Computational grid size

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Maco-Micro Modeling Simple methods for
incorporating small scale effects into large
scale solidification models Vaughan Voller,
University of Minnesota
Scales in a simple solidification process model
Computational grid size
Enthalpy based Dendrite growth model
Can we build a direct-simulation of a Casting
Process that resolves to all scales?
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Scales in Solidification Processes
(after Dantzig)
Can we build a direct-simulation of a Casting
Process that resolves to all scales?
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Well As it happened not currently Possible
1000 20.6667 Year Moores Law
2055 for tip
Voller and Porte-Agel, JCP 179, 698-703 (2002)
Plotted The three largest MacWasp Grids (number
of nodes) in each volume
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Scales in Solidification Processes
(after Dantzig)
To handle with current computational Technology
require a Micro-Macro Model
See Rappaz and co-workers
Example a heat and Mass Transfer model Coupled
with a Microsegregation Model
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Solidification Modeling
Process
REV
representative ½ arm space
solid
g
sub-grid model
50 mm
Micro segregationsegregation and solute
diffusion in arm space
5 mm
0.5 m
Computational grid size
from computation Of these values
need to extract
--
--
--
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Primary Solidification Solver
g
Transient mass balance
g
model of micro-segregation
Iterative loop
Cl
T
(will need under-relaxation)
Give Liquid Concentrations
equilibrium
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Micro-segregation Model
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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 on 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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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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Constant Cooling of Binary-Eutectic Alloy With
Initial Concentration C0 1 and Eutectic
Concentration Ceut 5, No Macro segregation
, k 0.1
Use 200 time steps and equally increment 1 lt Cl
lt 5
Calculating the transient value of g from
Remaining Liquid when C 5 is Eutectic Fraction
Parameter or Profile
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Results are good across a range of conditions
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Predictions of Eutectic Fraction With constant
cooling
Co 4.9 Ceut 33.2 k 0.16
Comparison with Experiments Sarreal Abbaschian
Met Trans 1986
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Parabolic solid growth No Second Phase No
Coarsening Use 10,000 equal of Dg C0 1,
k 0.13, a 0.4
Use
To calculate evolving segregation ratio
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Performance of Models under parabolic growth no
second phase
in last liquid to solidify
Prediction of segregation ratio
(fit exponential through last two time points)
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My Method of Choice
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I Have a BIG Computer Why DO I need an REV and a
sub grid model
solid
50 mm
5mm
(about 106 nodes)
.5m
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Application Inverse Segregation in a binary
alloy
Shrinkage sucks solute rich fluid toward chill
results in a region of ve segregation at chill
100 mm
Fixed temp chill results in a similarity
solution
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Comparison with Experiments
Ferreira et al Met Trans 2004