Numerical Simulation of Dendritic Solidification

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Presentation transcript:

Numerical Simulation of Dendritic Solidification

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

Problem Statement T < TM TW < TM TW < TM T > TM Solid Liquid T < TM TW < TM TW < TM S Liquid Solid T > TM 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 away from the interface and the latent heat of fusion released at the interface:

Finite Element Formulation For the energy equation in each domain, the Galerkin Weighted Residual finite element method formulation can be expresses as:

Finite Element Formulation - Continued Since a moving mesh is used to track the interface, the motion of the coordinate system must be incorporated: The FEM formulation of the heat equation can be rewritten as:

The time derivative is treated in standard finite difference form, and the heat equation is evaluated at time: with:

Finite Element Formulation - Continued The interface motion balances the heat diffused from the front and the heat liberated at the front: Numerically:

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 y Free surface x T = 0 T = DT Frozen region Unfrozen region

Simulation Results Sequences of Dendrite Growth from a Crystallization Seed

Finite Element Mesh Liquid Solid

Numerical Technique -- Interface node adjustment Coarse Interface Resolution Refined Interface Resolution

Numerical Technique --Remeshing Distorted Mesh Remeshed Mesh

Numerical Technique -- Solution Mapping k P i j

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

Simulation Results -- Isotherm Coarsening o C .01 -.02 -.06 -.09 -.12 -.16 -.19 -.23 -.26 -.29 -.33 -.36 -.40 -.43 -.47 -.50 I J K 0.1 mm

Side Branch Steering Steering direction 0.02mm

Simulation Results Scaling Relationship of Dendrite Size with the Critical Instability Wavelength --- Dimensional 0.1 mm DT: 5.92 DT: 10

Simulation Results Scaling Relationship of Dendrite Size with the Critical Instability Wavelength --- Non-dimensional 10 DT: 5.92 (DU: 0.25) DT: 10 (DU: 0.42)

Simulation Results Remelting Remelted Branch Tip

Simulation Results -- Dendrite Tip Velocity and Tip Radius Relationship 1 Experimental Data 2 Simulated Data 1 - 1 2 1 - 2 Dendrite Tip Velocity (cm/s) 2 1 - 3 2 1 - 4 Simulated undercooling range: 1.45-10 oC Experimental undercooling range: 0.0043-1.8 oC 2 1 - 5 1 - 4 2 3 4 5 1 - 3 2 3 4 5 1 - 2 2 3 4 5 1 - 1 Dendrite Tip Radius (cm)

Conclusions 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.