Heat Transfer with Change ofHeat Transfer with Change of Phase in Continuous Casting
Ernesto Gutierrez MiraveteErnesto Gutierrez‐Miravete
Rensselaer at Hartford
ANSYS Users Group Meeting
September 28, 2010p ,
OutlineOutline
• Continuous Casting Processes• Continuous Casting Processes• Physics and Mathematics of Heat yConduction with Change of Phase and Mass Transportand Mass Transport
• Finite Element Formulations
• Illustrative Examples
Continuous Casting ProcessesContinuous Casting Processes
M t l P i ft i l M lt• Metal Processing often involves Molten Metals
• Molten Metals must be Solidified to produce Bulk Solid Specimensp
• Metal Solidification for the Production of Bulk Specimens is carried out in PracticeBulk Specimens is carried out in Practice either in Batches (Ingot or Shape Casting) or Continuosly (Continuous Casting)Continuosly (Continuous Casting)
Mathematical Formulation of Heat Conduction with Change of Phase Problems
• Differential Thermal Energy Balance EquationDifferential Thermal Energy Balance Equation Inside the Bulk Phases (Energy Conservation)
• Heat Flux‐Temperature Gradient RelationshipsHeat Flux Temperature Gradient Relationships Inside the Bulk Phases (Fourier “Law”)
• Differential Thermal Energy Balance EquationDifferential 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 ChangeLatent Heat of Phase Change
Enthalpy (H)
Hf ∂H/∂T = Cp
Tf
Temperature (T)
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, l d P i i f M h N dunrelated to Position of Mesh Nodes
t = f(t)l b• Grid Peclet Number Constraint
V L Cp/2k < 1
Finite Element Formulation of Heat Conduction with Change of Phase Problems in CC
• Variational Statement of the ProblemVariational Statement of the Problem
• Galerkin’s Method
i S i• Time Stepping
• Handling of the Stefan Condition– Enthalpy Method
– Effective Specific Heat Method
• Effect of Mass Transport
Illustrative ExamplesIllustrative Examples
• Continuous Casting in 2D (a Useful• Continuous Casting in 2D (a Useful Toy Model)
• Direct Chill Continuous Casting Model
• Thin Slab Continuous Casting Model• Thin Slab Continuous Casting Model
Continuous Casting in 2DContinuous Casting in 2D
Effect of M (kg/min) and q (W/m2) on T‐z Curve along Slab Centerline
Effect of M Effect of q
1100
1200
Effect of M
1100
1200
Effect of q
800
900
1000
T (C)
900
1000
600
700
800T (C)
600
700
800T (C)
400
500
0 0.2 0.4 0.6 0.8 1400
500
600
0 0 2 0 4 0 6 0 8 1z (m)
0 0.2 0.4 0.6 0.8 1
z (m)
Predicted Metallurgical Length zmd l2D Model
M (kg/min) Zm (m) Q (W/m2) Zm (m)M (kg/min) Zm (m)
24 0.52
Q (W/m ) Zm (m)
‐0.8e5 1.10
27 0.63 ‐0.9e5 0.84
30 0.72
33 0 83
‐1.0e5 0.72
1 1e5 0 6633 0.83
36 0.98
‐1.1e5 0.66
‐1.2e5 0.57
Direct Chill Continuous CastingDirect Chill Continuous Casting
2D DC CC Model (Slab Detail)2D DC CC Model (Slab Detail)
2D DC CC Model (Mold 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 (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 CastingThin Slab Continuous Casting
Thin Slab Continuous Casting MoldThin Slab Continuous Casting Mold
This Slab CC Mold Heat Flux(Measured)
Thin Slab CC Mold Model (Mesh)Thin Slab CC Mold Model (Mesh)
Thin Slab CC Mold Model(Predicted Temperature and Displacements)
In ClosingIn Closing• Heat Conduction with Change of Phase is the Si l M d l f S lidif i SSimplest Model of a Solidifying System
• Additional Important Issues and Future Goals– Thermo‐Mechanical Effects
– Liquid Metal Flow Effects
– Solidified Microstructure Development
– Solid State Phase Changesg
– Optimal Heat Extraction Practices
– Comprehensive, Push‐Button ModelsComprehensive, Push Button Models