Numerical Simulation of Dendritic Solidification

Outline

• Problem statement• Solution strategies• Governing equations• FEM formulation• Numerical techniques• Results• Conclusion

Problem Statement

Solid

Liquid

Liquid

Solid

T > TM

TW < TM

T < TM

TW < TMS

S

When the solidification front grows into a superheated liquid, the interface is stable, heat is conducted through the solid

When the solidification front grows into an undercooled melt, the interface is unstable, heat is conducted through the liquid

Solution Strategies

Most analytical solutions are limited to parabolic cylinder or a paraboloid of revolution

Experimental studies are usually limited to transparent materials with ideal thermal properties such as succinonitrile

Numerical solution can change the various material characteristics and modify the governing

equations to gain more insight to the actual underlying physics

Governing Equations

Heat conduction is the dominant mode of thermal transport:

The interface temperature is modified by local capillary effects:

The second boundary conditions on the interface preserves the sensible heat transported awayfrom the interface and the latent heat of fusionreleased at the interface:

cT

tk T

TStT

LKm( ())

LV n k T k T ns L

[( ) ( ) ]

Finite Element Formulation

For the energy equation in each domain,

cT

tk T

the Galerkin Weighted Residual finite element method formulation can be expresses as:

WnTkNTWkWt

NTc jjii

jj )()(

c

T N

tW k W T N k T nWds

j j

i i j j i

( )( )

Since a moving mesh is used to track the interface, the motion of the coordinate system must be incorporated:

dt

dTNTNV

t

TN jjjj

j

}){(

)(

The FEM formulation of the heat equation can be rewritten as:

cT

tN W cV N T W k W T N k T nWds

j

j i j j i i j j i

( ) ( )

Finite Element Formulation - Continued

t

NTN

t

T

t

NT jjj

jjj

)(j

jj NVt

N

Dt

DN

0

jj NVt

N

The time derivative is treated in standard finite difference form,

dT

dt

T T

t t

k k

k k

1

1

and the heat equation is evaluated at time:

t t tk k k 1 1( )

with:

T T Tk k k 1 1( )

The interface motion balances the heat diffused from the front and the heat liberated at the front:

LVn kT kT ns L

( ) ( )

Numerically:

{ ( ) ( ) }LV n k T k T n Wdss L i

0

Finite Element Formulation - Continued

T / y 0

T / y 0

T =

0

T

Free surface x

y

Frozenregion

Unfrozenregion

n

Numerical Simulation Domain

• As the solidification front advances into the undercooled melt, unstable perturbations emerge and develop into dendrite

• Infinitesimal perturbation is seeded in the middle of the interface

• Boundaries are extended far enough to simulate a relatively unconfined environment

Sequences of Dendrite Growth from a Crystallization Seed

Simulation Results

Finite Element Mesh

LiquidSolid

Numerical Technique -- Interface node adjustment

Coarse Interface Resolution Refined Interface Resolution

Numerical Technique --Remeshing

Distorted Mesh Remeshed Mesh

i j

k

P

Numerical Technique -- Solution Mapping

Simulation Results --Thermal Field in the Vicinity of the Dendrite Tip

0.1 mm

I

J

K

-- Isotherm Coarsening

.01-.02-.06-.09-.12-.16-.19-.23-.26-.29-.33-.36-.40-.43-.47-.50

Simulation Results

o C

0.02mm

Side Branch Steering

Steering direction

0.1 mmT: 10T: 5.92 C C

Scaling Relationship of Dendrite Size with the Critical Instability Wavelength --- Dimensional

Simulation Results

10T: 10 (U:

0.42) CT:

5.92U: 0.25)

C

Scaling Relationship of Dendrite Size with the Critical Instability Wavelength

Simulation Results

--- Non-dimensional

Remelted Branch Tip

Simulation Results

Remelting

Dendrite Tip Radius (cm)

Den

drite

Tip

Vel

ocity

(cm

/s)

10-4 2 3 4 5 10-3 2 3 4 5 10-2 2 3 4 5 10-110-5

2

10-42

10-32

10-22

10-12

100Experimental Data

Simulated Data

Simulated undercooling range: 1.45-10 oCExperimental undercooling range: 0.0043-1.8 oC

-- Dendrite Tip Velocity and Tip Radius RelationshipSimulation Results

• A numerical model was developed for dendritic growth simulation during solidification based on the physics governing the conduction of heat.

• The model accounts for interfacial curvature, surface energy, and the latent heat of fusion released during solidification.

• The moving interface was tracked with moving mesh, automatic remeshing and refinement techniques.

• The simulated dendrite growth characteristics agree with experimental observation:

• dendrite growth tip velocity and radius relationship• dendrite size scaling relationship• side branch competition and remelting, isothermal coarsening.

Conclusions