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Ingot CastingContinuous CastingWelding & Laser RemeltingDirectional CastingShaped CastingSolidification Processing

R Tip Radius l2 Secondary Arm Spacing l1 Primary Arm SpacingDendritic Array Growth Temperature Gradient, G Growth Velocity, V Diffusion + Convection exist in the Melt

1

2

R

Modeling Dendritic Array GrowthExperimental modeling: TGS + Transparent MaterialsControlled G and VMinimum ConvectionNumerical modeling: Self-consistent model G/V DendritesG/V CellsSingle Cell/DendriteCellular/Dendritic Array

Cold

Hot

VV

Temperature Gradient Stage

Traction

Microscope

(

_1073374181.unknown

_1073375359.unknown

NH4Cl-70wt.% H2O

SCN-5.6wt.% H2O

SCN-4%wt.% ACT

200 m

Numerical Modeling of Cellular/Dendritic Array Growth(Diffusion Controlled Growth + No Convection concerned)

X

r

dCL/dX = G/m at X = 0

CL = C0 at X ( (

Solid

Ds = 0

Liquid

dCL/dr = 0

dT/dr = 0, dT/dX = G

X = (

X = 0

Numerical Modeling of Cellular/Dendritic Array Growth (Diffusion Controlled Growth + No Convection concerned)

X

r

dCL/dX = G/m at X = 0

CL = C0 at X ( (

Solid

Ds = 0

Liquid

dCL/dr = 0

dT/dr = 0, dT/dX = G

X = (

X = 0

Basic Equations

Solute Diffusion with moving interface:

Generrral: D(2C + VdC/dX = dC/dt (dC/dt = 0 for Steady State)

Local Interface: Vn(k0 1)CLi = DC/n

Interface Temperature:

T = TL - Ti = -m(CLi - C0) + ((/R1 + 1/R2) where ( = 1-15E4cos(4() --- Anisotropy

Numerical Method : 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

x

r

Enmeshment

Control Volume (

S

W

E

VK

N

VK+1

Spacing Adjustment of Array GrowthSpacing, l1 as Velocity, VMechanism of Spacing AdjustmentLower LimitUpper LimitV

1 mm

Array Stability CriterionUnstableStable

Result I: Shapes of Single Cell/Dendrite

Result I: Single CellGrowth in fine capillary tubes200 mmStable CellPerturbed Cell

Result II: Primary Spacing

Result II: Primary Spacing SCN 5.6 wt.% H2O System

Result II: Primary Spacing NH4Cl - 70 wt.% H2O System

Result III: Tip RadiusThe relation, R2V = Constant, is confirmed for all the cases examined in both experimental modeling and numerical modeling.

Result IV: Growth UndercoolingTLTi

k0 = CSi/CLi

T

m

CLi

TL

S

L

CSi

C wt.%

C0

S + L

Ti

T

Result V: The Effect of Temperature Gradient

Modeling Rapid Solidification Diffusion Coefficient Temperature Dependent: D as T D = D0exp[-Q/(RT)] Distribution Coefficient Velocity Dependent: k as V , Aziz (1988)

where Non-equilibrium vs. Equilibrium: Boettinger etc. (1986) G , V , DT Laser Remelting

k0 = CSe/CLe

k = CSi/CLi

T

me

CLi

TL

S

L

CSi

C wt.%

C0

S + L

Ti

T

m

CLe

CSe

Result VI: Rapid Solidification

Result VII: Global Structure PlanarCellularDendriticCellularPlanarV

Development of Semi-analytical Expressions (Hunt/Lu Model) Variables: Composition, C0, Liquidus Slop, m, Distribution Coefficient, k, Diffusion Coefficient, D, Gibbs-Thompson Coefficient, G, Surface Energy Anisotropy Coefficient, E4, Growth Velocity, V, Temperature Gradient, G, Primary Spacing, l, and Tip Undercooling, DT.Dimensionless Parameters:Temperature Gradient: G = GGk/DT02Growth Velocity: V = VGk/(DDT0)Primary Spacing: l = lDT0/(kG)Tip Undercooling: DT = DT/DT0 where DT0 = mC0(1-1/k)Properties of the Non-dimensionalization:G = V: Constitutional Undercooling Limit --- V = GD/DT0V = 1: Absolute Stability Limit --- V = DT0D/(kG)DT = 1: The undercooling with a planar front growth --- DT = DT0 = mC0(1-1/k)

k = CSi/CLi

T

m

CLi

TL

S

L

CSi

C wt.%

C0

S + L

Ti

T

T0

Result VIII: Semi-analytical Expressions (Hunt/Lu Model)Cellular Growth (Derived from the Array Stability Criterion): Undercooling: DT = DTs + DTr DTs = G/V + a +(1-a)V0.45 G/V[a + (1-a)V0.45] where a = 5.273 x10-3 + 0.5519k 0.1865k2 DTr = b(V G)0.55(1-V)1.5 where b = 0.5582 0.2267log(k) + 0.2034{log(k)]2

Cell Spacing: l1 = 8.18k-0.485V-0.29(V G)-0.3DTs-0.3(1-V)-1.4

Dendritic Growth:

Undercooling: DT = DTs + DTr DTs = G/V + V1/3 DTr = 0.41(V G)0.51

Primary Dendrite Spacing (Derived from the Array Stability Criterion): l1 = 0.156V(c-0.75)(V G)0.75G0.6028 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

Experimental Modeling: Effect of Deceleration on the Dendritic Array Growth (SCN - 5.5 wt.% H2O System)Tip Radius, R , Spacing, l1 as Velocity, VDecelerationRl1Tip Radius, R: Rapid response to velocity change. Every individual dendrite follows the Marginal Stability criterion approximately during deceleration.Primary Spacing, l1: 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)Secondary Arm, l2, Detached due to deceleration Accelerated ripening process. The fragmentation rate is proportional to the deceleration.