CONTROL OF ALLOY SOLIDIFICATION PROCESSES
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Deep Samanta
May 2006
ALLOY SOLIDIFICATION PROCESSES
Deep Samanta, Ph.D. Cornell University 2006
Solidification of most alloys is often accompanied by macrosegregation or large scale variations in the concentration of solute elements. Macrosegregation leads to numerous defects and non-uniform properties in the final cast alloy that severely af-fect its performance and suitability for different applications. In terrestrial gravity conditions, macrosegregation in solidifying alloys is primarily caused by thermosolu-tal buoyancy driven convection supplemented by thermocapillary or diffusocapillary convection and shrinkage driven flows. Understanding the effect of convection during solidification of alloys and exploring the role of magnetic fields in damping convec-tion and suppressing macrosegregaconvec-tion form the basic objectives of this thesis.
A single domain model based on volume-averaged transport equations for mass, heat, solute and momentum transport is used for simulating alloy solidification processes. As part of this thesis, stabilized finite element method based numerical models have been developed to simulate buoyancy driven convection and macroseg-regation during solidification of alloys that exhibit mushy zones. The two phase mushy zone consisting of both solid and liquid phases is modeled as a porous medium with either an isotropic or anisotropic permeability. The numerical model developed
lutal convection during solidification of aluminum alloys. The analysis is extended to study the role of mold surface topography on the deformation of the solid and mushy zones during early stage solidification of aluminum alloys. Optimal surface features are identified that reduce surface unevenness of the solid shell. The role of magnetic fields in suppressing convection and inhibiting macrosegregation during solidification of metallic alloys is explored. Lorentz force that occurs due to the interaction of a moving fluid with magnetic field lines is the main damping force. A finite dimensional optimization problem that involves designing the time history of the applied magnetic field is formulated using the continuum sensitivity method (CSM). The solution of the optimization problem yields coefficients that determine the time history of the applied magnetic field. The elimination of various segregation based defects is highlighted through the developed methodology. The thesis finally concludes with a summary of achievements and future extensions of the current work.
The author was born in Kolkata, formerly Calcutta, India in November 1979. After completing his high school education from Dayanand Anglo Vedic (D.A.V.) Senior Secondary School in Chennai, formerly Madras, the author was admitted into the Bachelor’s program in Mechanical Engineering at the Indian Institute of Technology Madras in 1997, from where he received his Bachelor’s in Technology degree in 2002. In January, 2002, the author entered the doctoral program at the Sibley School of Mechanical and Aerospace Engineering, Cornell University and was awarded a special Masters degree in May, 2005.
younger sister Moumita for their constant support and encouragement towards academic pursuits during my school and college years.
I would like to thank my advisor, Professor Nicholas Zabaras, for his constant sup-port, motivation and guidance over the last 4 years. I would also like to thank Professors Ruediger Dieckmann and Lance Collins for serving on my special com-mittee and for their encouragement and suggestions during the course of this work. The financial support for this project was provided by NASA (grant 98-HEDS-05) and the Department of Energy (grant DE-FC07-ID0214396). Partial support from the Alcoa Technical Center is also acknowledged. I would like to thank the Sibley School of Mechanical and Aerospace Engineering for having supported me through a teaching assistantship for part of my study at Cornell. The computing for this project was supported by the Cornell Theory Center during 2002-2005. In particular, I would like to acknowledge the support from the latest velocity cluster (V3) that helped in significantly accelerating our simulators.
Most of the computer codes associated with this project including the optimiza-tion simulator were written using the object oriented programming environment of Diffpack and the academic license that allowed for these developments is ap-preciated. The parallel simulators were developed based on a combination of lo-cally developed parallel solvers and an open source scientific computation package PETSc. I would like to acknowledge the effort of its developers. I also acknowledge
Baskar Ganapathysubramanian, with whom I closely worked during the course of this project and others in the group for their valuable contributions. Finally, my thanks are extended to the publishers John Wiley and Sons, Ltd. and Elsevier Ltd. for granting permission to reproduce figures from our papers [85, 90, 94, 99].
Table of Contents vii
List of Tables x
List of Figures xii
1 Introduction 1
2 Stabilized finite element based numerical model for alloy
solidifica-tion 12
2.1 Mathematical Model . . . 13
2.1.1 Governing Equations . . . 17
2.2 Thermodynamic Relations . . . 21
2.3 A stabilized finite element scheme for the momentum equation . . . . 25
2.4 Stabilized finite element scheme for energy and species equations . . . 34
2.5 Coupling of various sub-problems and time integration . . . 39
2.6 Numerical Example . . . 40
2.7 Summary . . . 45
3 Extension of the numerical model to include shrinkage and anisotropy in the mushy zone 46 3.1 Mathematical model . . . 48
3.2 Stabilized finite element for fluid flow . . . 51
3.3 Finite Element Scheme for energy and species equations . . . 57
3.4 Numerical Examples . . . 59
3.4.1 Example 1: Horizontal solidification of a lead-tin alloy . . . . 60
3.4.2 Example 2: Vertical two-dimensional solidification of a lead-tin alloy . . . 64
3.4.3 Example 3: Vertical three-dimensional solidification of a lead-tin alloy . . . 65
3.5 Summary . . . 69
aluminum alloys from uneven surfaces 73
4.1 Solidification on uneven surfaces . . . 73
4.2 Problem Definition . . . 76
4.3 Vertical solidification from uneven surfaces . . . 77
4.4 Horizontal solidification from uneven surfaces . . . 83
4.5 Summary . . . 89
5 A coupled thermomechanical, thermal transport and segregation analysis of aluminum alloys solidifying on uneven mold surfaces 91 5.1 Mathematical Model . . . 93
5.1.1 Description of the solidification problem . . . 93
5.1.2 Description of the deformation problem . . . 94
5.2 Numerical algorithm and computational strategies . . . 97
5.2.1 Coupling of the various sub-problems and time integration . . 99
5.3 Numerical Investigations . . . 100
5.3.1 Effect of different sinusoid wavelengths . . . 104
5.3.2 Effect of melt superheating . . . 107
5.3.3 Effect of different initial solute concentrations . . . 109
5.3.4 Effect of different mold materials . . . 110
5.4 Summary . . . 112
6 Control of macrosegregation during alloy solidification using mag-netic fields 116 6.1 Direct Problem . . . 119
6.1.1 Volume Averaged Potential Equation . . . 120
6.2 Design problem using tailored magnetic fields . . . 122
6.2.1 Continuum sensitivity method (CSM) . . . 123
6.3 Computational Techniques . . . 128
6.4 Numerical Examples - Direct Problem . . . 129
6.4.1 Validation problem for the MHD alloy solidification model . . 130
6.4.2 2D horizontal solidification of a metal alloy . . . 133
6.4.3 2D directional solidification of a metal alloy . . . 135
6.5 Numerical Examples - Optimization Problem . . . 137
6.5.1 Validation of the CSM method . . . 137
6.5.2 Optimization problem for 2D horizontal solidification of an alloy138 6.5.3 Optimization problem for 3D directional solidification of an alloy . . . 141
6.6 Summary . . . 143
7 Conclusions and suggestions for the future research 146 7.1 Multi-length scale framework for alloy solidification processes . . . 148
A Material properties 150
List of Tables
2.1 Governing volume-averaged transport equations for a binary alloy solidification system. . . 22 2.2 A comparison between maximum velocities calculated here and those
reported in [8] . . . 42 2.3 A comparison between stream function values in the pure liquid and
porous region calculated here, and those reported in [8] . . . 42 3.1 Governing equations for solidification of alloys with different phase
densities . . . 48 3.2 Comparison between maximum and minimum midplane
concentra-tion of tin calculated here and those reported in [44] for a similar example. . . 62 3.3 Comparison of maximum velocity magnitudes between present
cal-culations in Example 2 and those given in [20]. . . 66 3.4 Comparison of quantities obtained from current calculations for
Ex-ample 3 and those given in [27]. . . 70 4.1 Maximum velocity magnitudes, (|vmax|), at t = 84 s with varying
amplitudes and constant wavelength in Example 4.3 (λ = 10 mm). . 80 4.2 Maximum velocity magnitudes, (|vmax|), at t = 84 s with varying
wavelengths and constant amplitude in Example 4.3 (A = 0.5 mm). . 81 4.3 Comparison of GES with varying amplitudes in Example 4.4 (λ =
10 mm). . . 86 4.4 Comparison of ∆C with varying amplitudes in Example 4.4 (λ = 10
mm). . . 86 4.5 Comparison of GES with varying wavelengths in Example 4.4 (A =
0.5 mm). . . 87 4.6 Comparison of ∆C with varying wavelengths in Example 4.4 (A =
0.5 mm). . . 87 4.7 Maximum velocity magnitudes (|vmax|) with varying amplitudes at
t = 60 s in Example 4.4 (λ = 10 mm). . . 88 4.8 Maximum velocity magnitudes (|vmax|) with varying wavelengths at
t = 60 s in Example 4.4 (A = 0.5 mm). . . 89
5.3 Comparison of variation in solute concentration for varying wave-lengths at t = 100 ms (A = 0.232 mm, C0 =5% wt, ∆Tsup = 0 oC,
mold material - copper, N = number of nodes). . . 113
5.4 Comparison of variation in solute concentration for different initial solute concentrations (C0) (A = 0.232 mm, ∆Tsup = 0 oC, λ = 5 mm, mold material - copper, N = number of nodes). . . 113
6.1 Governing equations for solidification of alloys under the influence of magnetic field. . . 118
6.2 Continuum sensitivity equations for alloy solidification under the in-fluence of magnetic field. . . 124
6.3 Auxiliary sensitivity relations for sensitivity equations defined in Ta-ble 6.2. . . 125
6.4 Conjugate gradient algorithm for the optimization problem. . . 126
6.5 Dimensions (in mm) of the cavity in Example 6.4.1. . . 130
6.6 Properties of the mold material in Example 6.4.1. . . 130
6.7 Comparison between maximum velocity magnitudes at two different times for Example 6.4.1. . . 132
6.8 Comparison of maximum velocity magnitudes at two different times for magnetic fields of 0 T and 5 T in Example 6.4.2. . . 134
6.9 Comparison of ∆C at two different times for Example 6.5.2 with no and optimal magnetic fields (∆C = Cmax− Cmin). . . 141
6.10 Comparison of ∆C at two different times for for Example 6.5.3 with no and optimal magnetic fields (∆C = Cmax− Cmin). . . 145
A.1 Important physical parameters for ammonium-chloride water system [8]. . . 151
A.2 Important physical parameters for lead-tin alloys [28, 56]. . . 152
A.3 Important physical parameters for aluminum-copper alloys,[41, 43]. . 153
List of Figures
1.1 Different length-scales in a typical solidification process with a schematic of various convection patterns [3]. . . 2 2.1 Simplified equilibrium phase diagram for a binary alloy system. The
regions shown here are used in the update formulae. . . 24 2.2 Problem domain and boundary conditions for solidification of an
aqueous binary alloy. . . 41 2.3 Solidification of N H4Cl−H2O mixture (a) velocity and mass fraction
(b) streamfunction (c) temperature (d) liquid concentration at t = 71 seconds. . . 43 2.4 Solidification of N H4Cl−H2O mixture (a) velocity and mass fraction
(b) streamfunction (c) temperature (d) liquid concentration at t = 142 seconds. . . 44 2.5 Final macrosegregation patterns for two different mesh sizes (a) 50x50
mesh (b) 60x60 mesh. . . 44 3.1 (a) Domain and (b) finite element mesh for Example 1. . . 60 3.2 Midplane concentration of tin (Sn) at different times for Example 1:
(a) 25 s, (b) 50 s, (c) 100 s, (d) 200 s, (e) 400 s and (f) 600 s. . . 62 3.3 Distribution of tin in Example 1 at different times - (a), (b) and (c)
correspond to combined flow, (d), (e) and (f) correspond to buoyancy driven flow: (a) and (d) 50 s, (b) and (e) 200 s, (c) and (f) 600 s. . . 63 3.4 Mass fractions and velocity distribution in Example 1 at different
times - (a), (b) and (c) correspond to combined flow, (d), (e) and (f) correspond to buoyancy driven flow: (a) and (d) 50 s, (b) and (e) 200 s, (c) and (f) 600 s. . . 63 3.5 Isotherms in Example 1 at different times - (a), (b) and (c)
corre-spond to combined flow, (d), (e) and (f) correcorre-spond to buoyancy driven flow: (a) and (d) 50 s, (b) and (e) 200 s, (c) and (f) 600 s. . . 64 3.6 Domain and boundary conditions for vertical solidification of a
lead-tin alloy with anisotropic permeability (Example 2). . . 66 3.7 Velocity field and liquid volume contours for Example 2 at different
times: (a) 100 s, (b) 200 s, (c) 300 s, (d) 2000 s and (e) 3000 s. . . . 67
3.9 Average concentration of tin every 0.5 mm in the vertical direction for Example 2. . . 68 3.10 Domain and boundary conditions for vertical solidification of a
lead-tin alloy with anisotropic permeability in three dimensions (Example 3). . . 69 3.11 Macrosegregation profile (distribution of tin) at two different times
for Example 3: (a) 600 s (b) 1800 s. . . 70 3.12 Liquid volume fractions at two different times for Example 3: (a)
600 s (b) 1800 s. . . 71 3.13 Macrosegregation profile (distribution of tin) on x − y plane at
dif-ferent times for Example 3: (a) z = 0.005 m, 600 s (b) z = 0.015 m, 1800 s. . . 71 4.1 Top shows non-uniform front and undesirable columnar grain
struc-ture on bottom side of ingot, respectively (smooth mold surface). Below shows parabolic front and desirable equiaxed grain structure on bottom side of ingot, respectively (sand blasted mold) (courtesy ALCOA Corp.) . . . 75 4.2 A mold surface with periodic ‘groove’ topography to control heat
extraction during directional solidification (courtesy ALCOA Corp.) 76 4.3 Domain and the mesh for the solidification of an Al-Cu alloy. . . 77 4.4 (a) Isotherms (b) liquid volume fraction (c) liquid concentration lines
at t1 = 66 s and t2 = 121 s for solidification with shrinkage in Example 4.3 (λ = 10 mm, A = 0.5 mm). . . 78 4.5 (a) Isotherms (b) liquid volume fraction (c) liquid concentration lines
at t1 = 66 s and t2 = 121 s for solidification with shrinkage in Example 4.3. (λ = ∞, A = 0, perfectly rectangular cavity) . . . 79 4.6 Midplane (x = 0.005 m) solute concentration profiles at t1 = 66 s
and t2 = 121 s for different amplitudes at a fixed wavelength for solidification with shrinkage in Example 4.3 λ = 10 mm. . . 79 4.7 Midplane (x = 0.005 m) solute concentration profiles at t1 = 66 s
and t2 = 121 s for different wavelengths at a fixed amplitude for solidification with shrinkage in Example 4.3 (A = 0.5 mm). . . 80 4.8 Midplane (x = 0.005 m) solute concentration profiles at t1 = 66s
and t2 = 121s for different wavelengths at a fixed amplitude for solidification without shrinkage in Example 4.3 (A = 0.5 mm). . . . 83 4.9 (a) Isotherms (b) solute concentration distribution (c) liquid solute
concentration (d) liquid volume fraction and velocity distribution at time t = 66 s in Example 4.4 with λ = 10 mm and A = 0.5 mm. . . 84 4.10 (a) Isotherms (b) solute concentration distribution (c) liquid solute
concentration (d) liquid volume fraction and velocity distribution at time t = 121 s in Example 4.4 with λ = 10 mm and A = 0.5 mm. . . 85
concentration (d) liquid volume fraction and velocity distribution at time t = 66 s for a horizontal cavity with even left surfaces in Example 4.4. . . 85 4.12 (a) Isotherms (b) solute concentration distribution (c) liquid solute
concentration (d) liquid volume fraction and velocity distribution at time t = 121 s for a horizontal cavity with even left surfaces in Example 4.4. . . 85 5.1 Alloy solidification from a mold with sinusoidal topography. The
computational domain for the solid, mushy and liquid regions is only a small portion of the total domain considered due to emphasis on the early stages of solidification. . . 93 5.2 Domain of the casting and mold for the solidification of an Al-Cu
alloy. Also shown are the boundary conditions for the solidification problem. . . 101 5.3 (a) Temperature in K (b) solute concentration (c) equivalent stress
in MPa (d) liquid mass fraction and velocity vectors at (i) t = 5 ms (|v|max = 0.355 m/s) and (ii) t = 100 ms (|vmax| = 0.095 m/s) for λ = 5 mm, C0 = 5% Cu and no superheating (A = 0.232 mm, mold material: copper). . . 103 5.4 (a) Temperature in K (b) solute concentration (c) equivalent stress
in MPa (d) liquid mass fraction and velocity vectors at (i) t = 5 ms (|v|max = 0.342 m/s) and (ii) t = 100 ms (|v|max = 0.090 m/s) for λ = 3 mm, C0 = 5% Cu and no superheating (A = 0.232 mm, mold material: copper). . . 105 5.5 Variation of the maximum air-gap size as a function of time for
dif-ferent wavelengths (C0 = 5 %, ∆Tsup = 0 oC, mold material: copper).106 5.6 Variation of the maximum equivalent stress as a function of time
for different wavelengths (C0 = 5 %, ∆Tsup = 0 oC, mold material: copper). . . 106 5.7 Variation of the maximum air-gap size as a function of time for
differ-ent melt superheating values (λ = 5 mm, C0 = 5 %, mold material: copper). . . 108 5.8 Variation of the maximum equivalent stress as a function of time
for different melt superheating values (λ = 5 mm, C0 = 5 %, mold material: copper). . . 108 5.9 Variation of the maximum air-gap size at trough as a function of
time for different solute (Cu) concentrations (λ = 5 mm, ∆Tsup = 0 oC, mold material: copper). . . 109 5.10 Variation of the maximum equivalent stress as a function of time for
different solute (Cu) concentrations (λ = 5 mm, ∆Tsup = 0oC, mold material: copper). . . 110
5.12 Variation of the maximum equivalent stress as a function of time for different mold materials (λ = 5 mm, C0 = 5 %, ∆Tsup = 0 oC.) . . . 112 5.13 Maximum equivalent stress at roots of the dendrites for the
solidi-fying alloy at t = 100 ms (λ = 5mm, ∆Tsup = 0 oC, mold material: copper). . . 114 5.14 Maximum equivalent stress in the solidifying alloy and front
uneven-ness at t = 100 ms. (∆Tsup = 0 oC, C0 = 5 %). Following [95], the position difference of ǫl= 0.7 corresponding to the crest and the trough is used here as a measure of front unevenness. . . 114 6.1 Domain and boundary conditions for Example 6.4.1. . . 129 6.2 (a) Temperature (K) (b) Volume fraction and velocity (c) Solute
concentration fields at t = 210 s for Example 6.4.1. . . 131 6.3 Solute concentration of tin (a) 66x68 mesh (b) 71x78 mesh for
Ex-ample 6.4.1. . . 131 6.4 Domain and boundary conditions for the 2D Pb-Sn alloy
solidifica-tion problem in Example 6.4.2. . . 133 6.5 (a) Temperature (K) (b) Solute concentration (c) Liquid volume
frac-tion and velocity at t = 120 s for (i) No magnetic field and (ii) a magnetic field of 5 T in Example 6.4.2. . . 134 6.6 Domain and boundary conditions for the 2D directional solidification
problem in Example 6.4.3. G = 7700 K/m, r = 0.2 K/s, C0 = 10 wt. % Sn and T0 = Tliq. . . 135 6.7 No magnetic field - (a) Solute concentration (b) Volume fraction
and velocity. Magnetic field of 3.5 T - (c) Solute concentration (d) Volume fraction and velocity. . . 136 6.8 Sensitivity fields of (a) concentration (b) horizontal velocity (c)
verti-cal velocity (d) temperature at t = 120 s for (i) continuum sensitivity method and (ii) finite difference method with B0 = 2 T and ∆B0 = 0.5 T. . . 137 6.9 Cost functional versus CGM iteration for Example 6.5.2 (a) 4 design
variables (b) 5 design variables. . . 139 6.10 Plot of the optimal magnetic field for Example 6.5.2 (a) 4 design
variables (b) 5 design variables. . . 139 6.11 (a) Temperature (K) (b) Solute concentration (c) Liquid volume
frac-tion and velocity at t = 120 s (optimal time varying magnetic field) for Example 6.5.2. . . 140 6.12 Problem domain for Example 6.5.3 involving 3D directional
solidifi-cation of a Pb-Sn alloy. G = 1000 K/m, r = 0.0167 K/s, C0 = 10 wt. % Sn and T0 = Tliq. . . 141 6.13 Cost functional versus CG iteration for Example 6.5.3. . . 142 6.14 Plot of the optimal magnetic field for Example 6.5.3. . . 143
(optimal time varying magnetic field) for Example 6.5.3. . . 144 6.16 (a) Solute concentration (b) Liquid volume fraction at t = 800 s (no
magnetic field) for Example 6.5.3. . . 144
Introduction
Solidification is one of the preferred methods for obtaining near net shaped objects in industry. A clear understanding of alloy solidification processes is essential in many industrial applications such as casting, welding and growth of single crystals. Dif-ferent applications impose difDif-ferent restrictions on the solidification process. Most alloys solidify with the formation of a two phase region known as mushy zone, which is composed of solid dendrites and interdendritic liquid. The morphology and size of the resulting solidification microstructure and properties of cast products depend on many factors including convection in the melt and mushy region during solidifi-cation. The most common causes of fluid flow in alloy solidification are thermal and solutal gradients, surface tension gradients, shrinkage and external forcing agents such as rotation, vibration, electromagnetic fields, etc. The difficulty associated with modeling solidification processes arises from the morphological complexity of the resulting microstructure and the variety of length- and time-scales in the system. Figure 1.1 shows two typical length-scales and some of the corresponding physical phenomena that are involved in solidification. At the macroscopic scale, fluid flow, convective-conductive heat transfer, macrosegregation, solid movement and
solid Mushy zone q liquid ~ 10-1 m (b) Microscopic scale ~ 10-4 – 10-5m solid liquid
(a) Macroscopic scale
Figure 1.1: Different length-scales in a typical solidification process with a schematic of various convection patterns [3].
mation are some of the mechanisms present. At the microscopic scale, interdendritic flow, latent heat release due to phase change, nucleation and microstructure forma-tion mechanisms are present. Other scales such as the solute diffusion length scale and the capillary length scale, [1]-[3], also exist in the solidification process. The mushy zone, where solid and liquid phases coexist, plays an important role in de-termining the final properties of a cast alloy. It is characterized by a highly diffuse solid-liquid interface and serves to reduce or eliminate regions of constitutional su-percooling in the system, [4]. The characteristic length scale of the mushy zone is at least two to three orders of magnitude smaller than the macroscopic length scale. The size and micro-morphology of mushy zones vary according to the chemical sys-tem being solidified, [4].
The main focus of this thesis is on the use of volume-averaged, single domain transport equations for modeling alloy solidification. Single-domain models, which
overcome many of the limitations of multidomain methods (like front-tracking meth-ods) emerged in the mid-1980s, [5]-[6], and have become useful tools for simulating solidification processes. These models consist of a single set of equations for momen-tum, energy and species transport in multi-constituent, solid-liquid, phase change systems, which are concurrently applied in all regions (solid, mushy and liquid). They require only a single, fixed numerical grid and a single set of boundary con-ditions to compute the solution. This brings a significant advantage in the numer-ical solution as there is no need to track the boundaries between phases. Instead, interfaces are implicitly defined by distributions of energy and composition deter-mined from solutions of the model equations (i.e. as post-processing operations). Some of the earliest continuum models for binary alloy solidification used classi-cal mixture theory to postulate macroscopic equations without reference to any microscopic equations [7]-[9]. Other developments in this direction can be found in [10]-[14]. A similar mathematical model in which, however, the macroscopic transport equations were derived from the classical microscopic transport equations using volume-averaging is presented in [15]. Models have also been presented in which conservation equations for each phase are solved separately with the aid of interphase transport methodologies [16, 17]. Almost all these works have used finite difference/finite volume methods for numerical implementation of the single domain model, [9, 8, 11, 17]. This thesis addresses the development of a computational framework based on a volume-averaged single domain mathematical model for alloy solidification in the presence of convection. Stabilized finite element methodologies are used for discretizing governing transport equations and their development in the context of alloy solidification problems is also discussed. Initial emphasis is on modeling solidification in the presence of strong thermosolutal buoyancy forces
and observing the influence of thermosolutal convection on macrosegregation. Ef-fects of shrinkage driven flows and anisotropy in the mushy zone permeability are incorporated later. The development of a generic model paves way for various ap-plications like studying the effect of uneven surface topography on convection and macrosegregation during alloy solidification.
Transport phenomena occurring during solidification of alloys is a major cause of casting defects such as segregation, micro-voids, hot tears, porosity, internal and surface cracks. Heat flow across metal and mold surfaces directly affects the phase change process and plays an important role in determining freezing conditions within the metal. Solidification of most alloys often results in macrosegregation or large scale variations in the concentration of solute elements. Inter-dendritic melt flow caused by thermal and solutal buoyancy forces, capillary forces and shrinkage in the mushy zone, arising from density differences between individual phases, is one of the leading causes of macrosegregation. Understanding the interaction between melt convection and solidification is of paramount importance since convection transports heat that affects the rate of solidification and transports solute elements leading to macrosegregation and redistribution of alloy constituents. Macrosegregation leads to non-uniform properties and various defects in the cast alloy that severely de-teriorate its performance and suitability in many applications. In metallic alloys, macrosegregation manifests itself in the form of defects such as freckles, channels, bleed bands, centerline segregates, A- and V-segregates etc. Removal of most of these defects either lead to significant material and monetary losses or render the casting unusable for further applications. In the aircraft industry for example, al-most 40 % of the directionally solidified single crystal blades are lost during castings. Box 1 summarizes few typical casting defects caused by segregation.
• Freckles and Channels • Centerline segregates • A- and V-segregates
• Inverse and Surface segregates • Bleed bands and Cold shuts • Subsurface segregates • Non-uniform microstructure
Box 1: Some casting defects caused by segregation.
Thermosolutal buoyancy forces are the primary cause of convection in alloys solidifying in terrestrial gravity conditions. These forces may be supplemented by thermocapillary or diffusocapillary forces, if surface tension effects are significant, and shrinkage, arising from density differences between solid and liquid phases. Shrinkage in a solidifying alloy caused by volume change during solidification re-sults in the casting being pulled away from the mold wall and induces defects such as hot tears and porosity. For an alloy solidifying upwards in a vertical cavity, shrinkage driven flows cause redistribution of solute in the melt and mushy zones. As a consequence, concentration of solute is higher near the bottom surface from where solidification initiates followed by a zone that is depleted of solute elements. This phenomenon is called inverse segregation and is a leading cause of defects in castings solidified from below. The role of thermosolutal convection in macrosegre-gation has been highlighted through experiments and numerical simulations starting from Flemings [1] and Kurz [2] to several other researchers. In recent years, Heinrich and co-workers in [18]-[20], Beckermann et al. in [17, 21, 22] and Incropera et al. in [8, 10, 11, 12], have focussed on the simulation of macrosegregation in binary alloys
leading to defects like freckles, channels and inverse segregates in alloys. Macroseg-regation caused by multicomponent thermosolutal convection has been simulated in solidifying steel alloys by Beckermann et al. in [23]-[24] and in Nickel base alloys by Felicelli et al. in [25]-[26]. Three-dimensional simulations of freckle formation have also been carried out in recent years by various researchers [27]-[29]. Criteria for predicting freckle formation during directional solidification of alloys have been derived by some researchers in recent years, [30]-[32], based on thermal or solutal Rayleigh numbers. The dominant role played by thermosolutal convection in causing macrosegregation was highlighted in these and several other works.
Events occurring during early stages of solidification play an important role in determining the final microstructure and solid macro-morphology. One of the im-portant objectives of the current study is to obtain a detailed understanding of these events and their role in the surface-defect formation process in cast aluminum alloys. In particular, the role of tuned mold surface topographies to minimize or eliminate these defects is explored. Very often during casting of metals or alloys, air-gaps form at the metal-mold interface leading to a non-uniform heat transfer rate into the cast-ing. This in turn affects other transport phenomena and stress-development in the solid-shell. Semi-analytical studies of air-gap nucleation during solidification of a pure metal on sinusoidal surface topographies were carried out in [33]-[35] using a thermo-hypoelastic perturbation theory neglecting plastic deformation. Gap nucle-ation times for different sinusoidal wavelengths and for different mold-shell material combinations were obtained in [33]-[35] and the concept of a critical wavelength was introduced to classify different surface topographies based on air-gap nucleation lo-cations. The deformation of solidifying bodies has been studied in [36]-[38] using a hypoelastic rate-dependent small-deformation model. The total strain was
ex-pressed as a sum of elastic, thermal and plastic strains. The mold surface here was however assumed to be planar and air-gap formation was not modeled. In [39], a thermo-mechanical analysis of solidification to predict the air-gap thickness at the metal-mold interface was presented, but segregation and solute transport were not modeled. In vertically solidified casts, inverse segregation is commonly observed at the bottom due to exudation and shrinkage driven flow. Exudation is the process of the interdendritic liquid being forced through the solid shell, past the original casting surface and into the air-gap [40]-[42]. This is usually caused either by the remelting of the evolving solid shell or due to the metallostatic pressure of the liquid column. This leads to the inverse segregation phenomenon and results in a heteroge-neous solute distribution leading to non-uniform mechanical properties and defects such as bleed bands, cold shuts and segregates in the cast product which increase its susceptibility to failure during further mechanical operations. In [40]-[42], a model of surface segregation in aluminum alloys driven by exudation and solidification shrinkage was presented. Air-gap formation in their model was expressed mathe-matically through a variable convective heat transfer coefficient at the boundary and not modeled dynamically. Effect of shrinkage driven fluid flow on segregation, arising during solidification of alloys, has also been modeled in [20, 43, 44]. Inverse segregation was shown to be primarily caused by shrinkage driven flows. Of special interest is the deformation and stress development in the mushy zone that with progressive solidification leads to residual stresses in the solid. This is important while studying the non-uniform contact and air-gap formation that occurs near the metal-mold interface during early stages of solidification. In [45]-[46], the effect of strain-rate relaxation on the stability of the solid front growth morphology, during solidification of pure metals on uneven surfaces, was studied using an experimentally
determined creep law. The development of constitutive relations for deformation of the mushy-zone has been addressed in [47]-[49] using both experimental and numer-ical investigations. In [47], a new hot-tearing criterion for metal alloys was proposed and a critical deformation rate introduced beyond which nucleation first started. In [48], a continuum model was presented for an isotropic two-phase mushy-zone and hot-tearing criteria for metal alloys were determined from variation in para-meters like casting speed, solidification interval and cooling contraction of the solid phase. Constitutive models for viscoplastic deformation and thermal strain in so-lidifying aluminum alloys were developed in [49] based on experimental studies and thermomechanical simulations. This constitutive model is used in our numerical model to study the effect of uneven mold surface topography on deformation of the mushy zone. The numerical model, developed for alloy solidification, is extended to model coupled phase change and deformation during the early stages of solidification of aluminum alloys. A parametric analysis is carried out to determine the effect of different process variables like melt superheat, mold surface wavelength, mold mate-rial and concentration of solute element on stresses developing in the solid shell and air-gaps nucleating at the mold-metal interface. The main aim here is to identify optimal surface features that help in reducing surface defects like unevenness in the solid shell.
Damping thermosolutal convection is indispensable for suppressing and mini-mizing macrosegregation in cast alloys. Box 2 lists different methods of controlling convection during solidification. Of these, the use of electromagnetic fields is of special interest to us. Magnetic fields have been extensively used to damp convec-tion in electrically conducting melts. The convecconvec-tion damping is achieved through the Lorentz force that is produced by induced currents in the moving fluid
inter-acting with the magnetic field lines. Magnetic fields have been extensively used for flow control in crystal growth processes involving semiconductor melts to suppress temperature and concentration fluctuations, [50]-[53]. Ben Hadid et al. in [54]-[55] analyzed the effect of a magnetic field on combined thermal buoyancy and thermo-capillary driven convection in horizontal Bridgman configurations in two and three dimensions to determine important scaling laws for Hartmann flows. In [56], Incr-opera et al. studied the effects of low magnetic fields on convection and macroseg-regation during solidification of a metallic alloy and concluded that the intensity of the magnetic field would have to be sufficiently increased to damp thermosolutal convection effectively.
• Control of the boundary heat flux
• Multiple-zone controllable furnace design • Rotation of the furnace
• Micro-gravity growth • Electromagnetic fields
Box 2: Different methods of controlling convection during solidification. Rotating magnetic fields have also been used to control convection during crystal growth and solidification, [57]-[58]. Zabaras et al. in [59] have studied the effect of magnetic fields on combined thermosolutal and thermocapillary convection dur-ing the solidification of alloys. Sampath and Zabaras in [60]-[61] solved an adjoint based inverse problem to design the boundary heat flux during the solidification of metals and alloys in the presence of magnetic fields. Gunzberger et al., [62], solved a design problem to determine the optimal magnetic field required for the suppression of turbulent flow in the melt during crystal growth processes. Recently,
Evans at al. in [63] have demonstrated the use of a magnetic gradient combined with a magnetic field to damp convection in both electrically conducting and non-conducting materials. Tagawa et al. in [64]-[65] have used magnetic gradients to control convection in oxygen and water. The use of magnetic field combined with gradients to control convection during crystal growth and solidification processes was highlighted by Zabaras and Ganapathysubramanian in [66]. In [67] and [68], an optimization problem was addressed by the same authors where the main objective was to design the time history of the imposed magnetic field in the presence of a con-stant magnetic gradient to control convection in solidifying melts. However, while solving such optimization problems, all authors have considered dilute alloys with negligible mushy zones and front tracking methods were used to describe the solid and liquid regions separately. In this thesis, an optimization problem is formulated based on the continuum sensitivity method (CSM) developed for alloy solidification with mushy zones. The magnetic field is assumed to be spatially constant but vary-ing in time. This eliminates the need to maintain a constant high magnetic field throughout the solidification process.
The organization of this thesis is as follows. In Chapter 2, the governing model for alloy solidification consisting of single domain volume-averaged transport equa-tions is first described. Details about mushy zone modeling are also described here. Stabilized finite element (FE) techniques for dicretizing governing equations are then described along with a discussion on the stabilized finite element method for the flow equation that takes into account mushy zone permeability. A numerical example is then worked out to validate the code and demonstrate convergence. In Chapter 3, the stabilized FE method is extended to incorporate the effect of shrinkage in the solidifying alloy and anisotropic permeability in the mushy zone. Additional,
relevant numerical examples in both two and three dimensions are addressed. In Chapter 4, solidification of aluminum alloys on uneven surfaces in the form of sinu-soids is simulated using the numerical model developed herein. Solidification in both horizontal and vertical configurations is analyzed to study macrosegregation caused by different types of convection. The effect of uneven surface topography on fluid flow and macrosegregation is observed. In Chapter 5, early stage solidification of aluminum alloys is modeled after incorporating deformation of the mushy zone and air-gap formation at the metal-mold interface. A parametric analysis is first car-ried out to determine the effect of various process parameters on stress development in the solid shell and air-gap formation at the metal-mold interface. An optimal mold surface wavelength is then determined that minimizes growth front uneven-ness and residual stresses in the solid shell. In Chapter 6, the effect of magnetic field on solidification of alloys with mushy zones is studied with the main aim of damping convection and suppressing macrosegregation. Sensitivity of field variables with respect to the magnetic field is then evaluated using the continuum sensitivity method. A design problem based on CSM is formulated for determining the optimal time varying magnetic field that helps in suppressing macrosegregation and related defects in solidifying alloys. Both two and three dimensional optimization problems are addressed. Finally, in Chapter 7, conclusions of this work and suggestions for future research are summarized.
Chapter 2
Stabilized finite element based
numerical model for alloy
solidification
1 This chapter discusses single domain volume-averaged governing transport equations used for simulating alloy solidification with mushy zones. Stabilized finite element methodologies used for discretizing governing equations are also described here. Other computational details like time integration are also provided followed by some numerical examples. The organization of this chapter is as follows. Section 2.1 describes briefly the volume averaging procedure along with governing equations and assumptions invoked in the model. Section 2.3 describes the stabilized finite element
1
Reproduced with permission from the International Journal for Numerical Methods in Engi-neering, Vol. 60, N. Zabaras and D. Samanta, “A stabilized volume-averaged finite element method for flow in porous media and binary alloy solidification processes”, 1103-1138, Copyright (2004) John Wiley and Sons Inc.
technique for momentum equations that is an extension of the SUPG(stream line upwind Petrov Galerkin)-PSPG (pressure stabilizing Petrov-Galerkin) technique, [69]-[71], and incorporates the effect of the mushy zone permeability. Section 2.4 describes stabilized finite element techniques for discretizing the thermal and solute species transport sub-problems. Coupling of various sub-problems and the time in-tegration procedure are described in Section 2.5. A numerical example is described in Section 2.6 followed by a summary in Section 2.7
2.1
Mathematical Model
For all examples described in this work, we consider a two-component, two-phase (solid and liquid) binary alloy system. The macroscopic transport equations can be obtained by averaging the microscopic transport equations over a finite size control volume that contains both phases. The averaging volume is defined such that the scale it represents is small enough to capture the global fluid flow motion, heat trans-fer, and species distribution, but large enough to smoothen out the details of the morphological complexities, inter-dendritic fluid flow, latent heat release and species redistribution. For such systems, like in solidification processes, the averaging vol-ume can track the overall nature of the liquid, solid and mushy regions without accounting for any details of the solidification microstructure. Under typical solidi-fication conditions, the system and interfacial structures are of the orders of 10−1 to 100 m and 10−5 to 10−4 m, respectively, so the size of the averaging volume in solid-ification varies between 10−3 to 10−2 m. The resulting averaged transport equations need to be supplemented by constitutive relations that describe morphological char-acteristics and interactions between both phases. The volume-averaging technique shows how various terms in the macroscopic equations arise and how the resulting
macroscopic variables are related to the corresponding microscopic variables, [72]. This gives considerable insight into the formulation of constitutive relations and is important for incorporating the evolution of the solid structure and transport phenomena at the micro level into a macroscopic model.
The microscopic (exact) mass, momentum, energy and species transport equa-tions for phase k (here k = s, l) are given by:
M ass ∂ρk ∂t + ∇ · (ρkvk) = 0 (2.1) M omentum ∂ρkvk ∂t + ∇ · (ρkvkvk) = ∇ · σk+ bk (2.2) Energy ∂ρkhk ∂t + ∇ · (ρkhkvk) = −∇ · qk (2.3) Species ∂ρkCk ∂t + ∇ · (ρkCkvk) = −∇ · jk (2.4) where σ is the stress tensor, b is the body force, q is the heat flux and j is the species diffusion flux. v, h, C and ρ denote the velocity, enthalpy, solute concentration and density, respectively. The detailed expressions for these terms are supplied by constitutive equations for specific cases. The energy equation is here written in terms of the total enthalpy. For simplicity, viscous heat dissipation, compression work, and volumetric energy and species sources are not included. Eqs. (2.1)-(2.4) above representing microscopic transport in each phase k take the general form:
∂Φk
∂t + ∇ · (Φk vk) = ∇ · Jk+ Sk (2.5)
for appropriate selection of the fields Φ, J and S. In order to provide some physical insight of the mathematical models examined in this thesis, we herein provide a brief review of the foundations of volume-averaging techniques and for more details the reader can consult references [72]-[73]. Let us introduce the phase function νk taking the value 1 in phase k and zero elsewhere. We can then define the volume fraction ǫk of phase k as follows:
ǫk = 1 dV Z dV νk(x, t)dv = dVk/dV (2.6)
where dVk is the portion of dV that is occupied by phase k. The volume-averaged quantity < Ψk> of any quantity Ψ(x, t) in phase k over the entire averaging volume dV can now be introduced as:
< Ψk>= 1 dV Z dV Ψkνk(x, t)dv (2.7)
Similarly, one can introduce the intrinsic volume-averaged quantity < Ψk>k (aver-aged value of Ψ(x, t) in the control volume dVk) as:
< Ψk >k= 1 dVk Z dV Ψkνk(x, t)dv = < Ψk > ǫk (2.8) When Ψk is uniformly distributed in dVk, then < Ψk >k= Ψk. The fluctuating component ˆΨk is commonly introduced to represent the deviation of Ψk from the intrinsic volume-averaged < Ψk>k. It is given by:
ˆ
Ψk = (Ψk− < Ψk>k)νk (2.9)
In phase k, ˆΨk is zero only when Ψk is uniformly distributed. Various volume-averaging formulae important to transport phenomena have been derived in [74]-[77]. They include, for example, a relation for the average of a product of two fields, the average of the time derivative in terms of the time derivative of the average, the average of the spatial derivative in terms of the spatial derivative of the average, etc. Multiplying each side of Equation (2.5) representing microscopic transport by νk, integrating it over the averaging volume dV and applying averaging formulae such as ones mentioned above, we obtain the following macroscopic transport equation for phase k:
∂< Φk >
∂t + ∇ · ǫk< Φk> k< v
+∇ · 1 dV Z dV (−cΦkcvk)dv + 1 dV Z dAk Jk· nkdA + 1 dV Z dAk Φk(wk− vk) · nkdA (2.10) where dAk is the interfacial area of phase k with the other phase, n is the outward unit normal of the infinitesimal element of area dA of phase k, and w is the velocity of the microscopic interface. Compared with the exact microscopic equations (2.5), three extra terms ID
k , IkJ and I Q
k appear from the volume averaging procedure of the form: IkD ≡ ∇ · 1 dV Z dV (−cΦkcvk)dv (2.11) IkJ ≡ 1 dV Z dAk Jk· nkdA (2.12) IkQ ≡ 1 dV Z dAk Φk(wk− vk) · nkdA (2.13)
At the microscopic scale characterizing the two phase region (e.g. the mushy zone in solidification processes), there always exist species, temperature, and velocity gra-dients in the liquid. Despite this fact, almost all models reported in the literature neglect the ID
k term [7, 76, 77]. The same approximation is considered here as well. The term IkQ accounts for the interfacial transfer due to phase change, whereas IJ k represents the transport phenomena between phases within dV by diffusion and is related to the gradients of microscopic velocity, temperature and species concentra-tion on each side of the solid/liquid interface dAk, [17].
In this work, the averaged macroscopic equations from different phases are added within the averaging volume dV , therefore, detailed modeling of the interfacial trans-fer terms IJ
k and I Q
k can be avoided. The heat or mass lost from one phase is gained by other phases, i.e.
X k
IkJ = 0 and X
k
The model discussed here is unsuitable for modeling interfacial behavior. Interfacial phenomena can be modeled with the help of two phase models that involve detailed balance of interfacial fluxes between individual phases [17, 77]. In the following section, we will utilize this model in a single fixed domain and present the averaged macroscopic equations of mass, momentum, heat and species in the context of a binary alloy solidification system. Enhancement of this model to allow modeling of binary alloy solidification is provided in Section 2.2. Detailed discussion on more general volume averaging models is provided, for example, in [3], [78].
2.1.1
Governing Equations
The volume averaging techniques developed in the earlier section are herein applied to the analysis of double-diffusive convective flow and of the associated heat and mass transfer that occur in a binary alloy solidification system. To arrive at a model tractable for computation, we assume that only the solid and liquid phases may be present, that is, ǫl+ǫs= 1. In addition, the variations of material properties in dVk are neglected, although globally they may vary, that is
Assumption 1: < ρk >k= ρk, < µk >k= µk, < kk >k= kk, < Dl >l= Dl, Ds= 0, the last condition implying negligible species (solutal) diffusion in the solid phase.
Assumption 2: All phases in the averaging volume are assumed to be in thermo-dynamic equilibrium, i.e. < Ts>s=< Tl >l and the liquid in the averaging volume is solutally well mixed, that is, < Cl >l= Cl.
For the derivation of the macroscopic equation of mass conservation using Equa-tion (2.10), we substitute Φ = ρ, J = 0, and S = 0. By writing and adding the individual macroscopic mass conservation equations for the two phases, neglecting
the microscopic deviation term and considering the interfacial mass flux balance, i.e. IlQ+ IQ
s = 0, and the conditions of Assumption 1, we obtain: ∂ρ
∂t + ∇ · (ρv) = 0 (2.15)
where we have further defined:
ρ = ǫl ρl+ ǫs ρs (2.16)
v = fl < vl >l= ρl
ρǫl < vl >
l (2.17)
where fk (k = s, l) denotes mass fraction of phase k, i.e. ρfk= ρkǫk.
Assumption 3: Note that we have assumed that < vs >s= 0, or the solid is stationary. In the context of solidification processes, such assumption for example will be appropriate for columnar but not equiaxed growth.
For deriving the macroscopic equation of momentum conservation from Eq. (2.10), we take Φ = ρv and S = b. Furthermore, we assume the liquid to be Newtonian and flow laminar. The viscous-stress in terms of the rate of deformation is therefore given as,
σ = −plI + µl[∇vl+ (∇vl)T] (2.18) As discussed earlier, we consider that ID
l = IsD = 0 and approximate < µl >l= µl. The interfacial momentum fluxes due to solidification balance each other, that is, IlQ+ IQ
s = 0. However, IlJ + IsJ = σχ, where σ is the surface tension, assumed to be constant, and χ is the mean curvature of the interface. Flow through a mushy zone consisting of a continuous solid structure (here assumed as columnar dendritic crystals), is usually very slow due to the high value of the interfacial area concentration. Therefore, the dissipative interfacial stress may be modeled in analogy with Darcy law as follows [3, 76]:
IlJ = −ǫ 2
l µl < vl>l K(ǫl)
where K(ǫl) is the permeability of the mushy zone. Values of the permeability have been reported for a columnar dendritic structure in the literature [79]. The permeability is commonly taken as isotropic and approximated using the Kozeny-Carman equation:
K(ǫl) =
K0 ǫ3l (1 − ǫl)2
(2.20) where K0 is a permeability constant depending on the morphology of the two-phase mushy region. K0 is usually approximated as K0 = d2/180, where d denotes the dendrite arm spacing. Using the previous assumptions and the definition of ρ and v given earlier, the final form of the macroscopic transport equation of momentum conservation then yields:
∂(ρ v) ∂t + ∇ · (ρ vv fl ) = − ǫl∇ < pl >l+∇ · " µl ∇ ρ ρl v + ∇ ρ ρl v T!# − ǫlµl ρ ρl v K(ǫl) + ǫlρlg (2.21)
where g is the gravity vector. The change in liquid density is here expressed using the Boussinesq approximation ρl = ρ0[(1 − βC(Cl− Cl0) − βT(T − T0)] and it appears
only in the body force term. The term −ǫl∇ < pl >l in Eq. (2.21) is written as follows:
−ǫl∇ < pl >l= −∇ < pl > +
< pl > ǫl
∇ǫl (2.22)
This modification is better suited for CFD applications with the second term on the right hand side treated as a source term.
For deriving the macroscopic equation of energy conservation we take Φ = ρh, where h represents the total enthalpy. In addition, we consider S = 0 and utilize Fourier’s law J = −k∇T . We also define the following:
ρh = ρlǫl < hl>l +ρsǫs < hs >s (2.23)
Eq. (2.10) then yields the following: ∂(ρh)
∂t + ∇ · (ρ < hl>
l v) = ∇ · (k∗∇T ) (2.25)
For arriving at the macroscopic equation of species conservation, we note that for this case, Φ = ρC, where C represents solutal concentration (per unit mass) and S = 0. Furthermore, we utilize Fick’s first law for species diffusion flux, that is, J = −ρD∇C. The macroscopic transport equation of species conservation can be derived from Eq. (2.10) as follows:
∂(ρC) ∂t + ∇ · (ρ < Cl> l v) = ∇ · (ρ lD∗∇ < Cl>l) (2.26) where ρC = ρlǫl< Cl>l +ρsǫs < Cs>s (2.27) D∗ = ǫlDl (2.28)
Assumption 4: To further simplify the above equations, we will assume that the densities of the two phases are constant and equal, i.e. < ρs >s= ρs = ρl =< ρl >l= ρ = ρ0, thus fs = ǫs and fl = ǫl. In the context of solidification, such an assumption will imply no solidification-induced shrinkage. Moreover, pore formation is not modeled here and the solidifying domain is always assumed to be occupied by the liquid and/or solid phases.
Assumption 5: Solidification microstructures are not modeled here and empirical relations are used to incorporate the effect of microstructure in the mushy zone permeability. Nucleation is also not modeled here.
The main focus here is to model solidification processes characterized by strong thermosolutal convection. The thermal and solutal Rayleigh numbers, defined by RaT = βT∆T |g|L3/νlαl and RaC = βC∆Cl|g|L3/νlαl, respectively, are the most
important parameters characterizing convection during the solidification process. Here, αl, νl, βT and βC denote the thermal diffusivity, kinematic viscosity, thermal expansion coefficient and solutal expansion coefficient of the liquid phase, respec-tively. ∆T = Ti − T0, ∆C = Cli − Cl0, L and g denote the problem dependent
characteristic temperature and concentration difference, characteristic length scale and magnitude of the gravity vector, respectively. The governing equations for the transport of mass, heat, momentum and solute during the solidification of a binary alloy are summarized in Table 2.1.
The binary mixture is confined in mass impermeable walls. For simplicity, a uniform initial temperature Ti and concentration Ci are assumed here. The region of interest is denoted by Ω ∈ ℜnsd, where n
sd is the number of the space dimensions. The region Ω has a piecewise smooth boundary Γ which consists of ΓT (boundary with prescribed temperature) and Γq (insulated boundary).
2.2
Thermodynamic Relations
For complete description of the single-domain volume averaged numerical model, supplementary relationships are required to describe the evolution of volume frac-tions, temperature and solute concentration fields in individual phases during so-lidification. These are based on the linear phase diagram, shown in Fig. 2.1 where both the solidus and liquidus lines are assumed to be straight lines with constant slopes, msol and mliq, respectively. The general expression for mixture enthalpy and concentration is expressed in terms of individual phase enthalpy and concentrations as
Table 2.1: Governing volume-averaged transport equations for a binary alloy solid-ification system. ∇ · v(x, t) = 0, (x, t) ∈ Ω × [0, tmax] (2.29) ρ ∂v(x, t) ∂t + v(x, t)∇ · v(x, t) ǫ(x, t) = −∇p(x, t) + p(x, t) ǫ(x, t)∇ǫ(x, t) − ǫµ K(ǫ)v(x, t) + ∇ · h µ∇v(x, t) + ∇ (v(x, t))Ti −ǫ(x, t)ρ0g[βT(T (x, t) − T0) + βC(Cl(x, t) − Cl0)] eg, (x, t) ∈ Ω × [0, tmax] (2.30) ρ ∂h(x, t) ∂t + v(x, t) · ∇hl(x, t) = ∇ · [(ǫ(x, t)kl+ (1 − ǫ(x, t))ks)∇T (x, t)], (x, t) ∈ Ω × [0, tmax] (2.31) ∂C(x, t) ∂t + v(x, t) · ∇Cl(x, t) = ∇ · (ǫDl∇Cl(x, t)) , (x, t) ∈ Ω × [0, tmax] (2.32) Initial conditions: v(x, 0) = 0, h(x, 0) = hi, C(x, 0) = Ci, x∈ Ω (2.33) Boundary conditions: v(x, t) = 0, x∈ Γ (2.34) h(x, t) = hs, x∈ ΓT (2.35) ∂h ∂n(x, t) = 0, x∈ Γ q (2.36) ∂C ∂n(x, t) = 0, x∈ Γ (2.37)
C = Clǫ + Cs(1 − ǫ) (2.39) where h and C denote the enthalpy and solute concentration, respectively, with the subscripts s and l denoting corresponding variables for the solid and liquid phases. By assuming that all phases in the averaging volume are in thermal equilibrium, that is, Tl = Ts= T , where T denotes the temperature, hl and hs are expressed as
hs = cpsT (2.40)
hl = cplT + (cps− cpl)Te+ hf (2.41) where specific heats of solid and liquid phases, cpsand cpl, are assumed to be constant and do not vary with temperature. In Eq. (2.41), Te denotes the eutectic temper-ature and hf the latent heat. The mixture enthalpy in terms of the temperature is now given by
h = [ǫ cpl+ (1 − ǫ) cps] T + ǫ(cps− cpl)Te+ ǫhf (2.42) The ratio of the slopes, mliq and msol, is denoted by the constant κp, the partition coefficient of the alloy. We assume that the liquid within the averaged volume is solutally well mixed, that is, Cl = Cliq. In the mushy zone the temperature is therefore, related to Cl through the liquidus line as
T = Tm+ mliqCl (2.43)
where Tm is the melting temperature. The liquidus temperature is defined as, Tliq = Tm + mliqC. The evolution of the volume fraction can be determined using two different expressions given below [1, 2]:
Lever rule : ǫ = 1 − 1 1 − κp T − Tliq T − Tm , (2.44) Scheil rule : ǫ = T − Tm Tliq− Tm 1 κp−1 , (2.45)
C Ceut Tm Teut Region 2 (Mushy) Region 3 (Solid) Eutectic Point T Region 1 (Liquid)
Figure 2.1: Simplified equilibrium phase diagram for a binary alloy system. The regions shown here are used in the update formulae.
The Lever rule, given by Eq. (2.44), assumes infinite solute diffusion in the solid phase. The Scheil rule, also called the non-equilibrium lever rule and given by Eq. (2.45), assumes zero solute diffusion in the solid phase while the solute in the liquid phase is assumed to be well mixed. In our numerical model, we use the Scheil rule (Eq. (2.45)) for calculating liquid mass fractions from temperature and solute concentration in the liquid phase. Cs is calculated as
Cs = 1 1 − ǫ Z 1 ǫ kpCl dǫ = I 1 − ǫ, (2.46)
where I is the required integral updated at a particular time step using the following relation:
In+1 = In+ 0.5κp(Cl,n+ Cl,n+1)(ǫn− ǫn+1). (2.47) After substituting Eqs. (2.41) and (2.42) into Eq. (2.31), the governing equation for energy in terms of enthalpy is transformed in terms of temperature as
ρc∗p(x, t)∂T (x, t)
= ∇ · (k∗(x, t)∇T (x, t)) − ρS∗(x, t)∂ǫ(x, t)
∂t (2.48)
with S∗(x, t), c∗
p(x, t) and k∗(x, t) defined as follows:
S∗(x, t) = (cpl− cps)(T (x, t) − Te) + hf (2.49) c∗p(x, t) = ǫ(x, t)cpl+ (1 − ǫ(x, t))cps (2.50) k∗(x, t) = ǫ(x, t)kl+ (1 − ǫ(x, t))ks (2.51) The update formula for Cl and ǫ is now given in terms of T and C as follows:
• If T > Tliq, then the averaging volume is occupied by pure liquid and ǫ = 1 and C = Cl.
• If T < Te, then the averaging volume is occupied by solid and ǫ = 0 and Cl = Ce.
• If Te < T <= Tliq, then the averaging volume is occupied by the mushy zone ǫ = T −Tm
Tliq−Tm
1 κp−1
and Cl = T −Tmliqm.
• If T = Te, then a eutectic reaction occurs with Cl = Ce and ǫ = ǫe = Te−Tm Tliq−Tm 1 κp−1 .
2.3
A stabilized finite element scheme for the
mo-mentum equation
Let us define the function spaces Sv and Sp as follows:
Sv def= {v|v ∈ (L2(Ω))nsd, divv ∈ L2(Ω), v = 0 on Γ} (2.52) Sp def = {p|p ∈ L2(Ω), Z Ω pdΩ = 0} (2.53)
The classical Galerkin formulation for the flow problem in Table 2.1 can be stated as follows: Find V def= {v, p} ∈ Sv × Sp such that for all W
def
= {w, q} ∈ Sv × Sp the following holds:
B(W , V ) = L(W ) (2.54) where B(W , V ) = Z Ω w· ρ∂v ∂t + ρv · ∇ v ǫ + (1 − ǫ) 2 ǫ2 µ K0 v dΩ − Z Ω p∇ · wdΩ + Z Ω µ ∇w · ∇v + (∇v)TdΩ + Z Ω q ∇ · vdΩ L(W ) = Z Ω p ǫw· ∇ǫdΩ − Z Ω w· ǫρ0g(βT(T − T0) + βC(Cl− Cl0))egdΩ (2.55)
The above formulation theoretically works only for certain velocity and pressure interpolations. In the finite element implementation of the Navier-Stokes equa-tions, stabilizing techniques are needed to accommodate equal-order-interpolation velocity-pressure elements. The most common stabilization methods are the SUPG (Streamline-Upwind/Petrov-Galerkin) and the PSPG (Pressure-Stabilizing/Petrov-Galerkin) formulations proposed by many researchers (e.g. see Tezduyar et al. [69]-[71]). These stabilizing terms can be obtained by many approaches including by minimizing the squared residual of the momentum equation, i.e. following a least squares FEM approach, [80]. As shown in [81], a proper stabilization term for pure Darcy flows is needed that differs in sign from that obtained by regular least squares approach. In introducing the FEM used here, let us first define a modified pressure space S′ p as follows: Sp′ def= {p|p ∈ H1(Ω), Z Ω qdΩ = 0} (2.56)
The stabilized weak form proposed here is the following: Find V = {v, p} ∈ Sv ×S′ p such that for all W = {w, q} ∈ Sv × Sp′ the following holds:
where: Bstab(W , V ) = B(W , V ) + Z Ω F(v, p) · G(w, q)dΩ (2.58) Lstab(W ) = L(W ) + Z Ω {p ǫ∇ǫ − ǫρ0g(βT (T − T0) + βC (Cl− Cl0))eg} · G(w, q)dΩ (2.59) where we defined: F(v, p) = ρ∂v ∂t + ρv∗· ∇ v ǫ + ∇p + (1 − ǫ) 2 ǫ2 µ K0 v− µ∇2v (2.60) G(w, q) = τ1v∗· ∇ w ǫ − τ2 (1 − ǫ)2 ǫ2 µ K0ρ w− τ3 µ ρ∇ 2w+ τ4 ρ∇q (2.61) with v∗ a divergence-free velocity that in the implementation of Eq. (2.57) at a given time is usually taken as the known velocity at the previous time step. The stabilizing terms in Equations (2.58) and (2.59) can be derived by various techniques including a least squares procedure. However, note that the sign of the Darcy term used here is the reverse of that obtained by least squares [81]. The particular values of the parameters τ1, . . . , τ4 used in this work are introduced later in this section.
Let us now consider a given finite element partition ¯Ω =Snel
e=1Ω¯e. In this work, we assume that ǫ varies linearly within each element and is computed at the nodes as discussed in Section 2.2. To avoid high sensitivity of the fluid flow simulator on the variation of ǫ, a constant value of ǫ is taken for each element in the implementation of Eq. (2.57). Thus the value of ǫ used in the flow simulator is evaluated at the centroid of each element. With this assumption, the terms in Eq. (2.57) containing gradients of the porosity ǫ can be neglected. However, it is noted that a piece-wise linear variation of ǫ is maintained in the heat and mass transfer solvers.
We next define for the given finite element partition the spaces Sh
v and Sph′ as follows:
Svh def= {vh|vh ∈ S
Sph′ def= {ph|ph ∈ Sp, ph ∈ Co( ¯Ω), ph|Ωe ∈ P (Ωe), e = 1, 2, . . . , nel} (2.63)
The final FEM is then posed as follows: Find Vh = {vh, ph} ∈ Sh
v × Sph′ such that for all Wh = {wh, qh} ∈ Sh
v × Sph′ the following holds:
Bstabh (Wh, Vh) = Lhstab(Wh) (2.64) with Bstabh (Wh, Vh) = Bh(Wh, Vh) + nel
A
e= 1 Z Ωe Fh(vh, ph) · Gh(wh, qh)dΩe (2.65) Lhstab(Wh) = Lh(Wh) + nelA
e= 1 Z Ωe −ǫρg (βT (Th− T0) + βC (Clh− Cl0))eg· G h(wh, qh)dΩe (2.66)where we have defined Fh(vh, ph) and Gh(wh, qh) as: Fh = ρ∂vh ∂t + ρv h ∗ · ∇ vh ǫ + ∇ph+ (1 − ǫ) 2 ǫ2 µ K0 vh− µ∇2vh (2.67) Gh = τe 1vh∗ · ∇ wh ǫ − τe 2 (1 − ǫ)2 ǫ2 µ K0ρ wh− τe 3 µ ρ∇ 2wh+τ4e ρ∇q h (2.68)
Lh(Wh) is defined from Eq. (2.55) by neglecting the contribution of the pǫ∇ǫ term as discussed earlier. The stabilizing contributions from the advective, Darcy, viscous and pressure terms are expressed as:
(A) δh = 1ǫτe 1vh∗ · ∇wh (advection term) (B) γh = −τ2e (1−ǫ)2 ǫ2 µ ρK0w h (Darcy term) (C) ζh = −τe 3 ν∇2wh (viscous term) (D) ηh = τ4e ρ∇q h (pressure term)
It is easy to identify that the form of the stabilizing term δh corresponds to the classical SUPG stabilizer and the form of ηh to the classical PSPG stabilizer,
respectively. At the element interiors the contribution to the weighting function from the viscous term ζhis identically zero for bilinear elements used for all examples here. The last stabilizing term, γh, which we call DSPG, comes from the Darcy term and it is an important stabilizing term for the generalized Navier-Stokes/Darcy equations. The stabilizing parameters for the advective and pressure terms are selected as follows: τ1e = min x∈Ωe τSU P G, ǫ2(x) (1 − ǫ(x))2 K0ρ µ
for the convective term, (2.69)
τ4e= min x∈Ωe τP SP G, ǫ2(x) (1 − ǫ(x))2 K0ρ µ
for the pressure term (2.70) where τSU P G, τP SP G are defined as
τSU P G = ǫh 2kvhkz(Rev) (2.71) τP SP G = ǫh# 2kVhkz(ReV ) (2.72)
Here, Rev and ReV are the element Reynolds numbers, which are based on the local velocity vh and a global scaling velocity Vh and are given by
Rev = kv hkh 2ν (2.73) ReV = kV hkh# 2ν (2.74)
The element length h is computed by using the expression h = 2 nen X e=1 |s.∇Nαe| !−1 (2.75) where nenis the number of nodes in the element, Nαe is the basis function associated with the local node α, and s is the unit vector in the direction of the local velocity [69]-[71]. The element h# on the other hand is defined to be the diameter of the
circle which is area equivalent to the element. The function z(Re) is defined as follows: z(Re) = Re/3 0 ≤ Re ≤ 3 1 3 ≤ Re (2.76) Stabilization forms similar to that in Eq. (2.69) have been introduced earlier for the Stokes problem [82]-[84]. For the Darcy term, we select τe
2 such that the stabilization takes the form γh = −(1 − ǫ)wh, i.e.
τe 2 = ǫ2 1 − ǫ K0ρ µ (2.77)
Remark 1: The stabilization term γh induced by the Darcy flow varies linearly with ǫ transitioning from γh = 0 for a pure fluid phase (ǫ = 1) to γh = −wh towards a pure solid phase (ǫ → 0). Note also that for ǫ = 0 (pure solid), the advection stabilization term δhusing Eq. (2.69) becomes δh = limǫ→01ǫ
h ǫ2 (1−ǫ)2 K0ρ µ i = 0. Thus the selection in Eq. (2.69) allows for a stabilizing advection term that transitions from the classical SUPG values for a pure fluid to a zero contribution for a pure solid.
Remark 2: The stabilizing terms are linearized using quantities from the earlier time step or at the previous iteration at the current time step. For example, the advection stabilizing term at a particular time step is linearized using the velocity from the previous iteration denoted by v∗. The Darcy stabilizing function, γh, is linearized using the liquid volume fraction, ǫ, known prior to the fluid flow solution at a particular time step. The stabilizing parameters τe
1 and τ4edefined in Equations (2.69) and (2.70) are also linearized using the above mentioned procedures. Numer-ical simulations show that these linearizations do not affect the convergence rate in any significant way.
ordinary differential equations: [M + Mδ+ Mγ]{ ˙v} + [N(v) + Nδ(v) + Nγ(v)]{v} + [K + Kδ+ Kγ]{v} + [D+Dδ+Dγ]{v}− [G+Gδ+Gγ]{p} = {F(T, Cl)+Fδ(T, Cl)+Fγ(T, Cl)}, (2.78) Mη{ ˙v} + GT + Nη(v) + Kη+ Dη {v} + Gη{p} = {Fη(T, Cl)}, (2.79) where {v} is the vector of nodal values of vh, { ˙v} is the time derivative of {v}, {p} is the vector of nodal values of ph. The matrices M, N(v), K, D and G are derived, respectively, from the time-dependent, advective, viscous, Darcy and pressure terms. Note that only explicit dependencies on primary solution variables are shown. The vector F is derived from the buoyancy term. The subscripts δ, η and γ identify the SUPG, PSPG and DSPG stabilizing terms, respectively. The subscript p is used here to denote the global velocity degrees of freedom in the final system of equations that corresponds to the ith velocity degree of freedom of the elemental node α (α = 1, . . . , ne
nodes) in the eth element. Similarly, one can define q. We also
introduce r and s to denote the global pressure degrees of freedom. We also denote the SUPG contribution as Pe
α, the PSPG contribution in the ith (i = 1, ..., nsd) direction as Ee
αi and with Deα the DSPG contribution. They are defined as follows:
Pαe = 1 ǫτ e 1vh· ∇Nαe (2.80) Eαie = τ e 4 ρ N e α,i (2.81) Dαe = −(1 − ǫ)Nαe (2.82)
where ǫ in the above and following equations is referred to the constant value com-puted for the element e.
