” assumption, i.e. no net mass enters or leaves the domain during solidification. Some of the noteworthy models are shortly presented in the following section. 6.3.2. Models based on the “Closed System” Assumption The earliest description of solute redistribution during solidification by Scheil [4] involves several assumptions such as: negligible undercooling during solidification, complete solute diffusion in liquid, no diffusion in solid, no mass flow into or out of the volume element, constant physical properties, and fixed volume element (no dendrite arm coarsening). The equation that describes Scheil’s model is:

where is the interface concentration of solute in the solid phase, is the effective partition coefficient, is the solid fraction, and is the initial solute concentration.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

However, the diffusion of solute into the solid phase can affect microsegregation significantly, especially toward the end of solidification [5, 6, 7]. The models described in [5, 6] were used to explain microsegregation in Al-Cu and Al-Si alloys at cooling rates of up to 200 K/s [8]. Ohnaka [9] proposed models for plate (Fig. 6-4a) and “columnar” dendrites (Fig. 6-4b). Complete mixing in the liquid and parabolic growth was assumed. On the basis of an assumed profile, an equation for solute redistribution in the solid that includes the equation developed by Clyne and Kurz [7] was derived:

where is the interface concentration, is the primary dendrite arm spacing, n = 1 for plate, n = 2 for columnar, and is the local solidification time. Prior knowledge of the final solidification time is required. Note that for this equation reduces to the Scheil equation, and for it becomes the equilibrium equation. Ogilvy and Kirkwood [10] further developed the model developed by Brody and Flemings [6] to allow for dendrite arm coarsening in binary and multicomponent alloys. For binary systems the basic equation was:

Here, X is the distance solidified. Thus, The end term represents the increase in the size of the element due to arm coarsening, which brings in liquid of average composition that requires to be raised to the composition of the existing liquid. This equation was solved numerically under the additional assumptions of constant cooling rate and liquidus slope. Also, a correction factor for fast diffusing species was added. Kobayashi [11] obtained exact analytical solutions for the “plate” and “columnar” dendrite models. Diffusion in solid was calculated but complete liquid diffusion was again assumed. Solidification rate and physical properties, including partition coefficients, were considered constant. Linear solidification where is the final solidification time) was also assumed, which means that the motion of the interface was prescribed. The first order approximate solution derived by Kobayashi reduces to Ohnaka’s solution when applied to the interface. Calculations with the second order approximate solution were very close to the exact solution. The equation is:

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with

and The model has been further extended to include a thermal model of solidification, multicomponent alloys, and temperature dependence of diffusivity. Note that again for and Eq. (6-4) reduces to the Scheil equation. Also, for it becomes the equilibrium equation. One of the major disadvantages of Kobayashi’s solution is the large number of terms (20,000 for a required for convergence [13]. Matsumiya et al. [12] developed a 1D multicomponent numerical model in which both diffusions in liquid and solid were considered. Toward the end of solidification, especially for small partition ratios, lower liquid concentration than the analytical models was predicted. Yeum et al. [13] proposed a finite difference method to describe microsegregation in a “plate” dendrite that allowed the use of variables k, D, and growth velocity. However, complete mixing in liquid was assumed. Battle and Pehlke [14] developed a 1D numerical model for “plate” dendrites that can be used either for the primary or for the secondary arm spacing. Diffusion was calculated in both liquid and solid, and dendrite arm coarsening was considered. Further complications arise when multicomponent systems are considered. Chen and Chang [15] have proposed a numerical model for the geometrical description of the solid phases formed along the liquidus valley of a ternary system for plate dendrites. Constant growth velocity, variable partition ratios and the Brody-Flemings model for diffusion were used. A complete analytical model for ‘Fickian’ diffusion with temperatureindependent diffusion coefficients and zero-flux boundary condition in systems solidifying with plate, columnar or equiaxed morphology (Fig.6-4) was developed by Nastac and Stefanescu in [3]. The model takes into account solute transport in the solid and liquid phases and includes overall solute balance. The overall solute balance in integral form rather then the flux condition at the interface (time-derivative form) has been used for three reasons. First, it is more conservative than the interface mass balance, second, the solution does not require a prescribed movement of the interface, and third, unlike term-by-term differentiation of Fourier series, term-by-term integration is always valid. This model allows calculation of liquid and solid

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interface composition during and after solidification, for volume elements with plate, cylindrical or spherical geometry. Thus, it can be used to predict solute redistribution (microsegregation) for planar, columnar or equiaxed morphologies. The model in [3] and all other existing analytical microsegregation models are based on the “closed system” assumption, i.e. no net mass enters or leaves the domain during solidification. Such assumption may lead to erroneous calculations in many cases. Few models tackle coarsening and coalescence phenomena that occur during solidification. An analytical microsegregation model for open and expanding domains is summarized in the next section. The purpose of such a model was to establish the impact of the “open system” assumption (mass transport in and out of the element) and of the “expanding system” assumption (coarsening and coalescence) on microsegregation. 6.3.3. An Analytical Model for Estimation of Microsegregation in Open and Expanding Domains

Assumptions, governing equations and boundary conditions

Consider a macro-volume element within the solidifying metal. It can be for example, the mesh size of a macro-heat transfer model for casting solidification. Within this macro-element the temperature is assumed uniform and is obtained from the solution of the thermal field. The macroelement is further subdivided in a number of spherical micro-elements (Fig. 6-5a). Within each of these elements of radius a spherical equiaxed grain of radius

is growing until the whole volume is filled. If the particular

case of SG iron is considered, a graphite spheroid of radius within the spherical austenite grain of radius

The

is growing aggregate

solidifies by simultaneous growth of the graphite and austenite phases. The assumed geometry and the schematic solute concentration profiles developed in the solid and liquid phases for an element are presented in Fig. 6-5b. Micro-diffusion transport within the element starts concomitantly with solidification. Coarsening and/or coalescence are allowed to take place during solidification. The problem to solve is to calculate the composition profiles in both the solid and the liquid during solidification. Then, the microsegregation ratio

Chapter 6. Micro-Solute Transport Effects on Microstructure and Microsegregation

can be calculated. At the nucleation temperature,

83

the first solid

formed has a solute concentration k where k is the partition ratio. The liquid in the vicinity of the solid/liquid interface is either enriched in (if k < 1), or depleted of (if k>1) solute. The final liquid fraction will have a solute concentration that depends on the diffusion coefficients in the solid and liquid phases, growth velocity of the grain, coarsening and coalescence, as well as macro-convective flow through macrosegregation. The main assumptions of the model are as follows: (1) Solute transport in both phases is by diffusion with diffusion coefficients independent on concentration. Therefore, the double boundary problem must be solved for and The solute concentrations in the solid (S) and liquid (L) phases must satisfy Fick’s second law:

where, and are the diffusion coefficients in the solid and liquid phases, respectively, and m is an exponent (m = 2 for spherical geometry, m = 1 for cylindrical geometry, and m = 0 for plate geometry). (2) The material is incompressible and the densities in both phases are constant. (3) The solid-liquid interface is planar and under local equilibrium:

where the superscript * denotes values at the interface. From Eq. (2) two boundary conditions, unknown a priori, are obtained:

(4) There is solute flow into or out of the volume element considered (“open system”). Thus microsegregation calculation can include the contribution of mass transport by convective flow, and the effects of coarsening and coalescence. The overall mass balance for an open system can be written in integral form as:

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where

is the local mass average concentration over the volume

element, and is the volume of the element over which the mass balance is computed. Note that the volume of the element is varying in time. Coarsening and coalescence will impose an increase of this volume (“expanding system”). Assuming, for the sake of notation simplification, that the densities of the solid and liquid phases are not only constant, but also equal:

Then Eq. (6-9) is used to couple the concentration fields in both the solid and liquid phases. The boundary conditions for the finite open system are:

where is coarsening velocity. In the boundary condition for the liquid phase, the first term on the right hand side represents the influence of coarsening and the second term is the contribution of convective fluid flow on microsegregation. Due to the rapid liquid diffusion it is computationally convenient to include it in the flux boundary condition at rather than in the flux balance [16]. The flux boundary condition described by Eq. (610) is obtained by differentiating in time Eq. (6-9) and applying the Leibnitz’s integral formula for differentiation. In non-dimensional form, the flux boundary condition at is:

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(5) The initial concentration in the liquid, is constant for each volume element (micro-scale), but it is variable at the macro-scale level (casting). Finally, the solution of the coupled double boundary value problem can be obtained by solving Eqs. (1) with the boundary conditions described by Eqs. (6-6), (6-7) and (6-10). The liquid and solid solutal fields are coupled through Eq. (6-9). The following initial conditions are used:

where

is time when the local solidification starts.

Analytical solution

The solution of ‘Fickian’ diffusion for the interface solid concentration of a spherical element during solidification consists of the following equations:

with

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where:

where is the root of the equation The mathematical assumptions involved in derivation of Eqs. (6-13)-(6-16) are in line with those described in [3, 17]. They are valid for For the case of “closed system” and no coarsening, the microsegregation model (Eq. 6-13) reduces to that derived in [3]. Since no assumption on the evolution of the fraction of solid was used in the present derivation, any transformation kinetics model can be used to calculate the movement of the interface. Here, the fraction of solid is calculated through the heat transfer-transformation kinetics model for SG iron [18]. The radius of the austenite-liquid interface (austenite shell) is given by:

where

is the growth velocity of the austenite phase,

coefficient of carbon in austenite,

is the diffusion

is the concentration of carbon in

the austenite matrix at the graphite/austenite interface, is the concentration of carbon in the austenite matrix at the liquid/austenite interface, and

is the concentration of carbon in the liquid at the

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liquid/austenite interface. and are the radii of the graphite spheroid and of the austenite shell, respectively. They have been obtained in the previous time step of the numerical calculation. Coupling micro- and macro- segregation

The general methodology for coupling between micro- and macrosegregation (open system) is described in [19] for equilibrium (lever rule) and non-equilibrium (Scheil) cases. For limited diffusion in both solid and liquid phases, the link between micro- and macro- segregation done through use of the following equation:

where is the average macroscopic velocity over the volume element, and is the superficial velocity. Their origin is the shrinkage flow, the buoyancy flow (thermosolutal convection), and the relative motion flow of the liquid/solid interface. Eq. (6-18) assumes no diffusion at the macro-scale level. It is similar with that derived by Beckermann and Viskanta [20]. Note also that Eq. (6-18) represents an implicit link between micro- and macrosegregation. The concentration gradient in the liquid phase that contributes to the convective term (first term in Eq. (6-18)) is obtained from the macrosegregation model. The second term in Eq. (6-18) is used to solve the

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

microsegregation (see Eq. (11) in [23]) and is obtained through both the micro- and macro- segregation models. The governing equations for macroscopic transport include the conservation equations for mass, momentum, energies, and species for the geometry of interest. They are solved by the control volume method. Model evaluation

To evaluate the contribution of the “open system” and of coarsening on microsegregation a theoretical sensitivity analysis was performed. The variation of the interface liquid concentration and the fraction of eutectic were calculated for a hypothetical system using different process variables and assumptions with the model. The main data and variables used in calculation, and the calculated fraction of eutectics, are summarized in Tables 6-3 and 6-4. Constant growth and coarsening velocities for each case were assumed so that the solidification time, and the final grain radius, were the same. Other data used for calculation are: A linear variation with time for the local average concentration was assumed to allow for the solute to either enter or leave the domain. For instance, “Open+10%C+10%R” means that the local average concentration is linearly increased (solute enters the domain) from 100% (initial value at the onset of solidification) to maximum 110% (end of solidification), and 10% linear increase in the grain radius (coarsening) is allowed. The Scheil model (closed system, complete diffusion in liquid, no diffusion in solid) was used as a basis for comparison. It is apparent from Tables 6-3 and 6-4 and Fig. 6-6 that the results obtained with the Scheil model and the present model applied to the closed system are very close. Small discrepancies were observed at the end of solidification when the Scheil model predicted larger amount of eutectic than the proposed model. However, large differences in the prediction of both interface liquid concentration and fraction of eutectic were observed, when the “open system” assumption were used (Tables 6-3 and 6-4 and Fig. 6-7). This clearly emphasizes the importance of relaxing the “open system” assumption. When only coarsening and/or coalescence were introduced, small fluctuations in the results were observed. This is because both the size of the final domain and solidification time were maintained constant, such that the overall micro-diffusion was not much affected. Note that considering only coarsening and/or coalescence without allowing the solute to enter in the domain, may be misleading. Both coarsening and “open system” have to

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be considered. Then, the effect of coarsening and/or coalescence on microsegregation can be correctly analyzed. As shown in Tables 6-3 and 6-4, there are significant differences in the prediction of the fraction of eutectic between expandable and non-expandable domains. A possible explanation is that the partition coefficient may vary during solidification [22]. It may be dependent on composition and convection in the liquid phase. Thus, for accurate prediction of microsegregation it is necessary to have correct data for k.

Model validation

Validation of the model was performed using the experimental data of Boeri and Weinberg [21]. The diameter of the measured samples was 15 mm. The physical constants used for calculation are given in Table 6-5. The characteristic distance between equiaxed dendrite center and the last liquid to solidify, that is the final radius in this model is 75 mm, and the local solidification time is 30 s [21]. Since the eutectic temperature range is small, the diffusion coefficients were considered to be independent on

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

temperature. The diffusivity values at the equilibrium eutectic temperature were used. The redistribution of Mn and Cu in SG iron within a spherical element containing one graphite spheroid was calculated with the proposed model for the open and closed system cases. The results are compared with the experiments [21] in Fig. 6-8. Manganese produces normal microsegregation (k < 1), while copper produces inverse microsegregation (k > 1) during solidification of SG iron. For copper the model predicts lower concentration than the experimental values.

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91

As shown in ref. [22], much better agreement was obtained for copper redistribution when a variable k was introduced. Unfortunately, the equation for the variation of k [22] was strictly developed for the closed system assumption and could not be used here. As can also be seen from Fig. 6-8, similar results are observed for redistribution of Cu and Mn for both open and close system assumptions. This is because the convective flow, and therefore macrosegregation developed during solidification of SG iron thin castings (15 mm ID), were small, while the heat extraction rate was high. 6.3.4. Partition Coefficient Evaluation

Calculated microsegregation is very sensitive to the value of the partition coefficient used. Thus, for accurate prediction of microsegregation

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

it is necessary to have correct data for Unfortunately, limited experimental data on partition coefficients are available [24]. As emphasized by Kagawa and Okamoto [24], the partition coefficient is not constant during solidification. It is dependent on composition and convection in the liquid phase. Thus, the calculated equilibrium partition coefficient for silicon in the austenite-graphite eutectic (ternary system-Fe-C-Si) is expressed as follows [24]:

where is the silicon liquid concentration at the interface. Also, Burton et al. [25] have derived based on boundary layer theory the dependence between effective partition coefficient and growth rate under the assumption that the solid-liquid interface is planar. This is represented by the following equation:

where is the thickness of the boundary layer and it has to be determined from experiments. Kagawa and Okamoto [24] have experimentally evaluated the thickness of the boundary layer for silicon to be for and growth velocities of up to Thus, using in Eq. (6-21) and this experimental value for can be calculated with Eq. (6-22). It is true that can be directly calculated from experimental data if both the solid and the liquid concentrations are measured during solidification. Nevertheless, these measurements can be unreliable, in particular toward the end of solidification, because in complex alloys compounds precipitation is possible. From the experiments [21] used for validation in this study, only data on the interface solid concentration of silicon were available. The effective partition coefficient of silicon can be obtained through manipulation of Eq. (6-14). Thus, is the solution of the following quadratic equation:

The effective partition coefficient obtained with Eq. (6-22) is plotted in Fig. 6-9 against that obtained from experimental data [21] and Eq. (6-23). It

Chapter 6. Micro-Solute Transport Effects on Microstructure and Microsegregation

93

can be seen that the evaluated from experimental data on commercial SG iron is smaller than that calculated for ternary alloys. If the interface solid concentration is not available, the liquid interface concentration measured in the quenched region can be used. Then, can be calculated with [22]:

For example, in the case of copper redistribution in SG iron, the following regression was obtained with Eq. (6-24) when using the experimental data obtained by Boeri and Weinberg [21] for the liquid copper concentration at the interface,

Note that Eq. (6-25) is strictly valid only for the given experimental conditions. However, similar equations can be obtained with Eq. (6-24) for different conditions. Figure 6-10 shows a comparison between calculated with Eq. (6-25) and directly obtained from experimental liquid and solid interface concentrations. It is seen that linearly increases during solidification. As can be seen from Fig. 6-1 1a, when a constant obtained from the literature was used, the model predictions were in limited agreement with the experimental data. However, when calculated with Eq. (6-25) was introduced in the microsegregation model for SG iron much better agreement was obtained, as shown in Fig. 6-11b for copper redistribution.

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For systems where microsegregation results in the solidification of a eutectic, can be calculated from Eq. (6-14) as follows:

Chapter 6. Micro-Solute Transport Effects on Microstructure and Microsegregation

95

where eutectic fraction is and is the eutectic composition as given by the phase diagram. Then, if is experimentally evaluated, Eq. (626) can be used to calculate for the particular solidification problem corresponding to the experiment. For systems where no second phase precipitates, can be calculated from Eq. (6-24) for if the maximum interface composition at the end of solidification is known. The effective partition ratio for an Al-4.9 wt.% Cu alloy was calculated with Eq. (6-26) using the diffusivities in Table 6-6 and the experimental data of Sarreal and Abbaschian [26] given in Table 6-7. Plate geometry and constant growth velocity were assumed. is one half of the dendrite arm spacing. The results are given in Fig. 6-12 for various cooling rates. As expected, is not affected significantly in the range of low cooling rates On the contrary, for cooling rates above 1 K/s, increases rapidly.

6.3.5. Predictions of Microsegregation in Commercial Alloys A comparison of predictions of solute redistribution of niobium in Inconel 718 by various models is presented in Fig. 6-13. These models include the Scheil, Brody-Flemings, Clyne-Kurz and Ohnaka models for plate elements, the Ohnaka model for “columnar” dendrite, and the newly proposed model for spherical and plate geometry. Also, the time-dependent and quasi-steady state solutions of the proposed model are compared. The values of the physical parameters used in calculations are given in Table 6-8.

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For this case, the quasi-steady state solution seems to give poor results toward the end of solidification. However, it can be seen that up to 0.9 fraction of solid, the quasi-steady-state solution almost coincides with the time-dependent one. Different models predict different composition profiles and final fraction of Laves phases. Since experimental data on the composition profile were not available for this case, validation can be done only against the final fraction of Laves phase. The proposed model for spherical geometry predicts 1.2% Laves, which is in the experimental range of 0.38 to 1.44% by volume Laves phase, measured by Thompson, et al. [27]. All other models predict higher fractions. Thus, it appears that predictions by models assuming complete diffusion in liquid and plate element, applied to equiaxed geometry, give correct results only accidentally. To assess the validity of the new model not only in terms of final fraction of phase but also as to its accuracy in predicting the composition profile, calculated Nb redistribution for Inconel 625 was plotted in Fig. 6-14 against experimental data from directional solidified samples obtained by Sawai et al. [28] (see Table 6-8). Cylindrical geometry and constant growth velocity were assumed. Comparison was made with samples quenched from 1593 K, which is above the solidus temperature of 1523 K, and with samples quenched 2000 s after solidification, from 1423 K. Reasonably good agreement was obtained. It is considered that the agreement can be improved if an accurate transverse growth velocity can be used. Note that existing

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97

analytical microsegregation models cannot be used for calculation after the end of solidification, since they rely on solidification time. Additional validation was attempted for microsegregation of Mn and P in a plain carbon (0.13 wt.% C) steel directionally solidified with a columnar structure. Experimental data after Matsumiya et al. [12] were plotted in Fig. 6-15 together with calculated data using the Scheil equation and the proposed model. Cylindrical geometry and constant growth velocity were assumed for the model. The data used in calculation are given in Table 6-8. Experimental data were available for two different cooling rates. At the slower cooling rate of the dendrites in the microstructure did not exhibit clear patterns for secondary arms. Accordingly, calculations were performed for the primary arm spacing When the cooling rate was increased to the dendrites developed clear secondary arms. Thus, calculations with the model were performed for the secondary arm spacing according to the experimental measurements. Unlike the Scheil model, the data obtained with the proposed model fit well the experimental results. 6.3.6. Microsegregation Index (MSI) The microsegregation intensity (defined here as microsegregation index or MSI) will increase with solidification velocity and length scale and will decrease with solidification time liquid and solid diffusion coefficients, and with the partition coefficient (k) as follows [2]:

where is the Fourier number, is the cooling rate, A incorporates material properties, and a, b, c, and n are experimental parameters. Thus, the MSI defined by Eq. (6-27) includes both material and process parameters. The individual effect of grain size and solidification time on MSI number as well as their combined effects is illustrated in Fig. 6-16. As shown by Eq. (6-27), MSI is the product of cooling rates,

and

At low

is very small and the diffusion process is controlled by

the solidification time. Because increases with cooling rate, MSI also increases, which will result in increasing the amount of secondary phases (for example, Laves phase in alloy 718). At high cooling rates,

is very

small, and the diffusion process is mostly controlled by the grain size

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

variation. Hence, MSI will decrease with increasing cooling rate, which results in a lower amount of Laves phase. As also shown by Eq. (6-37) and Fig. 6-16, a critical cooling rate exists at which maximum segregation and thus maximum amount of secondary phases form. This has major practical importance because by manipulating process parameters, it should be possible to avoid significant amounts of secondary phases in the microstructure.

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99

6.4 DETERMINISTIC MODELING OF SECONDARY PHASES In this section, a deterministic approach to simulate the formation of secondary phases is presented. A stochastic modeling approach of secondary phases is described in chapter 9.3.9.

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The importance of modeling of NbC/Laves formation in Inconel 718 is broadly discussed in literature [29-36]. It is also known that the distribution of carbides, Laves phases, and microporosities in alloy 718 are affected by the solidification path. The volumetric solid fraction is a function of the local growth velocity, the solidification time, the solidus temperature, the local temperature gradient at the freezing rate, etc. The redistribution of elements strongly affects the phase evolution of common superalloys with respect to temperature, as well as their mechanical properties and surface stability at elevated temperature. Previous studies on alloy 718 showed that both NbC and Laves produce intergranular liquid films due to intergranular distribution of Nb and C [33, 34]. Also, the ability of Laves to promote intergranular liquation cracking (microfissuring and hot cracking) during heat treatment is much higher than that of NbC. This is because the temperature of Laves phase formation is usually lower than that of NbC, i.e. liquation initiates at the eutectic-Laves temperature. Thompson et al. [33, 34] demonstrated that the carbon content of alloy 718 directly affects the volume fraction of carbides. This is readily explained by a pseudo-ternary phase relationship during solidification [33]. A schematic of an isothermal section through the space diagram of the pseudo-ternary just above the ternary eutectic is represented in Fig. 6-17a. The alloy in as-cast condition could contain a higher volume fraction of NbC and Laves phase than what the phase diagram suggests due to the microsegregation during solidification. The relative volume fractions of NbC and Laves depend on the C/Nb ratio. Alloys with high C/Nb ratio will have a higher volume fraction of carbide than alloys with low C/Nb ratio. The possible solidification paths of the schematically presented in Fig. 6-17a, are shown in Table 6-9:

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101

The model for predicting the evolution of secondary phases is based on the following assumptions [37, 38]: instantaneous nucleation, carbides grow only in the liquid, negligible interference between growing carbides, carbides are pushed by the solid/liquid interface, volume diffusion limited growth of carbides, binary diffusion couple. The following steps are required to predict the NbC/Laves phase formation in IN718: (1) the nucleation and growth of NbC assuming that the slow step is the volume diffusion of carbon from the liquid to the NbC/L interface, (2) modeling of redistribution of Nb and C, (3) equiaxed dendritic growth, (4) growth of Laves phase, and (5) coupling between macro transport-solidification kinetics models. The growth mechanism of NbC is based on the carbon diffusion from the liquid to the NbC/L interface and the reaction kinetics between Nb and C. Applying the metastable progress variable approach, it can be shown that the chemical reaction rate is very high, i.e. the kinetics of reaction can be neglected (see Appendix I in [38]). Thus, carbon concentration is depleted at the NbC/L interface, and the amount of NbC is based on the volume diffusion of carbon from the liquid to the NbC/L interface.

The schematic representation of the diffusion pattern for NbC is shown in Fig. 6-17b and the main model assumptions are shown in Appendix II in [38]. It is assumed that the carbides instantaneously nucleate at the equilibrium solidification temperature of alloy 718 (T = 1609K). The solution of volume diffusion-limited growth is described by the averaging method:

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where liquid, and

where

is the volume of liquid,

is the number of carbides in the

is the solutal supersaturation defined as:

is the liquid interface concentration,

concentration, and

is the interface NbC

is the average liquid concentration.

The redistribution of both C and Nb are calculated with a modified version of the model described in [3]. The main assumptions are: solute transport is calculated in both solid and liquid phases assuming Fick’s law for binary systems in spherical coordinates, the solid/liquid interface is planar and under local equilibrium, no solute flow into or out of the volume element (closed system), constant initial liquid concentration, and the following overall mass balance:

with

Chapter 6. Micro-Solute Transport Effects on Microstructure and Microsegregation

103

where is the initial alloy concentration, is the actual volume of carbides, and is the volume of the element over which the mass balance is calculated. The overall mass balance is used to couple the concentration fields in both the solid and liquid phases. Note that the cross interdiffusion coefficients, as required in calculation of pseudo-ternary systems, are usually one order of magnitude lower than normal diffusion coefficients [41, 42] and are neglected in this analysis. The solution of this diffusion couple is a modified version of the model described in [3] and consists of the following equations:

where:

and Q,

and

and

with

Average concentration in the liquid phase is:

104

where

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

is the volume of the liquid phase, r is the radial coordinate,

the initial concentration, the eigenvalue, is the

is

is and

root of the equation

The coupling between macro transport–solidification kinetics models is accomplished through Latent Heat Method [46], where the heat source term is described as follows:

Laves phase starts to form when concentration of Nb reaches the eutectic-Laves composition which is 19.1 wt% Nb. Kinetics of Laves is very high due to its morphology (eutectic or globular type-divorced eutectic) and appearance (discontinuous thin film at the grain boundary). It is assumed that the amount of Laves phase is directly related to the Nb concentration [43-45]. The equiaxed dendritic growth is based on the alloy 718 pseudobinary phase diagram [29, 34], where the primary driving force for growth is the liquid Nb concentration. The model for equiaxed dendritic growth of alloy 718 is described elsewhere [43-45]. The physical properties used in calculations are shown in Table 6-10.

The measured grain equiaxed radius and carbide radius as a function of cooling rate are presented in Figs. 6-18 and 6-19, respectively. The experimental procedure is described in [44, 45]. Experimental and calculated amounts of NbC and Laves phase as a function of cooling rate for the initial carbon and niobium contents of 0.06 and 5.25 wt%, respectively, are shown

Chapter 6. Micro-Solute Transport Effects on Microstructure and Microsegregation

105

in Fig. 6-20. A typical example of calculation is given in Fig. 6-21. Note that as the cooling rate increases, the amount of precipitated NbC decreases, and falls short of the maximum (equilibrium) amount. The influence of cooling rate on the amount of NbC precipitated in cast Inconel 718, when the initial carbon and niobium contents were 0.125 and 5.25 wt.%, respectively, is presented in Fig. 6-21. Note in Fig. 6-21 that and are strongly dependent on the cooling rate. An optimum combination between C and Nb in function of cooling rate has to be predicted to minimize the amount of both NbC and Laves, in order to obtain a Laves free microstructure. This optimum combination may result in desirable mechanical properties, such as good stress-rupture ductilities due to the formation of a higher volume fraction of carbides (spherical shape), improved room temperature tensile strength and ductility due to elimination of Laves phase, etc. Experimental evidence demonstrates that the amount of NbC and Laves in cast alloy 718 is different from that predicted by phase equilibrium. The reason for this difference is that while in equilibrium processes mass diffusion transport is very fast compared with solidification kinetics (V << D/L), in casting processes solidification kinetics is much closer to diffusivity (V <= D/L). Thus, solidification kinetics cannot be ignored.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Chapter 6. Micro-Solute Transport Effects on Microstructure and Microsegregation

107

6.5 REFERENCES 1. T. Z. Kattamis and M. C. Flemings, Trans. TMS-AIME, Vol. 233, pp. 992, 1967. 2. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 27A, pp. 4061-4074 and 4075-84, 1996. 3. L. Nastac and D. M. Stefanescu, Met Trans, vol. 24A, pp. 2107-2118, 1993. 4. E. Scheil, Zeitschrift Metallkde, vol. 34, p.70, 1942. 5. J.A. Brooks, M. I. Baskes and F. A. Greulich, Met Trans, vol. 22A, pp. 915-26, 1991. 6. H.D. Brody and M.C. Flemings , Trans. TMS-AIME, vol. 236, p. 615, 1996. 7. T.W. Clyne and Kurz W., Met Trans, vol. 12 A, p. 965, 1981. 8. J.A. Sarreal and G.J. Abbaschian, Met Tran., Vol. 17A, pp. 2063-73, 1986. 9. I. Ohnaka, Trans. Iron and Steel Inst. of Japan, vol. 26, pp. 1045-51, 1986. 10. Ogilvy and Kirkwood, Applied Scientific Research, vol. 44, pp. 43-49, 1987. 11. S. Kobayashi, Trans. Iron and Steel Inst. of Japan, vol. 28, pp. 728-35 and pp. 535-42, 1988. 12. T. Matsumiya, H. Kajioka, S. Mizoguchi, Y. Ueshima and H. Esaka, Trans. Iron and Steel Inst. of Japan, vol. 24, pp.873-82, 1984. 13. K.S. Yeum, V. Laxmanan and D.R. Poirier, Met Trans. A, vol. 20A, pp. 2847-56, 1989. 14. T.P. Battle and R. D. Pehlke, Met Trans, vol. 21B, pp. 357-75, 1990. 15. S.W. Chen and Y.A. Chang, Met Trans, vol. 23 A, pp. 1038-43, 1992. 16. S. Sundarraj and V. R. Voller, HTD-Vol. 218/AMD-Vol. 139, ASME Conference, pp. 35- 42, 1992. 17. L. Nastac, S. Chang, D. M. Stefanescu, and L. Hadji, Proceedings of the Microstructural Design by Solidification Processing, Eds. E. J. Lavernia and M. N. Gungor, TMS, pp. 57- 75, 1992. 18. S. Chang, D. Shangguan, and D. M. Stefanescu, Met. Trans. A, vol. 23A, pp. 1333-46, 1992. 19. M. Rappaz and V. Voller, Met Trans A., Vol. 21A, pp. 749- 53, 1990. 20. C. Beckermann and R. Viskanta, Phys. Chem. Hydrodyn., vol. 10, pp. 195-213, 1988. 21. R. Boeri and F. Weinberg, AFS Trans., vol. 89, pp. 179- 84, 1989. 22. L. Nastac and D. M. Stefanescu, AFS Trans., Vol. 101, pp. 933- 38, 1993.

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23. L. Nastac, D. M. Stefanescu, and L. Chuzhoy, Proceedings of the Modelling of Casting, Welding, and Advanced Solidification Processes VII, Eds. M. Cross and J. Campbell, Engineering Foundation, London, UK, pp. 533-540, 1995. 24. A. Kagawa and T. Okamoto; Metal Science, pp.519-24, 1980. 25. J.A.Burton, R.C.Primm, and W.P.Slichter; J. Chem. Phys., vol 21, pp.1987-1993, 1953. 26. J.A. Sarreal and G.J. Abbaschian, Met Trans, Vol. 17A, 2063-73, 1986. 27. R.G. Thompson, D.E. Mayo, and B. Radhakrishnan, Met Trans, vol. 22A, p. 557-67, 1991. 28.T. Sawai, Y. Ueshima and S. Mizoguchi, ISIJ International, vol. 30, no. 7, pp 520-28, 1990. 29. H. L. Eiselstein: Advances in the Technology of Stainless Steels and Related Alloys, ASTM STP, ASTM, Philadelphia, PA, pp. 62-79, 1965. 30. M. J. Cieslak et. al., Superalloys 718 - Metallurgy and Applications, Ed. E. A. Loria, TMS, pp. 59-68, 1989. 31. R. G. Carlson and J. F. Radavich, Superalloys 718 - Metallurgy and Applications, edited by E. A. Loria, TMS, pp. 79-95, 1989. 32. M. J. Cieslak et. al., MeT. Trans A, Vol. 21 A, pp. 479-88, 1990. 33. B. Radhakrishnan and R. G. Thompson, Met Trans, Vol. 22 A, pp. 887902, 1991. 34. C. Chen, R. G. Thompson, and D. W. Davis, Superalloys 718, 625 and Various Derivatives, Ed. E. A. Loria, TMS, pp. 81 -96, 1991. 35. C. Peyroutou and Y. Honnorat, Superalloys 718, 625 and Various Derivatives, edited by E. A. Loria, TMS, pp. 309-24, 1991. 36. W. D. Cao, R. L. Kennedy, and M. P. Willis, Superalloys 718, 625 and Various Derivatives, edited by E. A. Loria, TMS, pp. 147-160, 1991. 37. D. Shangguan, S. Ahuja, and D. M. Stefanescu, Met Trans, Vol 23 A, pp. 669-80, 1991. 38. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 28A, pp. 1582-87, 1997. 39. T.Rosenqvist, Principles of Extractive Metallurgy, Edition, McGrawHill, N.Y., 1983. 40. H. B. Aaron, D. Fainstein, and G. R. Kotler, J. of Applied Physics, Vol. 41, No. 11, pp. 4405-09, 1970. 41. A. G. Guy, V. Leroy and T. B. Lindemer, Trans. of ASM, Vol. 59, pp. 517-34, 1966. 42. A. G. Guy and C. B. Smith, Trans. of ASM, Vol. 55, pp. 1-8, 1962. 43. L. Nastac and D. M. Stefanescu , Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes- VI, T. S. Piwonka et al. editors, TMS, Warrendale Pa., pp. 209-18, 1993. 44. L. Nastac, PhD Dissertation, University of Alabama, Tuscaloosa, 1995. 45. L. Nastac and D. M. Stefanescu, AFS Trans., Vol. 104, pp. 425-34, 1996. 46. D. M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Met Trans, Vol. 21A, pp.997-1005, 1990.

7

PROBABILISTIC (MONTE CARLO) MODELING OF SOLIDIFICATION STRUCTURE

7.1 FOURTH GENERATION OF SOLIDIFICATION MODELING

In this chapter, a new approach that is part of the “Fourth Generation of Modeling” (FGM) is introduced and applied for remelt ingots. FGM is entirely a probabilistic approach at both macro and micro-scales. The FGM is applied here to simulate the formation of solidification structures during casting of ESR processed alloy 718 ingots. The probabilistic microscopic approach to simulate the evolution of microstructure was described in the previous chapters. The probabilistic macroscopic approach is described in this chapter. A direct Monte Carlo technique (see references [1-6] for details) was developed and applied to describe the macro-transport phenomena in continuously solidifying cast ingots. The macro-model accounts for energy transport within the ingot and transport to the surroundings by conduction, convection, and thermal radiation. For a two-dimensional (2D) axisymmetric cylindrical geometry, the appropriate heat-transfer boundary conditions

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

(BCs) for remelt processes are: symmetry at the ingot center, convection and radiation at the ingot edge and ingot bottom, and process specific BCs at the ingot top to account for the heat input and heat loss. The BCs are described in details elsewhere [7–9]. In the presence of convective transport, the governing heat conduction equation in quasi-steady-state is given by

where r is radial coordinate, z is the axial coordinate in the moving coordinate system, is the thermal diffusivity, W is the withdrawal velocity, T is the temperature, is the liquidus temperature, is the solidus temperature, L is the latent heat of fusion and is the solidification interval. To solve Eq. (7-1), the MC scheme described in reference [2] for 2D Cartesian coordinates was used and adapted here for 2D axisymmetric cylindrical coordinates. The major advantages in using the MC probabilistic approach are emphasized in Refs. [1, 2]. Thus, the finite-difference representation of temperature (see Eq. (7-1)) at any node ( i , j ) becomes

Here, are the probability functions, is the mesh size (uniform and equal in r and z directions), and is the radial distance at node (i, j). Then, the temperature at the desired location is calculated with time (see references [1–6] for details of random walk procedures):

Chapter 7. Probabilistic (Monte Carlo) Modeling

111

where is the temperature at the terminal point of the random walk and N is the number of individual random walks. In the present work, it was found that N = 1000 provides accurate results while maintaining a small CPU-time (see also Ref. [1]). An example of a Monte Carlo simulation code is provided in the Appendix. It can be used to compute the temperature field in quasi-steady state of a 2-D solidifying domain with fixed step size and constant withdrawal velocity.

7.2 RESULTS AND DISCUSSION

Figure 7-1 shows the temperature results obtained based on the MC technique and the resulting simulated solidification macrostructure for an Electro-Slag Remelting (ESR) processed alloy 718 ingot (D = 432 mm, 272 kg/hr). The pool profile and the solidification predictions compared well with the experimental results in the steady-state region.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

7.3 REFERENCES 1. L. Nastac, Met. Trans., Vol. 29B, pp. 495-99, 1998. 2. A. Haji-Sheikh, Monte Carlo Methods, Handbook of Numerical Heat Transfer, Wiley-Interscience Publication, pp. 673-722, 1988. 3. A. Haji-Sheikh and E. M. Sparrow, J. of Heat Transfer, Vol. 89, pp. 121131, 1967. 4. A. Haji-Sheikh and F. P. Buckingham, J. of Heat Transfer, Vol. 115, pp. 26-33, 1993. 5. T. J. Hofmann and N. E. Banks, Nuclear Science and Engineering, Vol. 59, pp. 205-214, 1976. 6. A. Haji-Sheikh and E. M. Sparrow, J. SIAM Appl. Math., Vol. 14, pp. 370-389, 1966. 7. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang, Proceedings of the International Symposium on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, Eds. A. Mitchell and P. Aubertin, American Vacuum Society, pp. 145-165, 1997. 8. L. Nastac, S. Sundarraj, K. O. Yu, Proceedings of the Fourth International Symposium on Superalloy 718, 625, 706 and Various Derivatives, Ed. E. A. Loria, TMS, pp. 55-66, 1997. 9. L. Nastac, S. Sundarraj, Y. Pang, and K. O. Yu, Journal of Metals, TMS, No. 3, pp. 30-35, 1998.

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APPENDIX: MONTE CARLO PROGRAM FILE MONTE_CARLO_TEMP.C /* MONTE CARLO PROGRAM FOR HEAT TRANSFER SIMULATION: 2D RANDOM WALK WITH FIXED STEP SIZE, STEADY-STATE, VOLUMETRIC SOURCE, LATENT HEAT, WITHDRAWAL V */ #INCLUDE <STDIO.H> #INCLUDE <MATH.H> FILE *TEMP_1;FILE *TEMP_2;CHAR OUTPUT1[60];CHAR OUTPUT2[60]; #DEFINE PREFIX1 "TEMP" #DEFINE PREFIX2 "TAV" MAIN( ) { MONTE_CARLO(); } MONTE_CARLO() { STATIC INT FIRST_TIME=1,M1=54,N1=108,ML,NL,M,N,I,J,K; STATIC INT NSAMP,NCOUNT,NCHILL1,NCHILL2; STATIC DOUBLE XM,UMAX,VMAX,PII,ALFA,ALFAS,DXY,DL,DX,DY; STATIC DOUBLE TINIT,TCHILL,TD,TL,TR,TU,DD,SUMT,TEMP_AV; STATIC DOUBLE TEMP[55][109],GT[55][109],PR[55][109], LHFACTOR=0.25; STATIC DOUBLE TEMP_AVT, SUMG,X,PU[55][109],PL[55][109],PD[55][109]; IF(FIRST_TIME) {STRCPY(OUTPUT1, PREFIX1); STRCPY(OUTPUT2,PREFIX2);STRCAT(OUTPUT1, ".DAT"); STRCAT(OUTPUT2, ".DAT"); IF (( TEMP_1 = FOPEN( OUTPUT1, “W”) ) == N U L L ) { PRINTF( "\N\NUNABLE TO OPEN KINETIC OUTPUT FILE %S\N\N", OUTPUT1);EXIT(1); } IF (( TEMP_2 = FOPEN( OUTPUT2, “ W ” ) ) == NULL ) { PRINTF( "\N\NUNABLE TO OPEN KINETIC OUTPUT FILE %S\N\N", OUTPUT2);EXIT(1); } PII=3.1415926;UMAX=0;VMAX=4.3E-05;ALFA=5.2E-06;DXY=8E-03; NSAMP=2000;TINIT=600;TCHILL=600.0;TD=600;TL=600;TR=600;TU=1450; FOR (M=1; M<=M1; M++){FOR (N=1; N<=N1; N++){ TEMP[M][N]=TINIT;DD=ALFA*4.0-VMAX*DXY;PL[M][N]=ALFA/DD; PR[M][N]=ALFA/DD;PU[M][N]=ALFA/DD;PD[M][N]=(ALFAVMAX*DXY)/DD;}}FOR (M=1; M<=M1; M++){TEMP[M][1]=TD;TEMP[M][N1]=TU;PU[M][N1]=0.0; PD[M][N1]=0.5;PD[M][1]=0.0;PU[M][1]=0.5;} FOR (N=1; N<=N1; N++){TEMP[1][N]=TL;TEMP[M1][N]=TR;PL[1][N]=0.0; PR[1][N]=0.5;PR[M1][N]=0.0;PL[M1][N]=0.5;}NCHILL1=1;NCHILL2=N1; FOR (N=NCHILL1; N<=NCHILL2; N++){ TEMP[1][N]=TCHILL;TEMP[M1][N]=TCHILL;} /* GT IS THE SOURCE TERM IN W/M3 */ FOR (M=1; M<=M1; M++){FOR (N=1; N<=N1; N++){GT[M][N]=0.0;}} FIRST_TIME = 0;}/* START MONTE CARLO */

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

FOR (M=2; M<M1; M++){FOR (N=2; N=1180.0) && (TEMP[ML][NL]<=1330.0)){ ALFAS = LHFACTOR*ALFA; } ELSE { ALFAS = ALFA; } DD=ALFAS*4.0; VMAX*DXY;PL[ML][NL]=ALFAS/DD;PR[ML][NL]=ALFAS/DD; PU[ML][NL]=ALFAS/DD;PD[ML][NL]=(ALFAS-VMAX*DXY)/DD;}} FOR (ML=1; ML<=M1; ML++){TEMP[ML][1]=TD;TEMP[ML][N1]=TU; PU[ML][N1]=0.0;PD[ML][N1]=0.5;PD[ML][1]=0.0;PU[ML][1]=0.5;} FOR (NL=1;NL<=N1;NL++){TEMP[1][NL]=TL;TEMP[M1][NL]=TR; PL[1][NL]=0.0;PR[1][NL]=0.5;PR[M1][NL]=0.0;PL[M1][NL]=0.5;} /*START ITERATION */ LABEL1:;I=M;J=N;FOR (ML=1; ML<=M1; ML++){TEMP[ML][1]=TEMP[ML][2];TEMP[ML][N1]=TU;} FOR (NL=NCHILL1; NL<=NCHILL2; NL++){ TEMP[1][NL]=TCHILL;TEMP[M1][NL]=TCHILL;} /* UPDATE PROBABILITIES */ IF ((TEMP[I][J]>=1180.0) && (TEMP[I][J]<=1330.0)) {ALFAS = 0.25*ALFA;} ELSE { ALFAS = ALFA; } DD=ALFAS*4.0-VMAX*DXY;PL[I][J]=ALFAS/DD; PR[I][J]=ALFAS/DD;PU[I][J]=ALFAS/DD; PD[I][J]=(ALFAS-VMAX*DXY)/DD;LABEL210:;X=RAND(); IF(X<=PR[I][J]) {I=I+1;GOTO LABEL280;} IF(X<=(PR[I][J]+PU[I][J]) && X>PR[I][J]){J=J+1;GOTO LABEL300;} IF(X<=(PR[I][J]+PU[I][J]+PL[I][J]) && X>(PR[I][J]+PU[I][J])) { I=I-1;GOTO LABEL320;} IF (X>(PR[I][J]+PU[I][J]+PL[I][J]) && X<=1.0) J=J-1; SUMG=SUMG+GT[I][J];IF (J==1){SUMT=SUMT+TEMP[I][J]; GOTO LABEL340;} GOTO LABEL210;LABEL280:; SUMG=SUMG+GT[I][J];IF (I==M1) {SUMT=SUMT+TEMP[I][J]; GOTO LABEL340;} GOTO LABEL210;LABEL300:; SUMG=SUMG+GT[I][J];IF (J==N1) {SUMT=SUMT+TEMP[I][J]; GOTO LABEL340;} GOTO LABEL210;LABEL320:; SUMG=SUMG+GT[I][J];IF (I==1) {SUMT=SUMT+TEMP[I][J]; GOTO LABEL340;} GOTO LABEL210;LABEL340:;NCOUNT = NCOUNT + 1; IF (NCOUNT
8

MODELING AND SIMULATION OF SOLIDIFICATION STRUCTURE IN SHAPED AND CENTRIFUGAL CASTINGS

In this chapter, computer models of solidification structure in near-net shape and centrifugal castings are discussed. Simulation results for grain size, structural transitions (e.g., CET and gray-to-white transition [GWT]), and dendritic morphologies in as-cast alloys and metal-matrix composites are also presented.

8.1 SHAPED CASTINGS

8.1.1 Prediction of Grain Structure and of Columnar-to-Equiaxed Transition in Steel Castings

The effect of the melt superheat on CET is illustrated in Fig. 8-1. By changing the melt superheat from 35 K to 200 K, the macrostructures shown in Fig. 8-1 have changed from completely equiaxed to fully columnar. The melt superheat significantly influenced the nucleation and growth of equiaxed grains and, therefore, the location and size of CET. The simulated results in Fig. 8-1 agreed well with the experimental macrostructures provided in [1].

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117

8.1.2 Gray-to-White Transition in Cast Iron Introduction

One of the frequent reasons for rejecting gray iron castings is the occurrence of chill, in particular in the thin sections and at corners. The origin of this common defect is the inadvertent GWT, resulting from poor control of process or material variables. This is a structural transition from the stable austenite-graphite eutectic to the metastable austenite-cementite eutectic (ledeburite). The resulting microstructure may be entirely white, or a combination between white and gray phases, called mottled structure. The current understanding of the GWT is based on the interpretation of cooling curves [2, 3], summarized in Fig. 8-2. It is assumed that, if solidification occurs and is completed above the metastable eutectic temperature, the iron is completely gray. However, if any portions of the cooling curve lie under the metastable temperature white eutectic will form. If this happens before recalescence, the white regions are termed chill. If white eutectic forms at the end of solidification, mostly because of intergranular segregation of carbide promoting elements, is considered that intergranular carbides are formed.

The basic variables affecting GWT in cast iron are the cooling rates of the casting, the nucleation potential, and the chemical composition of the melt. Typically, GWT is favored by increased cooling rates, low nucleation potential, and low carbon equivalent and/or higher amounts of carbide promoting elements. In some cases, because of segregation during solidification, inverse chill may occur. These are regions of white structure formed in the last regions to solidify, which should in principle be gray because of their lower cooling

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rates. To model the GWT, the following phenomena must be described through appropriate physical and mathematical models [2, 3]: nucleation and growth of both the stable and metastable eutectics; growth competition between the same eutectics; change in equilibrium temperatures and solubility limits because of the macro- and micro-segregation of various elements. A model for the G/W transition that includes the effects of nucleation and grain growth for both eutectics, and the influence of microsegregation is presented in [3]. The model predicts formation of chill as well as of intergranular carbides. The mathematical details of the nucleation and growth models for both white and gray eutectics are presented in [2, 3]. Microsegregation of the third element and its influence on the G/W transition

The final structure results from the competition between the gray and white eutectic. This is directly influenced by the two equilibrium temperatures, since each phase can only start nucleating and growing under its equilibrium temperature. The equilibrium temperatures, and (in °C), for the gray and white eutectic, respectively are calculated as follows [3, 31]:

where is the intrinsic volume average liquid concentration. Consider a macro-volume element within the solidifying alloy. This macro-element can be for example of the mesh size of a macro heat transfer model for solidification of castings. Within this macro-element the temperature is assumed uniform, and is obtained from the solution of the thermal field. The macro-element is further subdivided in a number of spherical volume micro-elements. Within each of these elements, both gray and white eutectic can competitively grow into the liquid. The problem to solve is to calculate the volume average concentration of the third element in the liquid during solidification. The liquid concentration controls both the stable and metastable temperatures bellow which growth of gray and, respectively, white eutectic can occur. The final liquid fraction will have a volume average concentration that depends not only on the diffusion coefficients in the solid phases, but also on the growth velocities (or solid fractions) of the gray and/or white grains. The solute concentrations in the solid- gray,

and solid-white,

phases must each satisfy Fick’s second law in spherical coordinates:

Chapter 8. Modeling and Simulation of Solidification Structure in Shaped and Centrifugal Castings

where

and

119

are the diffusion coefficients in the gray and white

phases, respectively. The concept of this model is similar to the previously developed model for one solid phase [32]. However, because of the complications of tracking two different liquids, it was assumed that complete mixing in liquid occurs. Two liquid-solid interfaces must be considered: liquid-white eutectic and liquid-gray eutectic. The interface compositions of these interfaces are:

where

is the intrinsic volume average liquid concentration,

and

are the partition ratios between the liquid and gray and white, respectively, and the superscript * denotes values at the interface. Then, mass conservation is used to couple the concentration fields in the solid and liquid phases:

where

is the initial concentration of the alloy,

and

and are the fraction of gray, white, and liquid phases, respectively. The details of the derivation are given in Appendix I in [3]. The final solution that gives the intrinsic volume average liquid concentration is:

where and

and

have similar expressions and are expressed as:

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where t is the time, is the interface position, and the subscript i denotes either g or w subscripts. Model flowchart

The solidification kinetics model that includes the nucleation and growth competition between the gray and white eutectics was coupled to a macrotransport code through the latent heat method [33]. The flowchart describing the G/W transition is presented in Fig. 8-3. A particular issue in this calculation is the evaluation of the interface temperature. Using heat balance at the interface and overall balance within the control volume element, and assuming quasi steady state approximation and spherical geometry, the interface temperature, T*, can be calculated as:

where is the bulk temperature, V is the growth velocity (for gray or white eutectic), is the liquid thermal conductivity, L is the latent heat, is the density, R* is the interface radius, and is the final radius of the grain. Effects of silicon content Silicon is the main element that controls the transition in plain cast iron. Therefore, for the theoretical analysis of the model, silicon was used as a variable. The range was from 0.5 to 3.0 %. The other elements of the chemical composition were maintained constant. The main data used in computation are given in Table 8-1. The effect of silicon can be illustrated through the graphs in Fig. 8-4. From the cooling curves it may be implied that at 1% Si the iron solidifies completely white, since the solidification part of the cooling curve lies completely under the metastable eutectic temperature. However, from the graphs describing the evolution of the fraction of white and gray eutectic (Fig. 8-5) this may not be true. Indeed, a mottled structure is predicted with about 40% gray eutectic. The reason for this apparent discrepancy is that once nucleated, the gray eutectic can in principle grow under The final structure will result from the nucleation and growth competition between the gray and white eutectic.

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121

Unfortunately, because of the paucity of data related to the nucleation and growth of the white eutectic, an accurate prediction is difficult. To clarify this point, a sensitivity analysis will be presented later in this paper. When the Si content was raised to 2%, limited chill forms at the beginning of solidification, and massive intergranular carbides are produced at the end of solidification. This is evident from the fact that the cooling curve lies under for a limited time at the beginning of solidification, and for a long time

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

toward the end of solidification (Fig. 8-4b). Clear recalescence is observed on the cooling curve, as opposed to the 1% Si case. The overall effect of silicon is, as expected, to increase the amount of gray phase (Fig. 8-5b). The amount of chill is insignificant, but the total amount of white phase is still 20%. An interesting feature that is seen when comparing Figs. 8-4a and 84b, is that the two equilibrium temperatures diverge in time for the 1% Si sample, but converge for the 2% Si sample. This effect is a result of the evolution of the average silicon, manganese and phosphorus concentrations, the main effect being that of silicon. Indeed, as seen in Fig. 8-6, for 1% Si the silicon concentration increases continuously with time (fraction of solid). For 2% Si a reverse trend is observed. This is also true for 3% Si. This change in the redistribution of silicon results from opposite partitioning in the two eutectics, and the nucleation and growth competition between them. For the higher silicon content of 3% it is still possible to obtain intergranular carbides, as seen from the cooling curve in Fig. 8-4c, and from the evolution of fraction of solid in Fig. 8-5c.

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123

Combined influence of silicon and cooling rate

Fig. 8-7 summarizes the combined influence of silicon and cooling rate on the structure of a cast iron having 3.6% C, 0.5% Mn, 0.05% P, 0.025% S. Two structural transitions can be defined: a GWT resulting in a completely white structure, and a white-to-gray transition, resulting in a completely gray structure. In between these two, a mottled zone exists. The three structural zones on Fig. 8-7 can also be defined in terms of the type of carbide formation mechanism. A fully white structure is the result of chill formation. A mottled structure results when both chill and intergranular carbides are formed. It is seen that the size of the mottled zone increases with silicon and inoculation. It will also be affected by other elements such as chromium or aluminum and by inoculation.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

As expected, inoculation increases the gray region and decreases the white one (the curves are pushed upward on Fig 8-7b). The mottled region is also increased. Sensitivity analysis

For a better understanding of the G/W transition and its controlling variables, a sensitivity analysis was performed. The variables used in this analysis and the main results are presented in Table 2. For the reference parameters listed at the foot of Table 8-2 the fraction of white eutectic was 9%, and the fraction of gray eutectic was 91%. It is apparent that by selecting different literature values of the slope of the eutectic valley for silicon, the calculated fraction of white eutectic varies. On the contrary, choosing to use the bulk rather than the interface undercooling in calculation does not affect the outcome. This is because, as shown in Fig. 8-8, the maximum difference between the two temperatures is about 0.8 °C. While silicon changes the partition ratios between liquid and the white or gray phases, its overall influence is also limited. Indeed, the fraction of white phase is decreased by 4% from its original value of 9%. The most significant influence seems to be that of the growth coefficient for gray iron. When this coefficient was decreased three times, the amount of white phase increased by 80%, while when it was increased three times the iron became completely gray (8.9% decrease in The growth coefficient of the white eutectic seems to play a minor role on the outcome of the calculation. As shown in Table 8-2, decreasing or increasing its value three times changes only moderately. Increasing the number of nuclei for the white phase ten fold produces a higher amount of white phase, as expected, but the change is again minor. When increasing three times the nucleation of the gray phase, although the change is not spectacular, the influence is nevertheless very strong, since almost a complete gray structure is obtained. The importance of the nucleation potential of the gray phase was better illustrated by decreasing three times. In this case the amount of white phase increased dramatically by 49%. Thus, the major variables are the nucleation potential and growth coefficient of the gray phase. The fact that the solidification kinetics parameters of the white phase are not very well documented, may not be an insurmountable handicap in GWT modeling. Model application

An application of this model is shown in Fig. 8-9 for predicting GWT in a step casting. Figures 8-9a and 8-9b show a comparison between experiment [3] and simulated G/W transition for an initial Si content of 1.78 wt.%. The simulation closely matches the experiment.

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125

Note that the model also predicts the grain size in the gray region. The effect of the initial Si content on the GWT can be seen from Figs. 8-9b and 8-9c for an initial Si content of 1.78 wt.% and 2.5 wt.%, respectively. The Si macrosegregation map for an initial Si content of 2.5 wt.% is presented in Fig. 8-9d. The macrosegregation model is described in reference [6]. The effect of macrosegregation on the GWT is minor for the geometry, alloy, and processing conditions represented in Fig. 8-9.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

New experimental work on the fundamental understanding of the influence of various process parameters on the formation of GWT in lamellar, spheroidal, and vermicular graphite cast iron was performed at Kyushu University, Japan [7]. They have studied the effect of Si and of inoculation on GWT and on the formation of various phases in cylindrical castings. The inoculation was done with FeSi75 (2.39 % Ca and 1.47 % Zr). Some experimental results for gray iron that will further be used for validation of numerical modeling of GWT are presented in Fig. 8-10. The height and diameter of the samples in Fig. 8-10a were 200 mm and 12.7 mm, respectively. The samples were bottom-cooled by using a Cu chill. Insulating material was used on all the other sides of the castings. The carbon equivalent (CE) of the samples was kept constant at about 4.3 %. The GWT takes places at about 11 mm, 6, and 2 mm from the sample bottom for 1.8 wt.% Si, 2.5 wt.% Si, and 3.2 wt.% Si, respectively (see also the table in Fig. 8-l0a).

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127

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The grain density, N (number of eutectic was also measured as a function of the initial Si content (wt.%) and cooling rate , K/s) measured at the liquidus temperature of the alloy. The equation used to calculate grain density, where a, b, and c are shown in Table 8-3.

is

8.1.3 Modeling of Solidification Structure in Al-based Alloy Castings

A simulated evolution of equiaxed dendritic morphologies for Al-Si alloy system is presented in Figure 8-11. Coarsening of secondary dendrite arms and branching of secondary arms can be observed in Figure 8-11. Details related to thermophysical properties of the alloy and boundary and initial conditions used in simulations are presented in [8]. The grid size of the domain is This is fine enough to approximately resolve the dendrite tip radius that, in the present solidification conditions, is typically larger than Newton cooling boundary conditions were applied at the boundaries of the computational domain for multidirectional solidification simulations. Figures 8-12, 8-13, and 8-14 show comparisons of simulated and experimental microstructures for D357 alloy castings [9]. In these figures, the grain size prediction matches reasonable with experiments. It can also be noted from these pictures that the chill effect on grain size is significant. 8.1.4 Modeling of Solidification Structure in RS5 Alloys

Mesoscale modeling of microstructure of RS5 alloy

A recent modeling approach described in details in Ref. [8] was applied to simulate the solidification structure and segregation evolution in RS5 alloy, a Rolls Royce’s alloy [10]. More applications of this model are presented in [9, 10]. Figures 8-15 and 8-16 show the mesoscale simulations for the thin wall RS5 castings. It was found that a grain density, is required to obtain a fully equiaxed casting (see Fig. 8-15). The black color in Fig. 8-15a shows the potential location of the solidification shrinkage. The arm spacing selection and evolution of Nb concentration can clearly be seen in Fig. 8-16 for a thin wall (1 mm x 2 mm) RS5 casting.

Chapter 8. Modeling and Simulation of Solidification Structure in Shaped and Centrifugal Castings

129

130

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

In this case, a was used in order to achieve a complete columnar structure (see Fig. 8-15a). The primary dendrite arm spacing (PDAS) in Fig. 8-16 varies from an initial value of approximately to a final value of about at the end of solidification. Similarly, secondary dendrite arm spacing (SDAS) varies from an initial value of about to a final value of about (see Fig. 8-16). Figure 8-16 also shows the time-evolution of Nb segregation in these castings. Here, Nb strongly segregates from the surface to the center of the casting. Thus, Laves phases that are rich in Nb could easily form at the end of solidification in these castings.

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131

Numerical simulation of the bar casting experiments

In-house simulations and experiments were performed to determine a window of process and material parameters for obtaining the optimum microstructure (including secondary phases and CET) in the thermallycontrolled solidification (TCS) processed RS5 and IN718 castings. Bar casting experiments were designed to characterize the microstructure formation in thin wall castings during withdrawal and solidification. Figure 8-17a shows the stereo-lithography (STL) model of a bar casting used in the current simulations. Figure 8-17b describes the STL geometry of the final design used for performing the bar casting experiments. ProCAST withdrawal simulations (including radiation computations) of the bar casting experiments were performed to provide thermal data for microstructure predictions. Figures 8-18 to 8-21 show some of the results of this effort. The solidification time of the simulated bar castings with a withdrawal rate (W) of is about twice of the bar castings simulated without withdrawal (see Figure 8-18). Cooling rates in the mush were very similar for both types of withdrawal conditions while the mushyzone thermal gradients for the bar casting with withdrawal were about onehalf of the one without withdrawal.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Chapter 8. Modeling and Simulation of Solidification Structure in Shaped and Centrifugal Castings

133

Thus, the solidification front velocity of the simulated bar castings with a withdrawal rate of is about twice of that of the bar castings simulated without withdrawal (see Fig. 8-21). From Figs. 8-18 to 8-21 it can be concluded that the withdrawal rate has a significant effect on the solidification structure of these superalloys. For example, a high withdrawal rate favors CET formation in TCS-processed IN718 alloy.

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Modeling and Simulation of Micros tructure Evolution in Solidifying Alloys

Experimental work and comparison of experimental and calculated SDAS values

Results from a differential thermal analysis (DTA) investigation for the RS5 alloy are presented in Figure 8-22 for two cooling rates. The temperature values in Fig. 8-22 are for the liquidus, carbide, and solidus (or Eutectic-Laves phase) temperatures. Figure 8-23 presents a SEM micrograph of an as-cast RS5 sample continuously cooled from the liquid at 1 K/min. The solidification and post-solidification phases that were found in the ascast RS5 sample shown in Fig. 8-23 were: eutectic-Laves, Laves, MC-type carbides, porosity, and Table 8-4 shows the chemical composition of some of these phases. Thus, by using DTA measurements and techniques for metallographic characterization, it was demonstrated that secondary phases (carbides and Laves phases) are formed

Chapter 8. Modeling and Simulation of Solidification Structure in Shaped and Centrifugal Castings

135

in both alloys. Also, similar solidification paths occur in both alloys. The DTA measurements have also confirmed some of the theoretical calculations of the solidification characteristics of these superalloys.

Figure 8-24 presents a comparison between calculated and experimental values of SDAS. The SDAS was measured from the DTA samples and the solidification time for the experimental data was calculated from the cooling rates of the DTA samples and the solidification interval. In Fig. 8-24, SDAS was calculated with

where

Note the good

match between the experimental and the calculated results in Fig. 8-24.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

8.2 CENTRIFUGAL CASTINGS

8.2.1 Introduction

Centrifugal casting processing of metal-matrix-composites is a cost effective, innovative approach of producing locally reinforced components with improved hardness and wear resistance. A schematic of a vertical centrifugal casting device is shown in Figure 8-25. Three stages are involved during centrifugal casting: (1) pouring, (2) segregation, and (3) solidification. For example, for centrifugal casting of TiC/Bronze, these stages are described as follows. In the first stage, the molten bronze alloy/TiC particle melt is quickly poured into a preheated mold spinning at an angular speed equivalent to ~100G force (660 rpm).

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137

As the TiC/Bronze mixture hits the bottom plate of the mold, it picks up angular speed, climbs up the sidewall under the effect of the strong centrifugal force and forms a stable cylinder of fluid TiC/Bronze. In the second stage, segregation proceeds, due to the centrifugal buoyancy force. The less-dense TiC particles move towards the inner vertical surface of the TiC/Bronze cylinder, while molten bronze alloy continues to move towards the outer diameter. The net result is formation of a TiC-rich layer at the inner surface. In the final stage, solidification is completed and the basic shell of a TiC/Bronze composite drum is produced.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The centrifugal casting process is complex and difficult to control [1315] and therefore, numerical modeling is an important tool for understanding and improving the centrifugal casting of metal-matrixcomposites (MMC). Several analytical and numerical models were previously developed to predict the particle distribution during centrifugal casting of MMC [16-19] (see a more complete list of references including work on pushing/engulfment/entrapment transition in [13, 14]). Although the model in [13] could be applied to general centrifugal casting processes, it was specifically tailored to investigate the design and process parameters of the TiC/Bronze fiction drums. Objectives of this modeling effort were: (1) to provide an in-depth understanding of the casting process (e.g., prediction of the final particle distribution profile and determination of the influence of process conditions on the solidification); (2) to optimize the casting process for producing friction drums with desired quality; and (3) to reduce the need for expensive plant trials and to generate cost savings through model application. Next, the complex interaction between the solidification structure and the insoluble ceramic particles in centrifugallycast MMCs is investigated. Efforts were directed toward developing a comprehensive model, comprising of two components: (1) a macrotransport/solidification-kinetics (MT-SK) model to simulate heat transfer and solidification during the centrifugal casting, (2) a deterministic model to simulate the particle distribution in the casting after solidification; and (3) a stochastic model to simulate the microstructure formation in the presence of ceramic particles. A comparison between the calculated and the experimental results [16, 20] is also provided for a centrifugally-cast A356/SiC composite. 8.2.2 Model Description

1D numerical particle model

From a modeling point of view, the centrifugal casting process can be summarized in two important stages. The first is the pouring phase, which includes phenomena from the beginning of pouring to the end of mold filling. The second stage starts at the end of filling and continues to the end of solidification. It is assumed that the outer layer of metal is instantly picked up by the spinning mold. It is also assumed that during the pouring phase the incoming molten material is evenly distributed along the inside of the casting. These assumptions appear to be reasonable and valid, as demonstrated by an analysis described in Appendix A in Ref. [4]. Based on the above assumptions and the dimensional analysis described in the previous section, the particle model can be described as a 1-D axissymmetric system in cylindrical coordinates:

Chapter 8. Modeling and Simulation of Solidification Structure in Shaped and Centrifugal Castings

139

Particle Motion:

where and are the densities of the particle and liquid metal, respectively, is the mold spinning angular velocity, is the particle size, is the particle velocity, and is the liquid-metal effective viscosity including the effects of particle interaction and solid fraction (Eq. (8) in [13]). Particle Volume Conservation:

The initial and boundary conditions required to solve for Eq. (8-10) are as follows: Constant Initial Condition for Particle Volume Fraction:

Zero Velocity Condition for Particle Motion:

Zero-Gradient Condition for Particle Volume Fraction:

Particle clustering and agglomeration

Clustering and agglomeration of particles in the liquid change particle motion, usually making particles move faster. There are two key factors that should be considered. First, the particles in clusters or agglomerates move together. Therefore, the effective sizes of particles are much larger than those of individual particles. Secondly, when a cluster or agglomerate moves, the entrapped molten alloy inside the cluster or agglomerate moves as well. This modifies the effective density of the cluster or agglomerate.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

These two factors are taken into account in this work by using the following relations [13]:

Note that the “particle density”, in Eq. (8-14) is evaluated in Eq. (815), replacing particle material density with a value for the alloy/agglomerate combination. It is usually called “effective particle density.” Similarly, the particle size, in Eq. (8-14) is termed the effective particle size; it includes contributions of individual particles, clusters, and agglomerates. The effective particle size will be discussed in detail in the next section. In the Eqs. (8-14) and (8-15), is the particle materials density; k is the volume ratio of particles in clusters or agglomerates; and n is a constant, empirically determined to be 4.65 for most of the particle suspension systems [13]. The volume ratio of particles in clusters or agglomerates, k, is process and materials dependent. For the SiC/A356 alloy system, k varies from 0.18 to 0.65. In the most recent work, k is shown to be 0.42 for particles with an average effective size of For TiC/Bronze system, no data are reported in the open literature. In this work, a distribution of k is assumed and illustrated in Fig. 8-26 for both the SiC/A356 and the TiC/Bronze alloy systems. Particle size distribution

During the centrifugal casting process, the centrifugal force increases with effective particle size. Therefore, larger particles move closer to the inner or outer boundary of the fluid. In addition, the particle-free region increases with effective particle size. It was found that the largest particles (lowdensity case) were closest to the inner surface of the casting and smaller particles were further away. The effect of particle size distribution has been taken into account by the particle segregation model. A typical particle distribution as a function of effective particle size is shown in Fig. 8-27. Clusters and agglomerates are treated as single particles with the effective particle density specified by Equation (8-14). The term “particle size” is defined as the effective particle size including individual TiC particles, clusters, and agglomerates.

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Deterministic (MT-SK) modeling

In the absence of convective transport, the governing heat conduction equation for 1D axis-symmetric geometries is given by:

where T is the temperature, K is the thermal conductivity, is the density, is the specific heat, is the solid fraction, L is the latent heat of fusion. The last right-hand-side term is the source term associated with the change of the equiaxed dendritic and the eutectic phases, and describes the rates of latent heat evolution during the liquid/solid transformation. In developing a deterministic model for predicting the particle distribution profile after solidification, the following assumptions were made: (1) instantaneous nucleation of both primary and eutectic phases; (2) the interference between the moving particles was taken into account by a rheology-viscosity model; and (3) particles are engulfed/entrapped in the mushy region by the growing equiaxed structure. The model accounts for: (1) nucleation and growth of equiaxed dendritic and equiaxed eutectic grains; (2) the effect of particle size, particle concentration, and cooling rate on the final grain size; and (3) the impingement effect of particles on grain-growth kinetics.

The equiaxed dendritic model uses the Al-Si binary phase diagram, where the primary driving force for growth is silicon diffusion in the liquid phase.

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The models for equiaxed dendritic and eutectic growth of alloy A356 are described in [21, 22]. These models are applied to calculate the evolution of the primary and eutectic phases. To include the impingement effect of particles on grain growth of these phases, a correction factor, was added to the time-evolution of the fraction of solid as follows [23]:

where

is the time-evolution of the fraction of solid without the

presence of particles, is the liquid fraction, and is the volume fraction of particles. In the composite region, i.e., in the presence of particles, the average grain size, is calculated with [24, 25]:

where is the grain size and is the particle radius. In the particle-free region, the grain size is calculated as [22]:

where is the volumetric grain density, is the cooling rate computed at the liquidus temperature, and A, B, and C are experimentally determined constants. By using Eqs. (8-16) to (8-19), the interaction between the growing matrix phases and the ceramic particles is included in the model. Stochastic modeling

A comprehensive stochastic model was used [26] to simulate the microstructure evolution and segregation of ceramic particles. The stochastic model also accounts for redistribution of ceramic particles at the dendrite tip length scale during the solidification process. When the S/L interface velocity is larger than the minimum critical velocity of pushing/engulfment transition (PET), ceramic particles will be engulfed. Otherwise, they will be either pushed or engulfed (or entrapped) based on stochastic/probabilistic calculations. The model accounts for the effect of the S/L dendritic interface roughness and other mechanisms, such as sliding/rolling and solute redistribution, on PET (see more details in [27], where nucleation and growth as well as redistribution of NbC particles within a dendritic structure are modeled and compared with experiments for cast IN 718 alloys).

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8.2.3 Results and Discussion Deterministic modeling

Data used in the micro-modeling of alloy A356 are presented in Table 8-5, where is the solid diffusivity, is the liquid diffusivity, k is the partition ratio, is the eutectic composition, is the liquidus slope, is the initial composition, is the equilibrium liquidus temperature, and is the eutectic temperature. Figures 8-28(a) and 8-28(b) shows a comparison between the calculated particle distribution profile using the current model and the experimental particle distribution profile [16] for an A356/SiC centrifugally-cast composite. The initial particle concentration is 4.8% SiC. Calculated results with the present MT-SK model closely matched the experimental distribution profile, particularly the transition between the freeparticles region and the composite region. The effects of particle size, particle concentration, and cooling rate, on the final grain size is shown in Figures 8-28(c) and (d). In order to calculate the grain size distribution in the composite region, Eq. (A-2) from [13] was used. In the particle-free-region, the grain size was estimated [20]. Note that this estimate is in good agreement with the experimental measurements in [28]. In Figures 8-28(c) and (d), a sharp transition exists between the particle-free region and the composite region in terms of the grain size distribution. This is in agreement with the experimental results in [20], where the equiaxed structure is completely broken up by the presence of the particles in the reinforced zone. Stochastic modeling

A comparison between the calculated and the experimental results [16, 20, 30] in terms of particle distribution profile and grain size of a centrifugally-cast composite of A356 with diameter SiC particles is provided in Fig. 8-29. The initial particle concentration is 4.8 vol.% SiC. Results calculated with the present model matched approximately the experimental distribution profile and the grain size, particularly the sharp transition between the free-particles region and the composite region. Note that important phenomena in controlling grain size in the presence of SiC particles were solute redistribution and dendritic growth. These phenomena were correctly accounted for in the present stochastic model.

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145

Figure 8-30 shows typical macrostructures and microstructures of a TiC/Al-Bronze composite drum [13]. From the in-house metallurgical analysis, two distinct layers (i.e., a particle-rich region and a particle-“free” region) were noted visually across the thickness of the drum. The average size of individual particles was determined to be about Their shape was close to spherical. At mid-height of the drum, the volume fractions of TiC particles were determined at seven different radial locations, as shown in Figure 8-31. Also shown in Figure 8-31 are model predictions of the particle volume fractions. The following observations can be made from Figure 8-31: (i) the model correctly predicts the transition between the particle-rich and particle-“free” zones; and (ii) the model predictions in the particle-rich zone are in better agreement with the experimental measurements than in the particle-“free” zone. The latter discrepancy is explained by the model assumption of uniform particle size. Metallurgical analysis revealed that particles have a smaller size in the particle-“free” zone. The predicted thickness of the TiC rich layer is compared with the experimental measurements in Figure 8-32. The measured TiC thickness presented in the Fig. 8-32 is the TiC thickness averaged over 10 data points along the axial direction of the drum (longitudinal or vertical direction). One standard deviation of the 10 data points for each drum is also plotted in the figure (represented by error bars). This deviation arises from two sources: measurement error and non-uniformity of the TiC thickness along the axial direction. It was found that the non-uniformity of the TiC thickness is a major source of the deviation.

8.2.4 Summary of the Parametric Studies

Parametric studies were conducted using the present computer model to investigate the effects of various process parameters on TiC distribution [13]. The effects of process conditions and materials parameters are summarized as follows.

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Superheat. Superheat has some effect on the final particle distribution. The particle distribution near the designed wear surface zone will not change very much. However, a low superheat may cause machining problems in later manufacturing processes by entrapping particles in non-desired regions. Also, because the casting defects related to high superheat are not critical to the friction drum application, high superheat will help avoid the casting defects related to a low superheat, such as incomplete filling. Therefore, a relatively high superheat is better for both particle distribution and casting structure. A superheat greater than 100 K is recommended. Mold Coating. Mold coating and thickness significantly affect the final particle distribution through the effect on the casting interface/mold interface

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coefficient h. An alumina coating with a thickness of 0.03 in. (0.75 mm) is recommended. This will result in a moderate solidification time that allows completion of particle segregation before matrix solidification starts. Mold Preheat Temperature. Mold preheat temperature has little effect on the final particle distribution. However, a low mold temperature may decrease the life of steel or graphite molds due to thermal shock. Therefore, a moderate-to-high mold preheat temperature is recommended. No strict control of mold preheat is needed. Mold Materials. Selection of mold material has some effect on the final particle distribution. Sand, graphite, and steels are acceptable. Reinforced sand molds and graphite molds are recommended for general purposes. However, it may be more beneficial to use steel molds for large scale manufacturing because of their longer life. Mold Cooling Methods. Minor effect of mold cooling methods on final particle distribution was observed in the parametric study. Natural aircooling is recommended at both the inner surface and the outer surface of the mold holder. Mold Rotating Speed. Mold rotating speed significantly affects both particle motion and solidification. Higher mold rotating speed results in a stronger centrifugal buoyancy force and therefore gives rise to faster particle segregation. Higher mold rotating speed also generates stronger pressing force at the casting/mold interface, and therefore results in stronger interfacial heat transfer and quicker solidification. These factors significantly influence final particle distribution. A higher mold rotating speed is desired. A mold rotating speed equivalent to about 100G’s or larger is recommended. Initial Particle Volume Fraction. Initial particle volume fraction has a significant effect on the final particle distribution. An initial particle volume fraction in the range of 0.10-0.15 (10-15 vol. %) is recommended. Particle Size. The particle size significantly influences final particle distribution. Smaller particles, due to ineffective segregation, may be entrapped in non-desired region and cause problems in later machining process. Larger particles segregate faster but may distribute non-uniformly if they are bigger than the dendritic arm spacing. The use of particles with an effective size greater than but smaller than the dendritic arm spacing (about for aluminum bronze) is recommended. Minimum dwell-time and stirring are also recommended to limit the degree of agglomeration and clustering of the TiC particles.

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8.3 REFERENCES 1. Ozbayraktar, S. and Koursaris, A., Met Trans, Vol. 27B, pp. 287-96, 1996. 2. Nastac, L. and Stefanescu, D. M., Fifth International Symposium on Physical Metallurgy of Cast Iron, Nancy, France, 1995, Eds. Lesoult, G. and Lacaze, J., SCITEC Publishing, Switzerland, pp. 469-478, 1997. 3. Nastac, L. and Stefanescu, D. M., AFS Trans., Vol. 103, pp. 329-37, 1995. 4. Nastac, L. and Stefanescu, D. M., Modelling and Simulation in Materials Science and Engineering, Institute of Physics Publishing, Vol. 5, pp. 391-420, 1997. 5. Nastac, L., Sundarraj, S., Pang, Y., and Yu, K. O., J. of Metals, TMS, No. 3, pp. 30-35, 1998. 6. Nastac, L., Numerical Heat Transfer Part A, Vol. 35, No. 2, pp. 173-189, 1999. 7. Liliac M. and Ogi K., Private Communication, Kyushu University, Fukuoka, Japan, 2000. 8. L. Nastac, Acta Mater., Vol. 47, No. 17, pp. 4253-4262, 1999. 9. L. Nastac, M. Guclu, H. Dong, C. Wang, and M. Gungor, Aeromat-2000, ASM International, Seattle, WA, June 2000. 10. L. Nastac, J. J. Valencia, M. L. Tims, and F. R. Dax, International Symposium on “Superalloy 718, 625, 706, and Derivatives”, Ed. E. A. Loria, Pittsburgh, PA, June 2001. 11. L. Nastac, Keynote Lecture, International Conference on Modeling of Casting and Solidification Processes (MCSP-4), Eds. C. P. Hong et al., Yonsei University, Seoul, Korea, pp. 31-42, September 5-8, 1999. 12. L. Nastac, Proceedings of the Modelling of Casting, Welding, and Advanced Solidification Processes-IX, United Engineering Foundation, Eds. P. R. Sahm, P. N. Hansen, and J. G. Conley, Shaker Verlag, Aachen, Germany, pp. 497-504, August 20-25, 2000. 13. L. Nastac, J. J. Valencia, J. Xu, and H. Dong, A Computer Model for Simulation of Multi-Scale Phenomena in the Centrifugal Casting of MetalMatrix-Composites, published by TMS © 2000, Nashville, TN, March 2000. 14. H. Hao, X. Huang, L. Nastac, S. Sundarraj, and A. Simkovich, Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes-VIII, Eds. B.G. Thomas and C. Beckermann, TMS, Warrendale, PA, pp. 1015-1022, 1998. 15. D. Lewis III and M. Singh, Proceedings of the In-Situ Composites: Science and Technology, Eds. M. Singh and D. Lewis, TMS, 1994. 16. L. Lajoye and M. Suery, Proceeding of the Intern. Symposium on Advances in Cast Reinforced Metal Composites, Eds. S.G. Fishman and A.K. Dhingra, ASM Intern., pp. 15-20, 1988. 17. Y. Fukui, JSME International Journal, Series III, Vol. 34, No. 1, 1991.

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18. M. Mizuno et al., Novel Techniques in Synthesis and Processing of Advanced Materials, Eds. J. Singh and S.M. Copley, TMS, 1995. 19. C.G. Kang and P.K. Rohatgi, Met Trans, Vol. 27B, 1996, pp. 277-285. 20. M. Suery and L. Lajoye, Proceedings of the Solidification of Metal Matrix Composites, Ed. P. Rohatgi, TMS, pp. 171-79, 1990. 21. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 27 A, pp. 4061-74 and 4075-83, 1996. 22. D. M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Met Trans, Vol. 21A, pp. 997-1005, 1990. 23. C. F. Pezzee and D. C. Dunand, Acta Metall., Vol. 42, No. 5, pp. 150924, 1994. 24. D. J. Srolovitz et al., Acta Metall., Vol. 32, pp. 1429-38, 1984. 25.I. Andersen and O. Grong, Acta Metall., Vol. 43, No. 7, pp. 2673-88, 1995. 26. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, Institute of Physics Publishing, Vol. 5, pp. 391420, 1997. 27. L. Nastac, S. Sundarraj, and K.O. Yu, Proceedings of the Fourth Intern. Special Emphasis Symposium on Superalloy 718, 625, 706, and Derivatives, Ed. E. A. Loria, pp. 55-66, 1997. 28. M. Rappaz and Ph. Thevoz, Acta Metall., Vol. 35, No. 7, pp. 1487-97, 1984. 29. W. Kurz and D. J. Fisher, Fundamentals of Solidification, 3rd ed., Trans Tech Publications, Aedermannsdorf, Switzerland, 1989. 30. L. Lajoye, Ph.D. Thesis, Institut National Polytechnique de Grenoble, 1988. 31. A. Kagawa and T. Okamoto, The Physical Metallurgy of Cast Iron, H. Fredriksson and M. Hillert eds., Materials Research Society Symposia Proceedings, vol. 34, pp. 201-210, 1985. 32. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 24A, pp. 2107-18, 1993. 33. L. Nastac and D. M. Stefanescu, Proceedings of the Micro/Macro Scale Phenomena in Solidification, HTD-Vol. 218/AMD-Vol. 139, ASME, pp. 27-34, 1992. 34. R. A. Krivanek and C. E. Mobley, AFS Trans., Vol. 92, pp. 311-17, 1984. 35. G. Upadhya, D. K. Banerjee, D. M. Stefanescu and J. L. Hill, AFS Trans., vol. 98, pp. 699 706, 1990.

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9

MODELING AND SIMULATION OF INGOT SOLIDIFICATION STRUCTURE IN PRIMARY AND SECONDARY REMELT PROCESSES

In this chapter, computer modeling and analyses of microstructure evolution during the solidification of remelt and continuously-cast ingots, are presented. Simulation results of microstructure in as-cast ingots for aerospace and ground applications are also presented.

9.1 INTRODUCTION Quality is one of the major concerns in the manufacture of highperformance titanium (Ti) alloy (fan and compressor discs and blades) and nickel (Ni) superalloy (turbine and compressor discs) components for aerospace and other commercial applications. The materials for these components are made by a primary melting process, such as plasma arc cold hearth melting (PAM) or electron beam cold hearth melting (EBM), as well as a secondary remelting process, such as vacuum arc remelting (VAR) or electroslag remelting (ESR). To produce these ingots, these processes involve a continuous casting operation for PAM and EBM and a semicontinuous casting operation for VAR and ESR. The quality of cast ingots is governed by two factors: (i) grain structure formation (columnar and equiaxed morphologies as well as the structural columnar-to-equiaxed transition (CET)) and (ii) micro/macrosegregation phenomena (beta flecks in Ti-based alloys and freckles, Laves phases, and carbides in Ni-based superalloys). Process modeling has become a viable tool to optimize the

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EBM, ESR, and VAR processes and is currently being applied to the PAM process. A comprehensive modeling approach to simulate ingot solidification phenomena in primary melting and secondary remelting processes is depicted in Figure 9-1. Based on process parameters and material inputs, the deterministic model performs computations for macroscopic mass, heat transfer, fluid flow, electromagnetic, and species transport to provide temperature, velocity, and concentration fields as outputs. From the macroscopic temperature field the pool profile and pool size, as well as the shape and size of the mushy region can be determined. In addition, from the concentration field, macrosegregation related defects, such as beta flecks in Ti alloys and freckles and tree ring patterns in superalloys, could be obtained. The stochastic model uses results from the deterministic model to predict the evolution of grain structure in solidifying ingots. It can also compute the grain size and CET, as well as microsegregation related defects such as beta flecks in Ti alloys and Laves phases and carbides in superalloys. The relevance of the stochastic model is that the simulated structures can be directly compared with actual structures obtained from experiments. The computer becomes a “dynamic metallographic microscope.” The stochastic model is comprehensive because the competition between nucleation and grain growth kinetics of various phases is considered. Furthermore, grain impingement is directly accounted for in the model.

The effects of macroscopic transport of mass and energy on the evolution of the pool profile and grain structure formation during solidification of VAR, ESR, and PAM ingots were investigated. Also, a thorough investigation to analyze the effects of various nucleation and growth parameters on the formation of the solidification structure of PAM ingots is presented later. The model that follows the approach shown in Figure 9-1 consists of two essential components: (i) a deterministic macroscopic approach to model the mass and energy transport during alloy solidification

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and (ii) a stochastic microscopic approach to simulate the evolution of microstructure.

9.2 DESCRIPTION OF A MODELING APROACH FOR SIMULATION OF REMELT INGOTS 9.2.1 A Deterministic Macroscopic Model for Calculation of Mass and Energy Transport in Cast Ingots

In this section, a moving grid, fully implicit, control-volume approach for predicting the transient macro-transport phenomena in solidifying continuously cast ingots is presented [1]. This macro-model accounts for energy transport within the ingot by conduction and to the surroundings by conduction, convection, and thermal radiation. Also, the mass and energy additions associated with the continuously poured metal are included in the model. Two versions of the solidification model are shown below: (1) a solidification-kinetics model that includes the convective fluid flow effect and (2) a simplified solidification model that assumes a linear evolution of solid fraction and no convective fluid flow. More details about this model can be found in chapter 2. (1) In the presence of convective transport, the governing energy transport equation for a two-dimensional (2-D), axisymmetric, cylindrical geometry is given by [ 1 ]:

where t is time, T is the temperature, K is the thermal conductivity, is the density, is the specific heat, is the source term associated with the change of equiaxed and columnar phases, which describes the rates of latent

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heat evolution during the liquid-to-solid transformation, L is the latent heat of fusion,

and

are the solid fractions of the equiaxed,

columnar, and eutectic phases, respectively, is the liquid fraction, u and w are the superficial velocity components, and are the solid velocities of the equiaxed and eutectic phases, and Z(t) is the expanding domain length or the cast ingot length. The governing equations are written in terms of superficial velocities defined as:

Note that, for columnar solidification or after the occurrence of coherency in equiaxed solidification, the solid velocities are null. The continuum thermophysical properties and are weighted by the solid fraction as follows:

The other governing equations for conservation of mass, momentum, and species transport, are described in detail in [1]. The governing equations are in the conservative form as recommended by Patankar [2] for the numerical solution of heat, mass, and fluid flow problems. To solve these equations, it is required to know the competitive evolution of the solid fractions encountered in the present solidification system, i.e., eutectic, dendritic equiaxed, and dendritic columnar structures. The solidification-kinetics models for the calculation of the evolution of these solid fractions are summarized in the following paragraphs. The coupling between macrotransport and solidification kinetics is accomplished through the fraction of solid evolution that is described at the microscopic scale [3, 4]. This must be done for dendritic columnar, dendritic equiaxed, and eutectic equiaxed growth. For dendritic growth, the model recently developed by Nastac and Stefanescu [3, 4] is used. For dendritic growth (both columnar and equiaxed), the evolution of the fraction of solid is expressed as [3, 4]:

Chapter 9. Modeling and Simulation of Ingot Solidification Structure in Primary and Secondary Remelt Processes

where

is the liquid fraction,

155

is the solid fraction of the phase p, p

denotes a dendritic columnar, a dendritic equiaxed, or an eutectic phase, the geometrical factor is 3 for dendritic equiaxed growth and 2 for dendritic columnar growth, Eqs. (19)-(22) in [1]),

is the average growth velocity of the envelope (see is the equivalent dendrite envelope,

shape factor of the dendrite envelope, the solid instability,

is the

is the average growth velocity of

is the radius of the solid instability, and

is the

shape factor of the instability. (2) In the absence of convective transport, the governing heat conduction equation for a 2-D, axisymmetric, cylindrical geometry is given by:

where T is the temperature, is the liquidus temperature, is the solidus temperature, K is the thermal conductivity, is the alloy density, is the specific heat capacity, is the source term associated with the change of solid fraction, which describes the rates of latent heat evolution during the liquid-to-solid transformation, L is the latent heat of fusion, is the liquid fraction, and Z(t) is the expanding domain length or the cast ingot length. The fraction of solid is linearly dependent on the temperature in the mushy region. Although more sophisticated models [3-14] can be used to describe ingot solidification, the current approach not only describes both columnar and equiaxed solidification but also accounts for remelting phenomena. Expansion of the solidification domain in the withdrawal direction (the zcoordinate) is described through the addition of uniform volume elements. For a 2-D axisymmetric cylindrical geometry, the appropriate heat transfer

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boundary conditions (BCs) for these processes are: symmetry at the ingot center, convection and radiation at the ingot edge and ingot bottom, and process specific BCs at the ingot top to account for the heat input and heat loss. An important parameter is the heat transfer coefficient at the ingotmold interface, which was calculated by the method described in [15]. 9.2.2 A Stochastic Mesoscopic Model for Simulation of Structure Evolution in Solidifying Ingots

The present stochastic approach differs from the classical “Cellular Automata” technique [9, 16] in that it uses thermal history results from the deterministic model described in the previous section [17]. It is also based upon a Monte Carlo probabilistic approach required to calculate microscopic quantities [18, 19]. Development of the present stochastic model for grain structure evolution is described in detail in Chapter 4. This description includes nucleation and growth kinetics, as well as the growth anisotropy and grain selection mechanisms. The required input data for stochastic calculations are provided by the macroscopic model, and include: (i) local cooling rates calculated at the liquidus and solidus temperatures, (ii) timedependent temperature gradients in the mushy zone, also calculated at the liquidus and solidus temperatures, and (iii) local solidification start time and end time. Local cooling rates calculated at the liquidus temperature are used to compute the nucleation parameters (Eqs. (9-7) and (9-8) in this Section). Local average cooling rates and time-dependent temperature gradients in the mushy zone are used to compute the grain growth parameters [see Eqs. (9-8) to (9-11) in this Section]. During solidification of these continuously cast ingots, at least three grain morphologies are encountered: equiaxed grains, columnar grains solidified under a variable G/V ratio, and columnar grains solidified under a relatively constant G/V ratio, where G and V are the local temperature gradient and solid-liquid (S/L) interface velocity of the mushy region, respectively. All aforementioned morphologies as well as the columnar-to-equiaxed transition are driven more or less by the same solidification mechanism, that is, the nucleation and growth competition of various phases in the mushy region. The stochastic models for equiaxed and columnar grains solidified under a variable G/V ratio are given in Sections 9.2.2.1 to 9.2.2.5. The columnar structure solidified under a relatively constant G/V ratio, which is perhaps the most common morphology met during primary melting or secondary remelting, is described in Section 9.9.2.6.

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Description of a stochastic model for equiaxed and columnar grains solidified under variable G/V ratio

The entire casting is first enmeshed in macro-volume elements for which energy, mass, and momentum transport can be calculated by classic methods. Then, each macro-volume is subdivided into a number of cubic micro-volume elements that have a state index associated with them. In the time-step calculation, the average temperature of the specimen is calculated from an energy balance. When the average temperature is lower than the equilibrium (or nucleation) temperature, the nucleation and growth of grains begin. The structure of the stochastic micro-model consists of a regular network of that resembles the macro-volume element. The micro-model is characterized by: (a) geometry of the (b) state of the (c) neighborhood configuration, and (d) transition rules that determine the state of the Anderson et al. [20] compared triangular lattice with square lattice grain growth models. Their results suggest lattice-independent grain growth. Accordingly in this work, the geometry of is chosen to be cubic in 3-D and square in 2-D calculations. Each has two possible states: it is either liquid or solid. The ratio between the number of solid and the total number of represents the fraction of solid within the macro-volume element. A schematic 2-D representation of a macro-volume element is given in Figure 9-2a. The selected neighborhood configurations are also shown in Figure 9-2a. They are based only on the first- and second-order nearest neighbor Type “A” is the first-order configuration and it has four first nearest neighbors. Type “B” configuration has eight first nearest neighbors. Type “A” configuration is based on the cubic von Neumann’s definition of neighborhood, while type “B” configuration is based on Moore’s definition [21]. Type “C” hexagonal configuration has six first nearest neighbors. As shown in Figure 9-2b, the growth of a type “A” configuration reproduces rhomboidal grains, type “B” configuration gives square grains, while type “C” configuration leads to hexagonal grains, oriented at 45° with respect to the grid. Based on these configurations, other configurations can be obtained. If the two types of “C” configurations are utilized consecutively, the grains grow into octagons [16], while if they are sequentially combined, square grains can be simulated. Figure 9-2b also shows that each growing crystal is conserving its envelope although the growth directions are different. At the same simulation time, if only deterministic transition rules are applied, type “B” configuration will give the maximum occupancy. Maximum occupancy is defined as the pattern where all the solid within the square envelope (type “B” configuration) of the grain are occupied. Minimum occupancy is obtained for type “A” configuration.

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Intermediate occupancy is obtained with type “C” configurations. In fact, since the grain patterns are conserved during time calculations until grain impingement, the occupancy is also constant.

The occupancy level relative to type “B” configuration is 1, 0.5, and 0.75, for type “B”, “A”, and “C” configurations, respectively. All three selected configurations are totally deterministic. Probabilistic selection of the previously described deterministic configurations gives another growth configuration, named type “D” configuration. The probabilistic selection is

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based upon the crystallographic anisotropy of grain growth. This procedure will be described in the next section. Solidification behavior depends to a great extent on the transition rules. In the present model, the change of state of the from liquid to solid is initiated either by nucleation or by grain growth kinetics. For computational convenience, the models for the description of nucleation and grain growth kinetics are initiated based on the local solidification start time, which is calculated from the macroscopic model. The local solidification start times of were obtained by interpolating the local solidification start times of macro-volume elements.

Growth anisotropy and probabilistic selection of neighborhood configuration

At the beginning of the simulation, all the are liquid, and their state index is set to zero. As nucleation proceeds, some become solid, and their index is changed to an integer larger than zero, n. The in contact with the mold wall are identified with a different reference index, m. The index is transferred from the parent to adjacent as they become solid through growth. For the case of dendritic solidification, the integer takes into account the preferential growth of cubic crystals in the direction. When equiaxed eutectic solidification is considered, the integer shows the random location of various nuclei. For graphical representation each integer has a color associated with it, and each is a pixel on the computer screen. Both crystallographic orientation and random location of the new grains are chosen randomly among 255 orientation classes, which are the first 255 colors used for graphical representation. In 2-D calculations, the probability, that a newly nucleated grain has a principal growth direction in the range is given by [22, 23]:

where takes into account the four-fold symmetry of the crystal, i.e., the integral of from to is equal to unity. The probabilistic selection of the previously presented neighborhood configurations (type “D”) is based on the angle as shown in Table 9-1. Note that the integer i, calculated as a function of gives the neighborhood configuration rule. When type “C” configurations are selected. Since two types of “C” configurations are used in calculations, a

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further random selection of either one of the two type “C” configurations is chosen. As shown in Table 9-1, the selection of the type “C” configurations is arbitrary and is based on a random number, rand that is generated for each temporal and spatial calculation.

Nucleation

The number of grains, and that nucleate in the volume of the liquid and at the surface of the mold during one micro-time step, are calculated by using nucleation site distributions, and respectively. These distributions can be calculated by assuming some experimental approaches similar to those already used in deterministic models of solidification. The instantaneous nucleation model presented in [5] was used in this work to calculate the nucleation site distributions. Other nucleation models are described in [9, 16, 24]. Assuming no grain movement in the liquid, the grain density at any given location, N, can be expressed as a function of local cooling rate as:

where and experiments and solidification.

are the nucleation parameters determined from is the local cooling rate at the beginning of

Chapter 9. Modeling and Simulation of Ingot Solidification Structure in Primary and Secondary Remelt Processes

The probabilities,

or

for a

located in the bulk of the liquid

or at the surface of the mold to nucleate during the micro-time step,

where

and

161

are the number of

are:

in the bulk and at the

metal-mold interface, respectively. During each time-step calculation both metal-mold interface and bulk are scanned and a random number, rand, is generated for each of them The nucleation of a that is still liquid will occur only if or

Growth

It is assumed that a nucleus formed at a particular location (the center in each configuration of Figure 9-1) will grow based upon a growth kinetics model and a neighborhood configuration rule previously described. As nucleation proceeds, the becomes active (n or m > 0) and can grow over a distance, that is given by the following equation:

where is the initial time, t is the actual time, and V is the growth velocity. The initial time is defined as either the nucleation time, if growth is initiated through a nucleus, or the capturing time, if growth is initiated by capturing another At the capturing time, the initial on which the growth was initiated captures the nearest neighbors based on a selected neighborhood configuration. For the case of dendritic growth, this occurs when is equal to the distance:

where a is the size of the for the stochastic network and takes values from to At the capturing time, the initial becomes solid and its state index is transferred to the captured neighbors. For the case of eutectic

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solidification, growth is not controlled by Eq. (9-10) since crystallographic growth direction is irrelevant. Thus, In order to reflect the probabilistic nature of grain extension, the nearest neighbors are trapped by the active only when a randomly generated number, rand, is smaller than the capture probability, defined as follows:

Further, the same procedure is used until all

become solid.

Growth kinetics models used for calculation of With the present model, at least three grain growth morphologies can be simulated: equiaxed dendritic, columnar dendritic, and equiaxed eutectic. Nucleation and growth competition of the grain morphologies controls the distribution and amount of phases. For the case of columnar dendritic solidification, the growth kinetics of the dendrite tip, is calculated with the model developed by Kurz, Giovanola, and Trivedi [25] as follows:

where

is the liquid diffusivity,

is the Gibbs-Thomson coefficient, k is

the partition ratio, is the solidification interval, is the S/L interface undercooling, and is the actual cooling rate. For equiaxed dendritic solidification, the model developed by Nastac and Stefanescu [3, 26, 27] is applied. Thus, the growth velocity of the tip is described by:

where

is the liquidus slope,

is the density, L is the latent heat,

is the

liquid thermal conductivity,

is the liquid interface concentration, and the

stability constant

The melt undercooling for the system under

Chapter 9. Modeling and Simulation of Ingot Solidification Structure in Primary and Secondary Remelt Processes

consideration can be definition/assumption:

where

calculated

based

on

is the equilibrium liquidus temperature,

volume-averaged liquid concentration, and

163

the

following

is the intrinsic

is the bulk temperature

defined as the average temperature in the volume element.

and

are calculated with the microsegregation model described in [27]. The growth of equiaxed eutectic grains is calculated by [5, 6]:

Description of a columnar interface under a relatively constant G/V ratio

Tracking of the columnar front assumes that, at the columnar front, the growth velocity is equal to the interface velocity of the dendrite tip at the same location. For a 2D axisymmetric domain, the position of the columnar front at time can be iteratively computed by:

where the angle of growth,

is:

Here, is computed with Eq. (2-12) and and are the local temperature gradients in the mushy zone in the z and r directions, respectively. Since a square mesh is used in the present stochastic approach, a selective randomization procedure was developed to impose the primary growth direction of the columnar grains normal to the S/L interface. This is based on the fact that columnar grains will grow only when a randomly generated number, rand is smaller than the growth direction probability, defined as [28, 29]:

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where the parameters m and n control the growth direction of columnar grains. For m = 0.1 and n = 0.5, the randomization procedure works properly, i.e., the primary growth direction of the columnar grains will closely follow the movement of the S/L interface.

9.2.3 Computational Aspects for Modeling of Remelt Ingots

Following are some of the key issues related to stochastic modeling of remelt ingots: The time step, used in calculations is described by the Courant condition:

where a is the size of the micro-volume element and V is the interface velocity. Physically, Eq. (2-19) uses the same principle as applied in free surface fluid flow computations, i.e., growth is not allowed to take place for more than one half of during each time-step calculation. Note that, as shown in Eq. (2-19), the time step is linearly dependent on the grid size. It was found that the grid size should be between 100 and for simulation of both columnar and equiaxed grains. In this range the simulation results are stable and match experimental observations [30]. The computer memory and CPU-time requirements in the present stochastic calculations are discussed below. A RAM memory size of or 20 Kbytes/VE (one macro-volume element of 5 mm spacing with is needed in stochastic calculations. To simulate the structure of an ingot with H = 1.0 m and R = 0.25 m (axisymmetric calculations), a grid resolution of 4000x1000 (4 million of spacing each) is used. For this grid size, the time required to obtain an ingot simulated structure is on the order of 4 CPU hours on an SGI-200 MHz Challenger workstation. In addition, 200 Mbytes of RAM are needed. Note that, in the present approach, during a time-step calculation for any active both first- and second-order nearest neighbors are scanned. When a given and its neighbors become solid, the scanning procedure stops for that particular Thus, computer memory and CPU-time were decreased by at least one half.

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To date, there are also promising deterministic methods that can simulate the complex dendrite patterns. The Hele-Shaw problem [31] and the phase-field approach [32] constitute two of them. Although continuum equations are solved, their solutions are considered discretely integrated asymptotic solutions, i.e., the stability and CPU-time are much higher than the most efficient stochastic methods. However, to simulate a single dendritic equiaxed grain with the phase-field method, 7500 CPU seconds on the Cray Y-MP4E/232 was required [32], while the proposed stochastic model requires only 300 CPU seconds on the HP-755 workstation to simulate an entire specimen (grid resolution of 500x500) [9, 16]. 9.2.4 Primary and Secondary Dendrite Arm Spacings in Commercial Alloys (Deterministic Modeling)

The primary dendrite arm spacing (PDAS), for a multicomponent system is calculated with (see [33] for binary calculations) [34]:

where n is the number of alloying elements, is the liquidus slope, is the initial concentration, is the final concentration, is the liquid diffusivity, is the Gibbs-Thomson coefficient, k is the partition coefficient, and V and G are the growth velocity of the solid-liquid interface and liquid thermal gradient at the solid-liquid interface, respectively. The PDAS, for a pseudo-binary system can be calculated with:

The secondary dendrite arm spacing (SDAS), for a multicomponent system is calculated with (see [33] for binary calculations) [34]:

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The SDAS,

for a pseudo-binary system can be calculated with:

9.2.5 A Stochastic Model for Modeling Secondary Phases During Solidification of Alloy 718 Ingots

The importance of modeling the formation of NbC/Laves phases in alloy 718 has been discussed broadly in literature (see references in [27, 37, 51, 52]). It is also known that the distribution and amount of carbides and Laves phases in alloy 718 are affected by the solidification path. The volume fraction of solid is a function of local growth velocity, solidification time, solidus temperature, and local temperature gradient. The redistribution of elements strongly affects the phase evolution in common superalloys with respect to temperature, as well as their mechanical properties and surface stability at elevated temperatures. The primary goal of this work was to develop a solidification kinetics model to predict the evolution of NbC and Laves phases during ingot solidification. Previous studies on alloy 718 showed that both NbC and Laves phases produce intergranular liquid films due to the intergranular distribution of Nb and C [51, 52]. Also, the ability of Laves phase to promote intergranular liquation cracking (microfissuring and hot cracking) during heat treatment is much higher than that of NbC. This is because the temperature for the formation of Laves phases is usually lower than that for NbC (i.e., liquation initiates at the eutectic-Laves temperature). In [51, 52], it was demonstrated that the carbon content of alloy 718 directly affects the volume fraction of carbides. Note, that the as-cast alloy could contain a higher volume fraction of NbC and Laves phases than what the phase diagram suggests due to the microsegregation during solidification. The relative volume fractions of both NbC and Laves phases depend on the C/Nb ratio. Alloys with high C/Nb ratio will have a higher volume fraction of carbides than alloys with low C/Nb ratio. In developing a solidification kinetics model for predicting the formation of NbC/Laves phases in alloy 718, the following assumptions were made: instantaneous nucleation, carbide growth in the liquid, no interference between growing carbides, carbides are either pushed or engulfed by the S/L interface, volume diffusion-limited growth of carbides, and binary diffusion couple. The model accounts for: (1) nucleation and growth of columnar or equiaxed dendritic grains, (2) growth/remelting of spherical instabilities, (3)

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167

nucleation and growth of NbC assuming that the slow step is the volume diffusion of carbon from the liquid to the NbC/liquid interface, (4) redistribution of Nb and C concentrations, (5) redistribution of NbC particles between the solid and liquid phases at the spherical instability level, and (6) nucleation and growth of Laves phases. These features are further discussed below. (1) The dendritic growth (columnar or equiaxed) is based on the alloy 718 pseudo-binary phase diagram [3], where the primary driving force for growth is the Nb diffusion in the liquid. The models for equiaxed and columnar dendritic growth of alloy 718 are described elsewhere [2, 3, 11]. These are applied to calculate the evolution of the primary phase. (2) A coarsening (thickening) model for spherical instabilities was developed in [3, 4, 9]. It is assumed that the coarsening mechanism is analogous to Oswald ripening of precipitates. The model considers the dynamic nature of the spherical coarsening process through the evolution of the fraction of solid and time-variation of liquid concentration in the mush. The mechanism of coalescence of instabilities is ignored in this model. The final result is [3, 4, 9]:

Here, is the fraction of solid, is the S/L interface velocity of spherical instabilities, is one half of the average spacing between instabilities (at time t), instability radius at time

is the instability radius at time t,

is the initial

is a stability constant of the order of

is the intrinsic volume average liquid concentration, undercooling, and

where

is the liquid diffusivity,

is the is the

Gibbs-Thomson coefficient, is the liquidus slope, and k is the partition ratio. (3) The growth mechanism of NbC is governed by the carbon diffusion from the liquid to the NbC/L interface and by the reaction kinetics between Nb and C [8]. Accordingly, C is depleted at the NbC/L interface, and the amount of precipitated NbC depends on the volume diffusion of C from the liquid to the NbC/L interface. It is also assumed that the carbides instantaneously nucleate at the equilibrium solidification temperature of alloy 718. The solution of the spherical volume diffusion-limited growth is described by an averaging method provided in Ref. [37]:

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168

Here,

is the growth velocity of the NbC phase,

carbon diffusivity,

is the liquid

is the intrinsic average liquid carbon

concentration, is the carbon concentration in NbC, is the actual growing carbide radius, is the volume of liquid between instabilities, is the average number of NbC particles in

and

is the shape

factor of spherical NbC that considers the real geometry of the NbC (blocky type). The shape factor for NbC is assumed to be 0.64 [37]. (4) The redistribution of both C and Nb is calculated with the modified model described in Ref. [12]. The main assumptions are: solute transport is calculated in both solid and liquid phases assuming Fick’s law for binary systems in spherical coordinates, planar S/L interface in local equilibrium, closed system, and constant initial liquid concentration. The overall mass balance is used to couple the concentration fields in both the solid and liquid phases. The solution of this diffusion couple is a modified version of the Nastac and Stefanescu model [43]. It includes a sink term (Q in Eq. (2-26)) to account for the C and Nb depletion because of NbC phase growth. Thus, the interface concentration,

Here,

and

of both C and Nb is described by:

are described by Eq. (31) in Ref. [3, 4],

concentration of the alloy, k is the partition coefficient,

is the initial is the actual

carbide volume, is the volume of the element over which the mass balance is calculated, is the carbide density, is the alloy 718 density, is either C or Nb concentration in NbC, and is the average number of NbC particles in the liquid. The intrinsic average liquid concentration, required in Eqs. (2-24) and (2-25), is calculated with Eq. (32) in Refs. [3, 4]. (5) It has been generally accepted that there is a critical velocity for pushing/engulfment transition (PET) of dispersed second-phase particles by an advancing S/L interface [59, 60]. Non-wetting NbC particles are either

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169

pushed or engulfed by the S/L interface. In general, engulfment will lead to more uniform distribution of NbC particles, while pushing will result in segregation of NbC particles. Redistribution of NbC particles between the solid and liquid phases significantly affects not only the size and amount of NbC particles, but also the amount of Laves phases. Therefore, it is important to model particle redistribution during solidification of alloy 718. Unfortunately, there is no mathematical model that can completely describe this phenomenon. A comprehensive stochastic model that can describe the redistribution of NbC particles during the solidification of alloy 718 was developed and is outlined below. Similar to Scheil’s model for an open system [55], this present model assumes that there is no movement of particles in the solid, and that particles are uniformly distributed in the liquid. Accordingly, the following overall mass balance can be written:

Here,

is the average number of NbC particles in the volume

element,

is the number of NbC particles in the solid in the proximity

of the S/L interface,

and

are the density and final volume of the

NbC phase, respectively, and is the C concentration in the NbC phase. Following Hunt’s concept [14], an effective partition coefficient is introduced as:

Here, is a parameter that describes PET (0 for pushing and 1 for engulfment), is the particle velocity, and is the S/L interface velocity as described by Eq. (2-24). The particle velocity is dictated by the fluid flow around the particle. For a horizontal flow, the particle velocity is [53]:

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Here, g is the acceleration due to gravity,

is the liquid viscosity,

for a particle in an unbounded liquid (Stokes’s law), radius, and

is the actual NbC

for a particle approaching the S/L interface, where d

is the particle/interface separation distance behind the particle. is a function of the critical velocity of PET. If then otherwise, where rand is a computer-generated random number which takes values between zero and one. The probability of PET is:

Here, is the critical velocity of PET derived at [53] for a flat interface and a pure metal is the interfacial energy difference, is the atomic distance, and and are the thermal conductivities of the alloy 718 and NbC particles, respectively. (6) The Laves phase starts to form when the concentration of Nb reaches the eutectic + Laves) composition which is 19.1 wt.% Nb. The kinetics of Laves is very high due to its morphology (eutectic or globular type-divorced eutectic) and appearance as discontinuous thin films at grain boundaries. It is assumed that the amount of the Laves phase is directly related to Nb concentration [3, 4, 9, 37].

9.3 SIMULATION RESULTS FOR SOME COMMERCIAL APPLICATIONS 9.3.1 Modeling Parameters

A detailed description of the ESR and VAR processes is presented in Refs. [15, 30], while the PAM process is described in Ref. [35]. BCs, ingot dimensions, and control parameters of these processes are outlined in Table 9-2. Thermal properties of the materials used in the simulations are presented in Table 9-3 and the data used in the stochastic model are shown in Table 9-4.

Chapter 9. Modeling and Simulation of Ingot Solidification Structure in Primary and Secondary Remelt Processes

171

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The casting speed in Table 9-2 was calculated with:

where

is the mass flow rate of the pour stream (melting rate) and D is

the ingot diameter.

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173

9.3.2 Global Comparison of VAR, ESR, and PAM Processes

Grain growth direction and segregation defects, such as freckles and tree ring patterns in IN718 and beta flecks in Ti-base alloys, are directly related to the size and profile of the melting pool. In turn, the pool characteristics

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are mostly controlled by the casting rate and energy input. They are also strongly dependent on the ingot diameter. Grain size and shape as well as micro- and macro-segregation patterns are strongly influenced by the heat extraction rate at the metal-mold interface. Figure 9-3 shows simulated structures of various ingots made by different casting processes. In these pictures, the red (black) color represents the mushy zone, the yellow (light gray) describes the liquid region (molten pool), and the remaining area shows the solidified grain structure. The 255 colors in the solid region were used to describe the grain boundary of various structures. They also show the crystallographic orientation of columnar grains, which have nucleated at the ingot-mold interface. The nucleation and growth competition of columnar grains can clearly be seen in Figure 9-3. These grains randomly nucleate at the ingot-mold interface and grow toward the ingot center. Note that they also grow in a direction opposite to the withdrawal direction (upward solidification), closely following the direction of the mushy region gradients. The smallest pool depth is obtained with VAR

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175

The simulated ESR ingot, presented in Figure 9-3(c), has a V-shaped pool profile with non-uniform columnar grain size distribution from the ingot center to the outside edge. The simulated VAR (Figures 9-3(a) and 9-3(b)) and PAM (Figure 9-3(d)) ingots have U-shaped pool profiles with relatively uniform columnar grain size distribution. The grain growth direction in the ESR ingot is between 0° (at the ingot surface) and 45° (at the ingot center) with respect to the ingot vertical axis, while in the VAR and PAM ingots the grain growth direction is between 90° (at the ingot surface) and 0° (at the ingot center). A thorough discussion of ESR and VAR results as well as optimization procedures for VAR and PAM are provided below.

9.3.3 VAR Process Modeling

The simulated evolution of VAR ingot structures (case A in Table 9-2) is shown in Figure 9-4. The steady state in the VAR process (see Fig. 9-4(b)) is reached when the height of the ingot equals its diameter (H = D). An important characteristic of the solidifying VAR ingot structure is the columnar grain growth direction. It has angles between 90° (at the ingot surface) and 0° (at the ingot center) with respect to the ingot vertical axis. The smaller the growth angle at the ingot center, the lower the tendency for CET and major segregation zone to form at the ingot center. Note that the grain growth direction is mostly controlled by the power input and power distribution over the top surface of the ingot. Accordingly, a VAR ingot will have a very different grain structure from that of an ESR ingot. Figures 9-5 and 9-6 show comparisons between experiments and simulated results of VAR ingots cast without hot topping for two different melting rates. Note that after the power was shut off the ingot top region was allowed to solidify in vacuum (radiative heat loss over the ingot top area was considered in the numerical calculations). The complexity of the ingot structure is clearly seen from these pictures. There are three grain morphologies: columnar grains solidified under a relatively constant G/V ratio (the steady-state structure), columnar grains formed under a variable G/V ratio (from the top surface of the ingot), and equiaxed grains. Also, there are two transitions: the CET and the transition from small-to-large equiaxed grains. The former transition takes place because equiaxed grains nucleate and grow ahead of both columnar fronts. The latter transition occurs either because of grain sedimentation or a sudden change in the solidification conditions. In this model, the CET is driven by a critical temperature gradient below which equiaxed nuclei may form. The ability and tendency of the equiaxed grains to grow in the mushy region is directly considered in the model through the growth competition between columnar and equiaxed grains. If

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

equiaxed grain growth is slower than columnar grain growth, then columnar growth dominates and the small equiaxed grains can get stumped and trapped between the columnar grains. This aspect of CET modeling was successfully demonstrated in [9, 16, 23]. However, under the present solidification conditions, equiaxed growth is relatively fast compared to the columnar growth, and as soon as the equiaxed nuclei become active, they grow and initiate the CET (see Figures 9-5(b) and 9-6(b)). Also, from Figures 9-5(a) and 9-5(b) as well as 9-6(a) and 9-6(b), the steady-state columnar fronts (Figures 9-5(a) and 9-6(a)) advance at a much slower rate when compared to the final grain structures (Figures 9-5(b) and 9-6(b)). Overall, all the aforementioned micro- and macro-structural particularities, such as grain size, grain structure, grain growth pattern, and CET, are accurately predicted with the present stochastic model. The pool depth and pool profile match very well with the experiments. However, discrepancies in the grain growth direction close to the ingot surface for low melting rates can be observed from Figures 9-5(b) and 9-5(c). This demonstrates the fact that, although the macroscopic model predictions of the pool profile and pool depth can closely match experimental results, they may not be sufficient for process design analysis; the stochastic calculations could give indication of the correctness of the macroscopic-model calculations.

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177

The usefulness of the present state-of-the-art simulation tool is not only the capability of correct analysis of the structure evolution of cast ingots solidifying under various cooling conditions, but also the capability to verify the validity of the macro-model calculations.

9.3.4 ESR Process Modeling

The simulated evolution of ESR ingot structures (Case C in Table 9-2) is shown in Fig. 9-7. From these pictures, it can be observed that the pool size, pool profile, and macrostructure change with time. Similar to the VAR process, the size of the initial transient region in the ESR process equals the diameter of the ingot (see Fig. 9-7(b)). A comparison of ESR processes for two different melting rates (Cases C and D in Table 9-2) is presented in Figures 9-7(c) and 9-7(d). By doubling

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

the melting rate, the pool depth increases from 0.2 m (Case C) to 0.51 m (Case D). Also, the mushy zone is larger for the case of high melting rate. An ingot obtained with a high melting rate ESR process is prone to larger structure-segregation related defects than an ingot produced by a low melting rate ESR process.

Unlike the ingots obtained by the VAR process, where a U-shaped pool profile is observed with relatively uniform columnar grain size distribution from the ingot center to the outside edge (see Figures 9-4, 9-5, and 9-6), the ingots made by the ESR process have V-shaped pool profiles with nonuniform columnar grain size distribution (Figure 9-7). Comparisons between simulated results and literature experiments [42] are shown in Figure 9-8. Both experimental and simulated structures of the ESR top

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179

ingots (final transient regime) for two melting rates (see Table 9-2) are shown in Figures 9-8(a) to 9-8(f). Figures 9-8(b) and 9-8(e) show simulated results before stopping the power input, while in Figures 9-8(c) and 9-8(f) the simulated results were obtained after shutting off the power and allowing the top region to completely solidify. In this case, the slag acts as hot topping because its insulating capacity. Note that the shrinkage pipe formed at the ingot top region (see experiments-Figures 9-8(a) and 9-8(b)) is not simulated with the present model.

The model-predicted results such as pool profile and size, grain growth pattern, and shape and size of the columnar grains compared well with experiments. A comparison between experimental and simulated results after reaching steady state is presented in Figures 9-8(g) and 9-8(h). Although the calculated pool profile and size matched well with the experimental measurements, there is a small discrepancy in terms of grain growth direction. Certainly, accounting for the fluid flow in the calculations in the liquid and mushy region will reduce this mismatch. Initial transient, steadystate, and final transient regions are shown in Figure 9-9 for both simulated

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

and experimental results. The stochastic-model predictions of location and structure in these three regions are in reasonable agreement with the experiments. In Figure 9-10, a comparison between an experimental ESR processed alloy 718 ingot structure and calculated results with and without fluid flow is provided [48]. When fluid flow is considered (Fig. 9-10c), calculated grain structure closely matches the experiments (Fig. 9-10a) in terms of grain size, grain patterns, and growth direction.

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181

9.3.5 PAM Process Modeling

Comparisons of simulated and experimental PAM ingot solidification structures for high melting rate (550 kg/hr) and low melting rate (365 kg/hr) are presented in Figure 9-11. The melting rates of 365 kg/hr and 550 kg/hr correspond with withdrawal speeds of and respectively. The ingot diameter and height are 26 inches (650 mm) and 52 inches (1300 mm), respectively. The data presented in Tables 9-2 to 9-4 were used in these simulations. A Gaussian distribution with a maximum value of was assumed for the heat flux at the ingot top.

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Both the equiaxed and columnar solidification regimes were assumed in computing the ingot macrostructures in Figure 9-11. Note that the computed grain size in Figures 9-11(b), 9-11(c), 9-11(f), and 9-11(g) reasonably matches the grain size from the experimental macrostructures shown in Figures 9-11 (a) and 9-11(e). It appeared from Figure 9-11 that the melting rate has a very small effect on the ingot grain size and its morphology. A detailed sensitivity analysis regarding the effect of various nucleation and growth parameters on solidification structure is presented in the Appendix 1. Comparisons between predicted and experimental grain size are shown in Figure 9-12 for two melting rates. Measurements were taken at 1 m from the ingot bottom. The predicted grain sizes for both melting rates match the measured ones reasonably well. Figure 9-13 presents two microstructures taken from two different locations in a PAM ingot processed at 550 kg/hr. Dendrites can clearly be seen in these micrographs confirming that dendritic solidification occurs in PAM-processed Ti-17 ingots. Figure 9-14 shows two macrostructures at three different locations in a PAM ingot processed at 550 kg/hr. Both the columnar and equiaxed grains were formed in these ingots.

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9.3.6 Process Optimization

Two aspects are considered to improve the ingot quality: (1) the optimal melting (feeding) rate and (2) the hot topping procedure. (1) Optimal melting rates and energy input are related to the ingot diameter. A low melting rate may provide a macrostructure with columnar grains parallel to the ingot vertical axis. A high melting rate might improve ingot surface quality, help removal of white spots, and increase productivity. Figure 9-15 shows the effect of reducing the melting rate by one half for a PAM Ti-17 alloy cast ingot. A casting speed of (Case E in Table 9-2) which corresponds to the ingot casting speed in a normal PAM process was used in the simulation results presented in Fig. 9-15(a). A large equiaxed region is observed within the melting pool. Figure 9-15(b) shows an optimized PAM process, when the casting speed was decreased to (Case F in Table 9-2) while keeping other process and solidification conditions similar in both cases. It can be observed from the Case F that the pool depth and, therefore, the equiaxed region formed at the top of the ingot were considerably reduced. Overall, the columnar grain size (primary dendrite arm spacing) was also decreased for Case F as compared with Case E. However, the initial transient region shown in Case E is smaller than that in Case F. Also, note the sudden change in the grain structure at the ingot center (Case F) due to the abrupt change in the local solidification conditions. (2) An important step to reduce yield loss in these casting processes is the use of hot topping. Figure 9-16 shows a comparison between VAR ingot structures obtained without and with optimized hot topping (Figures 9-16(b) and 9-16(c)) for an IN718 alloy. When the hot topping was simulated, a sigmoidal decrease in the power input was considered (see final transient in Figure 9-16(a)), such that the temperature of the ingot top surface was

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always above the liquidus temperature of the alloy under consideration. This restriction has certainly prevented nucleation and, therefore, grain growth from the ingot top surface. In this case, this optimization will increase the ingot yield by 15%. The effects of a short-time power interruption and melt rate changes (from high to low power) on the macrostructures of VAR-processed alloy 718 ingots are shown in Figure 9-17. Two different equiaxed nucleation rates were studied. The columnar grain size after power interruption is larger when high equiaxed nucleation rate was assumed, suggesting a higher potential to freckling than in the low nucleation rate case.

9.3.7 Alloy Systems and Solidification Maps

A last issue that is discussed in this paper is the difference in the solidifying structures between IN718 and Ti-17 alloys. The solidification morphologies of IN718 and Ti-17 alloys, mapped with the solidification parameters (V and are shown in Figure 9-18. The boundaries of the GET region in Figure 9-18(a) were calculated with Hunt’s analytical model [44].

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185

The upper and lower bounds for the solidification parameters (i.e., and V) in the VAR process are shown as an operating window (see dotted square shown in Fig. 9-18 (a)), while the operating window for the PAM process is presented in Fig. 9-18(b). Figure 9-18 reveals that the tendency for the formation of GET is larger for IN718 (VAR) than for Ti-17 (PAM), even though more severe solidification conditions exist during the PAM process.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

High values of ratio favor columnar solidification, while low values of accommodate equiaxed solidification. For these reasons the feeding rate (or casting speed) of Ti-17 PAM ingots may exceed by several times the melting rate for producing IN718 VAR ingots. Also, for the solidification conditions in the PAM, VAR, and ESR processes, both alloy systems should solidify with dendritic (both equiaxed and columnar) structures, i.e., the cellular-to-dendritic transition would not occur. The S/L interface stability criterion (or constitutional undercooling criterion) may be applied to determine the stability of the S/L interface [33]:

where is the average temperature gradient in the mushy zone and is the nonequilibrium solidification interval. Equation (9-32) states that the S/L interface of the Ti-17 alloy should be more stable than that of the IN718 alloy since of IN718 is larger than that of Ti-17. The microstructure finesse depends on the growth velocity of the S/L interface, V. As shown in

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187

Eqs. (9-12) and (9-13), V is related to For the present solidification conditions, the higher the growth velocity the finer the microstructure. Calculated solidification maps [40, 46] are shown in Fig. 9-19 for IN718, Ti6Al-4V (Ti-6-4), and Ti-17 alloys.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Those solidification maps can be used to estimate the SDAS, porosity, and CET in these alloy systems (see Fig. 9-19) if process conditions are known. The modeling approaches for calculating the SDAS, CET, and porosity volume fraction (P) are described in [40]. 9.3.8 Prediction of Primary and Secondary Dendrite Arm Spacings

Figures 9-20 and 9-219 show the influence of solidification conditions on the PDAS and SDAS for the IN718 alloy and for the Ti-6-4 and Ti-17 alloys, respectively. The calculated values of PDAS and SDAS coefficients are shown in Table 9-5. For similar solidification conditions (e.g., solid-liquid interface growth velocity and temperature gradients at the solid-liquid interface), the PDAS in Ti-17 is approximately 50% larger than that in Ti-64, Figure 9-21(a). In this case, the lower the temperature gradient at the solid-liquid interface, the lower the PDAS. For the same solidification conditions (e.g., solidification time), the SDAS in Ti-6-4 is approximately two times larger than that in Ti-17, Fig. 9-21(b). The process windows shown in Figure 9-21 (see also the process window for PAM ingots shown in Fig. 11(b), p. 158, Ref. [34]) represent typical values of the PDAS and SDAS that should be found in 20-30 inch diameter PAM ingots.

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189

It is very difficult to detect and measure the primary and secondary dendrite arm spacings in Ti alloys [40]. Figure 9-22 shows the morphology (size and shape) of dendrites (dendritic crystals) found in a shrinkage cavity of a PAM processed Ti-17 ingot (see also the macrograph in Fig. 9-23). Both the PDAS and SDAS are clearly seen in Figure 9-22. The values of the PDAS and SDAS in Fig. 9-22 were estimated to be in the range of 1000 to and 150 to respectively. Since these spacings were found in shrinkage-cavity regions, which are relatively slowly cooled regions, it is believed that the dimensions of these spacings are somewhat higher than those in shrinkage-free regions. Comparisons of the simulated PDAS and SDAS values for the PAM ingots for high (1200 lbs/hr) and low (800 lbs/hr) melting rates are provided in Fig. 9-24. Similar values of the PDAS and SDAS can be seen from Fig. 9-14, meaning that the effect of changes in melting rate on these solidification characteristics is small.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Correct calculation of the evolution of the secondary dendrite arm spacing (SDAS) is a necessary condition for accurately predicting the formation of the secondary phases in alloy 718 remelt ingots. Figure 9-25 shows a comparison between calculated and experimental SDAS for alloy 718 VAR and ESR ingots. Two different melting rates are considered for each remelting process in Fig. 9-25. It can be seen that the calculated SDAS agrees closely with the experimental measurements. Some differences exist between the ESR and the VAR results in terms of SDAS. As expected, the SDAS decreases in size when the power input and melting rate are both increased. The predicted maps of both the primary and secondary dendrite arm spacings for a VAR processed alloy 718 ingot (melting rate of 172 kg/hr) are presented in Fig. 9-26. The primary dendrite arm spacing size varies between near the mold/metal interface and approximately along the ingot centerline. It is mostly controlled by the local gradients in the mushy zone and by the solidification rates. The size of the secondary dendrite arm spacing changes from 50 to over the ingot diameter. It is related to the local solidification times or the local cooling rates in the mushy zone.

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191

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

9.3.9 Stochastic Modeling of Secondary Phases

Previous studies on alloy 718 showed that an as-cast alloy could contain a higher volume fraction of NbC and Laves phase than what the phase diagram suggests due to the microsegregation during solidification [3, 4, 37. 50]. The

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193

relative volume fractions of NbC and Laves depend on the C/Nb ratio. Alloys with high C/Nb ratio will have a higher volume fraction of carbide than alloys with low C/Nb ratio (see details in references [3, 4, 37. 50]). A typical SEM micrograph shown in Fig. 9-27 [3, 4, 9] illustrates the morphology of Laves phases and NbC particles (blocky type) segregated in the interdendritic regions (between secondary dendrite arms) of alloy 718 at a cooling rate of 2 K/s. It can be seen from Fig. 9-27 that approximately 70% of NbC particles are located at the grain boundary, while the remaining 30% are found in the intradendritic regions. In the current model, Eqs. (9-27)–(930) were used to calculate the redistribution of NbC particles during solidification. When the S/L interface velocity is larger than the minimum critical velocity of PET, NbC particles will be engulfed. Otherwise, they will be either pushed or engulfed (or entrapped) based on stochastic calculations. Note that the stochastic model accounts for the effect of the S/L dendritic interface roughness and other mechanisms, such as sliding/rolling [60] and solute redistribution, on PET. Modeling Parameters. Data used in micro-modeling of alloy 718 are presented in Table 9-6. Data required to calculate the critical velocity of PET are: and [60]. The measured final NbC particle radius, expressed as [37]:

in

and

versus cooling rate is where

is the cooling rate

is in

Figure 9-28 shows the stochastic calculations of the redistribution of NbC particles between the solid and liquid phases for the case presented in Fig. 927. The final average liquid (particles located in the interdendritic regions) and solid (particles located in the intradendritic regions) NbC populations in Fig. 9-28 are about and respectively. The total number of NbC including both

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intradendritic and interdendritic regions, is approximately result matches closely with the experiment shown in Fig. 9-27.

This

Figure 9-29 depicts the average NbC population in the last liquid region as a function of the cooling rate with and without modeling of PET. Cooling rates were calculated at the beginning of solidification, and their range covers those encountered in VAR and ESR ingots. The average liquid NbC population increases monotonously when PET is neglected. However, when PET is considered, a decrease in the average liquid NbC population occurs for cooling rates above 5 K/s. This complex behavior is associated with the ratio between the critical velocity of PET and the S/L interface velocity that is, in fact, the PET probability. The higher the cooling rate, the bigger is the PET probability in Eq. (9-30) (i.e., entrapment/engulfment of particles is favored). This trend is also related to the variation of the final instability radius (secondary dendrite arm radius) with the cooling rate as presented in Fig. 9-30 and to the linear dependency of the final NbC carbide radius with the cooling rate as shown above.

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195

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The redistribution of both solid and liquid NbC populations during solidification is crucial since it will affect the size and amount of NbC particles and Laves phases. The influence of the cooling rate on the amount of eutectic + Laves) and NbC precipitated in alloy 718 ingots is presented in Figs. 9-31 and 9-32, respectively. As the cooling rate increases, the amount of precipitated NbC decreases, and falls short of the maximum (equilibrium) amount which is about 0.6 wt. %. On the contrary, as the cooling rate increases, the amount of Laves phases increases up to a maximum amount of 4.3 wt.% and then slightly decreases. Note that a calculation with Scheil’s model predicted 8.34 wt.% Laves phase, irrespective of the cooling rate. Since the amount of Laves phase is proportional with the intensity of Nb segregation, the maximum in Fig. 9-32 is connected to the evolution of the diffusional time in both solid and liquid phases, and related to the redistribution of NbC particles between the solid and liquid phases. The existence of a critical cooling rate at which maximum segregation, and thus maximum amount of Laves phases can form, has significant practical importance. By controlling process parameters, it might be possible to avoid significant amounts of Laves phases in the microstructure. The effects of redistribution of NbC particles during solidification (Fig. 9-29) on the amounts of NbC and Laves phases can be seen from Figs. 9-31 and 9-32, particularly for cooling rates below 5 K/s. The calculated amounts of Laves phase with and without PET modeling coincide for cooling rates higher than 5 K/s. In this case, the effect of the cooling rate on the formation of Laves phases prevails over the effect due to redistribution of NbC particles.

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197

Table 9-7 shows a thorough comparison between calculated and experimental results for alloy 718 VAR and ESR ingots. The results are presented in terms of pool depth, secondary dendrite arm spacing (SDAS), and amounts of secondary phases for two melting rates. The pool depth and the amount of Laves phases increase with power input and melting rate, while SDAS and the amount of NbC decrease. There are insignificant differences between ESR and VAR results in terms of size of SDAS, as well as size and contents of secondary phases. The calculated pool depth and SDAS agree closely with the experimental measurements. Under the present solidification conditions, when the C and Nb compositions of the melt are 0.06 wt. % and 5.25 wt. %, respectively, there is a small effect of the NbC content on the segregation and, therefore, on the amount of Laves phases.

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Table 9-8 presents calculated results of the effect of the melting rate and the initial C and Nb concentrations on the amounts of NbC and Laves phases precipitated at the interdendritic and intradendritic regions for the two VAR cases shown in Table 9-7. These results are shown only for the steady-state condition, when H = 3 R. Note that in the present calculations convection and macrosegregation phenomena are neglected. They may alter the present results, particularly for low cooling rates encountered in the center of the ingot. As shown in Table 9-8, mainly for large cooling rates encountered at the ingot edge, the amount of the Laves phases formed is strongly dependent on the local cooling rates and on the initial Nb content and less on the initial C content. A similar trend is observed for NbC, i.e., the amount of precipitated NbC is controlled by the initial C content and by local cooling rates, and is slightly affected by the amount of the Laves phases. The calculations in Table 9-8 reveal that combinations between C and Nb as a function of the cooling rate can be used to control the amount of both NbC and Laves phases. In VAR and ESR ingots, an initial Nb content of 4.75 wt.% and a minimum initial C content of 0.12 wt.% are required to obtain a Laves-free microstructure with maximum 1.25 wt.% NbC phase. Also, cooling rates below 5 K/s will favor the formation of low amounts of NbC and Laves phases, and NbC particles will be mostly segregated at the grain boundary. Results from this type of calculation can provide useful information for understanding the formation tendency of freckle-type defects in VAR and ESR ingots. Figure 9-33 illustrates the influence of Nb and C content on the amounts of NbC and Laves phases (VAR processed IN718 with 327 kg/hr). For example (see Fig. 9-33), for an initial composition of Nb=5.0 wt.% and C=0.05 wt.%, the amount of secondary phases that could form varies as follows: Laves = 3-3.3 wt.% and NbC = 0.53-0.6 wt.%.

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199

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Simulated results showing the distribution of NbC and Laves phases in a VAR processed alloy 718 ingot are shown in Fig. 9-34. The black color in Fig. 9-34 shows the probable location of NbC and eutectic phases. Figure 9-35 shows optical micrographs of carbides nucleated by TiN particles, eutectic, and globular Laves phases. Figure 9-36 presents an SEM micrograph of eutectic Laves phases. Microsegregation in an as-cast-alloy 718 ingot is presented in Fig. 9-37. The interdendritic region (Fig. 9-37a) is enriched in Nb and Ti and low in Fe and Cr (Fig. 9-37b).

Figure 9-38 presents another comparison between experiments and calculations for Nb microsegregation and secondary dendrite arm spacing in ESR processed alloy 718 ingots. The thermal history results were extracted from a nodal point located at the mid-radius of the ingot shown in Fig. 910(c). The macro-cell size is 1 mm. The microscopic simulation was performed in a 1 x 1-mm square box that resembled the macro-cell size. The micro-cell size is The white color in Fig. 9-38 shows the probable location of eutectic phase. The influence of solid mass diffusivity on Nb microsegregation and on Laves phase formation in as-cast alloy 718 is shown in Fig. 9-39. A significant effect of mass diffusivity on the Nb microsegregation can be seen from Fig. 9-39.

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201

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Chapter 9. Modeling and Simulation of Ingot Solidification Structure in Primary and Secondary Remelt Processes

203

9.3.10 Experimental Technique for Composition Measurements and for Estimating the Secondary Arm Spacing in Ti-17 Alloy

Two 1.5 inches by 1.5 inches samples (38 mm x 38 mm) were cut from the ingot surface and center, i.e., 1.5 inches (38 mm) and 10.25 inches (260 mm) from the ingot edge, respectively, and 7 inches (175 mm) below the top surface of the ingot slice, shown in Figure 9-23. The samples were crosssectioned and polished but were not etched. Composition measurements by EDS analysis were made for Al, Sn, Zr, Mo, and Cr elements [59]. The statistically significant experimental technique developed in [60] was used for investigating microsegregation in these samples. One hundred measurements for each of the above elements were carried out using a square-mesh systematic point count metallographic technique. The grid spacing (the distance between two consecutive measurements) was and the measurement area was The raw data (as-measured compositions from the EDS) are plotted in Figure 9-40 together with the ordered (ascending or descending order) data. The x-axis in Figure 9-40 shows the number of grid points or volume fraction of solid obtained by dividing each ordered data by the total number of data points (in this case, 100). Therefore, the number of points represented on the x-axis of Figure 9-40 will also describe the solid fraction (in vol.%) for insignificant diffusion of solute in the solid phase. The secondary dendrite arm spacing based upon the EDS measurements was estimated to be in the range of 75 to for this sample. Tables 9-9 and 9-10 summarize the processed data that include both the composition

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variations within the samples and the effective partition coefficients. and

in Tables 9-9 and 9-10 are the mean values of compositions that

are greater or smaller than the overall average composition of the sample. is calculated as the mean value of 100 data points. The effective partition coefficients in Tables 9-9 and 9-10 were calculated as the ratio between the minimum (if k < 1) or maximum (if k > 1) composition and Note that the values of the effective partition coefficients in Tables 3-7 and 9-10 are closer to one than the calculated values (equilibrium k) [59]. This difference is thought to be primarily due to the effects of fluid flow and macrosegregation, which were neglected in the theoretical calculations. From Figure 9-40 and Tables 9-9 and 9-10 it can be seen that Sn, Zr, and Cr segregate positively (k < 1) whereas Mo segregates negatively during the solidification of Ti-17 alloy. Aluminum can redistribute either negatively or positively (the value of k is close to one) during the solidification of Ti-17 alloy. The effect of cooling rates on the microsegregation profiles of Cr and Mo is shown in Fig. 9-41. As expected, larger microsegregation occurs in the ingot center (low cooling rates) as compared with ingot surface (high cooling rates). Also, the effect of cooling rate (i.e., location within the ingot) on the microsegregation and partition coefficients of Al, Cr, Mo, Sn, Zr, and Fe is shown in Tables 9-9 and 9-10. Segregation of these elements can strongly influence formation of and type II segregates in Ti-17 alloy [60].

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9.4 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

L. Nastac, Numerical Heat Transfer, Part A, Vol. 35, No. 2, pp. 173189, 1999. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, DC, 1980. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 27 A, pp. 40614074, 1996. L. Nastac and D. M. Stefanescu, Met Trans, Vol. 27 A, pp. 40754083, 1996. D. M. Stefanescu, G. Upadhya, and D. Bandyopadhyay, Met Trans, Vol. 21A, pp. 997-1005, 1990. M. Rappaz, International Materials Reviews, Vol. 34, No. 3, pp. 93123, 1989. L. Nastac, MS Thesis, University of Alabama, Tuscaloosa, AL, 1993. L. Nastac and D. M. Stefanescu, Proceedings of the Micro/Macro Scale Phenomena in Solidification, ASME, HTD-Vol. 218/AMDVol. 139, pp. 27-34, 1992. L. Nastac, Ph.D. Dissertation, University of Alabama, AL, 1995. D. M. Stefanescu, Proceedings of the Modeling of Casting, Welding and Advanced Solidification Processes- VI, Eds. T. S. Piwonka et al., TMS, Warrendale, PA, pp. 3-20, 1993. S. Sundarraj and V. R. Voller, International Communications in Heat and Mass Transfer, Vol. 21, No. 2, pp. 189-197, 1994. S. Sundarraj and V. R. Voller, Transport Phenomena on Solidification, ASME Winter Annual Meeting, Eds. C. Beckermann et al., Chicago, HTD-Vol. 284, pp. 29-42, 1994. S. Sundarraj, Ph.D. Dissertation, University of Minnesota, Minneapolis, MN, 1994. B. J. Yang and D. M. Stefanescu, Proceedings of the Advanced Technologies for Superalloy Affordability, Eds. K. M. Chang et al., Nashville, TN, March 12-16, pp. 175-183, 2000. K. O. Yu, Proceedings of the Vacuum Metallurgy Conference on Specialty Metals: Melting and Processing, pp. 83-92, 1984. L. Nastac and D. M. Stefanescu, Modelling and Simulation in Materials Science and Engineering, Vol. 5, No. 4, pp. 391-420, 1997. G. K. Upadhya, K. O. Yu, M. A. Layton, and A. J. Paul, Proceedings of the Modeling of Casting, Welding and Advanced Solidification

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APPENDIX: MODEL ANALYSIS

A9.1 SENSITIVITY ANALYSIS In this Appendix, a detailed sensitivity analysis of the ingot grain structure model in terms of various nucleation and grain growth parameters is presented. Comparisons of simulated results and experimental data (Figures 9-8 and 9-10) for PAM processed ingots are also provided. The variables studied include mesh resolution, impingement coefficient, nucleation rate, and time step. Each of the aforementioned parameters was studied for two melt rates (i.e., a high melt rate of 1200 lbs/hr and a base melt rate of 800 l bs/hr) and for two grain morphologies (e.g., columnar and equiaxed grain morphologies). The PAM ingot studied experimentally has dimensions of 650 mm x 2080 mm (26 inches diameter (D) x 82 inches height (H)), while the simulated PAM ingot has dimensions of 650 mm x 1300 mm (26 inches D x 52 inches H). Experimental grain sizes and simulated grain sizes were calculated by a line method [47]. Measurements were taken at five equally spaced positions each at heights of 1000 mm (40 inches), 1375 mm (55 inches), and 1500 mm (60 inches) and were assumed to contain only equiaxed/random structures.

A9.1.1 Melt Rate Effects

The following differences exist between the high and base melt rates: There is longer/larger grain growth along the surface edges, which increases with the mesh size. The center of the ingot becomes denser with increasing mesh size as well. Upon comparing the mesh resolution data of the model to the experimental data, it can be seen from Figures A9-3, A9-4, and A9-5 that the mesh size is important to matching the results of these two sets of data. The pool depth of the base-melt rate case is lower than that of the highmelt rate case. There is longer/larger grain growth along the surface edges, which increases with the mesh size.

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The center of the ingot becomes denser with increasing mesh size as well. The mushy region is smaller in the base-melt rate case than the high-melt rate case. The base-melt rate grains take longer to turn at the center of the ingot than the high-melt rate center grains. Both melt rates have a V-shaped liquid pool and mushy zone. Both appear to have the same growth angle and spacing of grains. A9.1.2 Mesh Resolution – Columnar Growth Mesh sizes chosen for modeling were 5x5, 10x10, and 20x20 (Figure A91). The mesh size used in the macroscopic computations was 6.6 mm x 6.6 mm. That means that the mesh size used in this microscopic analysis is 5, 10, or 20 times finer than that in the macroscopic computations. Both high and base melt rate cases have decreasing vertical grain growth at the bottom surface as the mesh size increases due to grain growth competition within the mesh size. As the mesh size increases so does the number of columnar grains. Grain size becomes smaller, and the average number of grains in a given area becomes larger. A9.1.3 Mesh Resolution – Equiaxed Growth Mesh sizes chosen for modeling were 5x5, 10x10, and 20x20 (Figure A92). Experimental data collected from the pouring side (0°) of the PAM processed ingot was compared with the data from the simulated ingot. There is longer/larger grain growth along the surface edges, which increases with the mesh size. The center of the ingot becomes denser with increasing mesh size. Upon comparing the mesh resolution data of the model to the experimental data, it can be seen from Figures A9-3, A9-4, and A9-5 that the mesh size is important to matching the results of these two sets of data. A9.1.4 Time Step Time step was changed from 10 sec to 1 sec for a 10x10 mesh (Figures A9-6(a) to A9-6(d) and Figures A9-7(a) to A9-7(d)) and a 20x20 mesh (Figures A9-6(e) and A9-6(f)). Grain growth competition seems to be stronger, resulting in larger grain size when the time step was decreased.

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When compared to the experimental ingot structure, the simulated ingot structure has longer vertical grain growth along the bottom edge. The differences previously mentioned also exist also for the high and base melt rate. Visually, it was almost no difference in the equiaxed grain morphology. However, Figure A9-8 shows some slight differences as compared to experimental data. A 20x20 mesh was also run for time step comparison. Grains are smaller in size and more plentiful than the 10x 10-mesh case, and increase in size as the time step is reduced.

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A9.1.5 Impingement Factor – Columnar Growth

An impingement factor of 1.0 was chosen for the control case, with 0.5, 1.0, and 1.5 used to evaluate the sensitivity of this parameter in the model (Figure A9-9).

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It was only a small visual difference in the columnar grain models. There were slightly finer grains as the impingement factor increased. However, a higher resolution would be needed to confirm this conclusion. The differences between the high and base melting rates are similar with those previously listed.

A9.1.6 Impingement Factor – Equiaxed Growth

An impingement factor of 1.0 was chosen for the control case, with 0.5, 1.0, and 3.0 used to evaluate the sensitivity of this parameter in the model (Figure A9-10). Differences from impingement results can be seen visually. A slightly larger grain size can be seen as impingement factor increases. Differences between high and base-melting rates are previously listed. Figures A9-11, A9-12, and A9-13 show the results when compared with experimental data. Grain size increases slightly with an increase in the impingement factor. A nucleation rate of and (see Eq. (9-7)) was chosen in order to determine sensitivity of impingement. At higher nucleation rates, impingement would be practically unnecessary because the nucleation rate would take precedence.

A9.1.7 Nucleation – Columnar Growth

A columnar nucleation rate of was chosen as the control case, with and used to evaluate the sensitivity of this parameter in the model. Very small differences were seen between the studied nucleation rates. A finer mesh resolution is needed to compare model sensitivity at these rates, since nucleation rate is directly connected to mesh size. Differences between high and base melting rates are listed previously.

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A9.1.8 Nucleation – Equiaxed Growth

Three cases of nucleation for equiaxed macrostructure were studied within the high and base melt rate models. Case 1: varies from to and to (Figure A9-14). Case 2: varies from to and to Case 3: varies from 0 to and to Only the simulation results from Case 1, which is the most relevant case, are presented in Figure A9-14. The results obtained for Cases 2 and 3 are discussed later in the summary of this section. Visually, some differences can be seen from the results shown in Figure A9-14. As nucleation rate increases, grain size becomes smaller. Nucleation sensitivity can be seen in these figures and will make a significant difference in results just by changing the nucleation by a factor of 10. There is, however, a limit to nucleation since it is dependent on mesh resolution and size. After a high enough nucleation rate, there will be no change in results. Figures A9-15, A9-16, and A9-17 show comparisons with experimental data for Case 1. The numbers in the symbol column of each graph represent the values of nucleation coefficients. For example, 4-5-0 means The differences between the high and base-melting rates have previously been presented.

A9.1.9 Summary

This sensitivity analysis shows various results when the input values of the grain structure model are changed. Grouped by category, the conclusions are summarized in the following paragraphs:

Mesh resolution Both high and base melt cases have decreasing vertical columnar grain growth at the bottom surface as the mesh size increases, due to grain growth competition within mesh size.

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As the mesh size increases so does the number of columnar grains. Grain size becomes smaller, and the average number of grains in a given area becomes larger. There is longer/larger equiaxed grain growth along the surface edges, which increases with mesh size. The center of the ingot becomes denser with increasing mesh size as well. Upon comparing the equiaxed mesh resolution data of the model to the experimental data, it can be seen from Figures A9-3, A9-4, and A9-5 that the mesh size is important in matching the results of these two sets of data.

Time step Grain growth competition seems to be stronger, resulting in much larger grain size, when the time step was decreased. When compared to the experimental ingot, the simulated ingot has longer vertical grain growth along the bottom edge. Visually, there was almost no difference in the morphology of equiaxed grains. However, Figure 6-8 shows some slight differences as compared to experimental data. A 20x20 mesh was also run for time step comparison. Grains are smaller in size and more plentiful than those observed in the 10x10-mesh case, and also, they increase in size as the time step is reduced.

Impingement factor

Almost no significant visual differences in the columnar grain models were found. Differences from impingement results can be seen visually for equiaxed grains. A slightly larger grain size can be seen as impingement factor increases. Figures A9-11, A9-12, and A9-13 show the results when compared with experimental data. Grain size increases slightly with an increase in impingement factor. A nucleation rate of was chosen in order to determine sensitivity of impingement. At higher nucleation rates, impingement would be practically unnecessary because the nucleation rate would take precedence.

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Nucleation rate

Case 1 Visually, some differences can be seen from the pictures shown in Figure A9-14. As nucleation rate increases, grain size becomes smaller. Nucleation sensitivity can be seen in these figures and will make a significant difference in results just by changing the value in nucleation by a factor of 10. There is, however, a limit to nucleation since it is dependent on mesh resolution and size. After a high enough nucleation rate, there will be no change in results. Figures A9-15, A9-16, and A9-17 show comparisons with experimental data for Case 1. Case 2 The value does not have as much an effect on the grain size as value does. Nucleation sensitivity can be seen in these figures and will make a significant difference in results after changing the value in nucleation by a factor of 100. There is, however, a limit to nucleation since it is dependent on mesh resolution and size. After a high enough nucleation rate, there will be no change in results. Case 3 Very little difference can be identified visually. Therefore, analysis shows that the results are independent of the value for nucleation rate.

A9.2

GRAIN GROWTH ANALYSIS

The objective of this section is to study the effects of some growth parameters on the PAM ingot grain structure (see Equation (9-18)). A9.2.1 Growth Parameter m - Columnar The growth parameter n was kept at a constant value of 0.10 while the growth parameter m was varied (Figures A9-18(a) to A9-18(e)). For comparison purposes, Figure A9-18(f) shows the high melting rate, linear grain growth simulation with a 10x10-mesh resolution. As m increases, the bottom surface grain growth becomes more triangular in shape and grains become longer in length. Figure A9-18(a) is the closest match with the linear model. Grain size shown in these figures is larger than grain size from the linearmodel.

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When m increases, the columnar grains have more difficulty in smoothly turning upward as they collide at the center of the ingot. A9.2.2 Growth Parameter n - Columnar The growth parameter m was kept at a constant value of 0.10 while n was varied (Figures A9-19(a) to A9-19(e)). For comparison purposes, Figure A919(f) shows the linear model for high melting rate, columnar growth case, with a 10x10-mesh resolution. When n decreases, the columnar grains have more difficulty in smoothly turning upward as they collide at the center of the ingot. The n value had a greater effect on the grain growth results than the m value. An n value of 0.50 is the closest match to the linear model data in this category. Grain size decreases with increasing n values. Rigid triangular bottom grain growth becomes more rounded as n values increase. A9.2.3 Time Step In this case, the time step was changed from 10 sec to 1 sec to determine the effect on grain structure (Figures A9-20(a) and A9-20(b)) and compare the results to the linear model (Figure A9-20(c)). In simulating these ingot structures, m = 0.10 and n = 0.10 were used. As the time step decreases, results become more similar to the linear model. Grain size is still larger than that of the linear model. This suggests that m and n values are not yet optimized in the exponential model used for comparison to the linear simulation. The columnar grains of the exponential model are more uniform in size and turn upward better in the center as the time step decreases.

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A9.2.4 Growth Parameters for High Melt Rate

Since n = 0.5 matched the linear model the best, the growth parameter n was kept at a constant value of 0.5 while m was varied (Figures A9-21(a) to A9-21 (e)). For comparison, Figure A9-21 (f) shows the linear model for high melting rate columnar growth in a 10x 10-mesh resolution. As m increases, the bottom surface grain growth becomes more triangular in shape and grains become longer in length. Except for having a wide column in the center of the ingot, the exponential growth model parameters that are closest to the linear growth model results are m = 0.1 and n = 0.5. Grain size in all figures is larger than the linear model grain size. However, simulated grain size in the ingots shown in Figures A9-21(a) to A9-21(e) is closer to the grain size of the linear model ingot than the ingots shown in previous figures. This suggests that a higher n value will give smaller grain sizes. When m increases, the columnar grains have more difficulty in smoothly turning upward as they collide in the center of the ingot.

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A9.2.5 Growth Parameters for Base Melt Rate Since n = 0.5 matched the linear model the best, the growth parameter n was kept at a constant value of 0.5 while m was varied (Figures A9-22(a) to

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A9-22(e)). Figure A9-22(f) shows the linear model for base melting rate columnar growth in a 10x10 mesh for comparison. As m increases, the bottom surface grain growth becomes more triangular in shape and grains become longer in length. Figure A9-22(a) is visually the closest match with the linear model, except for having a wide growth column in the center of the ingot. This means that for a high melting rate, the exponential growth model parameters that are closest to the linear growth model results are m = 0.1 and n = 0.5. Grain size in all figures is larger than the linear model grain size; however, Figure A9-22e is much closer to the grain size of the linear model ingot than other figures. This suggests that a higher n value will give smaller grain sizes. When m increases, the columnar grains have more difficulty in smoothly turning upward as they collide at the center of the ingot. A9.2.6 Summary In this section, various grain growth parameters were studied for the ingot structure model. Grouped by category, conclusions are summarized in the following paragraphs:

Growth parameter m – columnar As m increases, the bottom surface grain growth becomes more triangular in shape and grains become longer in length. Figure A9-18(a) is the closest match with the linear model in this category. Grain size in these figures is larger than the linear model grain size. When m increases, the columnar grains have more difficulty in smoothly turning upward as they collide at the center of the ingot.

Growth parameter n – columnar When n decreases, the columnar grains have more difficulty in smoothly turning upward as they collide at the center of the ingot. The n value had a greater effect on the grain growth results than the m value. An n value of 0.5 is the closest match to the linear model data in this category. Grain size decreases with increasing n values.

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Rigid triangular bottom grain growth becomes more rounded as n values increase.

Time step

As the time step decreases, results become more similar to the linear model. Grain size is still larger than with the linear model. This suggests that m and n values are not yet optimized in the exponential model for comparison to the linear simulation. The columnar grains of the exponential model are more uniform in size and turn upward better in the center as the time step decreases.

Effect of melt rate on growth parameters

As m increases, the bottom surface grain growth becomes more triangular in shape and grains become longer in length. Except for having a wide column in the center of the ingot, the exponential growth model parameters that are closest to the linear growth model results are m = 0.1 and n = 0.5. The grain size of all simulated ingots is larger than the linear model grain size. However, simulated ingot grain size in Figures A9-21(a) to A9-21(e) is much closer to the ingot grain size of the linear model than all other simulated ingots. This may suggest that a higher n value will give smaller grain sizes. When m increases, the columnar grains have more difficulty in smoothly turning upward as they collide at the center of the ingot. The conclusions are the same for both the high and the base melt rate cases.

A9.3

CET ANALYSIS

The objective of this section is to analyze the effects of various nucleation and grain growth parameters on CET and then to compare experimental data with simulation results for the PAM Ti-17 ingot structure. The PAM ingot studied experimentally has dimensions of D = 26 inches and H = 82 inches.

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The PAM ingot modeled numerically has dimensions of D = 26 inches and H = 52 inches. Grain size in both experimental and modeled ingots (mm) was calculated by a linear method. Measurements were taken at five equally spaced positions each at a height of 40 inches and are assumed to contain CET grains. All simulated models used a 6.6 mm mesh size with a 10x10mesh resolution. A9.3.1 Melt Rate Some differences exist between the high and the base melt rates. They are summarized below. The pool depth of the base case is lower than that of the high-melt rate case. The mushy region is smaller in the base case than that in the high-melt rate case. The high melt rate ingot has a very sharp, vertical interface between the columnar and equiaxed center grains. However, for the base melt rate ingot, the interface appears to be wavy. This is demonstrated best in Figures A923(a) and A9-23(d). Both melting rates have a V-shaped liquid pool and mushy zone. The base case has slightly longer columnar grains, leaving less space for the equiaxed zone. This is because the thermal gradient for the base-melt rate is higher than that for the high melt rate. Exponential growth rate coefficients of m = 0.01 and n = 0.20 were used in all figures unless otherwise stated. The linear growth rate will give smaller grain sizes than needed. Since the exponential growth rate created stronger growth competition, coarser grains formed and could easily be compared to experimental data. A9.3.2 Thermal Gradient The thermal gradient was changed to determine the sensitivity of the value on the results for a PAM processed ingot. was chosen to be the standard case. Results using other values, and were compared to the standard case. The sensitivity of the model on the thermal gradient is noticeable. A change in the thermal gradient values of 500 K/m made significant differences (Figure A9-23).

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In Figures A9-23(c) and A9-23(f), the area containing equiaxed grains increased from previous figures. The boundary between columnar and

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equiaxed grains progressed outward and also downward from the center as the gradient value increased. Figures A9-24 and A9-25 show the tabulated results for and A9.3.3 Nucleation - Columnar An exponential nucleation rate was used where the columnar nucleation value for was changed to 1, 10, and 100 to determine the sensitivity of the simulation model. Results for equal to 1 and 100 are presented in Figure A9-26. The equiaxed nucleation rate was kept at constant values of and The exponential growth values were kept at a constant m = 0.01 and n = 0.10. Values in the symbol column of Figures A9-27 and A9-28 represent the values for nucleation. A linear nucleation rate was unable to grow columnar grains coarse enough to match experimental results well (Figures A9-26(d) and A9-26(h)).

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Columnar grain size decreases with increasing nucleation values. Differences exist between high and base melting rate cases as previously listed. As nucleation value increases, the difference in grain size becomes less noticeable. Since nucleation is dependent on mesh size, there will be a point at which the mesh will become saturated with nuclei. Figures A9-27 and A9-28 show the tabulated results. A9.3.4 Nucleation - Equiaxed An exponential nucleation rate was used where the equiaxed nucleation value for was changed to and to determine the sensitivity of the model (Figure A9-29). The columnar nucleation rate was kept at a constant value of and The exponential growth values were kept at a constant m = 0.01 and n = 0.10. Values in the symbol column of Figures A9-30 and A9-31 represent the value for equiaxed nucleation. Differences exist between the high and base melt rate cases as shown listed. Figures A9-30 and A9-31 show the tabulated results. Equiaxed grain size decreases with the increase in the nucleation intensity. Columnar grain size remains the same; however, columnar grains redistribute to accommodate the change in nucleation. As nucleation value increases, the difference in grain size becomes less noticeable. Since nucleation is dependent on mesh size, there will be a point at which the mesh will become saturated with nuclei. A9.3.5 Time Step The time step was changed from 10 sec to 1 sec for the base melt rate case and from 10 sec to 20 sec for the high melt rate case, to determine any impact on results (Figure A9-32). Grain growth competition seems to be stronger, resulting in larger grain size when the time step was decreased. A9.3.6 Comparison with Experiments Figure A9-35 shows a comparison of experimental and simulated results for Ti-17 PAM ingot solidification structure for both the high and base melt rate cases. Note that the experimental macrographs presented in Figure A935 show only the solidification structures in the top and bottom sections of the ingots. The simulated results are similar to those from Figures A9-29(b) and A9-29(e). They are the best matches in terms of CET and grain size. Figures A9-33 and A9-34 show the tabulated results.

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A9.3.7 Summary In this section, simulated and experimental results gathered from PAMprocessed ingots of high (1200 lbs/hr) and base (800 lbs/hr) melt rates, were compared. The conclusions, grouped by categories, are summarized in the following paragraphs:

Melt rate The pool depth of the base case is about 50% shallower than that of the high-melt rate case. The mushy region is smaller in the base case than the high-melt rate case. The high melt rate ingot has a very sharp, vertical interface between the columnar and equiaxed center grains. However, for the base melt rate ingot, the interface appears to be wavy. This is shown in Figures A9-23(a) and A923(d). Both melting rates have a V-shaped liquid pool and mushy zone. The base case has slightly longer columnar grains, leaving less space for equiaxed center transition. This would suggest that the thermal gradient for the base melt rate is higher than that for the high melt rate. Exponential growth rate coefficients of m = 0.01 and n = 0.20 were used in all figures unless otherwise stated. The linear growth rate will give smaller grain sizes than observed experimentally. Since the exponential growth rate created larger growth competition, coarser grains that could easily be compared to experimental data were formed.

Thermal gradient The sensitivity of the model on the thermal gradient is very noticeable. A change in the thermal gradient of 500 K/m will produce significant differences in results. In Figures A9-23(c) and A9-23(f), the area containing equiaxed grains increased from previous figures. The boundary between columnar and equiaxed grains progressed outward and also downward from the center as the gradient value increased (see also Figures A9-24 and A9-25).

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Nucleation - columnar

A linear nucleation rate was unable to grow columnar grains coarse enough to match experimental results well (Figures A9-26(d) and A9-26(h)). Columnar grain size decreases with increasing nucleation values. As nucleation value increases, the difference in grain size becomes less noticeable. Since nucleation is dependent on mesh size, there will be a point at which the mesh will become saturated with nuclei (see Figures A9-27 and A9-28).

Nucleation - equiaxed Equiaxed grain size decreases with increasing nucleation rate values. Columnar grain size remains the same, however, columnar grains redistribute to accommodate the change in nucleation (see also Figures A930 and A9-31). Small differences exist between high and base melting rate cases. As nucleation rate values increase, the differences in grain size become less noticeable. Since nucleation is mesh dependent, there will be a point at which the mesh will become saturated with nuclei.

Time step Grain growth competition seems to be stronger, resulting in larger grain size when the time step was decreased.

10

PRACTICAL TECHNIQUES WITH SIMULATION EXAMPLES FOR CONTROLLING THE SOLIDIFICATION STRUCTURE

In this chapter, some of the most relevant methods for improving the solidification structure are introduced. Among them, electromagnetic stirring and ultrasonic vibration are very powerful techniques for controlling the solidification structure in castings and they are covered in detail below.

10.1 INTRODUCTION Some of the common methods for promoting the formation of the columnar-to-equiaxed transition (CET) because of dendrite fragmentation, growth, and sedimentation as well as techniques for reducing the macrosegregation in solidifying alloys are shown below: Bulk micro-chilling Composition control (very low impurities) Control of the heat extraction rate from the metal-mold interface Electromagnetic stirring (mechanical breakage/remelting of dendrite tips by using single-frequency and multi-frequency pulsating magnetic fields) Ultrasonic vibration (e.g., 18-20kHz and amplitude of 2 microns) Mechanical vibration (low frequency oscillation) Mold rotation. Several techniques are used for producing in-situ surface and functional gradient composites based on particle segregation as follows: Centrifugal casting Casting infiltration Electromagnetic separation of phases.

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10.2 ELECTROMAGNETIC STIRRING In many of the electromagnetic stirring (EMS) applications for remelt and continuously-cast ingots, single-frequency alternating magnetic fields are used. The electromagnetic (EM) forces are created in the molten alloy due to the interaction of induced electric currents and magnetic fields. The value of the frequency f of the magnetic field will dictate the magnitude of the EM forces in the bulk and at the free surface of the melt [1–5]. For single-phase applications, the shield number, can be related to the electromagnetic skin depth,

by [6]:

where is the magnetic permeability, is the electrical conductivity, L is the radius of the pool, and is the angular frequency Significant bulk fluid flow can be achieved by using frequencies in the range of 50 Hz to about 1 MHz (where typically For frequencies less than 10 Hz (where when the magnetic field frequency is comparable to the eigenfrequencies of the free surface deformation, free surface instabilities can be created. This aspect has recently been studied by Perrier et al. [6] using the motion generated by two-frequency magnetic field at the free surface of a Gallium pool. Thus, they were able to investigate the effects of a periodic magnetic field pulsating at two-frequencies (a basic frequency of 14 kHz and a modulation frequency to 11 Hz) on the behavior of a free surface of liquid Gallium. Their main objective was to use magnetic fields to produce metal heating, bulk stirring, and surface motion. Pulsating alternating current magnetic fields were also used in continuous casting of steel where the applied electric current that oscillates with a basic frequency was pulsated at 10 Hz [7]. EMS has successfully been applied to produce aluminum, steel and copper ingots with superior surface quality and sound interior structure. EMS can also be applied during PAM processing to produce Ti alloys. However, Ti is very reactive and has substantially lower electrical conductivity than the aforementioned metals. Whether EMS may effectively alleviate the problem of surface finish and He porosity, or yield favorable grain structure, is currently unclear. Furthermore, the PAM process employs several high-power moving plasma torches as heating source to assist melting, refining and casting. The manner in which the magnetic field externally applied to the mold interacts with the EM field imposed by the

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torch onto the molten pool of the ingot is also unknown. Therefore it is necessary to study the feasibility of EMS for the PAM process. The results presented in this section provide useful insight to justify the application of EMS to optimize the single-melt PAM process. A description of the methodology developed for the EMS modeling during PAM-assisted casting of Ti-6Al-4V (Ti-6-4) alloy ingots is presented in the following section. The combined EM effects of the plasma torch and the magnetic stirrer are numerically and experimentally investigated. Initial subscale results for 5-inch and 8-inch diameter Ti-6-4 ingots produced without EMS are also presented.

10.2.1 Mathematical Formulation

A schematic representation of the ingot casting process with magnetic stirring is shown in Figure 10-1a. The setup for the stirring coil in the mold is presented in Figure 10-1b. A 3-D cylindrical, quasi-steady state model is considered. The melt is fed from the top, cooled when passing through the mold, and withdrawn from below. Induction coils are placed around the mold near the ingot top melt surface to induce fluid motion in the molten pool. The magnetically induced fluid flow here is intended to help produce ingots with sound internal structure and surface quality. The fluid flow and heat transfer in PAM casting of Ti alloy ingots under the influence of EMS are described by the Maxwell equations, the Navier-Stokes equation and the energy equation, along with appropriate boundary conditions.

Electromagnetism

The electromagnetic fields are described by Maxwell’s equations in differential form as follows [1, 2]:

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where E (V/m) is the electric field intensity, B (Tesla) is the magnetic flux density, D is the electric flux density, q is the electric charge density, and J is the electric current density. Each vector above may be a function of three space coordinates x, y, z and time t. Equations (10-2)–(10-5) can be solved by using the following constitutive relationships:

where H (A/m) is the magnetic flux intensity, (H/m) is the magnetic permeability, (F/m) is the permittivity, and (1/Ohm-m) is the electrical conductivity of the material. Equation (10-8) is known as Ohm’s law in the presence of a fluid velocity field u in a magnetic field B .

Fluid dynamics and heat transfer

The fluid flow phenomena are described by the continuity (mass conservation) and Navier-Stokes momentum equation [3], where the additional source term is the Lorentz force that is given by:

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Heat transfer is described by the energy equation where the additional source term is the rate of Joule heating:

It is assumed that the heat flux (q) and the electric current density from the stationary plasma torch have the following Gaussian distribution:

where is the location corresponding to 95% of the area under the Gaussian curve, is the torch efficiency, and V and I are the torch voltage and current, respectively. To account for the torch movement, the heat flux and electric current density described by Eq. (10-11) were time-averaged. Both the normal and the tangential components of the magnetic and electric fields are continuous at the boundary between He and the metal [1]. At the ingot walls, no-slip conditions are applied for the velocity components. At the ingot top the casting velocity is applied. The casting withdrawal velocity is also applied in the solid ingot region. The enthalpy-porosity technique was used in the solidifying region; here, the mushy zone is treated as a porous medium [4]. In the current model the free surface flow including surface tension induced fluid flow was not considered. This feature is required to model the meniscus formation, which is important in continuous casting of steel but may not be applicable to the PAM process.

10.2.2 Solution Methodology and MHD Model Validation

Solution methodology

Fluent™ computational fluid dynamics (CFD) software (developed by Fluent Inc., Lebanon, NH, USA) with the magnetohydrodynamics (MHD)-

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module analysis capability was selected and used to simulate the EM casting process. Two approaches are available in Fluent [4]: 1) the magnetic induction method and 2) the electric potential method. The second approach was applied in the current work. Specific boundary conditions and source terms were implemented into Fluent through a set of user-defined functions that were written in C++.

MHD model validation

The MHD model was validated by Fluent against 2D and 3D MHD Hartmann flow analytical solutions under a number of flow conditions [4]. Also, the MHD model was validated against experiments for the flow of liquid mercury in the presence of a static magnetic field [4]. The purpose of this experiment was to investigate the braking effect of the imposed magnetic field on liquid mercury flow. A good match between predicted and measured flow velocities downstream the nozzle was achieved. Finally, the MHD model was validated against experiments for turbulent flow in a continuous steel casting mold and under a traveling magnetic field [4]. Reasonable agreement between experiments and simulations in terms of steel velocity at the meniscus was attained for a large variation of the magnetic field strength.

10.2.3 Results and Discussion for PAM-Processed Ti-6-4 Ingots

MHD of EMS

The MHD flow created by the EM forces is described by Eq. (10-9). The electric current density has a component in the radial direction induced by the plasma torch (Fig. 10-1). The magnetic flux vector is in the z-direction (along the ingot height) and is created by the DC induction coils. The resultant EM force vector is in the azimuthal direction (ingot top; x-y plane in 3D-Cartesian). Thus, the EM forces will cause strong stirring in the azimuthal direction. The magnitude of the swirling velocity created by the MHD is larger than that induced by buoyancy. The flow is similar to that in rotary stirring that is commonly used in the steel industry. The flow pattern will be shown later for the 5-in ingot case.

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Metallurgy of EMS

One of the most commonly recognized beneficial effect of EMS is the promotion of the columnar-to-equiaxed transition (CET). Two mechanisms would favor the early formation of CET: (1) mechanical shearing and remelting of the dendrite tips (these crystallites will serve later as nuclei for the growth of the equiaxed zone) and (2) the dissipation of superheat through high convective heat transfer at the solidification front and by the remelting of the free crystallites. The liquid becomes slightly undercooled due to constitutional supercooling at the tip of the columnar dendrites. As a result, the free crystallites (newly formed nuclei) can start growing. When they have grown sufficiently, CET occurs [8-–10.]

Application for PAM: Ti-6-4 5-in diameter case (experiments and 3D computed results with and without EMS)

Table 10-1 shows the physical properties in SI units of the Ti-6-4 alloy used in the present calculations. Table 10-2 illustrates some of the boundary and initial conditions. Figure 10-2a shows the top view of the mesh and the temperature profile for the EMS case. Figure 10-2a shows the solidified shell of about 2 mm, which is approximately one half of the shell without EMS. Figure 10-2b presents the pool and temperature profiles with and without EMS. The liquid pool is slightly deeper for the EMS case. The incoming molten alloy has a superheat of 50 K. The pool height will increase by about 10% for a superheat of 150 K.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

In Figure 10-3 the profiles for the electric field, the rate of Joule heating, and the Lorentz force are shown. The magnitude of the Lorentz force reaches a maximum of near the ingot corner, which is several times larger than the buoyancy force. Figure 10-4 shows a 2-D slice of velocity contours and the top-view of the velocity vectors with and without EMS for the 5-in diameter ingot.

Chapter 10. Practical Techniques with Simulation Examples for Controlling the Solidification Structure

255

256

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Chapter 10. Practical Techniques with Simulation Examples for Controlling the Solidification Structure

257

258

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Chapter 10. Practical Techniques with Simulation Examples for Controlling the Solidification Structure

259

The magnitude of the swirling velocity generated by the MHD force is about one order of magnitude larger than the velocity driven by buoyancy force alone. Figure 10-5 illustrates results for the 8-in diameter ingot. Figure 10-5a shows a comparison for the temperature profiles with and without EMS. The liquid pool is slightly deeper and hotter for the EMS case. Figure 10-5b shows the comparison between the velocity contours with and without EMS. Again, stirring velocity is significantly higher for the EMS case. Figures 10-6a and 10-6b present a comparison of numerical results and experimental macrographs without EMS. Figure 10-6c shows a simulated macrostructure with EMS. Details about the macrostructure model are shown in Refs. [8–10]. The simulated macrostructure matches the experimental one reasonably well in terms of grain size and growth direction. For the EMS case, the CET takes place under the process conditions described in Table 10-2. The critical thermal gradient for CET is 3000 K/m. The predicted macrostructure in Fig. 10-6c is in line with experimental ones in steel, aluminum and copper ingots. Figure 10-7a shows the calculated CET with and without EMS. In the presence of EMS, the CET will form earlier than without stirring. This is due to induced equiaxed nucleation (e.g., dendrite fragmentation) and also because the melt superheat is removed faster in the presence of EMS. Figure 10-7b presents the calculated primary dendrite arm spacing with and without EMS. The decrease of and of the thermal gradient in the mushy zone are due to the EMS. The secondary arm spacing is slightly affected by the EMS. Details about the calculation of the solidification maps can be found in Ref. [9].

10.3 MICRO-CHILLING IN STEEL CASTINGS Columnar-to-equiaxed transition (CET) in plain carbon steel would occur for the following conditions: (a) G/V<5.5x10e+07; and (b) G<500-2000 °C/m; where G and V are the average thermal gradient in the mush and solidification velocity, respectively. The micro-chilling technique is presented below.

10.3.1 Use of Steel Powder (Micro-chills) for Efficient Superheat Removal

The volume of microchills to be added in any molten alloy for efficient superheat removal can be calculated as follows:

260

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

where is the density, V is the volume, is the specific heat, L is the latent heat, is the ambient temperature, is the solidus temperature, is the solid fraction; c and p subscripts are used for casting and microchill material, respectively. Thus, the amount of micro-chills (powder) required to remove about 5060 °C superheat from a plain steel alloy would be about 2-3% from the total volume of the alloy (see Eq. 10-12 with micro-chills will be melted completely

It is assumed that the in Eq. 10-11) and are made

from the same alloy as the cast (i.e., plain steel). By using 2-3% micro-chills, the solidification time of the plate will decrease by about 20-25%. The diameter of these micro-chills should be maximum 1-2 mm. These microchills will melt/dissolve in seconds. The solidification map of a plain steel alloy with 0.04 wt.% C with and without powder addition is shown in Fig. 10-8. The addition of powder will cause an earlier CET, shown in Fig. 10-9 for a superheat of 50 K. For comparison purposes, a simulation case without power addition with a superheat of 10 K is also presented in Fig. 10-9. Prediction of macrosegregation during solidification of castings with equiaxed (see the simulation case in Fig. 10-9c) or columnar morphologies (see the simulation in Fig 10-9a). As expected, addition of microchills will produce an earlier CET with small equiaxed grains that will help reducing segregation of C in this alloy.

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261

262

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

10.4 ULTRASONIC VIBRATION A description of the beneficial effects of ultrasound technology on the solidification of materials is presented in this section. For more information on the theory and practical applications of ultrasound technology, the reader is invited to read the book written by Abramov [13]. 10.4.1 Ultrasonic Vibration Effects in Fluids

The ultrasound vibration of solidifying metals and alloys could lead, if applied properly, to structural refinement and improvement of their properties. The main effects of ultrasonic vibration on the properties of ascast materials are: (i) reduction in grain size and primary and secondary arm spacings; (ii) promotion of CET; (iii) affects size, amount, and distribution of secondary phases; (iv) increase uniformity in the distribution of nonmetallic inclusions; and (v) reduction in dendritic segregation and improvement in the material homogeneity. The most important effect from practical viewpoint is the grain refinement. Abramov and Gurevich [14] used a 20 kHz transducer to vibrate solidifying pure metals with different crystalline structures. Their results are summarized in Table 10-3. Structural changes were compared with their mechanical properties, in particular hardness and tensile strength. Ultrasonically-induced grain refinement increased hardness and tensile strength considerably, independently of the crystal lattice type.

Chapter 10. Practical Techniques with Simulation Examples for Controlling the Solidification Structure

263

Another interesting application of ultrasound is the zone refining of metals and semiconductor materials. Here, ultrasound treatment has a positive influence on the solidification front and boundary layer. Also enhances the melt mixing. It was found [15] that the ultrasound treatment has reduced the effective distribution coefficient of Cd in Zn and therefore has intensified the refining. The effect was maximum was reduced by 25%) at a vibrational amplitude of Similarly, the ultrasonic zonal refining of Te at a vibrational amplitude of has reduced the distribution coefficient of Se and Cu in Te by about 25%, as compared with the coefficient obtained by conventional refining. Seeman and Menzel applied the ultrasound technology to continuously cast a 290 mm diameter Al ingot [16]. They used a 25 kW oscillator at 40 kHz. It was found that the ultrasonic treatment refined both the micro and macrostructure of the ingots and therefore improved their mechanical properties. It was shown in [13] that, for VAR and ESR processing of superalloys and specialty steels, ultrasonic vibrations could optimally be transmitted through the bottom section of the ingot and eventually to the solidification zone. The fragmentation/destruction of growing crystals by cavitation due to ultrasound intensity was studied by Abramov in [17]. For protruding crystals of length l and radius r, the necessary pressure for their destruction can be calculated with:

where is the material strength near the melting point. The ultrasound intensity necessary for the dispersion of the crystals increases with the parameter (see Table 10-4). The experiment was performed on a transparent naphthalene-based material. The ultrasound effect on the rate of nucleation was studied in Refs. [ 18– 20]. It was found that ultrasound decreases both the metastability threshold (undercooling) and the expected time for nucleation [Table 10-5]. In metal, the ultrasonic treatability would depend on the intensity of cavitation in the melt. In alloys, however, the dispersion of growing crystals is governed by both alloy strength and crystal morphology. The crystal morphology is also related to the size of the mushy zone. For example, Al-alloys with a lower strength of growing crystals and larger mushy zone are better treatable using the ultrasound and also have greater refinement capability than, for example, Zn-based alloys.

264

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

It is concluded that the ultrasonic treatability of materials would strongly depend on their physical and mechanical properties near their melting point.

10.4.2 Modeling of Ultrasonic Vibration in Fluids

For planar propagation in x-direction, the ultrasonic pressure, is given by:

The ultrasonic intensity,

(in

can be calculated with:

(in

Chapter 10. Practical Techniques with Simulation Examples for Controlling the Solidification Structure

265

In Eqs. (10-14) and (10-15), is the material density, is the angular frequency, k is a coefficient of proportionality, is the material viscosity, is the reference ultrasonic intensity at x = 0, f is the sound frequency, v is the velocity, a is the amplitude of the oscillations, is the absorption coefficient, C is a material constant, and x is the distance. For example, for metals at f = 20 kHz is of the order of The fluid flow phenomena with ultrasonic vibration can be described by the continuity (mass conservation) and Navier-Stokes momentum equation, where the additional source term is the volumetric force that is calculated as:

Heat transfer is described by the energy equation where the additional source term is the rate of ultrasonic heating:

10.5 MODELING OF ELECTROMAGNETIC SEPARATION OF PHASES TO PRODUCE IN-SITU COMPOSITES The most common casting processes for producing in-situ composites are centrifugal casting (see Chapter 8.2) and casting infiltration [21]. These processes are complicated and require expensive equipment. A method for production of in-situ surface and functional gradient composites by electromagnetic separation of phases (EM-SP) was proposed by Xu et al. [22]. This method is introduced in the following paragraphs.

266

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

The EM-SP method uses the differences in electrical conductivity between the primary phases and the melt. The EM field can be generated by a DC current passing through a casting placed within a crossed uniform magnetic field (see Fig. 10-11]). The Lorentz force and the Joule heating can be calculated with Eqs. (10-9) and (10-10) in Chapter 10.2. The electric current density has a component in the r-direction induced by the DC current. The magnetic flux vector created by the induction coils is in the z-direction. The resultant EM force vector is in the azimuthal direction. Thus, the EM forces will cause the electromagnetic separation of phases in azimuthal direction. The magnitude of EM force is much larger than that the buoyancy force and therefore, the gravity effects on the primary phase can be neglected. For spherical particles, the net force exerted by the EM field is calculated as [22–24]:

where is the particle radius and and are the electrical conductivities of the melt and primary phases, respectively. For a very low the net force is

Chapter 10. Practical Techniques with Simulation Examples for Controlling the Solidification Structure

267

The primary phase velocity ( v ) based on the EM field is calculated with [22]:

where is the dynamic viscosity of the melt. This method was applied by Xu et al. [22] to study the separation of primary phases in Al-15wt.% Si, Al-19 wt.% Si, Al-11.7 wt.% Si-1.2 wt.% Fe-1.6 wt.% Mn alloys. They have used two electromagnets with a magnetic flux density of 1 T and a DC current intensity varying from of 0 to For these parameters, will range from 0 to Also, under the same conditions, for Al-Si alloys (with

and

assuming a particle radius of the primary phase velocity relative to the melt velocity will vary from 0 to approximately 33 mm/s. Thus, segregation of primary phases can easily take place during the solidification of castings. Indeed, as shown in [22], a significant amount of primary Si was moved to the casting surface before the eutectic reaction occurred. By playing with melt superheat and cooling conditions, they were able to produce in-situ surface and functional gradient composite castings with variable hardness and increased wear resistance.

10.6 REFERENCES 1. M. V. K. Chari and S. J. Salon, Numerical Methods in Electromagnetism, Academic Press, USA, 2000. 2. P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Edition, Cambridge University Press, UK, 1996. 3. K. A. Hoffmann, S. T. L. Chiang, M. S. Siddiqui, M. Papadakis, Fundamental Equations of Fluid Mechanics, Engineering Education System, USA, 1996. 4. Fluent 6.0 User’s Guide Supplement, Magnetohydrodynamics Analysis, Fluent Inc., Lebanon, NH, USA, March 2002. 5. R. Moreau, Magnetohydrodynamics, Kluwer Academic Publisher, 1990. 6. D. Perrier, Y. Fautrelle, and J. Etay, Met Trans, Vol. 34B, pp. 669-678, 2003.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

7. Y. Chino, K. Iwai, and S. Asai, Proceedings of the 3rd International Symposium On Electromagnetic Processing of Materials, The Iron and Steel Institute of Japan, Sendai, pp. 49-54, 2000. 8. L. Nastac, S. Sundarraj, K. O. Yu, and Y. Pang,, J. of Metals, TMS, pp. 30-35, March 1998. 9. L. Nastac, Y. Pang, and C. E. Shambleen, AFS Trans, pp. 7-42, 2001. 10. L. Nastac, Proceedings of the International Conference on Modeling of Casting and Solidification Processes (MCSP-5), Nagoya, Japan, 2002, in the International Journal of Cast Metals Research, March 2003. 11. L. Nastac, Y. Pang, C. Wang, F. Spadafora, O. Yu, and D. Winterscheidt, Proceedings of the Modelling of Casting, Welding, and Advanced Solidification Processes X, United Engineering Foundation, San Destin, Florida, May 25-30, 2003. 12. L. Nastac, F. Spadafora, and E. M. Crist, Proceedings of the CFD Modeling and Simulation of Engineering Processes, Eds. L. Nastac et al., TMS Annual Meeting, Charlotte, NC, USA, March 14-18, 2004. 13. O. V. Abramov, High-Intensity Ultrasonics: Theory and Industrial Applications, Gordon and Breach Science Publishers, 1998. 14. O. V. Abramov and Ya. B. Gurevich, Fiz. Khim. Obrab. Mater., 3, p. 18, 1972. 15. G. I. Eskin et al., Metallurgiya, Moscow, p. 197, 1974. 16. H. Seeman and H. Menzel, Zs. Metall., 1, p. 318, 1947. 17. O. V. Abramov, Metallurgiya, Moscow, p. 326, 1966. 18. J. D. Hunt and K. A. Jackson, J. Appl. Phys., 37, p. 254, 1966. 19. I. I. Frawley and W. J. Childs, Trans. Metal Soc. Of AIME, 242, p. 736, 1968. 20. O. V. Abramov, Izdat. AN Belorus. SSR, p. 358, 1962. 21. Metals Handbook, Vol. 15 (Castings), Ed. D. M. Stefanescu, ASM Metals Park, OH, pp. 419-425, 1988. 22. Z. Xu, T. Li, and Y. Zhou, Met Trans, Vol. 34A, pp. 1719-25, 2003. 23. D. Leenov and A. Kolin, J. Chem. Physics, Vol. 22, pp. 683-688, 1954. 24. P. Marty and A. Alemany, Proceedings of the Symposium of the IUTAM, Swed Alloys Society, Goteborg, pp. 245-259, 1984.

Subject Index

A

alloy A356 138, 140–144 Al-Cu 50, 60, 62, 63, 80, 95 Al-Si 80, 128, 129, 141, 267 binary 6, 32, 61–63, 65, 72, 80, 101, 102, 104, 141, 165, 166–168, 173 bronze 136–138, 140, 141, 145–147 cast alloy 29, 105, 115, 166, 192, 194–196, 200 commercial 95, 165 complex 92 Cu-Ni 31 D357 128–130 dilute 53, 63 Fe-C-Si 58, 59, 61, 65 IN718 42–46, 48 liquid 63, 68 magnesium 23 molten 136–139, 174, 248, 249, 253, 259 multicomponent 80, 81, 165

Pb-Sn 42, 46–48, 66–70 RS5 128, 131, 132, 134, 135– 137 Sn-Bi/Bi-Sn 58, 60, 61, 64, 65 solidifying 5, 6, 53, 78, 118, 247 steel 23, 97, 99, 107, 115, 116, 147, 210, 248, 251, 252, 259, 260, 268 superalloy 23, 52, 72, 98, 100, 108, 112, 133, 135, 148, 149, 151, 166, 207, 209, 263 systems 41, 42, 65, 140, 184, 186, 188 ternary 13, 81, 92, 93, 100, 103 Ti-17 203 Ti-6-4 187, 249 titanium 23, 151 analysis design 176 dimensional 138 DTA 135 EDS 203 kinetic 63 metallurgical 145

270

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

microscopic 212 sensitivity 88, 121, 124, 182, 211, 222 stability 64, 65 theoretical 120 transient 65 anisotropy 31, 32, 39, 48, 159 austenite 58, 59, 76, 82, 86, 92, 117 B

boundary boundary conditions 14, 32, 36, 37, 40, 66, 82–85, 109, 128, 139, 156, 171, 249, 252, 253 boundary layer 56, 62, 92, 263 boundary problem 83 boundary value problem 85 grain boundary 54, 104,174, 193, 198 moving boundary 104, 174, 193, 198 outer boundary 140 solutal boundary layer 56 Bousinesq approximation 7 branching 43, 128 Brody-Flemings model 81 buoyancy 6, 7, 11, 87, 137, 147, 252, 254, 259, 266 C

cast iron 13, 16, 65, 117, 120, 123, 126, 127 casting bar 131–134 centrifugal 115, 136–138, 140, 148, 265, 287 continuous 151, 248, 251

cylindrical 126 D357 128 defects 146 EM 252 equiaxed 128 functional gradient composite 267 IN718 77, 131 infiltration 265 net-shaped/shaped 115 PAM 249 process 105, 136, 138, 174, 183, 249, 250, 265, 266 rate/speed/velocity 172, 174, 183, 186, 251 RS5 128, 131 solidification 26, 77, 82 steel 115, 259 step 124, 126 structure 146 thin wall 131 cell 35–38,44, 50, 200 cellular 29, 44, 51, 156, 186 cellular growth 44 closed system 26, 41, 79, 82, 86, 88, 90, 91, 102, 168 coalescence 82, 88, 167 coarsening coalescence 82–84 dendrite arm coarsening 43, 79–81, 128 grain 89, 91 mechanism/process 167 microsegregation 88 model 167 velocity 84 columnar cellular 44, 46 CET 2, 29, 73, 115, 151, 156, 247, 253

Subject Index

dendritic 8, 11, 23, 24, 43–45, 70, 154, 155, 162, 167 front 44, 46, 163, 175, 176 grain 70, 99, 116, 156, 157, 163, 164, 174–176, 178, 179, 181, 183–185, 212, 219, 222, 225, 227, 230, 233–235, 237, 238, 245, 246 grain growth 175, 176, 222 grain size 175, 178, 183–185 growth 24, 155, 176, 227, 230, 233 model 219, 225 morphology 44, 99, 260 nucleation 219, 237–239 phase 10, 153 solidification 10, 47, 53, 79, 154, 182, 186 structure 97, 130, 156, 213, 216, 218, 228, 229–232 composite 137, 138, 142, 143, 145 conservation 5, 6, 8, 10, 34, 88, 119, 154, 250, 265 constitutionalsupercooling 61, 63, 253 undercooling 34, 41, 43, 61, 186 convection effects 33 mode 68, 70 reduced/negligible 54, 59, 60, 62, 65, 69 thermal 27 thermosolutal 5, 6, 69, 70, 87 cooling Curve 13, 20, 48, 50, 77, 78, 117, 120–122 criterion Biot 14 CET 173 constitutional undercooling 61, 186

271

stability 15–18, 20, 186 critical cooling rate 98, 196 Rayleigh number 6, 70 temperature gradient for equiaxed nucleation 173, 175, 259 threshold value for gravity 70 velocity from stable to unstable growth conditions 62 velocity of PET 170, 193, 194 crystal crystalline 78, 262 crystallites 253 cubic 35, 159 dendritic 47, 189 equiaxed 9 globulitic 8 lattice type 262 crystallographic anisotropy 159 effects 30 growth direction 48, 162 orientation 34, 35, 39, 48, 131, 159, 174 curvature average 47 effects 63 interface 38 mean 32, 34, 38 model 44, 46 undercooling 34, 46 D

deterministic approach 3, 4, 30, 99 configurations 158 methods 165 model 3, 4, 23, 30, 138, 141, 152, 156, 160, 285

272

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

modeling 285 techniques 29 transition rules 157 diffusion back-diffusion 54, 69, 76 binary 101, 166 coefficient 54, 59, 65, 81, 83, 86, 89, 103, 118, 119 complete 88, 96 couple 103, 168 field 25, 26, 56 limited 87 liquid 62, 80, 84 model 76 process 97 solid 10, 60, 62, 65, 69, 97 transport 82, 105 volume 101, 166, 167 directional solidification 41, 53, 58, 60, 65, 69, 97, 186 drag 7, 8 E

engulfment 169, 194 enthalpy 16, 17, 20, 251 entrapment 194 equiaxed casting 128 CET 2, 29, 70, 151, 156, 247, 253 crystals 9 dendrite 43, 44, 46, 50, 89, 189 dendritic 8, 11, 23, 24, 29, 41, 44, 46, 76, 101, 104, 128, 129, 141, 154, 155, 162, 165, 166 grain 30, 47, 61, 82, 115, 116, 156, 164, 175, 181–183, 211, 217, 225, 236, 245, 260

grain size 47, 61 growth 24, 32, 47, 155, 176, 225 morphology 42, 46, 48, 81, 98, 151, 261 nucleation 173, 184, 185, 237, 238, 241, 259 nuclei 175 region/zone 183, 235, 253 solidification 7, 10, 47, 48, 50, 61, 79, 154, 155, 186 structure 141, 143, 214, 217, 220, 222, 223 equilibrium distribution coefficient 54, 65 equation 80, 81 eutectic 90 lever rule 87 liquidus temperature 25, 34, 40, 143, 163, 195 local 34, 54, 83, 102, 168 non-equilibrium (Scheil) 87 nonequilibrium solidification 186 partition coefficient 62, 92 phase equilibrium 105 processes 105 solidification temperature 101, 167 equivalent binary approach 173 carbon equivalent 117, 126 dendrite envelope 24, 25, 155 specific heat 18 eutectic austenite-cementite 117 austenite-graphite 58, 59, 92, 117 composition 67, 95, 143 divorced 104, 170 equiaxed 23, 154

Subject Index

fraction 95 grain 25, 141, 163 gray eutectic 118–121, 124 gray iron 20 growth 142 lamellar 13 Laves 200, 202 metastable 117, 118, 120 recalescence 19 solidification 7, 20, 159, 162 eutectic temperature 89, 143 ternary 13, 100 undercooling 20 eutectic valley 124 white eutectic 117–121, 124 explicit FDM 19, 21 numerical analysis 15 schemes 17, 20 tracking scheme 37 F

feeding 183, 186 finite 12, 36, 37, 39, 81, 84, 110, 267 floating zone 54, 58, 65 fluid dynamics 251 flow 1, 11, 47, 54, 63, 70, 71, 84, 152–154, 164, 169, 179, 180, 204, 248–251, 265 motion 249 velocity 250 fourier 81, 97 G

gradient average temperature 18 6 concentration 55–58, 87

273

critical temperature 173, 175 functional gradient composite 247, 265, 267 local 190 mushy region 174 solutal 70 temperature 53, 70, 100, 156, 163, 166, 188 thermal 44, 46, 61, 69, 70, 131, 165, 235–237, 245, 259 Zero-Gradient Condition 139 grain boundary 104, 170, 174, 193, 198 characteristics 3, 30 coarsening 89, 91 density 15, 25, 70, 128, 142, 160 extension 30, 162 growth kinetics 70, 141, 152, 159 hexagonal 157 impingement 15, 152, 158 lattice grain growth models 157 morphologies 24, 156, 162, 175, 211 movement 160 patterns 158, 180 refinement 262 rhomboidal 157 selection 30, 70, 156 gravity 6, 30, 47, 66–70, 170, 266 growth angle 30, 34, 175, 212 anisotropy 30, 34, 156 coefficient 67, 124 competition 24, 31, 43, 118, 162, 175, 212, 222, 225, 235, 238, 245, 246 competitive 47

274

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

constant 15 crystal 53 dendrite/dendritic 23, 29, 30, 32, 33, 43, 44, 46, 78, 104, 143, 154, 161, 167 diffusion-limited 167 direction 35, 39, 47, 70, 157, 159, 163, 164, 173, 175, 176, 179, 180, 259 equiaxed 24, 32, 47, 155, 176 kinetics 16, 24, 39, 48,159, 161, 162 laws 15 model 118, 230, 233, 234 morphologies 24, 47, 49, 162 parameter 125, 152, 156, 182, 211, 226–234 rate 62, 92, 235, 245 unstable/stable 63 unsteady 61 velocity 15, 24–27, 34, 36, 37, 44, 46, 81, 83, 86, 92, 95–97, 100, 118, 120, 155, 161–163, 165, 166, 168, 186, 188

columnar 163 composition 82, 95, 119 concentration 25, 57, 79, 80, 93, 102, 162, 168 curvature 38 mass balance 81 metal-mold 14, 161, 174, 247 planar 62, 65 solid/liquid 24, 29, 32, 34, 37, 39, 46, 53–56, 60–62, 64, 65, 70, 83, 87, 101, 102, 142, 162– 164, 166–170, 186, 193, 194 stable/unstable interface transition 64 internal fraction of solid 76, 77 interphase 8, 9 isothermal 66, 71, 100 Ivantsov 26 K

kinetic 34, 63, 113 L

H

heat conduction 110, 141, 155 heat flow 16, 18 heat flux 171, 181, 251 hemispherical approximation 26 I

impurity 26 inoculation 123, 124, 126 instabilities 8, 25, 166–168, 248 interdiffusion 33, 103 interface cell 37

latent heat evolution 10, 141, 154, 155 fusion 10, 14, 110, 141, 154, 155 LHM 16, 20, 104, 120 MLHM 17, 20 solidification 33 laves eutectic 100, 104, 134, 166 formation 100, 200, 203 globular 200, 202 kinetics 104 Nbc/Laves 100, 101, 166, 286 phases 96, 97, 100, 101, 104, 105, 130, 134, 151, 166, 167,

275

Subject Index

169,170,192–194, 196–198, 200–203 temperature 100, 166 ledeburite 117 length scale 2, 6, 25, 69, 97, 142 M

macrostructure columnar 213, 216, 218, 228– 232 equiaxed 214, 217, 220, 222, 223 experimental 115, 182 ingot 182 simulated/predicted 259 solidification 69–71, 111 microsegregation index 97 intensity 97 inverse 90 model 11, 25, 76, 82, 86, 93, 97, 163 normal 90 patterns 2, 30, 129 microstructure evolution 2, 142, 151 experimental 128 formation 131, 138 management and control 4 modeling 128, 261 simulated/predicted 131 solidification 48, 50, 144 mold filling 138 graphite 147 material 147 metal 14, 161, 174, 247 preheated 136 sand 147 spinning 139

steel 147 surface 36 temperature 147 multidirectional solidification 41, 44, 45, 128 mushy-region 2, 44, 46, 68, 70, 141, 152, 155, 156, 174, 175, 179,212,235,245 mushy-zone 8, 9, 69, 156, 163, 174, 178, 186, 190,212,235, 237, 245, 251, 259, 263 N

Navier-Stokes 249, 250, 265 Neumann 35, 157 nucleation coefficients 222 columnar 219, 237, 238, 239 competition 30, 120, 122, 156, 174 equiaxed 173, 184, 185, 237, 238, 241, 259 heterogeneous 30 instantaneous 101, 141, 160, 166 intensity 238 kinetics 18, 156 parameters 156, 160 potential 117, 124 probabilities 35 nucleation-rate 35, 184, 185, 211, 219, 222, 225, 226, 237, 238, 240, 242, 246 site distributions 35, 160 temperature 83 time 161 nucleus 36, 44, 161 numerical ID model 81

276

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

model 29, 53, 65, 70, 71, 81, 126, 138 modeling 53, 126, 138 particle model 138 procedures 34 results 66, 70, 71, 259 solution 11, 154 O

open domain 82 squares 63 system 82–84, 87, 88, 90, 169 P particle ceramic 138, 142 concentration 141, 143, 144 density 140 distribution 138, 140, 141, 143, 144, 146, 147 interaction 139 model 138 motion 139, 147 NbC 142, 167–170, 193–196, 198 nonspherical/spherical 7, 266 segregation 140, 147, 247 size 139–141, 143, 145, 147 size/radius 142, 193, 266, 267 TiC 137, 140, 145, 147 TiN 200, 201 velocity 139, 169 volume fraction 141, 145, 147 partition coefficient 67, 173 partitioning 122

ratio 8, 24, 81, 83, 85, 95, 96, 119, 124, 143, 162, 167 Pe number 26, 27 Péclet number 26 permeability 8, 9, 248, 250 phase field 29, 31 phenomena fluid flow 250,265 local 56 macrosegregation 198 macro-transport 109 micro/macrosegregation 151 microscopic 17 physical 31 remelting 155 solidification 1, 152 phenomenon 19, 29, 47, 63, 69, 169 porous 7, 251 primary dendrite arm spacing 29, 47, 80, 97, 130, 165, 183, 190, 258, 259 driving force 104, 141, 167 growth direction 163, 164 melting 151, 152, 156 phase velocity 267 phases 266, 267 probabilistic approaches 3, 4, 30, 109, 110, 156 calculation 142, 160 macroscopic 109 microscopic 109 Modeling 4 models 4 Monte Carlo Modeling 109 nature 30, 162 selection 158, 159

Subject Index

Q

quasi-steady state calculation 57 gradient 61 growth 63 model 249 solution 57, 95, 96 theory 54 R

radiation 110, 131, 156, 171 radius carbide 104, 105, 168, 194 equiaxed 104 grain 15, 25, 88, 89, 91, 105 ingot 191,199 instability 8, 167, 194, 196 interface 120 NbC 170 particle 142, 193, 266, 267 secondary dendrite arm 194, 195, 196 tip 26, 27, 29, 41, 64, 128 rate casting 174 chemical reaction 101 cooling 13, 14, 20, 24, 26, 70, 76–80, 95–97, 100, 104–106, 117, 118, 122, 123, 128, 134, 135, 137, 141–143, 156, 160, 162, 187, 193–198, 204, 206 feeding 186 freezing 100 growth 62, 92, 235, 245 heat extraction 18, 44, 91, 174, 247 Joule heating 251, 254 latent heat evolved 14 mass flow 172

277

melt/melting 172, 175–177, 179, 181–186, 189–191,197, 198, 200, 211–224, 226–246 nucleation 35, 184, 185, 211, 219, 222, 225, 226, 237, 238, 240, 242, 246, 263 quenching 71 recalescence 20 solidification 190 ultrasonic heating 265 withdrawal 131–133 recalescence 13, 17, 18, 20, 48, 77, 117, 122 S

scale length scale 2, 6, 25, 69, 97, 142 logarithmic 63, 64 mesoscopic/mesoscale 2, 3, 29, 30, 39, 128 multi-scale 26 Scheil equation 80, 81, 97 model 88, 97 segregation alloy 47 Carbon 43, 260 defects 173 dendritic 262 intergranular 117 model 88 particle 140, 147, 247 patterns 43–46, 48–50, 174 primary phases 267 Sn 46, 48, 69 zone 175 shape factor 7, 24, 155, 168 shrinkage 3, 5, 87, 128, 179, 189, 190

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

solidification characteristics 4, 135, 189 columnar 10, 47, 53, 79, 154, 182, 186 defects 30 dendritic 7, 29, 32–34, 65, 159, 182 equiaxed 7, 10, 47, 48, 50, 61, 79, 154, 155, 186 eutectic 7, 20, 159, 162 kinetics 3, 11, 13, 15, 17,18, 23, 30, 71, 104, 105, 120, 124, 138, 153, 154, 166 multidirectional 41, 44, 45, 128 non-isothermal 26 unidirectional 43–47, 49, 53, 54, 61, 71 solute balance 81 concentration 41,47, 65, 79, 82, 83, 118 diffusion 33, 79 flow 83, 102 redistribution 31, 41, 43, 53, 58, 65, 79, 80, 82, 95, 142, 143, 193 transport 5, 81, 102, 168 source term drag 8 heat 15, 16, 104 constitutional undercooling diagram 62, 63 stability analysis 64, 65 condition 17, 63 constant 8, 64, 162, 167 constitutional undercooling criterion 62, 63, 65 criterion 15–18, 20, 186 function 63

morphological 61, 63, 64 surface 100, 166 steady state approximation 120 CS criterion 62, 65 growth 61 stochastic CA models 51 mesoscopic model 29 micro-model 157 microscopic approach 3, 153 network 161 T

thermal conductivity 10, 14, 25, 33, 120, 141, 153, 155, 162 convection 27 data 131 diffusivity 6, 110 expansion 6 field 13, 82, 118 gradient 44, 46, 61, 69, 70, 131, 165, 235–237,245,259 history 156, 200 radiation 109, 153 shock 147 thermosolutal convection 5, 6, 69, 70, 87 transient analysis 65 final 179, 183 initial 47, 59, 60, 65, 70, 177, 183 macro-transport phenomena 153 region 60 cellular-to-dendritic 186

Subject Index

columnar-to-equiaxed 2, 30, 70, 151, 156, 247, 253 transition gray-to-white 115, 118, 120, 121, 124 layer 145 morphological 65 morphology 29 pushing/engulfment 142, 168 pushing/engulfment/entrapment 138 rules 35, 157, 159 small-to-large equiaxed grains 175 structural 115, 117, 123 white-to-gray 123 transport convective 110, 141, 153, 155 diffusion 82, 105 energy 9, 109, 152, 153 interphase 8, 9 macro transport–solidification kinetics models 3, 101 macroscopic 6, 23, 88, 109, 120, 138, 152 momentum 157 solute 5, 55, 59, 81, 102, 168 species 1, 6, 10, 152, 154 U

undercooling bulk 35, 125 constitutional 34, 41, 43, 61, 186 curvature 34, 46 interface 24, 124, 162 kinetic 34 melt 25, 162 total 34 unidirectional

279

solidification 43–47, 49, 53, 54, 61, 71 transient solutal transport 54 unsteady growth rate 61 interface concentration gradient 61 species transport 57 unidirectional solidification 65 V

viscosity dynamic 267 effective 139 function 8 kinematic 6 liquid 170 material 265 rheology-viscosity model 141 volume average 118 conservation 139 Control Volume Method 13, 16, 88 diffusion 101, 166, 167 elemental 11, 14, 66 fraction 100, 105, 142, 145, 147, 166, 187, 188, 192, 203 intrinsic 8, 25, 26, 118, 119, 163, 167 macro-volume element 4, 82, 118, 157, 159, 164 micro-volume element 4, 157, 164 ratio 140 unit volume 15, 18, 195 volume-surface mean diameter 7

280

Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

W

wetting 168 Z

zone equiaxed 235, 253 floating 54, 58, 65 gray 127 mottled 123

mushy 8, 9, 69, 156, 163, 174, 178, 186, 190, 212, 235, 237, 245, 251, 259, 263 particle- 145 particle-rich 145 refining 263 reinforced 143 segregation 175 solidification 263 structural 123 wear surface 146

About the Author

Dr. Laurentiu Nastac received the Diploma Engineering degree in Metallurgy and Materials Science from the University Politehnica of Bucharest, Romania in 1985 and the M.S. and Ph.D. degrees in Metallurgical and Materials Engineering from the University of Alabama at Tuscaloosa in 1993 and 1995, respectively. He has held various engineering, research, and academic positions in Romania and USA. Since joining

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Concurrent Technologies Corporation in 1996, he has conducted research primarily in the area of stochastic and deterministic modeling for predicting the formation of microstructure and defects in solidifying alloys, cast metalmatrix-composites, and remelt ingots. He has also performed research on material/process/properties relationships in structural alloys. Currently, his research work involves the CFD modeling of Ti-6-4/SiC EB-PVD coating process and of other processes related to manufacturing industry. Dr Nastac is a member of the TMS-Solidification Committee and a member of Sigma Xi, the Scientific Research Society. In 1999, in recognition of his work in solidification of remelt ingots (Ti-based alloys and superalloys), he received the prestigious “Bunshah Best Paper Award” from the American Vacuum Society, Vacuum Metallurgy Division. Dr. Nastac has made over 30 technical presentations and co-authored over 60 publications in the areas of computational materials science and solidification processes including 2 books, 21 refereed journal publications, 22 refereed conference publications, 31 conference presentations, 10 seminars, 22 reports, workshops, and private communications, 11 research proposals, 2 international patents, and 29 national patents. He was a technical reviewer for a variety of academic journals and served in organizing committees for several prestigious international conferences. Currently, he is a Key Reader for Metallurgical Transactions A and B and member of the Editorial Board for the International Journal of Cast Metals Research (IJCMR), UK. Laurentiu’s biography was published in Lexington and Marquis Who’s Who and in the “2000 Outstanding Intellectuals of the Century,” Edition, 2001, IBC, Cambridge, England. List of Relevant Publications Books

Multiphase Phenomena and CFD Modeling and Simulation in Materials Processing, Eds., Laurentiu. Nastac and Ben Li, Proceedings of the “CFD Modeling and Simulation of Engineering Processes “ and “Multiphase Phenomena in Materials Processing,” TMS Meeting, Charlotte, March 2004. Journals

L. Nastac, W. Hanusiak, H. Dong, and F. R. Dax, Computational Fluid Dynamics Modeling of the EB-PVD SiC/Ti-6Al-4V Coating Process: Research Summary, Journal of Metals, TMS, March 204.

About the Author

283

L. Nastac, F. R. Dax, and W. Hanusiak, Methodology for Modeling of the EB-PVD Coating Process, Journal de Physique IV, EDP Sciences, January 2004, Proceedings of the International Conference on Thermal Process Modelling and Computer Simulation", Nancy, France, March 2003. L. Nastac, Solidification Structure Modeling in Ingots Processed Through Primary and Secondary Remelt Operations, International Journal of Cast Metals Research, April 2003, Proc. of the MCSP-5, Nagoya, Japan, 2002. L. Nastac, Y. Pang, and C. E. Shamblen, Estimation of the Solidification Parameters of Titanium Alloys, AFS Transactions, 2001, pp. 27-42. L. Nastac, Numerical Modeling of Solidification Morphologies and Segregation Patterns in Cast Dendritic Alloys, Acta Materialia, Vol. 47, No. 17, November 1999, pp. 4253-4262. L. Nastac, Influence of Gravitational Acceleration on Macrosegregation and Macrostructure during the Solidification of Directionally Cast Binary Alloys: A Numerical Investigation, Numerical Heat Transfer, Part A, Vol. 35, 1999, pp. 173-189. L. Nastac, Analytical Modeling of Solute Redistribution During the Initial Unsteady Unidirectional Solidification of Binary Dilute Alloys: Comparison with Experiments, Scripta Materialia, Vol. 39, No. 7, 1998, pp. 985-989. L. Nastac, Analytical Modeling of Solute Redistribution During the Initial Unsteady Unidirectional Solidification of Binary Dilute Alloys, Journal of Crystal Growth, Vol. 193, No. 1-2, 1998, pp. 271-284. L. Nastac, On the Validity of the Quasi-Steady State Equation for Heat and Mass Transfer Problems with an Axially Moving Boundary, Internat. Commun. in Heat and Mass Transfer, Vol. 5, No. 3, 1998, pp. 407-416. L. Nastac, S. Sundarraj, Kuang-O Yu, and Y. Pang, The Stochastic Modeling of Solidification Structures in Alloy 718 Remelt Ingots: Research Summary, Journal of Metals, TMS, March 1998, pp. 30-35. L. Nastac, A Monte Carlo Approach for Simulation of Heat Flow in Sand and Metal Mold Castings (Virtual Mold Modeling), Metallurgical Transactions, Vol. 29B, 1998, pp. 495-99. L. Nastac and D. M. Stefanescu, Computational Modeling of NbC/Laves Formation in INCONEL 718 Equiaxed Castings, Metallurgical Transactions, Vol. 28A, 1997, pp. 1582-87. L. Nastac and D. M. Stefanescu, Stochastic Modeling of Microstructure Formation in Solidification Processes, Modelling and Simulation in Materials Science and Engineering, Vol. 5, No. 4, pp. 391-420, 1997. L. Nastac and D. M. Stefanescu, Simulation of Microstructure Evolution during Solidification of Inconel 718, AFS Transactions 1996, Vol. 104, pp. 425-34. L. Nastac and D. M. Stefanescu, Macro Transport--Solidification Kinetics Modeling of Equiaxed Dendritic Growth. Part I, Metallurgical Transactions, Vol. 27A, 1996, pp. 4061-74.

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Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

L. Nastac and D. M. Stefanescu, Macro Transport--Solidification Kinetics Modeling of Equiaxed Dendritic Growth. Part II, Metallurgical Transactions, Vol. 27A, 1996, pp. 4075-84. L. Nastac and D. M. Stefanescu, Prediction of the Gray-to-White Transition in Cast Iron through Solidification Modeling, AFS Transactions 1995, Vol. 103, pp. 329-37. L. Nastac and D. M. Stefanescu, Modeling of Microsegregation During Solidification of Castings, ATTR (Romania) 1995, ed. C. Cosneanu, Bucharest, 1995. L. Nastac and D. M. Stefanescu, A Model for Solute Redistribution during Solidification of Plate, Columnar or Equiaxed Grains, Metallurgical Transactions, Vol. 24A, 1993, pp. 2107-18. L. Nastac and D. M. Stefanescu, Modeling of Microsegregation in Spheroidal Graphite Cast Iron, AFS Transactions 1993, Vol. 101, pp. 933-38. Conference Proceedings

L. Nastac, F. Spadafora, and E. M. Crist, CFD Modeling and Simulation Applications for PAM-Assisted Casting of Ti-6A1-4V Ingots and Slabs, Multiphase Phenomena and CFD Modeling and Simulation in Materials Processing, Eds., L.. Nastac and B. Li, TMS, Charlotte, March 2004. L. Nastac, F. Spadafora, O. Yu, Y. Pang, C. Wang, Y. Pang, and D. Winterscheidt, Magnetohydrodynamics in PAM Processed Ti-6Al-4V Ingots, Modelling of Casting, Welding, and Advanced Solidification Processes X, United Engineering Foundation, San Destin, Florida, May 2003. L. Nastac, C. Wang, H. Dong, and Y. Pang, Solidification Structure in Ti5Al-2Sn-2Zr-4Mo-4Cr Ingots Processed by Plasma Arc Cold Hearth Melting, International Symposium on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, Santa Fe, NM, September 23-26, pp. 288-300. L. Nastac, J. J. Valencia, M. L. Tims, and F. R. Dax, Advances in the Solidification of RS5 and IN718 Alloys, Intern. Symposium on “Superalloy 718, 625, 706, and Derivatives, Ed. E. A. Loria, Pittsburgh, PA, June 2001. L. Nastac and M. Liliac, Recent Developments in Modeling of Microstructure Formation during Solidification Processes, The Science of Casting and Solidification, Brasov, Romania, May 2001. L. Nastac, A Stochastic Approach for Simulation of Solidification Morphologies and Segregation Patterns in Cast Alloys, Modelling of Casting, Welding, and Advanced Solidification Processes IX, United Engineering Foundation, Aachen, Germany, August 2000. L. Nastac, J. J. Valencia, J. Xu, and H. Dong, A Computer Model for Simulation of Multi-Scale Phenomena in the Centrifugal Casting of MetalMatrix-Composites, TMS, Nashville, TN, March 2000. L. Nastac, J. J. Valencia, T. C. Kiesling, and M. L. Tims, Advances in Solidification of Thin Wall Superalloy Castings, TMS, Cincinnati, OH, 1999.

About the Author

285

L. Nastac, Keynote Lecture: A New Stochastic Approach for Simulation of Solidification Morphologies and of Segregation Patterns in Cast Dendritic Alloys, International Conference on Modeling of Casting and Solidification Processes, Yonsei University, Seoul, Korea, September 1999. L. Nastac, J. S. Chou, and Y. Pang, Bunshah Best Paper Award-1999, Assessment of Solidification-Kinetics Parameters for Titanium-Base Alloys, International Symposium on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, Santa Fe, NM, February 1999, pp. 207-233. S. Sundarraj, L. Nastac, Y. Pang and K. O. Yu, Modeling of Macrosegregation During Solidification of Ti-6-4 and Ti-17 PAM Ingots, Modelling of Casting, Welding, and Advanced Solidification Processes VIII, Engineering Foundation, San Diego, CA, June 1998, pp. 297-304. L. Nastac, S. Sundarraj, and Kuang-O Yu, Stochastic Modeling of Solidification Structure in Alloy 718 Remelt Ingots, Fourth International Special Emphasis Symposium on Superalloy 718, 625, 706, and Derivatives, Ed. E. A. Loria, Pittsburgh, PA, September 1997, pp. 55-66. L. Nastac, S. Sundarraj, Kuang-O Yu, and Y. Pang, Stochastic Modeling of Grain Structure Formation During Solidification of Superalloy and Ti Alloy Remelt Ingots, Intern. Symp. on Liquid Metals Processing and Casting, Vacuum Metallurgy Conference, Santa Fe, NM, Feb. 1997, pp. 145-165. L. Nastac and D. M. Stefanescu, Simulation of Microstructure Evolution During Solidification of Inconel 718, AFS Congress, 1996. L. Nastac, D. M. Stefanescu, and L. Chuzhoy, An Analytical Model for Microsegregation in Open and Expanding Domains, the Modelling of Casting, Welding, and Advanced Solidification Processes VII, Eds. M. Cross and J. Campbell, Engineering Foundation, London, UK, 1995, pp. 533-540. L. Nastac and D. M. Stefanescu, Prediction of the GWT Transition in Cast Iron through Solidification Modeling, AFS Congress, 1995. L. Nastac and D. M. Stefanescu, Modeling of the Stable to Metastable Structural Transition in Gray Cast Iron, Fifth International Symposium on Physical Metallurgy of Cast Iron, Nancy, France, 1994, Eds. G. Lesoult and J. Lacaze, SCITEC Publishing, Switzerland, 1997, pp. 469-478. L. Nastac and D. M. Stefanescu, Modeling of Growth of Equiaxed Dendritic Grains at the Limit of Morphological Stability, Modeling of Casting, Welding, and Advanced Solidification Processes -VI, Engineering Foundation, Florida, 1993, 209-17. L. Nastac, S. Chang, D. M. Stefanescu, and L. Hadji, A Model for Microsegregation in Multicomponent Systems Solidifying with Equiaxed Morphology, Microstructural Design by Solidification Processing, TMS, Chicago, 1992, 57-75. L. Nastac and D. M. Stefanescu, Assessment of Various Methods for Coupling Heat Transfer and Solidification Kinetics Codes, Micro / Macro Scale Phenomena in Solidification, HTD-Vol. 218/AMD- Vol. 139, ASME, Aneheim, CA, 1992, pp. 27-34.

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MESOS-2D AND MESOS-3D SOFTWARE PROGRAMS: INSTALLATION GUIDELINES AND INSTRUCTIONS

About Mesos-2D and Mesos-3D Software Programs Dr. Laurentiu Nastac developed the Mesos-2D and Mesos-3D simulation programs as personal projects. The intended use of these simulation programs is for educational purposes only. These simulation programs can be used to simulate on the computer screen the evolution of 3D dendritic microstructures during solidification of alloys. The equations and algorithms used to create these simulation programs are described in detail in Chapter 4 of the book. Mesos-2D also performs fluid flow computations.

System Requirements Hardware: Intel chip-compatible personal computers running Microsoft Windows 98, Windows NT4.0, Windows 2000, Windows Me, or above. Software modules: 2D version: Mesos_2D-Fluid-Flow-981x981.exe, INPUT.INP 3D version: Mesos_3D-131x131x131.exe, Mesos_3D-211x211x51.exe, Mesos_3D-451x451x11.exe, INPUT.INP Notes (please see also readme.txt files in the “2D” and “3D” directories): The above software programs were written in Fortran 90 and compiled with Compaq Visual Fortran 6.6 as Win32, standard graphics, singlewindow applications. For Mesos-2D, maximum number of cells in each direction is 981. For Mesos-3D, X, Y, and Z in the file name represent the maximum number of cells in x, y, and z directions; for example, when using Mesos_3D-211x211x51.exe program, maximum number of cells in x, y, and z directions are 211, 211, and 51, respectively. Minimum screen resolution is 1280x1024 with minimum 256 colors. To check/change your computer screen resolution go to “Start,” “Setting,” “control panel,” “Display,” Settings,” “screen area” and then set the

“screen area” to 1280 by 1024 pixels. Also, the user should use minimum 256 colors showed under the same “Display: Settings” option.

Software Installation and Instructions Create a folder titled “Microstructure Simulation” anywhere in the local disk. Copy all the 2D or 3D software modules in the “Microstructure Simulation” directory. Create a shortcut of the chosen application Modify the input variables to describe the simulation parameters, such as the geometry, the boundary conditions (Newton cooling using heat transfer coefficients and zero-flux solute boundary conditions are applied at all boundaries of the computational domain), the initial conditions (ambient and melt temperatures as well as solute concentration), the alloy’s physical properties, and the optional controlling parameters (screen plotting, output files, microchilling option, seed control, control of nucleation boundary conditions, control of growth type, etc.) in the INPUT.INP file. Note that the INPUT.INP file contains the description of all the input variables. Save the modified input file as INPUT.INP file. Note that the INPUT file must have the extension .INP and cannot be saved as a document or text file. Example of INPUT.INP files are provided for IN718-5 wt.%Nb, Ti-6 wt.%Al-4 wt.%V, and Fe-0.05 wt.%C alloys. Launch the application by clicking the shortcut icon to start. A new window will appear on your computer screen showing the input data information. At the bottom of this window the following message will appear “Fortran pause – enter command or
DISCLAIMER Copyright© 2004, Kluwer Academic Publishers All Rights Reserved This DISK (CD ROM) is distributed by Kluwer Academic Publishers with *ABSOLUTELY NO SUPPORT* and *NO WARRANTY* from Kluwer Academic Publishers. Use or reproduction of the information provided on this DISK (CD ROM) for commercial gain is strictly prohibited. Explicit permission is given for the reproduction and use of this information in an instructional setting provided proper reference is given to the original source. Kluwer Academic Publishers shall not be liable for damage in connection with, or arising out of, the furnishing, performance or use of this DISK (CD ROM).