Numerical Simulation of Steel Ingot Solidification

Overview

This project implements a 1D finite-difference numerical model for simulating the solidification process of a steel ingot during casting. The model follows the methodology presented in Radovic & Lalovic (2005) published in the Journal of Materials Processing Technology.

The simulation captures the complex physics of solidification including:

• Phase change in the mushy zone (Ts < T < Tl)
• Latent heat release during crystallization
• Lattice defect energy contributions
• Temperature-dependent material properties
• Time-varying boundary conditions (mould-gap formation)

Physical Problem

Solidification Physics

When molten steel above the liquidus temperature (Tl = 1663 K) is poured into a mould and allowed to cool:

1. Surface Cooling: The outer surface loses heat through convection and radiation to the mould
2. Mushy Zone Formation: A solidification front propagates inward where Ts < T < Tl
3. Latent Heat Release: Steel releases L = 270 kJ/kg during crystallization
4. Lattice Defects: Additional energy from condensation of lattice defects modifies the energy balance
5. Complete Solidification: Process continues until the entire ingot reaches T < Ts

Material Studied

• Alloy: 165XCrMoW high-alloy tool steel (DIN specification)
• Ingot Dimensions: 150 mm × 150 mm × 400 mm
• Simulation Domain: 70 mm (quarter-section due to symmetry)
• Liquidus Temperature: Tl = 1663 K
• Solidus Temperature: Ts = 1371 K
• Initial Pouring Temperature: Ti = 1663 K

---

Results

Key Findings

The simulation successfully reproduces the solidification behavior reported in the reference paper:

1. Surface Solidification: Surface cools rapidly and solidifies first
2. Inward Propagation: Solidification front moves progressively toward the center
3. Temperature Gradients: Steep thermal gradients near the surface gradually diminish
4. Complete Solidification: Core reaches solidus temperature last (~1000-1200s)

Generated Outputs

All results are automatically saved to group_12_results/ directory:

• fig3.png: Solid fraction vs. time at different depths
• fig4.png: Temperature distribution vs. distance at selected times
• fig5.png: Temperature field contour (x vs. t)
• fig6.png: Temperature vs. time at multiple positions
• fig7.png: Cooling curves at surface, mid-point, and center

Model Validation

The simulated results show strong agreement with the reference paper [Radovic & Lalovic, 2005], capturing:

• Overall solidification progression
• Temperature-time curves at different depths
• Solid fraction evolution profiles
• Spatial temperature distributions

Minor discrepancies are observed due to:

• Lack of exact polynomial coefficients for λ(T) and cp(T) in the paper
• Potential timing differences in reference data
• Approximations in material property correlations

---

Detailed Documentation

For comprehensive details including:

• Complete governing equations with derivations
• Numerical scheme implementation
• Material property functions
• Visualization code
• Result interpretation

Please refer to the implementation notebook and project presentation.

---

References

Main Paper

Radovic, Z., & Lalovic, M. (2005)
Numerical simulation of steel ingot solidification process
Journal of Materials Processing Technology, 160, 156-159.
DOI: 10.1016/j.jmatprotec.2004.07.094