ingot casting continuous casting welding & laser remelting directional casting shaped casting...
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Ingot Casting
Continuous Casting
Welding & Laser Remelting
Directional Casting
Shaped Casting
Solidification Processing
1
2
R
R – Tip Radius
2 – Secondary Arm Spacing
1 – Primary Arm Spacing
Dendritic Array Growth
Temperature Gradient, G
Growth Velocity, V
Diffusion + Convection exist in the Melt
Modeling Dendritic Array Growth
Experimental modeling: TGS + Transparent Materials
Microscope
Cold Hot
VV
Traction
Temperature Gradient Stage
NH4Cl-70wt.% H2O
SCN-5.6wt.% H2O
Controlled G and V
Minimum Convection
Numerical modeling: Self-consistent model
SCN-4%wt.% ACT
200 m
G/V Dendrites G/V Cells
A. Single Cell/Dendrite
B. Cellular/Dendritic Array
Numerical Modeling of Cellular/Dendritic Array Growth(Diffusion Controlled Growth + No Convection concerned)
X
r
dCL/dX = G/m at X = 0
CL
= C
0 at
X
Solid Ds = 0
Liquid
dT/dr = 0, dT/dX = G
dCL/dr = 0
X = 0 X =
Basic Parameters Given:
Materials Properties: C0 , mL , k , DL , (/S), E4 , Solidification Condition: G and V
Unknown: R, 1 , T (Ti)
T Ti
S + L
C0
C wt.%
CSi
L
S
TL
CLi m
T
k = CSi/CL
i
X
r
dCL/dX = G/m at X = 0
CL
= C
0 at
X
Solid Ds = 0
Liquid
dT/dr = 0, dT/dX = G
dCL/dr = 0
X = 0 X =
Numerical Modeling of Cellular/Dendritic Array Growth (Diffusion Controlled Growth + No Convection concerned)
Basic Equations Solute Diffusion with moving interface:
Generrral: D2C + VdC/dX = dC/dt (dC/dt = 0 for Steady State) Local Interface: Vn(k0 – 1)CL
i = DC/n Interface Temperature:
T = TL - Ti = -m(CLi - C0) + (/R1 + 1/R2) where = 1-15E4cos(4) --- Anisotropy
Numerical Method
x
r
Enmeshment
VK+
1
N VK
E W
S
Control Volume
:
Solute Flow:
i+1Ci+1 - iCi = AN(VNC + DdC/dr)Ndt – AS(VSC + DdC/dr)Sdt
+ AE(VEC + DdC/dx)Edt – Aw(VWC + DdC/dx)wdt
Spacing Adjustment of Array Growth
1 mm
Spacing,1 as Velocity, V
Mechanism of Spacing Adjustment
Lower Limit Upper Limit
V
Array Stability Criterion
Unstable
Stable
Solute
Solute
Result I: Shapes of Single Cell/Dendrite
Result I: Single Cell
Growth in fine capillary tubes
200 m
Stable Cell Perturbed Cell
Cell Width (m)
-200 -150 -100 -50 0 50 100 150 200
Cel
l Len
gth
(m
)
-300
-200
-100
0
SCN-4.8 wt.%Salol, E4 = 0.002, V = 0.12 m s-1
Hunt/Lu Model
Measureed result (Trivedi and Liu)
Result II: Primary Spacing
Result II: Primary Spacing – SCN – 5.6 wt.% H2O System
Growth Velocity, V (m s-1)
0.01 0.1 1 10 100 1000
Pri
mar
y S
pac
ing
, 1 ( m
)
10
100
1000
Minimum Spacings MeasuredMaximum Spacings MeasuredStable Range Predictedby Hunt/Lu Model
SCN-5.6 wt.% H2O ko = 0, mL = -3.56 K (wt.%)-1
G = 4.6 K mm-1
Succinonitrile - Water System
Composition, wt.% H2O
0 20 40 60 80 100
Tem
pera
ture
, °C
-10
0
10
20
30
40
50
60
70
-1.26 °C
18.82 °C
T (°C) = 58.01 - 6.9671C + 0.1733C2 + 0.0145C
3 (wt.%)Liquidus for C < Cm:
Cm
L1 + L2L2
L1
SCN + L2
SCN + Ice
Result II: Primary Spacing – NH4Cl - 70 wt.% H2O System
Growth Velocity, V (ms-1)
0.01 0.1 1 10 100
Prim
ary
Spa
cing
, 1
(m
)
100
1000
10000
Minimum Spacing MeasuredMaximum Spacing MeasuredStable Range Predictedby Hunt/Lu Model
NH4Cl -70 wt.% H2O
G = 2.5 K mm-1
ko = 0, mL = -4.8 K (wt.%)-1
wt. % H2O
40 50 60 70 80 90 100
Te
mp
era
ture
(°C
)
-20
0
20
40
60
80
100
- 16 °C
LNH4Cl + L
Ice + LNH4Cl + Ice
Ammonium chloride - Water System
Result III: Tip Radius
20 m
Growth Velocity, V (ms-1)
1 10 100
Tip
Rad
ius,
R ( m
)
2
3
4
5
6
7
89
1
10
6.5 wt.% H2O, Measured
4.5 wt.% H2O, Measured
5.6 wt.% H2O, Measured
5.6 wt.% H2O, Predicted (Hunt/Lu)
SCN - H2O System
R2V = 125.9 m3s-1
G = 4.6 K mm-1
k0 = 0, mL = -3.56 K (wt.%)-1
The relation, R2V = Constant, is confirmed for all the cases examined in both experimental modeling and numerical modeling.
Result IV: Growth Undercooling
T Ti
S + L
C0
C wt.%
CSi
L
S
TL
CLi m
T
k0 = CSi/CL
i
T
TL
Ti
Result V: The Effect of Temperature Gradient
Modeling Rapid Solidification
T
Ti
S + L
C0
C wt.%
CSi
L
S
TL
CLi
me
T
k0 = CSe/CL
e
k = CSi/CL
i
m
CLe
CSe
Diffusion Coefficient – Temperature Dependent: D as T
D = D0exp[-Q/(RT)]
Distribution Coefficient – Velocity Dependent: k as V , Aziz (1988)
)1(}
1
)]/ln(1[1{
00
00
e
iL
eL kV
V
k
kkkkCC
100/)1(/1
/
0
0
Le
e
CkDVa
kDVak
where
}1
)]/ln(1[1{
0
00
k
kkkkmm e
eS
eL
eC
Ckk
100
)100(0
Non-equilibrium vs. Equilibrium: Boettinger etc. (1986)
G , V , T
Laser Remelting
Result VI: Rapid Solidification
Result VII: Global Structure
PlanarCellularDendriticCellularPlanar
V
Development of Semi-analytical Expressions (Hunt/Lu Model)
1. Variables: Composition, C0, Liquidus Slop, m, Distribution Coefficient, k, Diffusion Coefficient, D, Gibbs-Thompson Coefficient, , Surface Energy Anisotropy Coefficient, E4, Growth Velocity, V, Temperature Gradient, G, Primary Spacing, , and Tip Undercooling, T.
2. Dimensionless Parameters:
Temperature Gradient: G’ = Gk/T02
Growth Velocity: V’ = Vk/(DT0)
Primary Spacing: ’ =DT0/(k)
Tip Undercooling: T’ = T/T0 where T0 = mC0(1-1/k)
3. Properties of the Non-dimensionalization:
G’ = V’: Constitutional Undercooling Limit --- V = GD/T0
V’ = 1: Absolute Stability Limit --- V = T0D/(k)
T’ = 1: The undercooling with a planar front growth --- T = T0 = mC0(1-1/k)
T Ti
S + L
C0
C wt.%
CSi
L
S
TL
CLi m
T
k = CSi/CL
i
T0
Result VIII: Semi-analytical Expressions (Hunt/Lu Model)
1. Cellular Growth (Derived from the Array Stability Criterion):
Undercooling: T’ = T’s + T’r T’s = G’/V’ + a +(1-a)V’0.45 – G’/V’[a + (1-a)V’0.45]
where a = 5.273 x10-3 + 0.5519k – 0.1865k2
Tr’ = b(V’ – G’)0.55(1-V’)1.5
where b = 0.5582 – 0.2267log(k) + 0.2034{log(k)]2
Cell Spacing:
’1 = 8.18k-0.485V’-0.29(V’ – G’)-0.3T’s-0.3(1-V’)-1.4
2. Dendritic Growth:
Undercooling: T’ = T’s + T’r T’s = G’/V’ + V’1/3
T’r = 0.41(V’ – G’)0.51
Primary Dendrite Spacing (Derived from the Array Stability Criterion):
’1 = 0.156V’(c-0.75)(V’ – G’)0.75G’–0.6028
where c = -1.131 – 0.1555log(G’) – 0.7598 x 10-2[log(G’)]2
* Expressions are developed with the Array Stability Criterion
Experimental Modeling of Grain Formation in Casting
Tip Radius, R , Spacing,1 as Velocity, V
Time after Deceleration, sec
0 500 1000 1500 2000 2500 3000
Min
imu
m P
rim
ary
Arm
Sp
acin
g, m
0
200
400
600
800
1000
1200
1400
Tip
Rad
ius,
m
0
1
2
3
4
5
6
7
Minimum primary arm spacingTip radius
Deceleration
Experimental Modeling: Effect of Deceleration on the Dendritic Array Growth
(SCN - 5.5 wt.% H2O System)
R
1
• Tip Radius, R: Rapid response to velocity change. Every individual dendrite follows the Marginal Stability criterion approximately during deceleration.
• Primary Spacing, 1: Slow response to velocity change.The array is unstable and is in transient condition during deceleration.
Experimental Modeling: Effect of Deceleration on the Dendritic Array Growth – Fragmentation
(SCN - 5.5 wt.% H2O System)
Continuous Deceleration, a = -1.0 ms-2
High Velocity Low Velocity
• Secondary Arm, 2, Detached due to deceleration – Accelerated ripening process. The fragmentation rate is proportional to the deceleration.