The Pennsylvania State University The Graduate School Department of Mechanical Engineering DEVELOPMENT OF MACROSEGREGATION DURING SOLIDIFCATION OF

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1 The Pennsylvania State University The Graduate Shool Department of Mehanial Engineering DEVELOPMENT OF MACROSEGREGATION DURING SOLIDIFCATION OF BINARY METAL ALLOYS A Thesis in Mehanial Engineering by Byungsoo Kim 2002 Byungsoo Kim Submitted in Partial Fulfillment of the Requirements for the Degree of Dotor of Philosophy Deember 2002

2 We approve the thesis of Byungsoo Kim. Date of Signature Fan-Bill Cheung Professor of Mehanial Engineering Thesis Advisor Chair of Committee Anil Kulkarni Professor of Mehanial Engineering Dhyshyanthan Sathianathan Assoiate Professor of Engineering Design Stefan Thynell Professor of Mehanial Engineering Robert Voigt Professor of Industrial Engineering Rihard Benson Professor of Mehanial Engineering Head of the Department of Mehanial and Nulear Engineering

3 iii ABSTRACT A ombined experimental and theoretial study of the onvetive transport phenomenon during the solidifiation of a binary metal alloy is performed. Neutron radiography is onsidered as a means of extending and improving experimental verifiation methods for alloy solidifiation models and other experimental results. Calibration, film proessing, and digital image proessing proedures are developed in order to aurately quantify the marosegregation reorded on neutron radiographs. The method yields a highly resolved marosegregation field, rather than a few disrete measurements that an be used to help interpret measured ooling urves and infer thermosolutal onvetion patterns. In this study, a gallium-27 wt. pt. indium alloy was solidified in a square avity, hilled along one vertial side wall, the temperature of the alloy was measured during the solidifiation, and the marosegregation in the solidified ingot was determined using neutron radiography. The measured ooling urves revealed the presene of nonequilibrium phenomena during the early stage of solidifiation. The analysis of the ooling urves and marosegregation patterns showed where and how the indium-rih dendrite fragments and ool gallium enrihed liquid were transported in the mold avity by the thermosolutal onvetion. A ontinuum model was also developed in this study that was ast in dimensionless form and used to simulate thermosolutal onvetion during the alloy

4 iv solidifiation. The numerial preditions showed the ompliated onvetion flow patterns due to the interation between thermal and solutal buoyany fores, whih were not diretly observable in experiments and they were in fair agreement with the experimental results. The numerial and experimental results showed that during the early stage of solidifiation, solid partile transport and double diffusive onvetion due to the interation between thermal and solutal buoyany fores were the key auses of marosegregation. Results of the present study provided the new insights on marosegregation development during alloy solidifiation that would be useful to researhers and pratitioners working in metallurgy and related areas.

5 v TABLE OF CONTENTS List of Tables. List of Figures Nomenlature.... Aknowledgments. vii viii xii xviii 1. Introdution Bakground and Motivation The Physis of Alloy Solidifiation Review of Literature Experimental Investigations Mathematial Models Objetives of This Researh Experimental Apparatus and Proedures Introdution Phase Change Material Apparatus Diagnostis Temperature Measurement Marosegregation Measurements using Neutron Radiography Experimental Proedures Mathematial Modeling Introdution Marosopi Conservation Equations General Conservation in a Single Phase of Fluid Volume Averaging Tehnique Conservation of Mass Conservation of Momentum Conservation of Energy Conservation of Speies Supplemented Relationships for Closure Conservation Equations for Closure. 79

6 vi TABLE OF CONTENTS (CONT D) Thermodynami Relationships and Mirosegregation The Pressure in the Solid Phase The Dissipative Interfaial Stress on the Solid Phase Mirosopi Models The Interfaial Area Conentration Nuleation Model Growth Model of Grains Continuum Model and Numerial Solution Sheme Continuum Model Numerial Calulation Shemes Numerial Calulation Conditions Comparison of Finite Differene Algorithms Results and Disussion Introdution Case I: One side Cooling Case Case I: Experimental results Case I: Numerial simulations using the ontinuum model Case I: Comparisons between experimental results and numerial preditions Case II: One side Cooling and The Other Heating Case Case II: Experimental results Case II: Numerial simulations using the ontinuum model Case II: Comparisons between experimental results and numerial preditions Summary and Conlusions Reommendation for Future Work Bibliography Appendix: Listing of Computational Codes

7 vii LIST OF TABLES Table page 2.1 Physial properties of Ga-27 wt. pt. In The orresponding subsripts for soure terms Boundary onditions employed in numerial simulations Controlling dimensionless parameters. 114

8 viii LIST OF FIGURES Figure page 1.1 Marosegregation patterns in a killed steel ingot Typial phase diagram for a binary alloy system A-B Solutal underooling in alloys Neutron attenuation oeffiients of seleted elements Ga-In equilibrium phase diagram Experimental setup Shemati illustration of experimental test ell Loations of thermoouple probes in the test ell Illustration of neutron radiography method: (a) neutron beam attenuation and (b) film assette arrangement Calibration urve for the digital sanner Relationship between alloy omposition (volume fration) and film density Shemati illustration of the averaging volume The speies diffusion lengths The simplified onentration profile Shemati of shape fators for (a) an equiaxed dendriti envelop and (b) a square arrangement of olumnar dendriti envelope 90

9 ix LIST OF FIGURES (CONT D) Figure page 3.5 Illustration of the dependene of the shape fators on the solid volume fration Dimensionless equilibrium phase diagram One-dimensional, uniform, staggered grid Computational ontrol volume Problem domain Cooling urves for different numbers of grids Cooling urves for different time steps Numerial mesh for the alulation Solidifiation onditions predited by PDS at t=5: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Solidifiation onditions predited by CDS/QUICK at t=5: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Solidifiation onditions predited by PDS at t=25: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Solidifiation onditions predited by CDS/QUICK at t=25: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Solidifiation onditions predited by PDS at t=70: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Solidifiation onditions predited by CDS/QUICK at t=70: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Cooling urves for different runs at (a) x*=0.05, (b) x*=0.50, and () x*=

10 x LIST OF FIGURES (CONT D) Figure page 5.2 Cooling urves at different x* loations for y*= Neutron radiograph showing the marosegregation pattern (mass fration of indium) within the solidified ingot Cooling urves at different y* loations (0<t<600 s) for (a) x*=0.05, (b) x*=0.50, and () x*= Cooling urves at different y* loations (0<t<600 s) for x*=0.05 in the opposite diretion of oolant Convetion onditions of Case I at t=5: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomp Convetion onditions of Case I at t=20: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Convetion onditions of Case I at t=45: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Marosegregation patterns of Case I after t= (a) 5, (b) 20, () 45, (d) Measured and predited urves of Case I for (a) y*=0.25, (b) y*=0.50, and () y*= Marosegregation patterns of the fully solidified ingot for Case I: (a) experimental result (b) preditions at t= Cooling urves at different x* loations (0<t<15000 s) for y*=0.50 (Case II) Neutron radiograph showing the marosegregation pattern (mass fration of indium) within the solidified ingot (Case II) Cooling urves at different y* loations (0<t<15000 s) for (a) x*=0.05, (b) x*=0.50, and () x*=0.95 (Case II)

11 xi LIST OF FIGURES (CONT D) Figure page 5.15 Convetion onditions of Case II at t=5: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomp Convetion onditions of Case II at t=40: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Convetion onditions of Case II at t=80: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Convetion onditions of Case II at t=170: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Convetion onditions of Case II at t=360: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps Predited urves of Case II at y*= Marosegregation patterns of the fully solidified ingot for Case II: (a) experimental result (b) preditions at t=

12 xii NOMENCLATURE A b d D D α surfae area body fore per unit volume speifi heat harateristi length of the solid phase density of X-ray film mass diffusion oeffiient of speies α Da Dary number (K o /L 2 ) E f f g g Ga Gr h In J heat soure mass fration fator funtion volume fration gravity vetor Gallium Grashof number enthalpy Indium Interfaial speies flux

13 xiii NOMENCLATURE (CONT D) k K k p l L M &M k thermal ondutivity onstants in nuleation model, permeability equilibrium partition oeffiient diffusion length mold width interfaial momentum transfer prodution of speies per unit volume m l n N q Q p p Pr S S liquidus slope number density of grains buoyany parameter heat flux vetor interfaial energy transfer pressure kinemati pressure Prandtl number interfaial area onentration / rate of generation Shmidt number

14 xiv NOMENCLATURE (CONT D) SD Ste t T T U U v w x density of a sanned image Stefan number time temperature underooling temperature / temperature rise of the oolant overall heat transfer oeffiient veloity vetor in mirosopi sale veloity vetor in marosopi sale veloity of the interfae thikness of the test ell Greek β χ ε φ Φ dimensionless parameter saled speies onentration volume fration solid shape fator neutron flux

15 xv NOMENCLATURE (CONT D) γ onstant in equation (3.31) η κ ν κ s λ Γ dimensionless enthalpy flow partition oeffiient dimensionless thermal ondutivity of solid phase dendrite arm spaing rate of total mass generation / Gibbs - Tompson oeffiient µ kinemati visosity θ ρ Σ τ ψ dimensionless temperature density marosopi apture ross-setion visous stress tensor general salar quantity Subsripts d e oolant (old) interdendriti liquid phase envelope / east boundary

16 xvi NOMENCLATURE (CONT D) E f Ga h In k kj l N o P s S sp st t w east node liquid phase / fusion Gallium heat exhanger (hot) Indium phase k k phase at k-j interfae extradendendriti liquid phase north node referene present node solid phase south node maximum paking of the solid phase standard total west boundary

17 xvii NOMENCLATURE (CONT D) Supersripts W west node 1 primary 2 seondary d α Γ τ dissipative α speies due to phase hange due to interfaial stress - average * effetive

18 xviii ACKNOWLEDGEMENTS The author is grateful to Professor Fan-Bill Cheung for is mentorship during the ourse of this work. In addition to his valuable advie, Professor Cheung provided me with enouragement to pursue the ompletion of this work and great liberty to pratie my ideas. Also, I am grateful to Dr. Patrik Presott for giving me this opportunity and ontinuous advie. Gratitude is also extended to Professors Anil Kulkarni, Dhyshyanthan Sathianathan, Stefan Thynell, and Robert Voigt for graiously serving as advisory ommittee members. The assistane from the entire tehnial staff in the Department of Mehanial Engineering and Penn State Breazeale Nulear Reator is gratefully aknowledged. Finanial support for this support was provided by National Siene Foundation through Award Number CTS , for whih I am very grateful.

19 1 Chapter 1 INTRODUCTION 1.1 Bakground and Motivation Solidifiation ours in many proesses, ranging in disparity from metal asting to rystallization of roks and minerals in the earth s magma. Due to its rihness in the physial phenomena, it is very diffiult to understand the solidifiation well and the phenomenon has hallenged researhers in many disiplines for several deades. In metal asting industry, the quality of the metal produt is very important and improving the quality requires tremendous amounts of researhes on the solidifiation proesses. Traditionally, the solidifiation has been examined in the view of miro-metallurgy. This researh onsiders the solidifiation phenomenon from an engineering of thermal siene perspetive and is ontributed to a better understanding of the physis of the solidifiation. The sope will be restrited to the solidifiation of binary metal alloys. The essential feature of solidifiation is a liquid-to-solid phase transformation along a moving interfae, whih is aompanied by the release of latent thermal energy and its transfer through both phases. Compliations arise from the evolution of the shape and the boundaries of the moving solidifiation front or mushy region, the interations among heat and mass transfer, fluid flow, and hemial reations, and the involvement of

20 2 two distint phases, whih have different thermophysial properties. Furthermore, the proess involves phenomena on different length sales, ranging from the moleular level at the liquid-solid interfae to the marosopi dimensions of the asting system. Moreover, phenomena at different sales influene eah other. The solidifiation of a multi-omponent alloy differs signifiantly from that of a pure metal, whih is understood well enough to adequately predit onventional proess harateristis. The study of multi-omponent alloy solidifiation should onsider additional phenomena, suh as phase hange over a temperature range, the formation of a two-phase mushy region, the redistribution of solutes, and double-diffusive onvetion. During the solidifiation proess of a multi-omponent alloy, the solutes are often redistributed non-uniformly in the asting and the resulting undesirable ompositional non-uniformity in the fully solidified ingot is referred to as segregation. Segregation ourring on a mirosopi sale (i.e., between and within dendriti arms) is known as mirosegregation and an be ontrolled or redued quite easily beause the diffusion distane for homogenization is suffiiently small. However, segregation ourring on a marosopi sale, alled marosegregation annot be eliminated. Figure 1.1 illustrates some marosegregation patterns in a killed steel ingot. The ause of marosegregation in the ingots is now understood to be physial movement of liquid and/or solid phases relative to eah other during solidifiation (Flemings, 1974). A signifiant driving fore for relative phase motion is buoyany, aused by thermal and ompositional effets on density. Thermal and solutal buoyany

21 Figure1.1 Marosegregation patterns in a killed steel ingot (Campbell, 1993). 3

22 4 fores may interat with eah other, and effet omplex, double-diffusive onvetion. Presott and Inropera (1996) reviewed the influene of the onvetive transport on solidifiation of alloys along with reent advanes in mathematial modeling. The past three deades have seen the advent of advaned mathematial models, suh as ontinuum (Bennon and Inropera, 1987) and volume-averaged (Ni and Bekermann, 1991) models. These models have been suessful at qualitatively prediting from basi priniples, the prevailing physial phenomena assoiated with alloy asting. However, the numerial preditions from suh models must be validated and an only be improved with proper experimentation. Most experimental validation studies have involved so-alled analog alloys, suh as aqueous salt solutions (Bekermann and Viskanta, 1988; Christenson et al., 1989), whih have been popular in laboratory experiments beause they freeze in a manner similar to metal alloys and their transpareny allows flow visualization. However, suh studies are of limited utility for understanding natural onvetion in solidifying metal alloys, beause the Prandtl number of aqueous solutions is two or three orders of magnitude larger than that of molten metal alloys. Thus, the behavior of molten metals in natural onvetion is signifiantly different from that of aqueous solutions. Experiments using metal alloys are sare (Shahani, 1992; Presott et. al. 1994), and they have relied on interpretation from temperature measurements and postexperimental omposition measurements at several disrete loations to infer onvetion patterns during solidifiation. The visualization of the flow in metal alloys during solidifiation and marosegregation patterns in solidified ingots, is possible through

23 5 radiographi tehniques. Even though the tehnique provides qualitative images in detail, the quantifiation of the results has been performed very rarely. One of radiographi tehniques uses neutron beam and the quantifiation of the image obtained from the tehnique is possible beause of the distintive neutron attenuation rate in different elements. Therefore, the quantitative investigation of the thermosolutal onvetion during the solidifiation of a metal alloy is proposed here through utilizing neutron radiography to measure omposition patterns at different stages of solidifiation. Also, the temperature of the alloy during solidifiation is measured and multiple numerial simulations are performed to understand the transient phenomena of the alloy solidifiation.

24 6 1.2 The Physis of Alloy Solidifiation Many phenomena play important roles in the alloy solidifiation with different length and/or time sales. In this setion, only a few basi onepts, whih are partiularly relevant to the topi under onsideration, are disussed. In ontrast to a pure substane whih hanges phase isothermally, alloys solidify over a temperature range, in whih the solid and liquid o-exist in thermodynami equilibrium. Figure 1.2 shows a typial equilibrium phase diagram of a fititious eutetiforming binary system A-B at a onstant pressure. At temperatures above the liquidus lines, a single liquid phase exists as a solution of onstituents A and B. Liquid and solid phases oexist in equilibrium over a range of temperatures between the liquidus and solidus lines, up to the euteti point. During the solidifiation of a binary alloy, the two phase region separates fully solidified and melt regions and it is known as the two-phase mushy zone. At the euteti point, a three phase mixture of a liquid phase and two solid phases designated as α and β, exists. When an initially superheated alloy ools, nuleation of small rystals ours at a temperature slightly below the liquidus temperature assoiated with its omposition. As the temperature of the solid-liquid interfae dereases, the ompositions of the solid and liquid at the interfae ontinually hange. As indiated in Figure 1.2, on the left side of the euteti point (hypoeuteti alloys), the solubility of speies B is lower in the solid than in the liquid, and the rejetion of the speies B from the solid leads to an inrease of

25 7 T i solidus liquidus liquid Temperature α solid α+liquid euteti point β+liquid α+β solid β solid A C αi C li Conentration B Figure 1.2 Typial phase diagram for a binary alloy system A-B

26 8 the B onentration in the liquid at the interfae. In most pratial asting proesses, this ondition of phase equilibrium at the solid-liquid interfaes is met. However, deviations from the phase diagram an take plae due to apillarity, pressure, and kineti effets (at high solidifiation rates) and they are disussed in several textbooks (Flemings, 1974; Kurz and Fisher, 1989). The formation of a solid is basially governed by the temperature and speies onentration at the interfae. However, the development of interfae shape is a more ompliated issue, involving stability and interfae urvature onsiderations. It is of utmost importane in solidifiation modeling to take into aount these mirosopi interfaial features, beause they ultimately determine the mirostruture, whih strongly affets the mehanial properties of the material. There are essentially two basi growth morphologies that an exist during alloy solidifiation. These are dendriti and euteti morphologies (Kurz and Fisher, 1989). Dendrites grow with a very large speifi surfae area and irregular solid-liquid interfae. In the marosopi sense, suh growth is virtually irresolvable. Hene, the mushy zone, whih is omprised of solid dendrites and interdendriti liquid, is treated as a porous solid struture whih is saturated with interdendriti liquid. The growth of eutetis is muh simpler than that of dendrites beause the euteti grain maintains a simple geometri interfae shape. Generally, both morphologies develop together. Due to the low mass diffusivity relative to the thermal diffusivity for metal alloys, the formation of the aforementioned mirosopi strutures is mainly dependent on the speies onentration gradient on eah side of the solid-liquid interfae. On the liquid

27 9 side of the interfae, most of the solute is rejeted from the solid, forming higher interfaial onentration than in the liquid away from the interfae. The differene between the onentrations at the interfae and nearby bulk liquid is usually referred to as the solutal underooling (Figure 1.3). In other words, the atual temperature in the liquid is below the liquidus temperature orresponding to the interfaial liquid onentration. The atual temperatures of the interfae and bulk liquid an also be different from eah other, and the differene is referred to as the thermal underooling. The thermal underooling is usually relatively small in metal alloys, beause of the high thermal diffusivity in metal alloys, and thus, sine the solute gradients are onfined to the small region near the interfae, onvetive influenes on the solute transport have been negleted traditionally. However, vigorous onvetion an severely alter the mirosopi onentration and temperature profiles, and hene the movement and shape of the interfae an be different from the traditional approah (Gliksman et al., 1986). The segregation of the hemial omponents at a solid/liquid interfae during solidifiation an also manifest itself on a marosopi sale. Movement of either the liquid or solid phases an indue ompositional inhomogeneities on the system sale, whih is termed marosegregation. The onvetive movement in the liquid phase an be indued by external fores, i.e., fored onvetion as disussed by Tangthieng (2002), surfae tension gradient at a free surfae and buoyany fores due to temperature and/or solute onentration gradients. In the absene of external fores, buoyany fores are often dominant. Sine solidifiation is indued by ooling through boundaries,

28 10 T solutal = T f α ( ) Tatual T(f α ) T atual f α f α (T atual ) (a) f α (T atual ) Solid Liquid k f α (T atual ) f α (b) Figure 1.3 Solutal underooling in alloys. (a) Phase diagram. (b) Temperature profile on the interfae.

29 11 temperature gradients are established in the system and beause fluids are sensitive to temperature hanges, the orresponding density gradients yield thermal buoyany fores, whih may indue fluid motion. Another onsequene of the temperature gradient is the hange of solubility in the solid phase, and thus, it indues a gradient in liquid omposition within the mushy zone. Sine the liquid density depends on solute onentration, solutal buoyany fores are established. Depending on the relative densities of the alloy onstituents and the solute rejeted to the interdendriti liquid, solutal and thermal buoyany fores may either augment or oppose eah other. If solidifiation ours along a vertial wall and the rejeted solute is less dense than the bulk melt, an upward solutal buoyany fore opposes the thermal buoyany fore. In ases where the rejeted solute is denser than the bulk liquid, the two fores augment eah other. Although the solutal buoyany fore arises from the mushy zone, its effet is not limited to this region, and the solutally driven flows interat with thermally driven flows in the bulk melt. The resulting transport is termed double-diffusive onvetion. When the less dense interdendriti fluids due to the solute rejetion are adveted upward along a old vertial wall and exit into the melt, a top layer of thermally heavier (older) and solutally lighter fluid is established, reating onditions for the formation of double-diffusive onvetion ells within the melt. The interdendriti fluid flow erodes hannels within the mushy zone, whih are subsequently overgrown as the solid region advanes, forming a slanted penil-like struture in the final ast. These penil-like regions are referred as A-segregates (Figure 1.1). If solidifiation is indued from the

30 12 bottom and the interdendriti fluid is buoyant, jets of the solutally buoyant fluid asend from the mushy zone. The jets are disharged from vertially oriented hannels and beome the last regions to solidify, reating a severe form of marosegregation, known as frekles. Remelting at the root of seondary dendrite arms due to thermo-solutal onvetion an also lead to frekle formation in diretionally solidified alloys (Shneider and et, 1997). Movement of the solid phase, typially in the form of equiaxed rystals, an also ause marosegregation (degroh III and Laxmann, 1988). Free equiaxed grains move mainly due to the density differene between the solid and liquid phases. The settling of equiaxed grains is known to be the ause of the bottom one shaped negative segregation (Figure 1.1). The grains may also move into a melt of different temperature and/or omposition and may be partially re-melted, ausing the release of liquid that has the same omposition as the remelted part of the grains. This phenomenon has reeived very little researh attention.

31 Review of Literature Sine marosegregation was found to be the result of fluid motion in the mushy zone (MDonald and Hunt, 1969), studies on the influene of onvetion during solidifiation have been reported in the literatures relevant to not just metallurgy but also fluid mehanis and heat and mass transfer. Various experiments have been performed to understand the mehanisms assoiated with solidifiation proesses and the development of marosegregation. At the same time, many mathematial models have been developed to predit these phenomena through analytial or omputational simulations. In the following setions, literatures that represent important ontributions toward understanding solidifiation phenomenon are reviewed. There are two setions; one overs experimental works and the other reviews mathematial models and their simulations Experimental Investigations Early experiments onerning the effet of onvetion during alloy solidifiation foused on the metallurgial grain struture. Cole and Bolling (1965) investigated the ontribution of the onvetion toward a transition from olumnar to equiaxed grain strutures during solidifiation of a Pb-Sb alloy, and demonstrated that the inhibition of onvetion using wire sreens resulted in the inrease of olumnar grain strutures in the asting. Follow-up studies dealt with the effet of enhaning onvetion using

32 14 eletromagneti field (Cole and Bolling, 1966) and a rotating mold (Cole and Bolling, 1967). It was onluded that enhaned fluid flow inreased thermal mixing, thereby reduing temperature gradients in the melt and promoting equiaxial grain growth. None of these studies onsidered marosegregation. One of the first investigations related to marosegregation due to onvetion was onduted by MDonald and Hunt (1969). They observed the formation of hannels leading to A-segregates in an aqueous ammonium hloride solution solidified from a sidewall, and they believed that the A-segregates were formed due to interdendriti flow driven by solutal buoyany fores within the mushy zone, rather than shrinkage effets (Flemings et al., 1968; Flemings and Nereo, 1967, 1968). In another experiment (MDonald and Hunt, 1970), zin hloride was added to the aqueous ammonium hloride solution in order to reverse the diretion of solutal buoyany fores, and hannels resembling inverted A-segregates were observed. Szekely and Jassal (1978) used dye traers and Shlieren photography to visualize flow strutures during the horizontal solidifiation of an ammonium hloride solution in a retangular avity. They observed the jets of solutally buoyant interdendriti fluids asending from the mush zone into the melt. It was onluded that these solutal plumes were responsible for the formation of A- segregates. The formation of frekles during unidiretional solidifiation of an aqueous ammonium hloride solution was observed by Copley, et al. (1970). Their experiments revealed jets of old interdendriti fluid exuding from the mushy zone due to its lower

33 15 salt onentration than the parent solutions. The jets lead to the formation of hannels in the mushy zone and the number of hannels was ontrolled by gravity and the heat flux. The influene of solutal buoyany fores is not onfined to the mushy region. Interation between flows in the fully melted and mushy zones results in double diffusive onvetion that has been the fous of more reent studies. Thompson and Szekely (1988) onsidered solidifiation of an aqueous sodium arbonate solution in a retangular mold, using dye injetion for the flow visualization. Solidifiation was indued along a sidewall while the opposite wall was maintained above the liquidus temperature. Sine the rejeted solute in the experiments was denser than the original melt, double-diffusive layers were observed to form from the bottom of the mold in the fully melted zone. Christenson and Inropera (1989) performed a series of solidifiation experiments with different melt ompositions of aqueous ammonium hloride and various thermal boundary onditions. The temperature and onentration measurements as well as flow visualization revealed many features of the thermally and solutally driven flow, inluding double diffusive onvetion. It was found that the relative strengths of thermal and solutal buoyany effets determined the solidifiation harateristis. Bekermann and Viskanta (1988) also performed solidifiation experiments with aqueous ammonium hloride in a square test ell. They used shadow graphi tehnique to reord doublediffusive layering and the haoti mixing, whih ours after the breakup of a doublediffusive interfae.

34 16 Solidifiation of an aqueous ammonium hloride solution flowing downward in a vertial retangular hannel was experimentally investigated by Bennon and Inropera (1989). Mixed onvetion onditions resulted from the interation of thermosolutal buoyany-indued flows with the fored through flow. The temperature and progressions of the liquidus front were reorded for different experimental onditions, and used to haraterize the influene of superheat, initial omposition, ooling wall temperature, and flow Reynolds number on the solidifiation proess. In a similar way, Lim and et al. (2001) examined systematially the effets of onstant wall temperature and initial onentration using aqueous ammonium hloride as binary alloy. They also determined the temperature of the mushy-liquid interfae by superimposing of the measured phase front profile and thus predited the onentration level at the interfaes. Although the experiments with the analog alloys have provided qualitative understanding of alloy solidifiation, it is important to realize that ertain parameters of analog alloys, suh as Prandtl number, are different from those of atual metal alloys, and thus, the behavior of molten metals in natural onvetion during the solidifiation is signifiantly different from that of analog alloys (Stewart and Weinberg, 1971a,b). However, experiments using metal alloys present diffiulties in handling the material and the high ost. Furthermore, due to the opaqueness of the metal alloys, even qualitative study and observation of flow patterns ourring during the solidifiation are far more diffiult than for analog alloys. It leads to a major diffiulty in the validation of numerial odes for liquid metal studies through omparison with liquid metal experiments during the solidifiation.

35 17 There are relatively few experimental investigations in whih marosegregation patterns were quantified. Mehrabian et al. (1970) studied solidifiation of an aluminumopper alloy from a sidewall in a retangular mold. Thermoouples were plaed at ertain strategi loations to monitor the solidifiation front, and marosegregation was quantified by X-ray fluoresene omposition analysis of ingot setions after the solidifiation was omplete. It was found that segregates formed as a onsequene of interdendriti fluid flow arising from a density gradient, and that a redution in ooling rates favored the formation of segregates. Ridder et al. (1981) used the same tehnique to quantify the marosegregation patterns in ylindrial lead-tin ingots. In their experiments, the sidewalls and a base of the mold were hilled, while a hot plate above the liquid surfae maintained a melt. After the omplete solidifiation, the radial marosegregation patterns were measured using X-ray fluoresene. The results agreed well with theoretial preditions, and it was onluded that, for the geometrial onditions in their experiments, onvetive motion in the melt had little effet on interdendriti fluid flow and marosegregation. Streat and Weinberg (1974) quantified the marosegregation in vertially solidified lead-tin ingots, using isotopes of tin and titanium as the radioative traers. After solidifiation, samples of the ingot were olleted from presribed loations, and the solute omposition was determined from the measured ativity and weight of eah sample. Also, to observe onvetion in the melt and mushy zone, the mold ontaining the molten alloy with radioative isotopes was hilled for one hour and then quenhed. Autoradiographs of ingot setions revealed fluid flow patterns from the solidifiation

36 18 proess. They onluded that the marosegregation was the result of onvetive flow aused by the solute rejetion and solutal buoyany fores in the mushy zone. Marosegregation in experimental ingots has been quantified through onventional post-experimental ompositional analysis. The tehnique has been used to validate numerial preditions of the marosegregation. Shahani et al. (1992) used this method when they studied the effet of natural onvetion during horizontal solidifiation of lead-tin alloys in a retangular avity. They measured transient mold wall temperatures, whih were used as the thermal boundary onditions for their numerial simulations, and the predited marosegregation was ompared to measurements made using atomi absorption spetrophotometry (AAS). Presott et al. (1994) performed experiments involving the solidifiation of a lead-tin alloy in an axisymmetri, annular mold. Marosegregation measurements were also made through AAS analysis of final ingots. The temperatures of the alloy at several loations were measured during the solidifiation and showed underooling and realesene, whih their numerial simulation did not predit (Presott and Inropera, 1994). Although these studies showed reasonable agreement between measured and predited marosegregation patterns, the omparison was limited only to a few disrete loations, and details of the predited transient onvetive phenomena ould not be validated. In order to visualize the flow pattern and omposition hange during solidifiation, various radiographi tehniques have been proposed for qualitative flow observation. Stewart and Weinberg (1969) studied natural onvetion in molten tin, using a radioative isotope of tin as a traer. After introduing the traer into the melt, the metal

37 19 was quenhed quikly, and a qualitative piture of the flow struture was obtained through the autoradiographs of slied setions of the ingot. They used the same tehnique for lead-tin alloy solidifiation to study the penetration of reirulating fluid from the melt into the mushy zone (Stewart and Weinberg, 1972). The similar tehnique was employed with muh suess to non-invasively visualize the morphology of melting and solidifiation interfaes (Kakimoto et al a,b and1991). One serious drawbak of above investigations of onvetion in metal systems is that the proess of solidifiation annot be followed from the beginning to the end. The proess has to be interrupted at an intermediate stage and a subsequent, postexperimental analysis must be performed. Reognizing the need for ontinuous observation of solidifiation, Bridge and Beeh (1983) performed experiments related to a solidifiation proess in aluminum-opper alloys, using an X-ray vidion (X-ray sensitive TV amera) and the proess was reorded on videotape. Still photographs were also taken every one or two minutes from the television monitor for subsequent analysis and study. A densitometer was used to examine the optial density of the photograph negatives. The images revealed that a large number of hannels originated at irregularities on growth front, resulting in the entrapment of a signifiant poket of liquid by the advaning front during the early stages of solidifiation. Kaukler and Rosenberg (1994) enhaned the X-ray real time radiographi tehnique to observe the morphologial features and partile-interfae interations in Al-1.5 Pb alloys. In reent years, sophistiated X-ray visualization tehnique has been developed that an be used to observe the onentration distribution in opaque liquid metalli

38 20 materials by reording the loal density hange (Campbell et al, 1997; Koster, et al., 1997). The density hange of the metals was alulated from the relation between the density and the monotoni gray density. Resolution of density differenes up to g/m 3 for pure gallium in a narrow vertial layer has been ahieved and onentration stratifiation of indium in liquid gallium was distinguishable down to g/m 3 (Koster and Derebail, 1997; Derebail and Koster, 1998). Using the real-time X-ray visualization, the ompositional segregation during the solidifiation was monitored for Ga-5 In alloys that were exposed to a horizontal temperature gradient (Koster, 1997). The experimental results showed quite different results than theoretial preditions due to more ompliated kineti effets during the solidifiation and the pronouned nonequilibrium solidifiation harateristis of the metal itself for this metal. Yin and Koster (1999, 2000) observed the solidifiation of Gallium-Indium melt under different transient horizontal temperature gradients. They were able to show the multiple onveting layers staggered on top of a melt layer in a ondutive state and onluded that the multiple layered onvetive flow determined the marosegregation, interfae morphology, and mirostruture of the final produt. Quantifying the image tehnique, they showed the omposition in 256 olored images in different stages of the solidifiation. They provided the good data for the omposition variations but laked the information for temperature and flow movements. More reently, another radiographi tehnique for flow visualization of nontransparent materials was developed using thermal neutrons (Cimbala et al., 1988). The neutron radiographi tehnique is similar to the X-ray radiography desribed by Bridge et

39 21 al. (1982). However, the use of thermal neutrons as a soure of radiation is a powerful tool beause attenuation of neutrons varies signifiantly with the elemental omposition of the material, while attenuation of X-rays hanges gradually with the atomi number inreases. Therefore, images obtained with neutron radiography (NR) ould have higher ontrast for ertain alloy ombinations, and thus, the details of a target struture ould be obtained. The foregoing review reveals that there is a pauity of experimental data orresponding to the transport phenomena, whih our during the solidifiation of metal alloys. While the diffiulty of obtaining experimental data for metal alloys renders numerial simulation attrative for resolving details of the onvetion onditions, it is imperative that models, on whih suh simulations are based, be validated with experimental data, preferably from experiments involving atual metal alloys. The validation requires omparison through marosegregation measurements within ingots. Suh validation an only be ahieved with detailed marosegregation measurements and/or good quality flow visualization during solidifiation Mathematial Models and Numerial Simulations The theoretial approah to the phenomena in the alloy solidifiation has many advantages ompared to performing experiments. For example, it is relatively easy to hange governing parameters and boundary onditions, and extrat useful insights to transport phenomena with numerial simulation. Therefore, over many years mathematial models have been developed by engineers and mathematiians, and

40 22 numerial alulations from suh models have predited many phenomena, at least qualitatively. However, theoretial models are onstrained by their assumptions, whih ultimately must be validated with experimental data. In the earliest models of marosegregation, the onvetion in the bulk melt and oupling between flows in solid, mushy, and melted zones were negleted. Flemings and o-workers (Flemings and Nereo, 1967, 1968; Flemings et. al., 1968) predited marosegregation using the so-alled solute redistribution equation, whih was an extension of mirosegregation analysis. Interdendriti flow was assumed to be indued by thermal ontration and solidifiation shrinkage and the solute was transported by fluid advetion. Their model required solidifiation rates, temperature field, and the fluid veloity field to be presribed. However, even with the limitation of the model, it was able to demonstrate that interdendriti flow was responsible for the marosegregation and it was verified with experiments (Flemings and Nereo, 1968). Later, the model was modified by inorporating a model for buoyany driven flow in the mushy zone (Mehrabian et al., 1970). The mushy zone was modeled as a porous medium, inorporating Dary s law and the permeability was desribed as a funtion of the liquid volume fration. The analysis still required solidifiation rates and temperature profiles to be presribed, and the mixture onentration was assumed onstant in the volume element. Fujii et al. (1979) further refined the model by solving the oupled mass, momentum, energy, and speies onservation equations for the mushy zone. They simulated the solidifiation of steel and investigated the effets of alloy ompositions on

41 23 marosegregation. A shortoming of the model was that onvetion in the fully melted liquid region was not onsidered. Szekely and Jassal (1978) solved for the first time the omplete set of onservation equations for both mushy and bulk liquid regions. Sine the momentum equations for the mushy zone (Dary s law) and the bulk liquid region (Navier-Stokes equations) were of different forms, they utilized a multi-domain approah, in whih the transport equations for eah region were written separately and oupled through ertain interfae onditions. However, marosegregation was not predited and the solutal buoyany effets were not onsidered in their analysis. Later, the first model to ouple the bulk melt and mushy regions, and also inlude marosegregation was reported by Ridder et al. (1981), who used a multi-domain model. The solutions of these models required expliit traking of the liquidus interfae. As the interfae beame more ompliated, the implementation of the numerial method beame more diffiult. Hene, the multi-domain models were not well suited for the ases with highly irregular interfae shape. In order to overome the diffiulty of the multi-domain approah, a number of single-domain models have been proposed in the reent years (Bennon and Inropera, 1987a; Voller and Prakash, 1987; Bekermann and Viskanta, 1988) and showed promise of beoming useful tools for simulating solidifiation proesses (Bennon and Inropera 1987b). Bennon and Inropera (1987a) utilized lassial mixture theory and presented a set of marosopi onservation equations, whih were onurrently appliable to all regions (solid, mushy, and liquid) and required only a single, fixed numerial grid and a

42 24 single set of boundary onditions. Hene, the solid, mushy, and liquid regions were impliitly oupled. The mixture model, whih was later larified by Presott et al. (1991), viewed the mushy zone as an overlapping ontinuum, whih was oupied by the solid and liquid simultaneously with marosopi properties. Another approah was motivated by theories of flow through porous media and of other multi-phase systems (Gray, 1975; Hassanizadeh and Gray, 1979a; Gray, 1983; Drew, 1983), whih utilized a volume averaging tehnique (Bekermann and Viskanta, 1988; Ganesan and Poirier, 1990). The marosopi onservation equations were rigorously derived from mirosopi (exat) equations, while the mixture theory assumed the validity of ertain ontinuum relations on a marosopi sale. In the approah, the mushy zone was oneived to onsist of two interpenetrating phases (the lassial onservation equations were applied only within eah phase but not over the entire mixture). Ni and Bekermann (1991) used the tehnique to develop the two-phase model for solidifiation of a metal alloy. In the model, separate volume-averaged onservation equations were derived for the solid and liquid phases, permitting a rigorous treatment of disparate solid and liquid veloities, thermal and solutal non-equilibrium, and interfaial momentum, heat and speies exhange. Mirosopi features ould be inluded through the interfaial transfer terms, nuleation models, and stereologial formulations, whih aounted for the geometry of mirosopi solid strutures (Feller and Bekermann, 1993). Volume-averaged models provide greater insight to the oupling between mirosopi marosopi sales than mixture theory (Hassanizadeh and Gray, 1990). Despite the differene of two approahes, the model yielded the same marosopi equations with the ontinuum model developed

43 25 by Bennon and Inropera (1987a) when they were simplified by reognizing the dominant physial mehanisms in the fluid flow (Presott et al., 1991). The models mentioned above have revealed physial insights to the alloy solidifiation through numerial simulations. The ontinuum model proposed by Bennon and Inropera (1987a) has been used for several studies of alloy solidifiation indued from a side wall in a retangular mold (Bennon and Inropera, 1987b; Christenson et al., 1989) or ylindrial avity (Presott and Inropera, 1994). Key features of the alloy solidifiation, suh as irregular liquidus front, double diffusive onvetion in the melt, development of flow hannels in the mushy zone, remelting of solid, and formation of marosegregation patterns, were predited. However, the numerial preditions have shown only fair quantitative agreement with experimental results and the disrepany has been attributed to the unertainties in presribed parameters in the model and assumptions. For example, most simulations assumed the permeability of the mushy zone to be isotropi. Yoo and Viskanta (1992) onsidered the effet of anisotropi permeability in a model and suggested that the effet should be onsidered to ahieve aurate simulations. Mat and Ilegbusi (Ilebugsi and Mat, 1997; Mat and Ilegbusi, 2002) extended the ontinuum model by onsidering the mushy zone as non-newtonian semisolid slurry below the ritial solid fration and a porous medium thereafter. The value of the ritial solid fration was hosen based on the experimental results (Amberg et al. 1993). Although the model desribed the physis more aurately than the onventional model, it brought in more unertainty in the parameters, whih depended on the boundary

44 26 onditions and the material. The predition showed little differene from the model viewing the mushy zone as a porous medium (Christenson et al., 1989). Unidiretional solidifiation from below has also been simulated using a ontinuum model (Neilson and Inropera, 1991, 1992), whih were able to predit the formation of frekles resulting from buoyany-indued fluid motion and above the mushy zone. The volume-averaged two-phase model of solidifiation was used to simulate equiaxed solidifiation of an aluminum-opper alloy in a retangular avity ooled from one vertial side wall (Bekermann and Ni, 1992). The model aounted for nuleation, growth, and advetive transport of grains and was able to predit not just evolution of marosegregation but also realesene whih ould not be shown by other simulations using a ontinuum model, due to the assumption of loal thermodynami equilibrium. However, quantitative validation was not performed. In the multiple omponent alloy solidifiation, the gravitational influene in the onvetion in the melt an be muh more ompliated than the binary alloy beause eah alloying element ontributes to the hange in the liquid density and the phase diagram of the alloy is muh more omplex. The ontinuum or volume average models have been used reently with different algorithmi strategies than the ones used for binary alloys due to the omplex solidifiation path (Shneider and Bekermann, 1995a and 1995b; Felielli et al., 1997 and 1998). Although the fluid flow during the solidifiation is driven by thermal and solutal buoyany, as well as by solidifiation ontration, in most studies the natural onvetion was onsidered as the only driving fore sine solidifiation ontration is more diffiult

45 27 to takle, beause it involves a domain hange. Therefore, not many studies with the solidifiation ontration are available. Tsai and o-workers (Diao and Tsai, 1993; Chen and Tsai, 1993) used the ontinuum model to predit marosegregation driven by solidifiation ontration during the unidiretional solidifiation of Al-Cu alloys, assuming the omplete diffusion both in the liquid and solid, whih was unreasonable for most asting solidifiation. Chang and Stefanesu (1996) extended Tsai s model by removing the assumption and allowing the visosity to hange dramatially when the dendrite ohereny was reahed. They showed that the solidifiation ontration had a stronger effet on the liquid flow in the mushy region than buoyany in some ases suh as the unidiretional solidifiation of plate asting. Despite their suess, ontinuum or volume averaged models are not well suited for inorporating mirostrutural features present in dendriti solidifiation, beause of the single-sale averaged desription of phase behaviors (Wang and Bekermann, 1992). Transport phenomena ourring on the various mirosopi sales differ from one another, and a single-sale model provides insuffiient resolution to apture dynami behaviors on several mirosopi length sales. Therefore, the resolution of multiple sales is required for the omplete inorporation of mirosopi effets in a marosopi model and the predition of mirostruture formation during the solidifiation. Reently, onsiderable progress has been made to aount for the nature of mirostrutures in soalled miro-marosopi modeling for both equiaxed (Dustin and Kurz, 1986; Rappaz and Thevoz, 1987a,b) and olumnar (Flood and Hunt, 1987; Giovanola and Kurz, 1990) dendriti solidifiation, but without onsidering onvetion. In these models, the

46 28 mirosopi growth kinetis of the dendriti tip was inorporated in a marosopi energy equation through a solid volume fration term, and the liquid phase in a ontrol volume was viewed as two distint fluids assoiated with two length sales. That is, the liquid within the dendriti struture (interdendriti liquid) and the liquid outside the dendrites (extradendriti liquid) were distinguished from one another. Rappaz and Thevoz (1987a) developed a solute diffusion model for equiaxed dendrites, aounting for nuleation and growth kinetis and introdued the idea of a spherial grain envelope that separated the interdendriti and extradendriti liquids. The interdendriti liquid was assumed to be solutally well mixed, and the dynamis of the envelope was determined by the growth kinetis of the dendriti tips. They also onsidered a simplified model (Rappaz and Thevoz, 1987b), in whih the solute diffusion equation outside of the grain envelope were replaed by a solute layer thikness, and the simplified model showed exellent agreement with their previous more ompliated model. Although their preditions for an aluminum-silion alloy agreed well with experimental measurements, nuleation was assumed to our at a single temperature and temperature was obtained from the experimental observation, rather than from basi physial priniples. Furthermore, it was assumed that the average onentration of the liquid outside of the grain remained at its initial value, whih is not valid in the presene of marosegregation. Giovanola and Kurz (1990) proposed an empirial approah to alulate the solid volume fration as a funtion of temperature and dendriti tip growth veloity. The model divided the mushy zone into the region around the dendriti tips and the rest of the mushy zone, in whih the liquid is assumed to be ompletely mixed. A urve-fitted

47 29 polynomial and the Sheil equation for the solid fration in eah of the two regions were utilized umbersomely. Other similar models were reviewed by Rappaz (1989). A new analytial model for alulation of the evolution of solid fration during alloy solidifiation with equiaxed dendritis using solidifiation kinetis-maro transport modeling was followed by Nasta and Stefanesu (1996a). The model relaxed some of the assumptions made in previous model by simultaneously relating the internal solid fration to the movement of the dendrite tips and to the dynami oarsening of dendrite arms. It also inorporated the mirosopi kineti model into a marosopi heat flow model by using two different time sales in omputation miro and maro time sales. The validation of the model was performed against experimental data for INCONEL 718 superalloy astings (Nasta and Stefanesu, 1996b). Although the preditions agreed well with experiments in mirosegreation, marosegregation was not predited due to the assumption of no onvetive transport. More reently, Wang and Bekermann (1993a, b and 1994) developed a unified model for both equiaxed and olumnar dendriti solidifiation, based on a multi-phase approah and volume averaging. The multi-phase model was an extension of the twophase model (Ni and Bekermann, 1991), and it onsidered three phases, i.e., solid, interdendriti liquid, and extradendriti liquid phases. The interfaes between the phases have different harateristi length sales, requiring two sets of interfaial speies diffusion length sales. The model was used to ompute one-dimensional mirosegregation and ooling urves without onvetion and was ompared to other models inluding those disussed above. Limited validation of the model was provided

48 30 for vertial solidifiation of an aqueous ammonium hloride solution (Bekermann and Wang, 1996) and there appeared to be good agreement between the measured and predited shapes of sedimented ammonium hloride at the bottom of the enlosure. Although the model was apable of prediting the phenomena better, it introdued many new parameters, whih requires extensive experiments. There were also still unertainties in the modeling, suh as the generation of equiaxed rystals and their growth in the onveting melt, and the model required many additional sub-models. The omputation using the model might have many diffiulties due to the omplexity of the model and might need very large omputing power in some ases like the solute redistribution during re-melting beause the history of the mirosegregation in the solid must be preserved at every node in the omputation domain (Felielli et al, 1993). The diffiulties lay on this miro-maro modeling of alloy solidifiation beause using multisales is a relatively new approah to simulate alloy solidifiation and it needs more development. Despite the diffiulties, the miro/maro solidifiation models demonstrated its ability through the verifiation, and are likely to beome useful tools in simulations.

49 Objetives of This Researh Through the review of existing literature, it is lear that detailed onvetive transport phenomena during solid-liquid phase hange of metal alloys, still need to be improved. Sine the transport of momentum, heat, and speies strongly influene the metallurgial struture, hemial homogeneity, and other properties of final metal produts, improvement in the phase hange proess of metal alloys depends on a detailed understanding of these transport phenomena. The solidifiation proess in the metal alloys is further ompliated by oupling between miro- and maro-sale phenomena, whih is to be examined in the present work. A ombined experimental and theoretial approah is proposed to study the transport phenomena in a binary metal alloy. The experimental study will utilize neutron radiography to determine marosegregation patterns in experimental ingot, and to quantify their severity. The theoretial study will fous on the extension of a ontinuum model and a multi-phase model to aount for stohasti behavior of the mushy region and nonequilibrium effets, suh as underooling and realesene. A numerial ode based on the ontinuum model will be generated and numerial omputations will be ompared to experimental results. The speifi objetives of the researh work are: 1. Design an experimental apparatus suitable for metal solidifiation and neutron radiography. 2. Develop a quantifiation proess of marosegregation patterns, using neutron radiography.

50 32 3. Perform a series of experiments whih quantifiably reveals the evolution of marosegregation during solidifiation of a binary alloy, in order to investigate the transport phenomena. 4. Critially assess proedures for modeling a binary solidifiation system and utilize ontinuum and multi-phase models, with mirosopi phenomena inluded. 5. Perform numerial simulations to demonstrate the nature, validity, and apabilities of the model and ompare the numerial results with experimental data.

51 33 Chapter 2 EXPERIMENTAL APPARATUS AND PROCEDURES 2.1 Introdution The experiments in this investigation foused on obtaining quantitative data assoiated with transport phenomena during the solidifiation of a binary metal alloy. The experimental apparatus whih was instrumented for temperature measurements and designed to aommodate neutron radiography, had been onstruted to study the effets of varying thermal onditions on solidifiation from a vertial surfae in a square avity. Temperature measurements in the alloy would provide the insight of transient phenomena during solidifiation, and stati neutron radiographi images of fully solidified ingots would provide detailed marosegregation measurements. In the following four setions, the seletion riteria for a phase-hanging material, as the working fluid, design of the experimental apparatus, diagnosti tehniques, and experimental proedures, respetively, are disussed.

52 Phase Change Material The most signifiant riteria used for seleting a binary metal alloy were the simpliity of an equilibrium phase diagram (Figure 1.2), a low melting temperature, different onstituent densities, and different onstituent attenuation oeffiients for neutrons. In using neutron radiography, the fourth riterion was ruial for obtaining lear images of marosegregation patterns, whih ould be quantifiably analyzed with reasonable unertainty. The low melting temperature metal alloys used in early experiments, suh as lead-tin (Shahani et al., 1992; Presott and Inropera, 1994) and aluminum-opper systems (Mehrabian et al., 1970; Bridge and Beeh, 1983) satisfied the first three riteria, but not the fourth (Figure 2.1). The attenuation oeffiients of aluminum and opper were of the same order and likewise for lead and tin. Possible alloy systems inluded aluminum-lithium, bismuth-indium, and gallium-indium systems, of whih the gallium-indium (Ga-In) system was seleted beause it melted at temperature near ambient (Figure 2.2). The neutron absorption oeffiient of indium was approximately twenty times that of gallium (Figure 2.1), providing images of neutron radiography with high ontrast. The equilibrium phase diagram for the Ga-In system is shown in Figure 2.2. The system is a simple euteti system without intermetalli ompounds. The euteti omposition and temperature are Ga-21.4 wt. % In and 15.3 C, respetively. As indiated by the phase diagram, when a hypereuteti Ga-In alloy is solidified along a vertial wall, the interdendriti liquid is enrihed with Ga, whih is less dense than In. Hene, solutal

53 Gd Total Cross Setion (m -1 ) B Li C Al Ni Fe Cu Cr Ga Zn Cd In Sn Hg Au Pb Bi Atomi Number Figure 2.1 Neutron attenuation oeffiients of seleted elements.

54 C L Temperature ( C) oC 0.00 (α Ga) C (In) Ga Weight Perent Indium In Figure 2.2 Ga-In equilibrium phase diagram.

55 37 buoyany fores oppose thermal buoyany fores in the mushy zone during solidifiation. Conversely, when a hypoeuteti Ga-In alloy is solidified, the interdendriti liquid is enrihed with In, and solutal and thermal buoyany fores augment eah other. Sine the ondition of opposing thermal and solutal buoyany fores was of partiular interest in this study, a hypereuteti alloy with omposition Ga-27 wt. % In was seleted. The liquidus and solidus temperatures for this alloy were 30 C and 15.3 C, respetively, and hene, temperatures required to melt and freeze the alloy were easily ahieved in the laboratory. The thermophysial properties of Ga and In are well doumented (Weast, 1975; Brandes, 1983), and representative property values are listed in Table 2.1.

56 38 Table 2.1. Physial properties of Ga-27 wt. pt. In Properties Values h f fusion enthalpy (J/kg) T f fusion temperature ( o C) s l k s k l solid speifi heat ( Jkg K) 243 liquid speifi heat ( Jkg K) 398 solid thermal ondutivity ( Wm K) 80 liquid thermal ondutivity ( Wm K) 25.5 µ dynami visosity ( kg m s) ρ density (kg/m 3 ) 6090 D l binary diffusion oeffiient (m 2 /s) β T thermal expansion oeffiient (K -1 ) β S solutal expansion oeffiient K 0 permeability oeffiient (m 2 )

57 Apparatus In order to aomplish the objetives of the proposed experiments, the test apparatus had been onstruted for horizontal solidifiation of a metal alloy in a retangular mold. Design of the test apparatus was based on several fators: 1. It must be apable of solidifying a metal alloy under ontrolled onditions. 2. The ross-setion of the freezing hamber must be similar in size to the neutron beam diameter and imaging system at the Penn State Breazeale Nulear Reator (PSBR). 3. It must to be easily disassembled for neutron radiographi analysis and reassembled for further experiments. 4. It must be amenable to instrumentation without signifiantly affeting the proess. The experimental setup onsisted of a test ell, onstant temperature bath irulators, and a data aquisition system (Figure 2.3). The test ell ontained the metal alloy and onsists of a Plexiglas mold and two opper heat exhangers. Solidifiation was initiated by pumping a oolant from a onstant temperature bath through one of the heat exhangers, while a warm fluid was pumped through the other heat exhanger. The test ell might also be used with the seond heat exhanger inative. The solidifiation test ell is illustrated in Figure 2.4. Copper heat exhangers

58 40 Data aquisition system HP 3852A IBM PC Constant temperature bath irulator Cold Hot Constant temperature bath irulator Test Cell Figure 2.3 Experimental setup.

59 Figure 2.4 Shemati illustration of experimental test ell. 41

60 42 formed the side walls, while all other walls were made of 25.4 mm thik Plexiglas. The test ell had internal dimensions of mm in height and width, and 10 mm in depth, whih was seleted to ensure full overage by the 229 mm diameter neutron beam from the PSBR neutron ollimator and to allow adequate ontrast in radiographs. The test ell was designed so that a Ga-In alloy ould be melted and solidified repeatedly in the mold. Moreover, the design permitted the front and bak walls of the test ell to be removed for the post-experimental, neutron radiographi analysis of the solidified ingot. Rubber gaskets were used to prevent leakage of the molten alloy from the solidifiation hamber. Beause neutrons annot penetrate Plexiglas, the front and bak walls must be removable for neutron radiographi analysis. Therefore, there were no permanent joints and the parts were seured to eah other with soket head srews and stainless steel tierods. A small hole, leading to a disharging port, had been drilled in the bottom mold wall (Figure 2.4b). The top part of the mold had a large opening filled with a plug, whih ould be removed easily to allow the harging the melt into the test ell. The large port in the top wall also provided aess for stirring with a stainless spoon to ensure homogeneity of the alloy prior to an experiment. After the harging or mixing proess, the port was losed with the plug. The heat exhangers were onstruted of 16 mm thik opper plates, with a fluid passage (10 mm wide mm long) milled into it to permit the irulation of fluid from the onstant temperature bath irulators. The heat exhangers were painted to

61 43 prevent orrosion, espeially from the molten alloy. Pure (200 proof) ethanol was used as a oolant in the hill heat exhanger and water is used in the hot heat exhanger. A Plexiglas over plate was seured to hot heat exhanger with soket head srews, and a nylon over plate was used for the old heat exhanger, beause Plexiglas was orroded by ethanol. Inlet and outlet ports drilled into the over plates were onneted to the onstant temperature baths through flexible tubing overed with Styrofoam insulation.

62 Diagnostis. In order to investigate the transient onvetive phenomena during the solidifiation of binary metal alloy, the temperature history was measured in different loations within the test ell. The marosegregation, whih results from onvetion, was measured using neutron radiography. The instrumentation and measurement tehniques used in this study are disussed below Temperature Measurement Several thermoouples were installed in the test ell in order to measure various temperatures during an experimental solidifiation run. The thermoouple voltages were measured and proessed by an HP 3852A data aquisition system. The temperature measurements through the system were olleted by an IBM personal omputer. Temperature measurements within the test ell were made using hromelonstantan (type E) thermoouples, whih were seleted beause of their relatively large temperature range (-200 to 900 C), high sensitivity, and ompatibility with existing HP data aquisition systems. Auxiliary experiments had been performed (Singh, 1994) and the thermoouples had been found to read onsistent with eah other within 0.1 C over the range of temperature in this study. The standard limits of error (Omega Engineering Temperature Handbook) for type E thermoouples was 1.7 C or 0.5% (1%) above

63 45 (below) 0 C (whihever is greater). The time onstant of thermoouples was estimated in the manufaturer s (Omega Engineering, In.) literature to be less than 0.29 seonds. Nine thermoouples were inserted through the front mold wall at loations orresponding to three horizontal arrays at y=2.5, 5.0 and 7.5 m and horizontal positions of x=0.5, 5.0 and 9.5 m (Figure 2.5). The probe tips were positioned lose (~0.5 mm) to the inside mold wall surfae in order to minimize the disturbane on the flow, and loked in their position using Swagelok fittings. Three opper-onstantan (type T) thermoouples were installed in eah opper heat exhanger to measure the opper plate temperature. They were inserted through the over plate and mounted to the outside surfae at the opper plate by thermally ondutive epoxy. They were loated in the same horizontal planes as type E thermoouples. The auray relative to one another was within 0.1 C (Singh, 1994) and the time onstant for the probes was less than 0.27 seonds (Omega Engineering Temperature Handbook). The temperature data were olleted through the data aquisition system whih onsisted of a Hewlett-Pakard 3852A data aquisition/ontrol unit with an HP 22701A integrating voltmeter, two 20-hannel HP 44710A multiplexer aessories, and interfaes to an IBM 80486DX personal omputer. The HP system had the apability to measure

64 Figure 2.5 Loations of thermoouple probes in the test ell. 46

65 47 voltages integrated over one power line yle, and thus, it had exellent normal mode noise rejetion. The inauray involved in the onversion algorithm of the HP data aquisition system had been measured for a temperature range of -50 to 300 C, and the error had been found to be within ±0.06 C (Presott, 1992). The software for temperature measurements was written in HP BASIC programming language Marosegregation Measurements using Neutron Radiography Marosegregation measurements of a fully solidified ingot was made by using neutron radiography, whih is a method of non-destrutive testing and proven to be a useful tool for visualizing flow within metal housings (Cimbala et al., 1988). Compared to other radiographi tehniques, neutron radiography an provide greater ontrast in images as the omposition of a subjet hanges, whih is very important for quantifying marosegregation. The fundamentals of neutron radiography and their appliation to marosegregation measurement are desribed below. The basi operating priniple of neutron radiography is illustrated in Figure 2.6 (a). A ollimated beam of thermal neutrons with intensity I o is inident upon an objet whih is semi-transparent to the neutron beam. Hene, a neutron beam with intensity less than I o (I< I o ) exits the objet. The intensity of the exiting neutron beam is determined from the following relation. II 0 = exp( Σ t x) (2.4)

66 48 Neutron Beam Objet I Io Io (a) Film Neutron Beam Objet X-ray I Gadolinium I o I o (b) Figure 2.6 Illustration of neutron radiography method: (a) neutron beam attenuation and (b) film assette arrangement.

67 49 where Σ t is the total (absorption plus sattering) marosopi apture ross-setion of the objet material for thermal neutrons and x is the thikness of the objet. If the objet in Figure 2.6 (a) is nonuniform in thikness or apture ross-setion (perpendiular to the neutron beam path), the intensity of the exiting neutron beam will be non-uniform, and an image from a nonuniform beam an be reorded on X-ray film, as shown in Figure 2.6 (b). The film and a gadolinium foil are plaed in an aluminum assette behind the test speimen. The nonuniform neutron beam passes through the aluminum housing and film, with virtually no effet, and is absorbed by the Gd foil, whih emits eletrons in proportion to the flux of the neutron beam. The emission of eletrons is responsible for exposing the X-ray film. The density of a developed film D is a linear funtion of the total neutron exposure Φt, where Φ is neutron flux and t is exposure time. Hene, D = Φ I = exp( Σ D o Φ o I t x) (2.5) o When a neutron radiograph is obtained from an ingot of uniform thikness, equation (2.5) an be used to determine variations in Σ t from measured variations in film density. Sine Σ t depends on the omposition of the alloy (integrated through the thikness of the ingot), Σt = g In( Σt, In Σt, Ga) + Σt, Ga (2.6) the film density measurements an be used to determine volume fration indium, g In In order to obtain highly resolved marosegregation patterns, the X-ray film image was digitally proessed with a 30 bit Relisys olor sanner (RELI4830T), whih

68 50 measured gray sale values between 0 and 255 in inrements of The relationship between the film density and the gray sale value measured by the sanner was found using Kodak projetion print sale film, whih had 16 gray sales. The density of eah seleted gray area was measured using a densitometer. Densitometer measurements are plotted against sanned image density in Figure 2.7, whih also ontains a alibration urve. The alibration equation was, D = log 1 SD 255 (2.7) where SD is the density of the sanned image, and its unertainty is ±6.3 perent. It should be noted that the alibration equation (2.7) only applies for the partiular sanner used. The aforementioned method of reording neutron radiographs yielded distint images of marosegregation patterns on X-ray film. Ideally, equations (2.4)-(2.7) ould be used to quantify the distribution of alloy onstituents from the digitized image. However, the raw image was tainted by gamma rays whih affet the film exposure. Gamma ray noise was present in the neutron beam from the ollimator, and it was also emitted by enrihed indium in the test speimen. In order to aount for this unwanted signal, eah radiographed speimen was exposed twie, and the seond exposure was reorded without the Gd onverter foil. Thus, the seond exposure, whih was idential in neutron flux and exposure time as the first, yields an image of the unwanted gamma ray noise, whih ould be digitized and subtrated from the digitized image of the first exposure. The orreted image was then analyzed using equations (2.4)-(2.7).

69 D= log(1-sd/255), R= Sanned Image Density, SD Figure 2.7 Calibration urve for the digital sanner. Film Density, D

70 52 An additional auxiliary experiment was performed to asertain the relationship between film density and alloy omposition. Sine sattering usually has the effet of reduing the effetive Σ t (Berger et al., 1985), the use of referene values for Σ t,ga and Σ t,in (i.e. from Figure 2.2) in equation (2.6), would introdue an error in the alulation of the volume fration of indium. The auxiliary experiment was onduted with a small test ell made of opper and with internal dimensions of mm 3 and wall thikness of 1.6 mm. The test ell was used to obtain neutron radiographs of Ga-In samples over a range of ompositions. Initially, it ontained Ga-27 wt. pt. In alloy, and the alloy was gradually diluted with pure gallium (99.99%). Figure 2.8 shows the measured relationship between volume fration of alloy and film density. The density of the image was determined by a densitometer and the volume fration was estimated from measurements of the sample mass. The differene between the atual indium and gallium total ross setions (Σ t,in -Σ t,ga ) was found to be ± m -1, whih is 36.5 % smaller than that determined from the referene value of m -1. It was also found that a great amount of unertainty ame from the film proessing, partiularly from the freshness of the developer. As the developer ages, it tends to produe denser (darker) films. In order to minimize this unertainty, the neutron radiograph of this small test ell whih ontains Ga-27 wt. % In was taken at the same time with other radiographs and all measured image densities were ompared with the density of the small test ell. The relationship between volume fration and film density ould be expressed as:

71 D/Dst= exp( gin), gin), R= R= Volume Fration of In Figure 2.8 Relationship between alloy omposition (volume fration) and film density. D/Dst

72 54 g In = g st 1 ( Σ Σ ) In Ga st D D ln = gst ln (2.8) x D D st where x is the thikness of the test ell (10 mm). The variation of the volume fration due to the unertainty in the film density was ± Using Equations (2.7) and (2.8), the omposition of the ingot was found from the image on X-ray film. It should be noted that the expeted range of volume fration of indium was between 0.21 and 0.31 and the measured range was between 0.11 and Thus, the alibration equation (2.8) would be extrapolated to the expeted range of the volume fration of indium and it might introdue larger unertainty in resulting data of marosegregation

73 Experimental Proedures The experimental proedure began with heating the Ga-In alloy to approximately 60 C by immersing the bottle ontaining the alloy in a hot water bath. The bottle was agitated to ensure that all solid was dissolved and the omposition beomes uniform throughout. The alloy was harged into the test ell through a funnel while one of the heat exhangers was heated by irulating hot water (about 50 C) from a onstant temperature bath. In the ase of repetition of the experiment, the test ell whih ontained the fully solidified ingot, was heated by irulating hot water, whose temperature was maintained at 50 C, for about 7-8 hours and the alloy was stirred frequently to ensure uniform omposition. During this operation, the data aquisition system sanned all thermoouple hannels and reported temperatures on the display unit. When all the thermoouples attained a onstant temperature, the irulation of hot water was stopped, and the liquid alloy was allowed to ool slowly while the onstant temperature baths were prepared at temperature of 35 and -15 C. The temperature variations in the melt dereased as it ooled down and eventually the melt beame isothermal to within 0.5 C when the melt approahed the intended initial temperature (35 C). Solidifiation is initiated by ativating the flows from the irulators through the heat exhangers. Simultaneously, the data aquisition system is triggered to start olleting and storing thermoouple readings. After a ertain time, whih would be a ontrolled parameter in the experiments, the irulation of hot water was stopped to fully solidify the ingot. Data was olleted for twelve hours for eah ase or until the solidifiation is ompleted.

74 56 The desired thermal onditions in the test ell were the onstant temperatures of oolant above the temperature of liquidus line for one side and under the euteti temperature for the other, in order to ensure the alloy near the old wall to be fully solid and near the hot wall to be fully melted. These onditions leaded the extension of solidifiation time and helped the observation of the phenomena during the solidifiation in details. The set temperature was determined, based on not only the phase diagram of Ga-In alloy (Figure 2.2) but also the temperature measurements during trial solidifiation experiments due to the underooling phenomena of gallium phase. It turned out that the atual temperature of oolant in the old wall must be lower than -13 C to initiate the euteti solidifiation. Thus, the temperature of the irulator for the old wall was set to -15 C. The temperature of the other side wall just needed to be above the liquidus temperature at the initial omposition beause the melt near the hot wall was expeted to be gallium-rih and onsequently, the liquidus temperature was lower than one of the initial omposition. Therefore, the temperature of the irulator for the hot wall was set to 35 C, whih was muh higher than 30 C to be on the safe side. Plexiglas, a hydrogen-arbon-oxygen-based ompound has a very high absorption oeffiient for the neutron. Therefore, after the ingot was ompletely solidified, the test ell was prepared for neutron radiography by replaing the front and bak panels with thin (1.59 mm) opper plates, whih were virtually transparent to the neutron beam. The test ell was exposed for 40 minutes to a uniform neutron beam produed by PSBR operating at 750 kw. Sine the ingot was muh older than the surrounding, it was found that some ondensation ould appear on the surfaes of the opper plates and ould

75 57 distort the measurement. Therefore, the surrounding air of the ingot was hilled by blowing old vapor of dry ie through a fan while the ingot was exposed to the neutron beam. Neutron radiograph images of the ingot were reorded on Kodak SR X-ray film, and as mentioned previously, two radiographs were taken with and without the gadolinium foil, in order to aount for gamma ray noise. In eah exposure, the small test ell, whih ontained a Ga-27 wt. pt. In sample, was inluded as a referene. The image on X-ray film was arefully developed and proessed by the digital sanner with a resolution of 200 dpi. The densities of the sanned image were alibrated and the volume frations in the image were alulated using Equations (2.7) and (2.8). The proedures desribed above were repeated for several ases with different durations of hot water irulation. In the first run, the hot and old temperatures on the side walls were maintained until the temperatures in the alloy reah a quasi-steady state, after whih the alloy was fully solidified by stopping the irulation of hot water. After analyzing the temperature measurements, additional runs were planned. The additional runs were interrupted (i.e., heating terminated) at various times before reahing a quasisteady state, for the purpose of olleting marosegregation data from neutron radiography at various stages of the solidifiation proess. Thus, the evolution of the final marosegregation proess was well resolved. The results of the experiments were supposed to provide a sequene of solidifiation stages and reveal quantitative insights to the proess. They would also show the formation of solid phase and movement of the interfaes. Furthermore, the

76 58 interpretation of temperature measurements during the solidifiation would be enhaned and justified by the neutron radiography measurements of these experiments.

77 59 Chapter 3 MATHEMATICAL MODELING 3.1 Introdution Modeling of transport phenomena during solidifiation of a binary alloy is a hallenging problem due to the ompliated phenomena ourring over different length sales. In the past several deades, the modeling on a single sale, marosopi or mirosopi levels, has been very sophistiated individually and generated suessful results. Reent interest in solidifiation modeling fouses on the oupling of mirosopi and marosopi models (Rappaz, 1989; Ni and Bekermann, 1991; Wang and Bekermann, 1992; Tangthieng, 2002). The urrent study follows the steps of the reent researhes and this hapter onerns with inorporating models of fundamental mirosopi phenomena, suh as nuleation, underooling, and grain growth, into models of marosopi transport phenomena. However, numerial simulations using the newly developed models were not performed due to being able to resolve the instability in handling the step hange of some equations. Therefore, the following mathematial model is a suggestion or guideline to the future researh in alloy solidifiation. In the numerial simulation, the onventional ontinuum model was used and it is desribed in Chapter 4.

78 60 Figure 3.1 shows a small volume element whih ontains several dendriti strutures ourring during the solidifiation. In olumnar growth the solid is attahed to a ooled wall and the speed of the solid front is onstrained by the movement of the isotherms. In equiaxed growth, on the other hand, the rystals grow radially into an underooled melt and the latent heat is removed through the liquid melt. As disussed in Setion 1.2, there are two distint interfaial sales in the volume element (Wang and Bekermann, 1992): one is the sale of the interfae between the solid and interdendriti liquid phases, and the other is the larger sale assoiated with the interfae between the extradendriti and interdendriti liquid phases. The size of the volume element is hosen suh that it is larger than all interfaial length sales, but small ompared to the overall asting system. The representative physial property of the volume element an be found by properly averaging the property belonging to the mirosopi sale over the volume element. However, sine there are two different length sales, averaging a property over the volume assoiated with a large sale, requires knowledge of the property averaged over a volume of the smaller sale. Hene, in order to find a mirosopi property for the phase in the smaller sale, the mirosopi property must be spatially averaged suessively over two averaging volumes of different sizes. This tehnique is known as dual-sale averaging and was developed by Wang and Bekermann (1993). As mentioned above, the volume element in Figure 3.1 an be onsidered to onsist of three different phases: the solid phase and two liquid phases. The two liquid phases are haraterized by different length sales and thus transport phenomena in eah

79 61 Equiaxed dendrite Liquid melt Columnar dendrite Figure 3.1 Shemati illustration of the averaging volume.

80 62 phase are onsidered separately. Therefore, the marosopi transport equations are formulated for three phases and the mirosopi transport equations are linked through interfaial transfer terms in the marosopi equations. Details of derivation for marosopi equations using the multiphase approah, are well doumented (Bekermann and Wang, 1996) and for ompleteness, the formulation is reviewed in this hapter. In the development of models, the marosopi onservation equations are developed in the form of an existing mixture model (Bennon and Inropera, 1987a) by summing the marosopi equations for eah phases, beause the formulation is amenable to standard, single-phase numerial solution proedures. A similar approah was implemented by Ni and Bekermann (1995a) for two phase models.

81 Marosopi Conservation Equations General Conservation in a Single Phase of Fluid Transport phenomena ourring during solidifiation of alloys an be desribed by using a set of mirosopi onservation equations. The derivation of equations are readily found in many textbooks (Bird et al., 1960) and the equations an be desribed in terms of a general variable, φ, as the following. For a flux J of a property φ per unit volume of a fluid element of volume V moving at veloity U, the net flow aross a losed surfae is given by J n da. The A time rate of hange of the property φ in this elemental volume is ( ddt) φ dv. The rate V of generation of this property per volume is denoted as S, and within the volume is SdV V. Sine the hange within the volume is a onsequene of net flow aross the surfae and the generation, d φ dv S dv dt φ dv = φ V t da da V + U n = A J n + A (3.1) V Applying the Gauss divergene theorem gives, φ + φ + SdV V U J t = 0 (3.2) Sine the volume element is arbitrary, the general onservation equation is given as

82 64 φ t + φu + J S = 0 (3.3) The onservation of mass is given by setting φ=ρ k, J=0, and S=0: ρ t k ( ρ ) + U = 0 (3.4) k k where ρ k is the density of phase k, U k is the veloity. The onservation of momentum is given by φ=ρ k U k, J=P k I-τ k, and S=b k : ρku t k ( ρ U U ) + = P + τ + b (3.5) k k k k k k where P k is the stati pressure, τ k is the visous stress tensor and b k is the body fore per unit volume ating on phase k. The energy onservation is given by φ=ρ k h k, J=q k, and S=E k : ρkh t k ( ρ h ) + U = q + E (3.6) k k k k k where h k is the enthalpy per unit mass, q k is the heat flux vetor and E k is the heat soure per unit volume inside phase k. The speies onservation is given by φ=ρ k f α k, J= ρ α α D f, and S= M & k : k k k ρkf t α k α α α ( ρ ) ( ρ ) + U f = D f + M& (3.7) k k k k k k k where f k α is the α speies mass fration, D k α is the mass diffusion oeffiient for speies α and & M k is the prodution of speies per unit volume inside phase k.

83 Volume Averaging Tehnique Due to omplex interfaial struture in the solidifiation alloys, detailed solutions in mirosopi sale are not feasible. A realisti approah is to express the essential physial quantities of the system in terms of averages. The so-alled volume averaging tehnique has been popular for deriving marosopi onservation equations from mirosopi equations (Hassanizadeh and Gray, 1979 a, b; Drew, 1983) in multiphase systems. The volume averaging tehnique shows how the various terms in the marosopi equations arise and how the marosopi variables are related to the orresponding mirosopi variables. The size of averaging volume must properly represent the physial size of the phase. In alloy solidifiation, it is adequate to average the mirosopi quantities over two averaging volumes of different sizes as mentioned in the previous setion. If the smaller averaging volume is assumed to be spatially independent inside the large volume, the averaging theorem is idential to the onventional volume averaging method. Therefore, the onventional volume averaging method will be implemented for eah phase. This setion provides a brief desription of the tehnique, while the details of the tehnique have been well doumented (Drew, 1983; Soo, 1989). Consider phase k whih oupies a variable volume V k with the interfaial area A k in an averaging volume, V o as shown in Figure 3.1. The volume average of a quantity ψ k is defined by an extensive average: ψ 1 = V x ψ dv k 0 V k k o (3.8)

84 66 where x k is a phase funtion, equal to unity in phase k and zero elsewhere. When the quantity is averaged over V k, ψ k k = 1 V x ψ dv k V k k o (3.9) is alled an intrinsi average. The two averages are related by volume fration of phase k, i.e., g k Vk = (3.10) V o and ψ = g ψ (3.11) k k k The averaging theorems of derivatives (Whitaker, 1969; Slattery, 1967), as applied to the system in Figure 3.1 are summarized as k ψ ψ k k 1 = ψ k k kda t t V w n (3.12) A k 1 ψ k = ψk + V o o A k ψ n da (3.13) k k 1 ψ k = ψ k + ψ k n A kda (3.14) V k where w k is the veloity of the interfae and n k is the outwardly direted unit normal vetor on the interfae. Appliation of the volume averaging theorems given by equations (3.12) - (3.14) to the onservation equations (3.4) - (3.7) results in the volume averaged marosopi onservation equations. For onveniene of writing equations, the intrinsi o

85 67 volume-averaged quantity will be used without the brakets in the following setions, suh as ρ k for <ρ k > k, v k for <v k > k, h k for <h k > k and so on Conservation of Mass The volume averaged mass onservation equation is given by averaging the ontinuity equation (3.4): t ( g ρ ) ( g ρ v ) + = k k k k k kj jj, k Γ (3.15) where Γ kj denotes the rate of total mass generation of phase k at k-j interfae due to phase hange, and an be modeled as the produt of the interfaial area onentration, Skj = Akj Vo and a mean interfaial flux, using the mean value theorem for the integral. It is given by Γ kj 1 = k( k k) A kda = Skj k w nkj V ρ v w n ρ (3.16) kj o where w nkj is defined as the average interfae veloity, relative to the averaged veloity of the phase. The volume averaged mixture mass onservation equation an be obtained by summing individual onservation equations with the interfaial mass balane ( Γ + Γ = 0 ) as follows: kj jk ρ t ( ρ ) + v = 0 (3.17) where the mixture density and veloity are defined as

86 68 ρ = g k ρ k k v = f k v k k (3.18) (3.19) and the mass fration, f k is defined as f k = g k ρ ρ k (3.20) Conservation of Momentum The averaged momentum onservation equation is given by averaging equation (3.5): t ( ρ ) ( ρ ) ( ) g v + g v v = g p + + M + g bk (3.21) k k k k k k k k k τ k kj jj, k where the term M kj is the interfaial momentum transfer at k-j interfae, whih onsists of the interfaial momentum transfer rates due to phase hange and interfaial stress, i.e., k Γ M = M + M τ kj kj kj (3.22) The interfaial transfer rate due to phase hange is given by 1 M Γ kj = kvk( vk wk) n A kda V ρ (3.23) kj o Using the mean value theorem for the integral, the term is modeled as M Γ kj v kj = k( vk wk) nkda v A kjγkj V ρ = (3.24) kj o

87 69 where v kj is the average interfaial veloity over the interfaial area, A kj. In many solidifiation systems, the volume hange upon phase hange is relatively small, so that the interfaial momentum transfer due to phase hange may be negleted, ompared to one due to dissipative interfaial stress. Therefore, the term will not appear in the final equation. The interfaial momentum transfer due to the interfaial stress is given as τ 1 d Mkj = ( k pki) nkda = p M A kj gk + k V τ (3.25) kj o d where M kj is the dissipative part of the interfaial stress and p kj is the average interfaial pressure, whih is equal to the intrinsi average pressure of phase k, p k beause of the d instantaneous mirosopi pressure equilibration. The term M kj ontains the dissipative interfaial momentum transfer due to visous and form drag and its model is disussed later. Assuming Newtonian-fluid, the marosopi visous stress an be written as 1 * µ ( g v ) vknkda = µ ( g v ) v g A kj * τ k = k k k + jk, jvo { g v ( v v ) g } = µ + * k k k k k k { } k k k k k (3.26) where µ k * is the marosopi visosity, whih may be different from the mirosopi visosity, and v k is the average interfaial veloity over the entire interfae area. The seond term on the right-hand side of the last form is non-zero only in the mushy zone, and sine the gradient of the term is negligible ompared to the total dissipative interfaial momentum transfer in equation (3.25), the term will be ignored. Substituting equations (3.26) into (3.21) and onsidering the previous disussion, the momentum onservation equation beomes

88 t * ( ρ ) ( ρ ) ( µ ) d gk kvk + gk kvkvk = gk pk + kgk vk + Mkj + g b (3.27) k jj, k Summing the individual phase momentum equation and imposing Newton s third law ( M d kj + M d jk =0) and b k =ρ k g, a mixture momentum equation an be derived as k 70 ( ) ( ) ρ ρ g p t µ * v + vv = k k + g v k k k k k gkρk( vk v)( vk v) + ρg k (3.28) In order to simply the third term on the right-hand side of equation (3.28), a flow partition oeffiient κ v (Wang et al., 1995), whih is defined as the ratio of the liquid mass flux through the porous dendrites to the total liquid, is introdued and then, the relative portions of the flow in the interdendriti (d) and extradendriti (l) regions an be quantified by and g ( v v ) g ( v v ) ρ = κ ρ (3.29) d d d s v f f f s ( v v ) = ( ) g ( v v ) g ρ 1 κ ρ (3.30) l l l s v f f f s where g f and v f denote the total liquid fration and mixture veloity for both liquid phases, respetively, i.e., g f = g d + g l and g fρfvf = g dρdvd + g lρlvl. Note that if κ v = ρ d g d /ρ f g f, the veloities of inter- and extra-dendriti region beome equal. Its orrelation has been disussed in Wang et al.(1995). Using equations (3.29) and (3.30), the following expression is obtained. k= d, l g ρ ( v v)( v v) = γρ g ( v v )( v v ) k k k k f f f s f s (3.31)

89 71 where γ is given by γ 2 κ v = 1 ρ fg f + ρdgd ( 1 κ ) ρ g l l v 2 (3.32) Using equation (3.31) and ( ) of equation (3.28) an be rewritten as v = v v f + v, the third term on the right-hand side f s f s ρ gkρk( vk v)( vk v) = ( fs γ )( v vs)( v v s) (3.33) f k This term is non-zero only within the mushy zone and it represents a measure of the differene between the total net advetion of momentum from all phases individually and the advetion represented by the mixture parameters on the left-hand side of equation (3.28). Assuming loal mehanial equilibrium between inter- and extra- dendriti liquids, i.e., p d = p l = p f and that thermophysial properties are the same for both liquid phases, equation (3.28) beomes f ( ) ( ) ρ v + ρ vv = g ρ f pf gs ps + g t ρ ( f γ )( )( ) µ ρ * s v vs v vs + f v fl ρf ρ µ µ ρ fs ρfs + f v f vs µ fv ρl ρf * * * : s: ρl + µ g g v v v s 2 * f * f v f d f s s s gg d l g f l ( κ g g )( ): + ( g µ ) (3.34)

90 72 The solid marosopi visous stress has been expressed in terms of the derivative of the solid veloity and an effetive solid visosity, µ s *, whih, from the rheology of suspensions, an be modeled as (Ni and Bekermann, 1993) µ * s = ( 1 g ) µ µ g s s * f = 25. g sp g s 1 g g 1+ sp g s s µ * f (3.35) where g sp is the maximum paking fration of the solid phase. If the phase densities are assumed to be quasi-equivalent, i.e., ρf ρs 1, the sixth term on the right-hand side of equation (3.34) vanishes. The ninth term is removed beause usually κ v is lose to ρ d g d /ρ f g f (Wang et al., 1995). The fourth and eighth terms are non-zero in the two-phase region, where they are negligible in omparison to the drag interation term whih is impliitly expressed in the solid pressure gradient term (equation (3.21) for solid phase). Sine in dispersed flow whih ours in the beginning of the solidifiation, the volume fration of the solid phase is low ( g s 0), the effetive solid visosity is approximately equal to the liquid visosity and the seventh and tenth terms anel eah other for onstant density. For a stationary solid struture, the solid phase an be assumed to be 2 rigid ( = v s 0 ), and thus, the seventh and tenth terms beome negligible. For intermediate states, the roles of two terms are unknown, and they are assumed to approximately balane eah other. Therefore, the equation is redued to the following form. ( ) ( ) ( ) ( ) ρ v + t ρ vv = p g p p ρ µ f + s f s + g + * f v (3.36)

91 73 If the liquid density varies with temperature and/or speies onentration, the Boussinesq approximation an be adopted to express the buoyany fore in terms of thermal and solutal expansion oeffiients. The third term on the right-hand of equation α α (3.36) then redues to gsρs + glρl[ + βt T Tref + β ff ff ref ] 1 ( ) ( ), where β t and β are the thermal and solutal expansion and T and f f α are the loal liquid temperature and speies onentration. For a dispersed flow with a moving equiaxed rystal, the pressures of solid and liquid may be assumed to be equal to the equilibrium pressure (Drew, 1983), in whih ase, the seond term on the right side of equation (3.36) is zero. When the solid phase is stationary, the momentum equation for the solid phase with zero veloity is redued to d g p = M g ρ g (3.37) s s s The pressure within the solid phase is balaned with the interfaial momentum transfer and gravity potential. Substituting equation (3.37) into (3.36) and using D Ary s law for the momentum transfer due to interfaial interations between the liquid and solid, equation (3.36) beomes idential to the model developed by Presott et al. (1991). The transition of the solid pressure from a dispersed system to stationary solid is a ritial issue. To aommodate the transition, a onstitutive model for the phase pressure differene is introdued in the form of Bernoulli s equation (Givler and Mikatarian, 1987; Roo, 1993) as follows: 1 p s ps pf = f ( gs)( f s) ( f s) 2 ρ f v v v v (3.38) s s

92 74 where f(g s ) is a funtion of solid volume fration, whih aounts for the interations among the mirosopi solid phases in the volume element. Equation (3.38) proposes that the solid phase pressure exeeds the liquid pressure by an inertial orretion. In order to determine the preise form for the fator f(g s ), the behavior of marosopi suspension of fine partiles in the liquid needs to be studied. The behavior of the funtion and the model used in this study will be disussed in the later setion. It should be noted that in order to use equation (3.38), the veloity of liquid or solid phases must be known. Therefore, in addition to the mixture momentum equation, the momentum equation for solid phase must be solved, simultaneously, or the relationship between the veloities in the solid and liquid phases must be assumed Conservation of Energy The averaged energy onservation equation in terms of enthalpy is given by t ( ρ ) ( ρ v ) g h + g h = q + Q + g E k k k k k k k k kj j, j k k k (3.39) where q k is the marosopi heat flux whih an be desribed by Fourier s law, i.e., q k k k T k = and Q kj is the interfaial energy transfer due to phase hange and interfaial heat flux at k-j interfae. Assuming that all phases are in loal, thermal equilibrium with eah other (T k =T for any k) and defining the following mixture variables, h = f h (3.40) k k k

93 75 k = g k (3.41) k k k E = g E (3.42) k k k it follows that ( h) ρ t [ s s l s ] ( ρ ) ( ) ρ( )( ) + Vh = k T f V V h h + E (3.43) The temperature term an be replaed by the enthalpy from equation (3.43), using the following relationship, T = h k = h + k k k k ( h h) (3.44) where the subsript k represents an arbitrary phase with onstant speifi heat. Therefore, energy equation (3.43) beomes where there is no heat soure, ( h) ρ t k + = k s * ( ρ ) h + ( h h) vh * * s [ fsρ( v vs)( hl hs) ] s (3.45) where * s represent a onstant speifi heat of solid phase. The marosopi thermal ondutivity, k k may be different from the mirosopi property (Soo, 1989; Ni and Bekermann, 1991). In general, it depends on the volume fration of phase k, and mirosopi morphologial features of the interfae as well as the moleular thermal ondutivity. However, sine the relation between two is unknown, the marosopi thermal ondutivity is assumed to be equal to its mirosopi ounterpart.

94 Conservation of Speies The averaged speies onservation equation is given by averaging equation (3.7), t ( ρ α ) ( ρ v α ) ( ρ α α ) g f + g f = g D f + J k k k k k k k k k k k kj j, j k (3.46) where D k α is the marosopi speies diffusivity of phase k and is assumed to be same with the mirosopi ounterpart like marosopi thermal ondutivity. J kj is the interfaial speies flux at the interfae, whih ontains two terms, i.e., Γ J = J + J d kj kj kj (3.47) where J Γ kj is the interfaial speies transfer rate due to phase hange and J d kj is the interfaial speies flux. The interfaial speies transfer rate due to phase hange an be modeled in a manner similar to that used in the momentum equations and is given by J Γ kj 1 = kfk ( k k) kda f V ρ α v w n = α A kjγkj (3.48) kj o where f α kj is the average speies onentration over the interfaial area. The interfaial speies flux is given by J d kj 1 α α = kdk fk kda V ρ n (3.49) A kj o The integral an be modeled as J α α ( kj kj ) S ρ D = f f l d kj k k kj kj (3.50)

95 77 where f α kj is the average speies onentration at the interfae k-j and l kj is the diffusion length, whih is illustrated in Figure 3.2. In general, the length an be obtained by performing a mirosopi analysis on the sale of the averaging volume. Summing the individual phase speies equation, a mixture speies equation an be obtained in a manner similar to the development of the momentum equation. where ( ) ( ) ( ) ( ) ρ α ρ α ρ α ρ α α f + vf = D f + t kgkdk fk f k fs α α - ρ ( f fs )( v vs) ff (3.51) ρfg f α α ( ρdgd κvρfgf)( ff fd )( vf vs) ρlg l f α D α = f f (3.52) k k k = f D α (3.53) k k s Sine κ v is usually lose to ρ d g d /ρ f g f as mentioned before, the fourth term on the righthand side of equation (3.51) is negligible ompared to other terms. If the marosopi diffusion rate of speies through the solid phase is assumed to be negligible, the following mixture equation an be obtained. ρ ρ ρ t ρ f f α α α s α α ( f ) + ( vf ) = ( D f ) - ( f f )( v v ) f α α α α [ ρdf( fd ( fd f ) fl ( fl f ))] + + s s (3.54)

96 Figure 3.2 The speies diffusion lengths. 78

97 Supplemented Relationships for Closure Equations (3.17), (3.36), (3.45), and (3.54) are the governing equations for mixture variables, v, p l, h, and f α. These equations are mutually oupled to eah other for buoyany driven flow onditions, where temperature and liquid onentration gradients influene momentum through the prodution of vortiity. However, the equations inlude other variables and parameters besides the mixture variables, whih must be desribed for losure. This setion summarizes supplementary relationships, inluding additional onservation equations, thermodynami and other neessary desription for the additional parameters Conservation Equations for Closure In the previous setion, the onservation equations were developed in terms of mixture variables. However, individual phase properties were not totally removed from the final equations. Therefore, these properties must be evaluated through some other equations, and some of them require the onservation equations of their phases, although additional differential equations are not desired. The phase variable whih most frequently appears in the governing equations is the veloity of solid phase, v s. The veloity an be evaluated by solving the differential equation (3.27) for the solid phase; i.e.,

98 80 t * ( ) ( ) ( ) d g ρ v + g ρ v v = g p + µ g v + M + g ρ g s s s s s s s s s s s s sd s s (3.55) It is assumed that the solid phase interats only with the interdendriti liquid phase. In the ase of olumnar growth from a wall, the solid veloity an be assumed to be zero without the use of equation (3.55). However, in the ase of equiaxed growth, the veloity is expeted to be lose to the veloity of the liquid phase when the volume fration of the solid is very low (g s ~ 0), and it must approah zero when its volume fration approahes that of paking, beause of its high ollision frequeny between the mirosopi grains in the volume element, i.e., high solid visosity, whih an be found from equation (3.35). Between these limiting ases, there is no known simple relationship between the solid veloity and the mixture veloity, and hene, equation (3.55) should be used to determine v s. There are three speies onentration variables, besides the mixture onentration, appearing in equation (3.54), and only two of them are independent of mixture onentration. If it is assumed that interdendriti speies onentration an be desribed by a simple thermodynami relationship, whih is explained in the following setion, only one more speies onentration needs to be determined, for whih equation (3.46) is used. The speies onentration of extra-dendriti liquid phase is hosen here beause the parameters at the interfae, suh as the interfaial veloity are well defined. The equation is given by t ( ) ( v ) ( ) α α α α α g ρ f + g ρ f = g ρ D f + ρ f S w l l l l l l l l l l l l ld ld nld ρld l + S ld f ld f l ld α α ( ) l (3.56)

99 81 The last variable requiring a onservation equation is the volume fration of eah phase. Sine the volume fration of solid phase an be found from that of grains (gg = gs + gd) by a simple thermodynami relationship, only the volume fration of extra-dendriti liquid phase needs to alulated from equation (3.15), i.e., t ( ) ( ) g ρ + g ρ v = S ρ w (3.57) l l l l l ld l nld Without the use of equation (3.57), the advetion of grains and their stereologial properties ould not be onsidered Thermodynami Relationships and Mirosegregation If a binary alloy is in a state of thermodynami equilibrium, the equilibrium phase diagram (Figure 2.2) an be used to determine temperature, mass fration solid, and individual phase onentrations from the enthalpy and onentration of the mixture. However, it is well known that the assumption of loal equilibrium is valid only at the interfae between solid and liquid phases, that is, non-equilibrium onditions exist in the interdendriti liquid. This setion ontains a review of the thermodynami relationships and the treatment of non-equilibrium states in multiphase mixtures, using mirosegregation models. While a non-equilibrium state (referring to underooling) is present in the extradendriti liquid, omplete solute mixing in the interdendriti liquid is assumed for simpliity ( f α ds α = f ), as shown in Figure 3.3, and justified on the basis of a relatively d small length sale assoiated with transport in the interdendriti region. Therefore, the

100 82 onentration in the interdendriti liquid is equal to the equilibrium liquid onentration at the solid-liquid interfae, and it is related to the loal temperature by the liquidus line of the phase diagram. Furthermore, sine the speies diffusion oeffiient of the solid is assumed to be zero, the fration of the solid an be alulated from the onventional model of the mirosegregation known as Sheil equation, whih is, f f α d α g = 1 ( ) ( κ 1 f ) s 1 (3.58) where κ = f f α sd α d (3.59) ( ) f + f f = f f + f f α α α d s g d d s s (3.60) f d α α and f s are the onentrations defined, respetively, by the liquidus and solidus lines for the loal temperature on the equilibrium phase diagram and the temperature is a funtion of the enthalpy of the phase. Equation (3.58) an be solved for the solid fration, whih is given by f s f = 1 f α d α g 1 κ 1 (3.61)

101 Figure 3.3 The simplified onentration profile. 83

102 84 From equation (3.40), the mixture enthalpy is desribed by the enthalpies of the total liquid (h f ) and solid phase (h s ), i.e., ( ) h = h + f h h (3.62) f s s f The enthalpies are funtions of temperature and their respetive phase ompositions and an be written as h s = * T (3.63) s * h = T + h o l l l (3.64) where * s and * l are the averaged speifi heats of solid and liquid phases, respetively. h o l is the referene enthalpy for the liquid phase, whih aounts for the fusion enthalpy, as well as differene between the phase speifi heats. Substituting equations (3.63) and (3.64) into (3.62), the temperature is given by T = o h + fshl (3.65) * * * + f ( ) s s s l The temperature allows the onentration of the solid phase at the interfae and interdendriti liquid phase to be found from the equilibrium phase diagram. Assuming the liquidus line to linear, the onentration of interdendriti liquid is given by f α d = T Tf m l (3.66) where m l is the slope of the liquidus and T f is the virtual melting temperature of pure α speies. Therefore, using equations (3.60), (3.65), (3.66) and the following expression, ( ) ( ) α α α α α f = f f 1 f + f (3.67) g l l l

103 85 the mass fration of the solid phase is alulated. Note that if the temperature is lower than the euteti temperature, the speies onentration of interdendriti liquid is assumed to be the euteti onentration, instead using of that given by equation (3.66) The Pressure in the Solid Phase It is quite diffiult to interpret the pressure of a solid phase. In the flow of a mixture of solid grains and fluid, the solid grains have a variety of roles in transmitting momentum by diret ontat with fluid and ollisions between the grains or a grain and a wall. Therefore, the kinematial analysis of the solid phase may be neessary to desribe the solid pressure orretly. A model for a kinematial pressure of the solid phase, p was proposed (Givler and Mikatarian, 1987) and it was given by (3.38) whih is rewritten here for ompletion: 1 p s ps pf = f ( gs)( f s) ( f s) 2 ρ f v v v v (3.68) The funtion f(g s ) aounts for the inreased interations among the mirosopi solid grains as the onentration of the solid inreases, giving rise to an inrease in the kinematial pressure. Sine there is no model of the funtion for the metal alloy solidifiation, the model is developed by urve-fitting of the data from Kirkwood et al. (1950), who studied the kinematial pressure of the solid phase in dense gases. The funtion is in the form of ( ) f g s g s = log 10 1 g (3.69) sp

104 86 From the equations (3.38) and (3.69), it is noted that the kinematial pressure vanished when there is no relative motion between the phases, the solid pressure rises dramatially with inreasing onentration near the paking, whih eventually inhibits the motion of the solid phase The Dissipative Interfaial Stress on the Solid Phase There have been different approahed to modeling of the dissipative interfaial stress, M d sd. For high solid frations or stationary solid phase, the porous medium approah is adopted, using the permeability model. For low solid frations, the submerged objet model (de Groh et al., 1993) whih is the modifiation of the Stokes law for a single spherial partile, is often used. Reently, both approahes have been unified by Wang et al. (1995) and their model is utilized in this study and reviewed below. The dissipative interfaial stress on the solid has been modeled as 4βµ 2 d f 2 Msd = 2 g v v d s ( ) f f s (3.70) where d s is the diameter of an equivalent sphere (equiaxed) or ylinder (olumnar) having the same volume as the grain, and β is a dimensionless parameter that is only a funtion of the grain volume fration and its morphology and is given by β = β [ g + ( β β ) ] n n g d / l d 2 12 n (3.71) where

105 β d = 3 5 Ss 32 (3.72) φ S ( 1 g ) si e e 87 β l = 9 g 2 g η β 1 β β η + 3η 2η C φ 2β + 31 β β p ( ) e ( tanh d d) ( tanh ) d d 1 2 (3.73) n = log β (3.74) d g si g s = (3.75) g g η=g g 13 (3.76) φ e is the shape fator of the interdendriti envelope whih aounts the irregularity of the envelope shape and it will be defined in follow-up setion. The funtion C p (φ e ) aounts for the effet of non-spherial or non-ylindrial shapes. The Kozeny-Carman equation whih is valid for irregular shapes, suggests that ( φ ) 2 C p e φ e = for 0.7 > g l > 00. (3.77) However, the shape funtion has not been evaluated for high liquid volume fration and the only empirial orrelation available in the limit of a single equiaxed dendrite as proposed by de Groh et al. (1993). The relationship is ( φ ) φ 126. log 10 for 1 > g > l (3.78) e C p e = It is noted that this drag model aounts for the multiple length sales present in the dendriti struture. The model has been validated against various experimental data available in the literature (Wang et al., 1995).

106 Mirosopi Models The volume averaging tehnique eluidates the oupling between mirosopi and marosopi transport variables led through the interfaial transfer terms. The phenomena are related to the morphology and dynamis of the interfae. This setion overs the mirosopi models for the parameters related to the interfaial phenomena whih appear in the marosopi equations The Interfaial Area Conentration Previous setions show that the interfaial area onentrations are important parameters in the modeling of the interfaial transfer terms, and they are losely related to omplex mirosopi phenomena. In general, they are funtions of the grain volume fration, geometrial parameters in the mirosopi sale suh as, dendrite arm spaing, and the number density of nuleus. The inverse of the area onentration represents an aurate measure of the length sale of a mirostruture. The area onentration of the interfae between the solid and interdendriti liquid, S s has been modeled by assuming a simple one-dimensional geometry for seondary dendrite arms (Wang and Bekermann, 1992). This analysis showed that the mean harateristi length and the area onentration are given by d s = g sλ 2 (3.79) g g

107 89 S s = 2 λ 2 (3.80) where λ 2 is the seondary dendrite arm spaing. It is noted that the results are appliable to equiaxed and olumnar strutures. Substituting equation (3.79) into (3.80), the relationship between the harateristi length and the area onentration is given by S s g s = 2 (3.81) gd g s The numerial fator of two in the equation (3.81) an be adjusted for other hoies of the geometry. The area onentration of the interfae between the inter- and extra- dendriti liquid (the dendriti envelope), S e is more diffiult to desribe due to its ompliated geometri shape. However, it an be modeled by onsidering the envelope as an equivalent simple geometry dendrite envelope and modifying the area onentration of the hosen geometry by a shape fator in order to reflet the atual geometry (Wang and Bekermann, 1992). The envelope shape fator is defined as φ e A eq = (3.82) A at A eq is the surfae area of a geometry whih has the same volume as the atual envelope. Equiaxed grain envelopes are desribed by equivalent spheres, while equivalent ylinders are hosen for the olumnar grains as shown in Figure 3.4. A shape fator lies between zero and unity before paking between grains ours and if the shape of the envelope is preserved during the growth, the shape fator an be onsidered as a onstant. The basi

108 90 (a) (b) Figure 3.4 Shemati of shape fators for (a) an equiaxed dendriti envelope and (b) a square arrangement of olumnar dendriti envelop.

109 Figure 3.5 Illustration of the dependene of the shape fators on the solid volume fration. 91

110 92 dependene of the shape fator on the solid volume fration is illustrated in Figure 3.5. Sine the nuleus in the very beginning of the solidifiation proess is lose to the spherial shape, the shape fator an be approximated to be unity. During the intermediate stage, the shape fator beomes less than unity due to the development of fine dendriti struture. At the final stage of the solidifiation, it beomes very large beause of paking between the grains. The alulation of the shape fator still needs more analytial and empirial attention, and in this researh, this behavior is modeled by a simple paraboli equation and the equation similar to the one found in Feller and Bekermann (1993) before and after the impingement ours, respetively. φ e = g s g s 1+ ( 1 φ ) g min 2 g sp g for g s < sp 1 gsp 13 gsp g s 1 g s φ g g min sp g for g s > 1 sp sp sp (3.83) where φ min is a minimum value of the shape fator whih is between one and zero. For equiaxed growth, the following relation for S e, an be written as (Wang and Bekermann, 1995) S ( ) e = π n gg (3.84) φ e where n is a number density of grains and must be alulated from a nuleation model, whih is desribed in the following setion. Assuming a square pattern of the olumnar dendrites on a transverse ross-setion for olumnar growth, the envelope area onentration beomes

111 93 S e = 1 12 ( π) λ 1 gg (3.85) φ e where λ 1 is the primary dendrite arm spaing. The primary dendrite arm spaing depends mainly on the olumnar front veloity and the temperature gradient and an be alulated by an appropriate model (Flemings, 1974;Kurz and Fisher, 1989). Equations (3.84) and (3.85) indiate that the area onentration inreases from zero, eventually reahes a maximum value, and then dereases again to zero, beause impingement ours, i.e., the shape fator approahes infinity Nuleation Model In order to evaluate the envelope area onentration, the number density of grains is needed. The number density of grains depends not only on the rates of nuleation (during the initial stage of solidifiation) and fragmentation, but also on the advetion of grains. This an be expressed by the following onservation equation (Ni et al., 1990). n t ( ) + v n n s = & (3.86) The right hand side represents a soure whih inludes the birth and death of grains due to nuleation, detahment of dendrite arms, lustering of nulei, total remelting, and other effets. Sine a model for the nuleation generation in the presene of onvetion and fragmentation is not available, a simple nuleation rate model will be used in this study. Aording to the Hunt (1984), the heterogeneous nuleation rate an be expressed as

112 n& = I = K ( ns n) exp 1 K 2 ( T) 2 (3.87) 94 where n s is the number density of substrates or the maximum possible number density and T is the underooling temperature in the liquid phase. In this study, only solutal underooling is onsidered and is given by α α ( ) T = m f f l l d (3.88) K 1 is a onstant related to the ollision frequeny of atoms of the melt with the nuleation sites of the heterogeneous partiles, and K 2 is onstant related to the interfaial energy balane between the nuleus, the liquid, and the foreign substrates on whih nuleation ours. It should be noted that the equation (3.87) is valid only when there exists the underooling. While K 1 and K 2 affet the nuleation rate, only n s will determine the final grain density. Thus, the dependeny between ooling rate and number density (Basdogan et al., 1982) needs to be refleted in a relationship between ooling rate and the number density of substrates. Stefanesu et al. (1990) suggests the following paraboli equation, n K K dt s = dt 2 (3.89) The onstants K 3 and K 4 an be determined experimentally for a given alloy. Note that the above nuleation models are not valid during remelting of the solid, beause the terms whih reflet the temperature dependeny are squared, and thus, the ase of temperature inrease ausing the remelting still inreases the number density. Therefore, if the loal temperature inreases or there is no underooling, the present nuleation model must be biased.

113 Growth Model of Grains The models of interfaial transfer due to phase hange in equations (3.56) and (3.79) require information of the envelope growth veloity w ne. If it is assumed that the mean dendriti tip veloity is equal to the envelope veloity, the model for the veloity an be supplied from various model in the literature (Kurz and Fisher, 1989). Generally, there may be onsiderable differenes in the speeds of the primary and seondary dendrite arm tips, resulting in the irregular shape of the envelope, whih an be aounted for through the use of envelope shape fators. The growth veloity has been extensively studied for different strutures (euteti or dendrites). Analytially, the growth model is obtained by onsidering solute transport near the tip and tip stability. Experimental and theoretial results suggest the following simple form. w ne ( T) 2 = (3.90) where the growth onstant, an be either alulated or determined from experiments and depends on the struture of grains. For example, Esaka and Kurz (1984) developed the following equation for dendriti growth, assuming that the growth of dendriti tip is ontrolled by solute diffusion and thermal underooling is negligible, w D f mf ( ) ( T ) 2 ne = 2 α (3.91) π Γ κ 1 l d where Γ is the Gibbs-Thomson oeffiient. A growth model of the euteti struture was developed by Jones and Kurz (1981) as the following.

114 96 w ne = φ 1 KK e φ 2 e + 1 r 2 2 ( T) (3.92) where the onstants, K and K r, an be alulated from the properties of the material and arise from the solute diffusion and apillary alulations, respetively (Jakson and Hunt, 1966) and φ e is the shape fator aounting for the irregular euteti morphologies.

115 97 Chapter 4 CONTINUUM MODEL AND NUMERICAL SOLUTION SCHEME 4.1 Continuum Model Although the models summarized in Chapter 3 have a potential of simulating many phenomena of the alloy solidifiation in different length sales, they introdued many new parameters, whih are unknown at this moment and some funtions are not so suitable for numerial omputations. Therefore, the ontinuum model for binary solidliquid phase hange (Bennon and Inropera, 1987a; Presott et al., 1991) is used to simulate the solidifiation of binary metal alloys. Following is the summary of the mathematial model. Continuum model equations for onservation of mass, x-momentum, y- momentum, energy, and speies are the following. Continuity ρ ( ρ ) t + V = 0 (4.1) x-momentum ( u) ρ t ( ρ ) µ ρ p µ l ρ + V u = l u ( u us) (4.2) ρ x K ρ l l

116 98 y-momentum ( v) ρ t ρ µ ρ + = l ρ p y µ ρ K ρ l ( Vv) v ( v v ) l [ ( ) (, )] + ρ g β T T + β C C l T o s l l o l s (4.3) Energy ( h) ρ t k + = k s [ ] * ( ρvh) h + ( h h) f ρ( V V )( h h ) * * s s s s l s (4.4) Speies ( C) ρ t ( ) ( ) ( ) [ l ] [ s ( s)( l s) ] + ρvc = ρd C + ρd C C f ρ V V C C (4.5) Several assumptions were made to obtain equations ( ), the most important of whih are (i) the solid phase forms a oherent mushy and remains stationary, (ii) Dary s law applies in the mushy zone, (iii) no gaseous phase trapped in the system, (note: void refers to the volume fration of interdendriti liquid in the mushy region), (iv) the mixture thermal ondutivity is given by k = k + ε ( k k ) l s l, and (v) the rate of mass diffusion through the solid phase is negligible (hene, D = f D ). The third terms on the right side of equations (4.2) and (4.3) are Dary damping terms. The permeability of the mushy zone is determined from the Blake-Kozeny equation. l l K = K o ( 1 ε) 2 ε 3 (4.6) where K o is the permeability onstant and ε is the volume fration solid. The seond and third terms on the right side of equations (4) and (5) are diffusion -like and advetion-like soure terms, respetively, that aount for the two phase nature

117 99 of the mushy zone. It is important that these terms are treated numerially in a manner onsistent with the treatments of the diffusion and advetion terms of the main (mixture) field variables. Equations ( ) an be ast in dimensionless form by introduing length, veloity, time, temperature, enthalpy, and speies onentration sales, in suh a way that the dominant terms are of order unity. Length and veloity sales are seleted as L and gβ TL, where T is an appropriate temperature sale, and are typial for thermal T buoyany indued onvetion problems. The time sale is the quotient of length and veloity sales LgβT T, and the enthalpy sale is l T. Although speies onentration is dimensionless, it too is saled with - T/m, where m is the slope of the liquidus line on the equilibrium phase diagram, whih simplifies the thermodynami relationships used for losure (see below). In dimensionless form, the governing equations are Continuity x-momentum ( V) = 0 (4.7) u 1 p ( 1 ε) ε 2 + ( Vu) = u t Gr x Da Gr 2 3 ( u ) u s (4.8) y-momentum

118 100 v 1 p ( 1 ε) ε 2 + ( Vv) = v t Gr y Da Gr 2 3 ( v v ) + θ( 1+ Nχ ) s l (4.9) Energy ( ) εκ ( ) η 1 εκs s + ( η) = η V ( ηl η) t Pr Gr Pr Gr (4.10) [ fsρ( V Vs)( ηl ηs) ] Speies χ 1 f 1 f + = t S Gr S Gr [ ] s 2 s 2 ( Vχ) χ+ ( χ χ) f ρ( V V )( χ χ ) l s s l s (4.11) where θ = T T T e, η= h l h e T mc, χ= T s, N = β m β, and κ s T k s =. k l In order to simplify the supplemental relations for losure, the following assumptions are invoked: (1) loal thermodynami equilibrium exists between the solid and liquid phases, (2) thermophysial properties of eah phase are onstant (but different from eah other), and (3) the liquidus and solidus lines on the equilibrium phase diagram are linear. Hene, the supplemental relations of Bennon and Inropera (1988) an be used. Figure 4.1 shows the relevant portion of the simplified equilibrium phase diagram in terms of θ and χ. Note that θ e = 0 and χ e = θ f, and the liquidus and solidus lines are desribed by the following equations. θ = θ χ (4.12) θ sol liq f χ = θ (4.13) f k p

119 101 θ θ f L S+L S θ =0 e 0 χ s,a = k χ p e χ e χ Figure 4.1 Dimensionless equilibrium phase diagram.

120 102 Dimensionless enthalpies assoiated with solidus and liquidus temperatures are, respetively, η sol s = θsol (4.14) l η = θ + (4.15) liq liq Ste where Ste = h f T l (4.16) The loal state of the alloy is determined from its mixture ompositions χ, on whih η sol and η liq depend, and the mixture enthalpy (η=f s η s +f l η l ). If η > η liq : θ = η Ste (4.17a) f s = 0 (4.17b) If η < η< η : e liq θ= a a 2 a a a (4.18a) f s 1 θ θ = 1 k θ θ p liq f (4.18b) where a 0 Steθ = ( η Ste) θf 1 k liq p (4.19a) a 1 1 k s l p = θ k k Ste liq θf η (4.19b) 1 1 p p

121 103 a s l 1 = + 1 k 2 1 p (4.19) If η < η< η : sol e θ = θe = 0 (4.20a) f s ( fs e) = 1 1, η η η η e sol sol (4.20b) where f se, = η e e χ e χ ( k p) χ 1 ( f s e ) (4.21) = Ste 1, (4.22) If η < η sol : η θ = s l f s = 1 (4.23a) (4.23b)

122 Numerial Calulation Shemes The marosopi onservation equations of the mixture models, (3.17), (3.36), (3.45), and (3.54) and the governing equations in the ontinuum model, (4.7) through, (4.11) were deliberately ast into a general advetion-diffusion form in order to utilize the ontrol-volume based, finite differene SIMPLER algorithm (Patankar, 1980). The general form is ( ) ρφ t ( ρvφ) ( ) ( S Spφ) + = Γ φ + + (4.24) where φ is a general salar quantity, Γ is the diffusion oeffiient, and (S +S p φ) is a linearized soure term. For the disretization of the advetion term, Patankar (1980) suggested a power law differene sheme, whih is a variation of the first-order upwind differene sheme. However, it has been found that for the natural onvetion problems of low Prandtl number fluids, the first-order aurate shemes for the advetion terms introdue exessive false diffusion (Mohamad and Viskanta, 1989) and thus dampen dynami behaviors suh as osillatory states observed in Hurle et al. s (1974) experiments. On the other hand, the seond-order aurate entral differene sheme (CDS) for advetion terms has suessfully predited the dynami harateristis of an osillatory onvetion state (Cless and Presott, 1996). Hene, a sheme higher than firstorder should be used to orretly predit the behavior of a liquid metal. Sine it has been known that the numerial instabilities of the entral differene sheme are great for the solidifiation problem, whih has large soure-driven liquid

123 105 onentration gradients (Presott and Inropera, 1991), rather than using the CDS, an alternative seond-order finite differene sheme, alled QUICK (Leonard, 1979), was implemented for the energy and speies equations. However, sine the momentum equation requires speial, individual treatment in the SIMPLER algorithm, it was disretized, using the entral differene sheme. Hene, the numerial implementation used here was a ombination of entral differene and QUICK shemes and is referred to as CDS/QUICK. The QUICK sheme was implemented in the manner proposed by Hayase et al. (1992) and the implementation is explained in below. Furthermore, to be onsistent with the disretization of the main advetion term, interpolation was used to evaluate the advetion-like soure term in equations (4.10) and (4.11), with the upstream diretion determined from the mixture veloity omponents. The QUICK sheme employs a three-point upstream-weighted quadrati interpolation tehnique of a ontrol-volume approah for alulating on a staggered grid. Figure 4.2 shows the one-dimensional staggered grid. The evaluation of the salar φ at the ontrol volume surfae an be done by fitting a parabola to the values of at three onseutive nodes, i.e., the two nodes loated on either side of the surfae of interest, plus the adjaent node on the upstream side. In order to ensure the final disretized equations to satisfy the rules given by Patankar, the surfae values are evaluated as following (Hayase et al., 1992).

124 Figure 4.2 One-dimensional, uniform, staggered grids. 106

125 107 For u e > 0, u > 0, w φ φ = φ + S + e i e = φ + S + w i 1 w (4.25) For u e < 0, u < 0, w φ φ = φ + S e i+ 1 e = φ + S w i w (4.26) + + where the soure terms S e, S w, S e, and S w are generally written as S = aφ1 + bφ2 + φ3 (4.27) a = b = = ( xu x2)( xu x3) ( x1 x2)( x1 x3) ( xu x1 + x2 x3)( xu x2) ( x2 x3)( x2 x1) ( xu x1)( xu x2) ( x x )( x x ) (4.28) The orresponding subsripts for eah soure term are shown in Table 4.1 and x is the oordinate of the variable φ. In the ase of uniform grids, the soure terms beome S S S S = φ φ + φ = φ φ + φ = φ φ + φ = φ φ + φ e i 1 i i+ 1 + w i 2 i 1 i e i+ 2 i+ 1 i w i+ 1 i i 1 (4.29)

126 108 Table 4.1 The orresponding subsripts for soure terms Soure term u S e + e i+1 i i-1 S w + w i i-1 i-2 S e e i i+1 i+2 S w w i-1 i i+1

127 109 Using the QUICK sheme, the equation (4.24) an be disretized over a omputation ontrol-volume as shown in Figure 4.3 and it results a P a a a a b a α φ φ φ φ φ α = α 1 P E E W W N N S S Pφ * P (4.30) where a E, a W, a N, a S are advetion-diffusion oeffiients, b is a soure/transient oeffiient, and a P is a ombined advetion-diffusion and soure/transient oeffiient. Also, α represents a relaxation oeffiient and φ P * represents the value of φ P from the previous iteration. The oeffiients are exatly same as the first-order upwind sheme exept b and a P b = S F S F + S F + S F S F S F + S F + S F e e e e w w w w n n n n s s s s + S x y + a φ o P o P (4.31) o ap = a + ap + aq SP x y (4.32) where a = F + F F F + F + F F F Q e e w w n n s s (4.33) ( ) F + = max ρu, 0 A (4.34) e e e ( ) F = min ρu, 0 A (4.35) e e e The other advetion oeffiient F s are defined similarly. The quantity a o P refers to values at the previous time step.

128 Figure 4.3 Computational ontrol volume. 110

129 Numerial Calulation Conditions All the simulation studies were arried out on a non-uniform grid of a omputational domain, whih was patterned after the experimental apparatus for whih H=L=0.1m. Figure 4.4 illustrates the problem domain and the boundary onditions employed in numerial simulations are summarized in Table 4.2. All four walls of the avity are impermeable and slip free. The alloy is initially in a quiesent, isothermal molten state and is at a temperature higher than the liquidus temperature of the initial alloy omposition. At time t=0, the thermal onditions for left and right side walls are applied while other walls maintain adiabati. Note that T is less than the euteti temperature and T h is higher than the liquidus temperature of the initial alloy omposition. The equilibrium phase diagram for the gallium-indium system is shown in Figure 2.2 and the main properties are shown in Table 2.1. The properties and the initial and boundary onditions are onverted to dimensionless parameters in order to inorporate the equations in Chapter 4.1 and they are shown in Table 4.3. The temperature sale used in the simulation is the differene between the liquidus and euteti temperatures (i.e. T=14.7 K). The hot and old side temperatures are equivalent to 30 and 15 C. The omputation domain has been disretized to grid lines spaed as the following relations,

130 Figure 4.4 Problem domain. 112

131 113 Table 4.2 Boundary onditions employed in numerial simulations. Case Left wall Right wall Top wall Bottom Wall I T=T adiabati adiabati adiabati II T=T T=T h adiabati adiabati

132 114 Table 4.3 Controlling dimensionless parameters. Parameters Value Gr Grashof Number Pr Prandtl Number S Shmit Number 150 Da Dary Number Ste Stefan Number 11.8 N Buyoany Parameter s / l Speifi Heat Ratio κ s Thermal Condutivity Ratio 3.14 k p Equilibrium Partition Ratio 0 θ f Fusion Temperature θ i Initial Temperature 1.34 θ h Hot Side Temperature 1.34 θ Cold Side Temperature χ i Initial Composition 13.04

133 x i i 1 = W N 2 1 x (4.36a) y i j 1 = L N y (4.36b) where i=1,2,, N x /2, j=1,2,, N y /2, and x i and y i are the loations of the grid lines from the boundary to the enter of the omputational domain, while N x and N y are the total number of grid lines, in the x- and y- diretions respetively. In order to determine the number of grids, Case I in Table 4.2 was simulated with different numbers of grids. A time step of was used in the alulations. Figure 4.5 shows the alulated dimensionless temperature at x*=0.50 and y*=0.05 between dimensionless time unit 0 and 10 with different number of grids. All temperature profiles followed the same trend. However, sine the temperature was averaged from temperature at the surrounding grids around the loation, the temperature varied with the number of grids. Considering 1% variation as a threshold, the number of grids for the alulation should be greater than 50 by 50. In a similar way, Case I was simulated again with different size of time steps, to determine a proper time step for the alulation. A mesh of 50 by 50 has been used in alulation. Figure 4.6 shows the alulated dimensionless temperature at x*=0.50 and y*=0.05 between dimensionless time unit 0 and 10 with different time steps. Unlike the temperature variations in different grid size, the temperatures were expeted to be the same if the time step was small enough. In Figure 4.6, the alulation result of t=0.01 starts to deviate from others, indiating that the time step was too big to resolve the

134 Figure 4.5 Cooling urves for different numbers of grids. 116

135 Figure 4.6 Cooling urves for different time steps. 117

136 118 transient system. Therefore, the time step that an be used in the alulation should be less than 0.01 units. From the dependeny study results shown in Figures 4.5 and 4.6, a time step of units and grid dimensions of 50 by 50 were hosen to resolve the system transients of a ontrol-volume-based finite differene method with the least amount of alulation power. The seleted numerial mesh for the disretization is shown in Figure 4.7.

137 Figure 4.7 Numerial mesh for the alulation. 119

138 Comparison of Finite Differene Algorithms Traditional implementations of the ontinuum model have utilized either upwind (UPS) or power-law (PDS) differene shemes and fully impliit time marhing (Bennon and Inropera, 1987b; Presott and Inropera, 1994), whih are known to dampen dynami behavior. Hene, the higher order sheme, CDS/QUICK is implemented in this study. In this setion, the results of CDS/QUICK are ompared with those from traditional alulation proedures and demonstrate better ability of resolving dynami features of thermosolutal onvetion during solidifiation of metal alloys. The solidifiation of Ga-In binary metal alloys for ase I was simulated using both first-order (PDS) and seond order (CDS/QUICK) methods. Calulated results for ase I are shown in Figures , whih ontain plots of veloity vetors, streamlines, isotherms, and liquid isoomps. Figures 4.8 and 4.9 show onditions predited at t=5 by the PDS and CDS/QUICK shemes, respetively, and are representative of the early solidifiation period. The strutures of thermosolutal onvetion patterns predited by the two shemes at t=5 are nearly idential, but isotherms and espeially liquid isoomps are more greatly distorted by the fluid motion in the CDS/QUICK simulation, Figures 4.8 (d) and 4.9 (-d), beause false diffusion effets are redued. The redution in false diffusion (visosity) in the CDS/QUICK sheme also aounts for the 11% inrease in maximum veloity predited at t=5, Figures 4.8 (a) and 4.9 (a).

139 121 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 4.8 Solidifiation onditions predited by PDS at t=5: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

140 122 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 4.9 Solidifiation onditions predited by CDS/QUICK at t=5: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

141 123 Figures 4.10 and 4.11 show solidifiation onditions predited by the PDS and CDS/QUICK shemes, respetively, at t=25, when a transition from thermally- to solutally-dominated onvetion is ourring. In both ases, the upflow of interdendriti liquid and its disharge from the mushy zone are hilling the top portion of avity and responsible for slightly enhaned growth of the fully solid zone near the top, as shown in Figures 4.10 (a) and () and 4.11 (a) and (). The growth of the mushy zone and the liquid onentration gradients therein inrease the strength of the solutal reirulation ell, but its ability to derease the strength of the thermal ell is redued in the CDS/QUICK simulation, beause artifiial visosity has been redued. Hene, the initial thermal ell remains stronger and ative for a longer period of time in the CDS/QUICK simulation. Furthermore, the struture of the thermosolutal onvetion predited by the CDS/QUICK sheme, Figure 4.11 (b) is more omplex than that predited by the PDS, Figure 4.10 (b), whih is due in part to the greater degree of distortion in temperature and liquid onentration fields by advetion, Figure 4.11 () and (d), whih, through their effet on buoyany fores, affet vortiity prodution in the liquid. The ompetition between thermal and solutal ells predited by the CDS/QUICK sheme results in the impingement of warm liquid in the liquidus interfae near y=0.5 and retard the growth of the mushy zone relative to regions above and below the impingement region. Eventually, solutal fore arising within the mushy zone dominated over thermal buoyany fores. Figures 4.12 and 4.13 show solidifiation and onvetion onditions predited by the PDS and CDS/QUICK shemes, respetively, at t=70, when a single solutal ell oupied the mushy and melted zones, Figures 4.12 (b) and 13 (b).

142 124 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 4.10 Solidifiation onditions predited by PDS at t=25: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

143 125 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 4.11 Solidifiation onditions predited by CDS/QUICK at t=25: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

144 126 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 4.12 Solidifiation onditions predited by PDS at t=70: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

145 127 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 4.13 Solidifiation onditions predited by CDS/QUICK at t=70: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

146 128 Differenes in prior onvetion onditions are manifest as small differenes between plots in Figures 4.12 and 4.13, but overall, onvetion onditions predited by the two shemes at t=70 are very similar to eah other. As time ontinues, differenes beome even less notieable, beause the effet of the Dary damping terms on fluid flow inreases, while advetion effets derease, as the mushy zone grows. The effet of using higher-order disretization shemes for advetion and transient terms is most pronouned during early and intermediate stages of solidifiation. When thermal and solutal buoyany fores interat, there is signifiant flow oblique to the grid (whih enhanes false diffusion), and advetive transport is yet ontrolled by Dary damping effets. It should be noted that, if the transport of dendrite fragments had been onsidered, differenes between the two shemes may have been greater, sine Dary damping effets would have been absent for a longer period of time. In later stages of solidifiation, when solutal onvetion ours exlusively and is limited by Dary damping in the mushy zone, differenes between the first and seond order integration shemes are insignifiant. Sine the flow dynamis in the early and intermediate stages is very important during the alloy solidifiation, the seond order integration sheme is utilized to resolve dynamis features of thermosolutal onvetion.

147 129 Chapter 5 RESULTS AND DISCUSSION 5.1 Introdution The solidifiation of a gallium-27 wt. pt. indium metal alloy from a vertial sidewall in a square mold was investigated. Two sets of boundary onditions were onsidered. In the first set of boundary onditions, the solidifiation was initiated from the left vertial wall, while other walls maintained the adiabati ondition. This ondition simulated a regular asting proedure and it was expeted that the onvetion flow would have a great influene only in the early stage of the solidifiation, ausing some marosegregation in the solidified ingot. In the seond set of boundary onditions, the solidifiation started from the left vertial wall like the first set but the right wall maintained the temperature above the liqidus point, while other horizontal walls were insulated. This ondition would reate higher and longer horizontal thermal gradients in the alloy so that the influene of the onvetion flow would be more pronouned. The seond set would not our in the real ast proess. However, it would be interesting to see how the marosegregation would be hanged over the long period of onvetion in the alloy.

148 130 Although neither the experiments nor the numerial simulations of this study ould fully determine the phenomena during solidifiation of a binary metal alloy, there was still muh to be learned. Shortomings of the proposed experimental proedures ould be olleted and they required areful and aution interpretations. On the other hand, numerial simulations relied on simplifying assumptions and were limited by omputing resoures. The material in this hapter is presented in hronologial order in whih results were obtained. Namely, experimental results are presented first and are followed by the numerial simulation. The subsequent omparison between experiments and preditions shows the adequaies of the preditions and gives further understanding of the solidifiation proess in a binary alloy.

149 Case I: One Side Cooling Case Case I: Experimental results. Several experiments were performed for the first set of boundary onditions. The proedure is desribed in Setion 2.5. The temperatures at ertain strategi loations were measured, in order to understand solidifiation harateristis of the metal alloys, and quantified maro-segregation patterns of fully solidified ingots were obtained by neutron radiography. The initial alloy temperature was 35 C and a old left wall temperature was -15 C, while all other surfaes were insulated. The reproduibility of the experimental runs is illustrated in Figures 5.1 (a-), whih show measured temperature histories at the mid-horizontal plane of the avity for x*=0.05, 0.50 and 0.95 respetively for three different runs. Differenes between ooling urves at x*=0.05 (Figure 5.1 a) are less than 0.7 C. Although run-to-run differenes are somewhat greater at x*=0.50 and 0.95 (Figures 5.1 b and ), idential trends our in all runs and the overall repeatability is ahieved. The differenes an be attributed to minor variation of initial onditions and random phenomena during the solidifiation, suh as nuleation of a solid phase and fraturing of dendrites. Cooling urves at y*=0.50 and three horizontal loations are plotted in Figure 5.2 whih shows that the entire solidifiation proess ours in six stages. Initially, the

150 132 (a) Figure 5.1 Cooling urves for differene experimental runs at (a) x*=0.05, (b) x*=0.95, and () x*=0.95.

151 133 Figure 5.1 Continued. (b)

152 134 Figure 5.1 Continued. ()

153 Figure 5.2 Cooling urves at different x* loations for y*=

154 136 superheated melt ools rapidly and beomes underooled below the liquidus temperature of 30 C (Stage I). Within approximately 300 s, a realesene assoiated with the nuleation and growth of primary indium-rih grains, ours. The brief realesene period (Stage II) is followed by more than 1800 s, during whih the primary solid phase grows, and the interdendriti liquid beomes underooled below the euteti temperature of 15.3 C (Stage III). At t 2700 s, a seond distint realesene ours due to the nuleation and growth of the seondary, gallium-rih solid phase (Stage IV). Solidifiation ontinues with the euteti interfae advaning toward the right (insulated) wall (Stage V), and finally for t > 5300 s, the fully solidified ingot ools further (Stage VI). Figure 5.3 shows the marosegregation pattern obtained from the fully solidified ingot using neutron radiography. Lines of onstant mass fration indium have been superimposed on the image of the neutron radiograph, and the isopleth orresponding to the nominal omposition of 0.27 mass fration indium is drawn with a thiker line. The irregularity of the isopleths in Figure 5.3 indiates the diffiulty with identifying trends from a few disrete omposition measurements along with regions of large omposition gradients (Shahani et al., 1992; Presott et al, 1994). The neutron radiograph shows that the regions near the bottom wall and lower half of the left (hilled) wall are enrihed with indium, while the top and middle-right regions are gallium-rih. The marosegregation pattern shown in Figure 5.3 is attributed to the advetion of gallium rih liquid and indium rih dendrite fragments during solidifiation, and helps to interpret measured ooling urves.

155 Figure 5.3 Neutron radiograph showing the marosegregation pattern (mass fration of indium) within the solidified ingot. 137

156 138 Cooling urves at three vertial loations during the first 600 s period, are plotted in Figures 5.4 (a-) for x*=0.05, 0.50, and 0.95, respetively. An interesting trend is seen near the hill plate for 15<t<180s in Figure 5.4 (a); the temperature at y*=0.75 is lower than those at y*=0.50 and This trend, whih ontradits an expeted positive vertial temperature gradient due to thermal onvetion in the molten alloy, is attributed to onjugate heat transfer between the oolant, hill plate, and alloy and to the release of the latent heat assoiated with the growth of primary phase dendrites entrained in thermal down flow near the hill plate. The oolant was introdued to the hill plate at the top and exited at the bottom. Hene, the ooling effetiveness was largest near the top, whih might explain why the temperature at y*=0.75 was the lowest in the early stages of solidifiation. Conjugate heat transfer effets assessed by reversing the diretion of measured at x*=0.05 during this additional experimental run are shown in Figure 5.5. The temperatures at y*=0.25 are generally lower in Figure 5.5 than in Figure 5.4 (a), while the temperature at y*=0.75 are higher in Figure 5.5 than in Figure 5.4 (a), indiating the presene of a onjugate heat transfer effet. However, onjugate heat transfer does not ompletely aount for the aforementioned trend of relatively ool temperatures at y*=0.75 in Figure 5.4 (a) during the early stage of solidifiation. The high onentration of indium along the bottom wall and the lower half at the hilled (left) wall in Figure 5.3, ould be the result of indium rih dendrite fragments being transported from upper regions and settling during early stages of solidifiation. This pattern of transport would our beause thermal buoyany fores in the bulk melt would effet a ounter-lokwise reirulation with down flow along the left wall and

157 139 (a) Figure 5.4 Cooling urves at different y* loations (0<t<600 s) for (a) x*=0.05, (b) x*=0.50, and () x*=0.95.

158 140 Figure 5.4 Continued. (b)

159 141 Figure 5.4 Continued. ()

160 Figure 5.5 Cooling urves at different y* loations (0<t<600 s) for x*=0.05 in the opposite diretion of oolant. 142

161 143 beause indium rih dendrites are denser than the bulk liquid. This pattern is also onsistent with the measured ooling urves in Figure 5.4 (a), beause as the dendrites fragments are being transported downward, they would grow in underooled liquid by whih latent heat is released. Sine more latent heat would be released in the lower region, the upper region would ool more rapidly. The ooling urves at x*=0.50 and 0.95 for t<90 s, Figures 5.4 (b) and (), reveal typial temperature stratifiation of thermal onvetion in differentially heated avities (Gebhart et al., 1988), indiating that the effets of latent heat release due to grain growth are less signifiant away from the hilled wall. Past approximately 90 s, trends in the ooling urves hange, indiating hanging onvetion onditions. At x*=0.05, Figure 5.4 (a), the temperatures at y*=0.50 fall below those at y*=0.25 after 180s, indiating the possibility of upflow of interdendriti liquid near the hill plate. Figure 5.4 () shows a pattern of realesenes beginning at t 90 s, and the rossing of ooling urves at y*=0.25 and 0.50 (three times) is indiative of a transitional period, during whih omplex heat transfer onditions exist. Furthermore, the variability in ooling rates seen in Figures 5.4 (b) and () for 90<t<390 s suggests that a flow ondition in whih dendrite fragments are transported by liquid onvetion exists during this period, while the smoothness of the ooling urves at later times is indiative of a oherent mushy zone in whih flow instabilities are damped. Beginning at t 390 s, temperatures at y*=0.50 drop below those at y*=0.25 at both x*=0.50 and 0.95, Figures 5.4 (b) and (), and the midplane temperatures remain lowest until the seond realesene ours at t 2700 s. Sine the marosegregation

162 144 pattern in Figure 5.3 shows gallium enrihment near the right-mid height region, the ooling urves indiate that ool, gallium rih liquid is transported rightward near y*=0.50 during the intermediate time period after about 390 s. The aforementioned trends suggest the following possible senario. Thermal onvetion begins soon after the onset of ooling, and following the nuleation of the primary solid phase, dendrite fragments are entrained in the down flow along the hilled wall. Indium-rih grains are aumulated along the bottom and ontinue to grow in the underooled liquid, while a portion of the dendrites is transported through the avity. Solutally driven flow begins to our within the array of fallen dendrites, and a onvetion pattern develops with a lokwise, solutal ell oupying the bottom portion of the avity and a ounterlokwise thermally driven slurry ell in the top. Hene, old gallium enrihed liquid is transported from left to right near y*=0.50. The thermally driven ell eventually deays as a oherent mushy zone forms and oupies the entire avity, and as the solidifiation proeeds further, the permeability of the mushy zone dereases and heavily damps solutally driven flow to nearly stagnant onditions Case I: Numerial simulation using the ontinuum model. The experimental test onditions using a Ga-27% In alloy in a square mold avity were numerially simulated, using the ontinuum model as reported in Chapter 4. The boundary onditions are desribed in Chapter 4. Convetion onditions of ase I are represented by field plots of veloity vetors, streamlines, isotherms, and liquid isoomposition lines in Figures These plots are drawn on an x-y plane. The

163 145 solidus and liquidus interfaes are represented by thik lines on the plots. Veloity vetors represent the mixture veloities and are saled aording to the urrent maximum dimensionless veloity indiated at the bottom of eah veloity plots. One unit of dimensionless veloity based on T=14.3 C and L=0.1m orresponds to 0.38 mm/s. Streamlines assoiated with lokwise irulation have negative values, while ounterlokwise irulation ells have positive values. Those values are dimensionless. Isotherms represent ontours of onstant temperature with the dimensionless temperature at the left hill wall equal to (-15 C). Liquid isoomposition lines are ontours of onstant saled gallium onentration in the liquid phase. One unit of dimensionless time is equal to 2.6 seonds. One ooling is initiated at the left wall, a ounterlokwise thermal onvetion ell is established in the melt, due to temperature gradient in x-diretion. As ooling ontinues, solid dendrites begin to preipitate at the left ooled wall, thereby forming a two-phase (mushy) zone, and the mushy zone grows with the liquidus and solid interfaes moving horizontally. Figure 5.6 represents the onvetion onditions at t=5, showing the solid and liquidus fronts are almost planar. Figure 5.6 (b) shows two reirulation ells, one due to thermal buoyany fores in the melt, and the other due to solutal buoyany fores in the mushy region. Isotherms in Figure 5.6 () are nearly vertial, indiating one-dimensional horizontal ondution. In order to satisfy phase equilibrium requirement, the preipitation of solid is aompanied by solvent (gallium) enrihment of interdendriti liquid in Figure 5.6 (d), whih indues solutal buoyany

164 146 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 5.6 Convetion onditions of Case I at t=5: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

165 147 fores ating upward on the interdendriti liquid. Therefore, solutal buoyany fores oppose thermal buoyany fores aused by horizontal temperature gradient and beause the density of gallium is signifiantly less than In, the solutal fores dominate within the mushy zone (N=-5.8). However, sine the gradient in solid volume fration is large in the mushy zone, Dary damping is very strong and hene, the solutal upflow is very weak. The thermally driven onvetion ell in Figure 5.6 (b) penetrates the liquidus interfae near the top and drives downward the interdendriti fluid esaping from the mushy zone at the top, thereby onfining the liquid omposition gradient within the mushy zone, Figure 5.6 (d). The fluid in the ell enters the mushy zone with nominal omposition and leaves with slightly higher gallium omposition at the bottom by the penetration. The high downward veloities in the thermally driven ell ours along the liquidus interfae as shown in Figure 5.6 (a). They are likely to shear In-rih dendrite tips and may transport them to the bottom of the avity as proposed in the experiments. As the mushy zone ontinues to grow, the influene of solutal buoyany gradually inreases and reates very dynami onvetion onditions. Figure 5.7 represents the initial stage of a transition from thermally dominated to solutally dominated onvetion at t=20. The solutal ell is growing in strength due to mushy zone expansion, while the thermal ell is weakening, due to shrinking of melted region and dereasing of temperature gradient. The solutally driven ell is disharging gallium-rih fluid into the bulk melt at the avity top (Figure 5.7 d). An aumulation of gallium rih fluid at the top

166 148 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 5.7 Convetion onditions of Case I at t=20: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

167 149 auses a depression in the liquidus temperature, whih retards the growth of liquidus front at the avity top. Thus, the solid and mushy zone thikness beome nonuniform. The strong interation between the two ells ours near the top of the avity, Figure 5.7 (a), x*=0.45 and y*=0.70. The strong solutally driven flow, enrihed with gallium, leaves the mushy zone and depresses the loal liquidus temperature, whih is favorable to remelting. In addition, the warm gallium-rih flow is adveted from the melt into mushy zone and remelting is enhaned. This ondition favors the development of a hannel. However, in this ase, the fast advanement of liquidus does not give enough time for full development of a hannel. The momentum assoiated with the thermal onvetion ell gradually dereases, as temperature gradients in the melt diminish and opposing solutal buoyany fores inrease. At t=45, the thermal ell is onfined to the bottom, right orner of the avity, Figure 5.8 (b), while the solutal ell enompasses the mushy zone and most bulks melt, whih has beome solutally stratified, Figure 5.8 (d). The thermal ell is ompletely extint by 50 time units, beyond whih a large solutally driven ell oupies the entire mold avity and provides for the reirulation of interdendriti liquid. Figure 5.9 shows the evolution of marosegregation patterns in the solidifying ingot at different time. The value of marosegregation orresponds to the mixture gallium mass onentration. As ooling is initiated, thermosolutal onvetion begins in the mushy zone and the melt. The onvetion onditions start to build up gallium-enrihed regions near the top of the avity and indium enrihed regions near the bottom (Figure 5.9 a).

168 150 (a) Veloity (b) Streamlines () Isotherms (d) Liquid Isoomps Figure 5.8 Convetion onditions of Case I at t=45: (a) veloity vetors, (b) streamlines, () isotherms, and (d) liquid isoomps.

169 151 (a) t=5 (b) t=20 () t=45 (d) t=180 Figure 5.9 Marosegregation patterns of Case I after t= (a) 5, (b) 20, () 45, (d) 180.

170 152 Gallium-rih interdendriti fluid in the upper middle portion of the ingot, flows upward into a warmer region of the mushy zone, partially remelting or dissolving the existing solid. Hene, a hannel is reated in the region and as interdendriti fluid is drawn into a hannel, it preipitates indium in a region to the right and below the hannel, leading to an A-segregate pattern (Figure 5.9 b) Marosegregation ontinues and solutal onvetion in the mushy zone enrihes the lower regions of the ingot with indium and transports gallium to the top and downward along the right wall. Thus, a gallium-rih wedge (analogous to a one in a ylindrial ingot) segregate is well developed by t=180 (Figure 5.9 d). Beyond t=180, marosegregation does not evolve signifiantly, beause the interdendriti fluid beomes nearly stagnant throughout most of the mold avity due to inreased solid fration and dereased permeability Case I: Comparisons between experimental results and numerial preditions. Measured and predited ooling urves of ase I are ompared in Figures 5.10 (a) -(), whih orresponds to the temporal temperature variations in y*=0.75, 0.50, and 0.25 respetively. Solid lines in the figures orrespond to measurements from the experiment of ase I, while dashed lines orrespond to predition of the equilibrium ontinuum model. Overall, the measured ooling rates in experiments exeed the predited ooling rate, and trends of numerial and experimental results are somewhat similar. The

171 153 (a) Figure 5.10 Measured and predited ooling urves of Case I for (a) y*=0.25, (b) y*=0.50, and () y*=0.75.

172 154 Figure 5.10 Continued. (b)

173 155 Figure 5.10 Continued. ()

174 156 disrepany between the experimental results and preditions may ome from the model assumptions and unertainties in presribing the model parameters, suh as the thermophysial properties of the working fluids. The other major disrepany is that underooling and realesene are absent in the predited ooling urves sine the numerial preditions were based on assumption of loal thermodynami equilibrium between solid and liquid phases and ontinuous stationary solid phase. Partiularly, the assumption of equilibrium auses the predited ooling rates to be lower than measured ones beause the experiments showed about maximum 30 C of underooling for a long time and thus, the release of latent heat is less in experiments during the earlier period of solidifiation. This disagreement is very important in regards to marosegregation, sine the realesene after the underooling during the early stages of solidifiation has a great influene on hannel formation and marosegregation patterns. Therefore, the predition without inluding proper modeling of non-equilibrium phenomena, auses a big disrepany with the experimental results as shown in the figures and the numerial results are not suitable for diret omparison. Figure 5.11 shows the predited marosegregation pattern at t=365 with the aptured marosegregation pattern of the fully solidified alloy through the neutron radiography. At t=365, the solidifiation was 98 perent ompleted. From the figures, one major differene is that muh broader area enrihed with gallium appears on the top and the minimum gallium region appears between the top and bottom areas in the experiment result exept the very top left small area. Comparing the ooling urves, the differene

175 157 (a) (b) Figure 5.11 Marosegregation patterns of the fully solidified ingot for Case I: (a) experimental result (b) preditions at t=365.

176 158 may be related to the unusually large underooling (approximately 25 C) in a long period of the time (approximately 900 s) after the first realesene related to the first nuleation during the experiments (Figure 5.2). During this period, the primary solid phase enrihed with Indium grows and the interdendriti fluid with higher ontent of Gallium esapes from the mushy zone to the top as shown in Figure 5.5. Sine this proess ours long period without ourrene of the seond phase in the experiments and the entire alloy temperature is below the liquidus temperature, the entire alloy ould beome slurry and the top area would ollet muh more gallium ontent than the predition. Consequently, the thermally driven onvetion ell might our in the uniform gallium-rih area for a long time due to the lak of solutal stratifiation and the area might go through a different solidifiation proess than the rest of the alloy. In the predition, the solutally driven onvetion ell enompasses the entire alloy very quikly due to the fast derease in the thermal gradient and the solutal stratifiation. Therefore, the top area above 0.07 meter might have an almost uniform omposition with higher gallium ontent and the minimum gallium region was formed between the top and bottom regions. Another differene between the experiments and preditions is that the experimental result does not show any hannel formation near the old wall while the predition shows two inompletely developed hannels. It should be noted from the preditions that the hannels appear only lose to the interfae between the mushy and solid regions. Sine the fully solid region did not appear for a long time due to the heavy underooling, the formation of the hannel might be oppressed. After the solidifiation of

177 159 seond phase is triggered, the solid front might advane very quikly and did not give enough time for the hannel formation. It also should be noted that the hannel formation was enouraged in the preditions beause the solid phase was assumed to be stationary. However, in the experiments the dendrites related to the first phase solidifiation ould be freely moved along the interdendriti flow and some of them might be aumulated in the bottom of the ingot as shown in Figure 5.11 (b).

178 Case II: One Side Cooling and The Other Heating Case Case II: Experimental results. Another set of experiments was performed with different thermal boundary onditions. The vertial right wall was replaed with another opper heat exhanger, whih served as a heat soure. Initially, the alloy was prepared at 42 C and thermal boundary onditions were imposed with the left old wall at -15 C and the right hot wall at 35 C after the alloy was ooled to 35 C. Beause the right wall temperature was above the liquidus one, thermal buoyany fores prolonged longer in the melt, and their interation with solutal buoyany fores lead to very ompliated results. Figure 5.12 shows the ooling urves at y*=0.50 and three horizontal loations. The solidifiation proess undergoes six sequential stages before it reahes a steady state. As the initial superheated melt ools down (Stage I), a realesene assoiated with the formation of primary indium-rih grains ours within 300s (Stage II), and the liquidus front advanes from the left to the right walls. It is followed by the interation between thermal and solutal buoyany fores in the melt, resulting in double diffusive onvetion while the primary solid phase grows and the liquid near the hill plate beomes underooled below the euteti temperature (Stage III). At t 10300s, a seond realesene related to the nuleation and growth of the seondary solid phase, ours (Stage IV). After the brief

179 Figure 5.12 Cooling urves at different x* loations (0<t<15000 s) for y*=0.50 (Case II). 161

180 162 distint realesene period, the ooling urves show periodi flutuation, assoiated with the breakage and re-melting of dendrites on the solidifiation front (Stage V). After t s, the flutuation disappears and finally at t 20300s, ingot reahes the steady state ondition (Stage VI). Figure 5.13 shows the marosegregation pattern of the quenhed ingot obtained from neutron radiography. Similarly to the previous set of experiments, the regions near bottom and left walls have high onentration of indium, while the top regions are enrihed with gallium. A distint hange of the omposition in the bottom half is shown and believed to be the fully solid front when the ingot reahes the steady state (Figure 5.13). The shape of the urved interfae is smooth and extends more horizontally at the bottom of the test ell. The right side of the interfae shows nearly uniform omposition enrihed with gallium, due to the onvetive mixing aused by the prolonged heat soure. Figures 5.14 (a-) show ooling urves at three vertial loations for x*=0.05, 0.50, and 0.95, respetively. In Figure 5.14 (a), the interesting trend of temperature profile near the hill plate observed in the previous set of experiments, is also shown here; the temperature at y*=0.75 is lower than those at y*=0.50 and In Figure 5.13, the region enrihed with indium, is at the very bottom of the test ell. The observations from two figures propose that very initially, some of primary phase dendrites were not firmly attahed to the heat exhanger, desended downward along the right wall, releasing latent heat, and were olleted at the bottom. After the melt ooled down, the mushy zone formed in the region next to the hill plate and the solutal buoyany fores

181 Figure 5.13 Neutron radiograph showing the marosegregation patter (mass fration of Indium) within the solidified ingot (Case II). 163