Heat Transfer with Change of Phase in Continuous Casting

Ernesto Gutierrez-MiraveteRensselaer at Hartford

ANSYS Users Group MeetingSeptember 28, 2010

Outline

• Continuous Casting Processes• Physics and Mathematics of Heat

Conduction with Change of Phase and Mass Transport• Finite Element Formulations• Illustrative Examples

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)

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

Latent Heat of Phase Change

Enthalpy (H)

Temperature (T)

Hf

Tf

∂H/∂T = Cp

Critical Issues in Numerical Solution ofHeat 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 Nodest = f(t)

• Grid Peclet Number ConstraintV L Cp/2k < 1

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

Illustrative Examples

• Continuous Casting in 2D (a Useful Toy Model)• Direct Chill Continuous Casting Model• Thin Slab Continuous Casting Model

Continuous Casting in 2D

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

Predicted Metallurgical Length zm 2D Model

M (kg/min) Zm (m)

24 0.52

27 0.63

30 0.72

33 0.83

36 0.98

Q (W/m2) Zm (m)

-0.8e5 1.10

-0.9e5 0.84

-1.0e5 0.72

-1.1e5 0.66

-1.2e5 0.57

Direct Chill Continuous Casting

2D DC CC Model (Slab Detail)

2D DC CC Model (Mold Detail)

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

3D DC CC Model (Slab CL Detail)

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

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

Thin Slab Continuous Casting

Thin Slab Continuous Casting Mold

This Slab CC Mold Heat Flux(Measured)

Thin Slab CC Mold Model (Mesh)

Thin Slab CC Mold Model(Predicted Temperature and Displacements)

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