1. Heat Transfer with Change of Phase in Continuous Casting Ernesto Gutierrez-Miravete Rensselaer at Hartford ANSYS Users Group Meeting September 28, 2010

2. Outline Continuous Casting Processes Physics and Mathematics of Heat Conduction with Change of Phase and Mass Transport Finite Element Formulations Illustrative Examples

3. Continuous Casting Processes Metal Processing often involves Molten Metals Molten Metals must be Solidified to produce Bulk Solid Specimens Metal Solidification for the Production of Bulk Specimens is carried out in Practice either in Batches (Ingot or Shape Casting) or Continuosly (Continuous Casting)

4. Mathematical Formulation of Heat Conduction with Change of Phase Problems Differential Thermal Energy Balance Equation Inside the Bulk Phases (Energy Conservation) Heat Flux-Temperature Gradient Relationships Inside the Bulk Phases (Fourier “Law”) Differential Thermal Energy Balance Equation at the Interface between Phases accounting for the Latent Heat of Phase Change (Stefan Condition) Boundary Conditions on External Boundaries

5. Latent Heat of Phase Change Enthalpy (H) Temperature (T) HfHf TfTf ∂H/∂T = C p

6. Critical Issues in Numerical Solution of Heat Conduction problems with Change of Phase in CC Stefan Condition makes problem Non- Linear even for Constant Properties T(x,t)   t   T(x,t) Interface Motion driven by Physics, unrelated to Position of Mesh Nodes  t  = f(t) Grid Peclet Number Constraint V L  C p /2k < 1

7. Finite Element Formulation of Heat Conduction with Change of Phase Problems in CC Variational Statement of the Problem Galerkin’s Method Time Stepping Handling of the Stefan Condition – Enthalpy Method – Effective Specific Heat Method Effect of Mass Transport

8. Illustrative Examples Continuous Casting in 2D (a Useful Toy Model) Direct Chill Continuous Casting Model Thin Slab Continuous Casting Model

9. Continuous Casting in 2D

10. Effect of M (kg/min) and q (W/m2) on T-z Curve along Slab Centerline

11. Predicted Metallurgical Length z m 2D Model M (kg/min)Z m (m) 240.52 270.63 300.72 330.83 360.98 Q (W/m 2 )Z m (m) -0.8e51.10 -0.9e50.84 -1.0e50.72 -1.1e50.66 -1.2e50.57

12. Direct Chill Continuous Casting

13. 2D DC CC Model (Slab Detail)

14. 2D DC CC Model (Mold Detail)

15. 3D DC CC Square Bar Model (Slab and Mold Temperatures)

16. 3D DC CC Model (Slab CL Detail)

17. 3D DC CC Model (Full Slab View from CL)

18. 3D DC CC Model (Full Slab View from Narrow Face)

19. Thin Slab Continuous Casting

20. Thin Slab Continuous Casting Mold

21. This Slab CC Mold Heat Flux (Measured)

22. Thin Slab CC Mold Model (Mesh)

23. Thin Slab CC Mold Model (Predicted Temperature and Displacements)

24. In Closing Heat Conduction with Change of Phase is the Simplest Model of a Solidifying System Additional Important Issues and Future Goals – Thermo-Mechanical Effects – Liquid Metal Flow Effects – Solidified Microstructure Development – Solid State Phase Changes – Optimal Heat Extraction Practices – Comprehensive, Push-Button Models Partial support from CCAT for the performance and presentation of this work is gratefully acknowledged