Numerical Simulation ofNumerical Simulation of Dendritic Solidification

Outline

• Problem statement• Solution strategies

i i• Governing equations• FEM formulation• Numerical techniquesq• Results• Conclusion

Problem Statement

Solid Liquid T < TM

Liquid

q

Solid

TW < TM

M

TW < TMS

T > TMS

When the solidification front grows When the solidification front growsWhen 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 Strategiesg

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

E i t l t di ll li it d t t tExperimental 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 underlyingequations to gain more insight to the actual underlying

physics

Governing Equationsg q

Heat conduction is the dominant mode of thermal transport:

The interface temperature is modified by

cTt

k T

The interface temperature is modified by local capillary effects:

T S tT

Km( ( ) )

The second boundary conditions on the interface preserves the sensible heat transported away

T S tL

K( ( ) )

preserves the sensible heat transported awayfrom the interface and the latent heat of fusionreleased at the interface:

L V n k T k T ns L

[( ) ( ) ]

Finite Element FormulationFinite Element Formulation

For the energy equation in each domainFor the energy equation in each domain,

cTt

k T

the Galerkin Weighted Residual finite element method f l ti bformulation can be expresses as:

( cT N

tW k W T N k T nWdsj j

i i j j i

( )( ) t i i j j i

Finite Element Formulation - Continued

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

Finite Element Formulation Continued

motion of the coordinate system must be incorporated:

NTNT jjjj )( jj NVNDN

0

dTTN )(t

NTN

tT

tNT j

jjjjj

)(j

jj NVtDt

0

NdtdT

NTNVt

TN jjjj

j

}){(

)( j

j NVt

N

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

cTt

N W cV N T W k W T N k T nW dsjj i j j i i j j i

( ) ( )

The time derivative is treated in standard finite difference formfinite difference form,

dTdt

T Tt t

k k

k k

1

1dt t tand the heat equation is evaluated at time:

t t tk k k 1 1( )t t t 1( )

with:with:

T T Tk k k 1 1( )

Finite Element Formulation - Continued

The interface motion balances the heat diffused from the

Finite Element Formulation Continued

front and the heat liberated at the front:

LV n k T k T ns L

( ) ( )

Numerically:Numerically:

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

0

Numerical Simulation Domain

T / y 0

Numerical Simulation Domain

• As the solidificationfront advances into they

Free surface x

y undercooled melt, unstable perturbations emerge and develop into

T =

0

T

x

dendrite

• Infinitesimal perturbationi d d i h iddl f

Frozenregion

Unfrozenregion

n is seeded in the middle ofthe interface

• Boundaries are extended far

T / y 0

region region • Boundaries are extended farenough to simulate a relativelyunconfined environment

Simulation Results

Sequences of Dendrite Growth from a Crystallization Seed

Finite Element MeshFinite Element Mesh

LiquidSolid qSolid

Numerical Technique -- Interface node adjustment

Coarse Interface Resolution Refined Interface Resolution

Numerical Technique --RemeshingNumerical Technique Remeshing

Distorted Mesh Remeshed Mesh

Numerical Technique -- Solution Mappingq pp g

k

i jP

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

-- Isotherm CoarseningSimulation Results Isotherm Coarsening

.01

Simulation Results

o C

I

-.02-.06-.09- 12I

J

K

.12-.16-.19-.23

26-.26-.29-.33-.36-.40-.43-.47

50

0.1 mm

-.50

Side Branch Steering

Steering direction

0.02mm

Simulation Results

Scaling Relationship of

Simulation Results

Dendrite Size with theCritical InstabilityWavelength

--- Dimensional

0.1 mmT: 10T: 5.92 CC

Simulation Results

Scaling Relationship of Dendrite Size with the Critical Instability Wavelength

--- Non-dimensional

10T: 10 (U: 0.42)CT: 5.92 U: 0.25)C

Simulation Results

Remelting

R l d B h Ti

Remelting

Remelted Branch Tip

-- Dendrite Tip Velocity and Tip Radius RelationshipSimulation Results

100

Experimental Data

Dendrite Tip Velocity and Tip Radius RelationshipSimulation Results

y (c

m/s

)

2

10-12 Simulated Data

p Ve

loci

ty

1032

10-22

endr

ite T

ip

10-42

10-3

De

10-4 2 3 4 5 10-3 2 3 4 5 10-2 2 3 4 5 10-110-52

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

Dendrite Tip Radius (cm)

Conclusions

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

• The model accounts for interfacial curvature, surface energy, gyand 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 withi l b iexperimental observation:

• dendrite growth tip velocity and radius relationship• dendrite size scaling relationship

id b h titi d lti i th l i• side branch competition and remelting, isothermal coarsening.