Download - Solidification Processing

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Page 1: Solidification Processing

Ingot Casting

Continuous Casting

Welding & Laser Remelting

Directional Casting

Shaped Casting

Solidification Processing

Page 2: 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

Page 3: Solidification Processing

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

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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

Page 5: Solidification Processing

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

Page 6: Solidification Processing

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

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Spacing Adjustment of Array Growth

1 mm

Spacing,1 as Velocity, V

Mechanism of Spacing Adjustment

Lower Limit Upper Limit

V

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Array Stability Criterion

Unstable

Stable

Solute

Solute

Page 9: Solidification Processing

Result I: Shapes of Single Cell/Dendrite

Page 10: Solidification Processing

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)

Page 11: Solidification Processing

Result II: Primary Spacing

Page 12: Solidification Processing

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

Page 13: Solidification Processing

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

Page 14: Solidification Processing

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.

Page 15: Solidification Processing
Page 16: Solidification Processing

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

Page 17: Solidification Processing

Result V: The Effect of Temperature Gradient

Page 18: Solidification Processing

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

Page 19: Solidification Processing

Result VI: Rapid Solidification

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Result VII: Global Structure

PlanarCellularDendriticCellularPlanar

V

Page 21: Solidification Processing

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

Page 22: Solidification Processing

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

Page 23: Solidification Processing

Experimental Modeling of Grain Formation in Casting

Page 24: Solidification Processing

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.

Page 25: Solidification Processing

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.