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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.