Computational grid size
l
TLe
~0.5 m
Process
~5 mm
REV
Maco-Micro Modeling—Simple methods for incorporating small scale effects into large scale solidification models– Vaughan Voller, University of Minnesota
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Can we build a direct-simulation of a Casting Process that resolves to all scales?
Scales in a “simple” solidification process model
~ 50 m
solid
representative ½ arm space
sub-grid model
g
Enthalpy basedDendrite growth model
chill
A Casting
The REV
NucleationSites
columnar equi-axed
The GrainEnvelope
The SecondaryArm Space
The TipRadius
The DiffusiveInterface
~ 0.1 m
~10 mm
~ mm
~100 m
~10 m
~1 nm
103
101
10-1
10-3
10-5
10-7
10-9
10-9 10 10-3 10-1
Length Scale (m)
interfacekinetics
nucleation
solute diffusion
growth
grainformation
casting
heat and mass tran.
Time Scale (s)
Scales in Solidification Processes
2 of 19
(after Dantzig)
Can we build a direct-simulation of a Casting Process that resolves to all scales?
1.0E+02
1.0E+04
1.0E+06
1.0E+08
1.0E+10
1.0E+12
1.0E+14
1.0E+16
1.0E+18
1.0E+20
1.0E+22
1.0E+24
1.0E+26
0 20 40 60 80 100 120
Year-1980
No
de
s
Well As it happened not currently Possible
1000 20.6667 Year
“Moore’s Law”
Voller and Porte-Agel, JCP 179, 698-703 (2002) Plotted The three largest MacWasp Grids (number of nodes) in each volume
2055 for tip6 decades
1 meter
1 micron
3 of 19
chill
A Casting
The REV
NucleationSites
columnar equi-axed
The GrainEnvelope
The SecondaryArm Space
The TipRadius
The DiffusiveInterface
~ 0.1 m
~10 mm
~ mm
~100 m
~10 m
~1 nm
103
101
10-1
10-3
10-5
10-7
10-9
10-9 10 10-3 10-1
Length Scale (m)
interfacekinetics
nucleation
solute diffusion
growth
grainformation
casting
heat and mass tran.
Time Scale (s)
Scales in Solidification Processes
To handle with current computational Technology require a “Micro-Macro” Model
See Rappaz and co-workers
Example a heat and Mass Transfer modelCoupled with a Microsegregation Model
4 of 19
(after Dantzig)
l
TLe
~0.5 m
~ 50 m
solid
~5 mm
Computational grid size
Process REVrepresentative ½ arm space
sub-grid model
]C[ll
0
ss C)g1(dCg
g
]H[ HTc)g1(Tcg llss
from computationOf these values need to extract
lCg g
0
s
s
s dCg
1C T
Solidification Modeling
-- -- --
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Micro segregation—segregation and solute diffusion in arm space
C
A
C
)C,....,C,C(GTrk
l
r2l
r1l
r
Primary Solidification Solver
H
Tc]H[)g1(
l
1rpr
Transient mass balance
equilibrium
g
ClTIterative loop
g
model of micro-segregation
(will need under-relaxation)
6 of 19
Give Liquid Concentrations
oldl
oldl
oldoldl C)g1(CC)g1(C
liquid concentration due to macro-segregation alone
Micro-segregation Model
oldg
g
In a small time step new solid forms with lever rule on concentration C
C
Q -– back-diffusion
Need an easy to use approximationFor back-diffusion
transient mass balance gives liquid concentration
QC)g1(C)gg(C)g1( slr
lloldr
sold
l
Solute mass densitybefore solidification
Solute mass densityof new solid (lever)
Solute mass densityafter solidification
7 of 19
g
sCQ
x,
t
t
f
2fDt
Solute Fourier No.
½ Arm space of length takestf seconds to solidify
)CC(gdt
dCgtQ old
lll
The parameter Model --- Clyne and Kurz,
)Scheil(0
ddg
g)lever(1
Ohnaka 8 of 19
2
1
d
dgg
21
g
For special case Of Parabolic Solid Growth
And ad-hoc fit sets the factor
2
2and
14
4
In Most other casesThe Ohnaka approximation
21
2
Works very well
QC)g1(C)gg(C)g1( slr
lloldr
sold
l
)CC)(1m(g
Qs
sl
The Profile Model Wang and Beckermann
baC ms
Need to lagcalculation onetime step andensure Q >0
9 of 19
g
0
m
s
s
s b1m
gadC
g
1C
QC)g1(C)gg(C)g1( slr
lloldr
sold
l
1
s
s
l
C
C2~m
m is sometimes take as a constant ~ 2 BUTIn the time step model a variable value can be use
Due to steeper profile at low liquid fraction ----- Propose
bagC ml
Coarsening Arm-space will increase in dimension with time
3/1n
This will dilute the concentration in the liquid fraction—can model be enhancing the back diffusion
g
sCQ
A model by Voller and Beckermann suggests 1m
n2
2
1mg 3
13
4
If we assume that solid growth is close to parabolic
1
s
s
l
C
C2~m
In profile model
m =2.33 inParameter model
1.0g 3
4
10 of 19
Remaining Liquid when C =5 is Eutectic Fraction
11 of 19
No Coarsening
0.10.110.120.130.140.150.160.17
0.001 0.01 0.1 1 10
Fourier No.
Eut
etic Numerical
Profile
Parameter
Coarsening
0.10.110.120.130.140.150.160.17
0.001 0.01 0.1 1 10
Fourier No.
Eut
ecti
c
QC)g1(C)gg(C)g1( slllold
soldl
oldl
Constant Cooling of Binary-Eutectic Alloy With Initial Concentration C0 = 1
and Eutectic Concentration Ceut = 5, No Macro segregation , = 0.1
Use 200 time steps and equally increment 1 < Cl < 5
Calculating the transient value of g from
Parameter or Profile
oldlCC
21
2
Coarsening
0.10.110.120.130.140.150.160.17
0.001 0.01 0.1 1 10
Fourier No.
Eut
ecti
c
No Coarsening
0.10.110.120.130.140.150.160.17
0.001 0.01 0.1 1 10
Fourier No.
Eut
etic Numerical
Profile
Parameter
Results are good across a range of conditions
0
0.020.04
0.060.08
0.10.12
0.14
0.001 0.01 0.1 1 10Fourier No.
Eut
etic
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.001 0.01 0.1 1 10Fourier No.
Eut
etic
Note
Wide variation
In Eutectic
12 of 19
0
2
4
6
8
10
0.01 0.1 1 10
Fourier Number
wt%
Co
nc
en
tra
tio
n
Approximation
Experiment
Numerical
No-Coarsening
Predictions of Eutectic FractionWith constant cooling
Co = 4.9
Ceut = 33.2
k = 0.16
Comparison with Experiments Sarreal Abbaschian Met Trans 1986
13 of 19
oldlCC
QC)g1(C)gg(C)g1( slllold
soldl
oldl
Parabolic solid growth – No Second Phase – No Coarsening Use 10,000 equal of g
C0 = 1, = 0.13, = 0.4
0
l
C
kC
Use
To calculate evolving segregation ratio
14 of 19
00.5
1
1.52
2.53
3.5
44.5
5
0.00010.0010.010.11
Liquid Fraction
Seg
rega
tion
Rat
io
Full NumericalSolution
Profile
Profile Fixed m=2
Parameter
Parameter
1
s
s
l
C
C2~m
2
2
14
4
21
2
)CC)(1m(g
Qs
sl
1
s
s
l
C
C2~m
QC)g1(C)gg(C)g1( slllold
soldl
oldl
Performance of Models under parabolic growth no second phase
0
l
C
kCPrediction of segregation ratio in last liquid to solidify(fit exponential through last twotime points)
15 of 19
2
2
14
4
)CC(gdt
dCgtQ old
lll
0102030405060708090
100
0.01 0.1 1 10Fourier No.
Seg
rgat
ion
Rat
io a
t g
=1 analytical
profile
paramter
0
2
4
6
8
10
0.01 0.1 1 10
Fourier No.
Seg
rega
tion
Rat
io a
t g
=1
)CC(g12
2tq old
ll
Parameter
Robust Easy to UsePoor Performance at very low liquid fraction— can be corrected
)CC)(1m(g
tqs
sl
1
s
s
l
C
C2~m
Profile
A little more difficult to use
With this Ad-hoc correctionExcellent performanceat all ranges
Two Models For Back Diffusion
A
C
]C[
]H[
C
Predict g
QC)g1(C)gg(C)g1( srl
rl
rl
oldrs
oldl
predict Cl
predict T
Calculate C
Transient solute balance in arm space
Solidification Solver 16 of 19
1mg 3
13
4
Account for coarsening
My Method of Choice
1.0E+02
1.0E+04
1.0E+06
1.0E+08
1.0E+10
1.0E+12
1.0E+14
1.0E+16
1.0E+18
1.0E+20
1.0E+22
1.0E+24
1.0E+26
0 20 40 60 80 100 120
Year-1980
No
de
s
1000 20.6667 Year
“Moore’s Law”
current for REV of 5mm
Voller and Porte-Agel, JCP 179, 698-703 (2002)
l
TLe
.5m
I Have a BIG Computer Why DO I need an REV and a sub grid model
~ 50 m
solid
~5mm(about 106 nodes)
17 of 19
2055 for tip
Model Directly
(about 1018 nodes)
Tip-interface scale
chill
solid
mushy
liquid
riser
y
Application – Inverse Segregation in a binary alloy
100 mm
Shrinkage sucks solute rich fluid toward chill – results in a region of +ve segregation at chill
Fixed temp chill results in a similarity solution
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
4.4 4.6 4.8 5 5.2 5.4
Concentration
Sim
ilari
ty V
aria
ble
back diff.
Scheil
lever
similarity(lever)
similarity(Scheil)
)CC(g12
2tq old
ll
Parameter
1mg 3
13
4
2fDt
Current estimate
empirical
18 of 19
Ferreira et al Met Trans 2004
Comparison with Experiments
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