Ingot Casting Continuous Casting Welding & Laser Remelting Directional Casting Shaped Casting Solidification Processing
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 Controlled G and V Minimum Convection Numerical modeling: Self-consistent model 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) Basic Parameters Given : Materials Properties: C 0, m L, k, D L, ( / S), E 4, Solidification Condition: G and V Unknown: R, 1, T (T i )
Numerical Modeling of Cellular/Dendritic Array Growth (Diffusion Controlled Growth + No Convection concerned)
Numerical Method : Solute Flow: i+1 C i+1 - i C i = A N (V N C + DdC/dr) N dt – A S (V S C + DdC/dr) S dt + A E (V E C + DdC/dx) E dt – A w (V W C + DdC/dx) w dt
Spacing Adjustment of Array Growth Spacing, 1 as Velocity, V Mechanism of Spacing Adjustment Lower Limit Upper Limit V
Array Stability Criterion Unstable Stable Solute
Result I: Shapes of Single Cell/Dendrite
Result I: Single Cell Growth in fine capillary tubes 200 m Stable CellPerturbed Cell
Result II: Primary Spacing
Result II: Primary Spacing – SCN – 5.6 wt.% H 2 O System
Result II: Primary Spacing – NH 4 Cl - 70 wt.% H 2 O System
Result III: Tip Radius 20 m The relation, R 2 V = Constant, is confirmed for all the cases examined in both experimental modeling and numerical modeling.
Result IV: Growth Undercooling TT TLTL TiTi
Result V: The Effect of Temperature Gradient
Modeling Rapid Solidification Diffusion Coefficient – Temperature Dependent: D as T D = D 0 exp[-Q/(RT)] Distribution Coefficient – Velocity Dependent: k as V , Aziz (1988) where Non-equilibrium vs. Equilibrium: Boettinger etc. (1986) G , V , T Laser Remelting
Result VI: Rapid Solidification
Result VII: Global Structure Planar Cellular Dendritic Cellular Planar V
Development of Semi-analytical Expressions (Hunt/Lu Model) 1. Variables: Composition, C 0, Liquidus Slop, m, Distribution Coefficient, k, Diffusion Coefficient, D, Gibbs-Thompson Coefficient, , Surface Energy Anisotropy Coefficient, E 4, Growth Velocity, V, Temperature Gradient, G, Primary Spacing,, and Tip Undercooling, T. 2.Dimensionless Parameters: Temperature Gradient: G’ = G k/ T 0 2 Growth Velocity: V’ = V k/(D T 0 ) Primary Spacing: ’ = DT 0 /(k ) Tip Undercooling: T’ = T/ T 0 where T 0 = mC 0 (1-1/k) 3.Properties of the Non-dimensionalization: G’ = V’: Constitutional Undercooling Limit --- V = GD/ T 0 V’ = 1: Absolute Stability Limit --- V = T 0 D/(k ) T’ = 1: The undercooling with a planar front growth --- T = T 0 = mC 0 (1-1/k)
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 = x k – k 2 Tr’ = b(V’ – G’) 0.55 (1-V’) 1.5 where b = – log(k) {log(k)] 2 Cell Spacing: ’ 1 = 8.18k V’ (V’ – G’) -0.3 T’ s -0.3 (1-V’) 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’ – where c = – log(G’) – x [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 Deceleration Experimental Modeling: Effect of Deceleration on the Dendritic Array Growth (SCN wt.% H 2 O 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 wt.% H 2 O 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.