Key Issues in Solidification Modeling

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Key Issues in Solidification Modeling Vaughan
Voller, University of Minnesota, Aditya Birla
Chair
Validation and Verification
Do governing equations model the correct physics?
Is approximate solution a solution of governing
Equations ?
Ferreira et al
How can we deal with this problem Micro-macro
models
How feasible is Direct Modeling of
Microstructure? What can it tell us about the
process Scale?
Prediction Of microstructure
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Scales An Example Problem MacrosegregationIngot
alloy solidification
Result after full solidification is macro-scale
areas with concentration above or Below the
nominal concentration see Flemings
(Solidification Processing) and Beckermann
(Ency. Mat)
solid crystals liquid mushy region
This solute is redistributed at process scale by
fluid and solid motions
shrinkage
solid
grain motion
liquid alloy
convection
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Key Scales in Macro-segregation
Computational grid size
Solute value in liquid phase controlled by local
diffusion in solid microsegregation
Solute transport controlled by advection
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Scales in General Solidification Processes
(after Dantzig)
Question for later Can we build a
direct-simulation of a Casting Process that
resolves to all scales?
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A solidification model has three components The
Domain The Grid The sub-Grid Examples
The Grid
Problem Domains
Sub-grid --Constitutive -- Controlled by
averaged Properties in REV
Realizations--Of multi-phase regions Element in
numerical Calculation ---REV State described by
averaged mixture values
Macro Process
Effect of morphology on flow
REV
METER
Meso
Microstructure
The Grain Envelope
Solid-liquid interface
A representative Arm spacing Form of
Constitutive model
TF(g)
f 1 f -1
The Diffusive Interface, e.g.
NANO-Meter
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Key Scales in Macro-segregation
Computational grid size
Solute value in liquid phase controlled by local
diffusion in solid microsegregation
Solute transport controlled by advection
Develop a Macro-Micro Model (Rappaz) Solve
transport equations at macroscopic scale
(MACRO) Use sub-grid model to account for
microsegregation (MICRO) a constitutive model
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Macro (Process) Scale Equations
Equations of Motion (Flows)
mm
REV
Heat
Solute Concentrations
Assumptions for shown Eq.s -- No solid motion
--U is inter-dendritic volume flow
If a time explicit scheme is used to advance to
the next time step we need find REV values for

• T temperature
• Cl liquid concentration

• gs solid fraction
• Cs(x) distribution of solid concentration

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The Micro-Macro Model
MACRO
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Primary Solidification Solver
g
Transient mass balance
g
model of micro-segregation
Iterative loop
Cl
T
(will need under-relaxation)
Gives Liquid Concentrations
equilibrium
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Micro-segregation Model
liquid concentration due to macro-segregation
alone
½ Arm space of length l takes tf seconds to
solidify
In a small time step new solid forms with lever
rule of concentration
Need an easy to use approximation For
back-diffusion
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The parameter Model --- Clyne and Kurz,
Ohnaka
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The Profile Model
Wang and Beckermann
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An Important wrinkle ---Coarsening
Due to dissolution processes some arms will
melt and arm-space will coarsen
Time 1
Time 2 gt Time 1
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Arm-space will increase in dimension with time
Coarsening
This will dilute the concentration in the liquid
fractioncan model be enhancing the


back
diffusion ?
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Verification of Micro Models Constant Cooling
of Binary-Eutectic Alloy With Initial
Concentration C0 1 and Eutectic Concentration
Ceut 5, No Macro segregation , k
0.1
As solidification proceeds the concentration in
liquid increases.
When the eutectic composition is reached
remaining liquid solidifies isothermally,
Eutectic Fraction
In model calculate the transient value of g from
Use 200 time steps and equally increment 1 lt Cl
lt 5
Parameter or Profile
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Verification of Micro Models Verify approximate
model for back-diffusion by comparing solution
with FD solution of Ficks equation in arm space.
Parameter or Profile
Remaining Liquid when C 5 is Eutectic Fraction
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Validation of Micro Model
Predictions of Eutectic Fraction With constant
cooling
Co 4.9 Ceut 33.2 k 0.16
Al-4.9 Cu
Comparison with Experiments Sarreal-Abbaschian
Met Trans 1986
X-ray analysis determines average eutectic
fraction
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My Method of Choice
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Modeling the fluid flow could require a Two
Phase model, that may need to account for Both
Solid and Liquid Velocities at low solid
fractions A switch-off of the solid velocity in a
columnar region A switch-off of velocity as solid
fraction g ? o.
An EXAMPLE 2-D form of the momentum equations in
terms of the interdentrtic fluid flow U, are
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Verification of Macro-Micro ModelInverse
Segregation in a Binary Alloy
Shrinkage sucks solute rich fluid toward chill
results in a region of ve segregation at chill
100 mm
Flow by simple app. of continuity
Fixed temp chill results in a similarity
solution
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Validation Comparison with Experiments
100 mm
Ferreira et al Met Trans 2004
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Direct Microstructure Modeling
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Example Growth of dendritic crystal in an
under-cooled melt
(seminar on July 14)
Solved in ¼ Domain with A 200x200 grid
Growth of solid seed in a liquid melt Initial
dimensionless undercooling T -0.8 Resulting
crystal has an 8 fold symmetry
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Grid independent results with correct dynamics
can be readily obtained
Tip Velocity
Interacting grains
grid anisotropy
prediction of concentration field
Scale of calculations shown
1 mm
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So can we use DMS to predict microstructure at
the process level?
Sub grid scale
For 2-D calc at this scale Will need 1018 grids
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For 2-D calc at this scale Will need 1018 grids
Voller and Porte-Agel, JCP 179, 698-703 (2002
1000 20.6667 Year Moores Law
2055
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CONCLUSIONS
Conclusion Can Currently Build Validated and
Verified Models that Can successfully model
across 4 decades Of length scales
Able to use Macro-Micro Approach To model all
scales of Heat and Mass Transport
Able to build Local Microstructure models
But a long way from DMS Direct microstructure
simulation at the process scale
In the meantime what Value Added can we get from
Local microstructure models
Use as generator for constitutive models Use in
volume averaging approaches