Continuous Casting — Arbitrary Lagrangian-Eulerian Method

This example simulates the process of continuous casting of a metal rod from a molten state (Figure 1) using the boundary condition. Continuous Casting — Apparent Heat Capacity Method is a variant of this model using the domain condition.

To optimize the casting process in terms of casting rate and cooling, it is helpful to model the thermal and fluid dynamic aspects of the process. To get accurate results, you must model the melt flow field in combination with the heat transfer and phase change. The model includes the phase transition from melt to solid, both in terms of latent heat and the varying physical properties.

Figure 1: Continuous metal-casting process with a view of the modeled section.

This example simplifies the rod’s 3D geometry in Figure 1 to an axisymmetric 2D model in the rz-plane. Figure 2 shows the dimensions of the 2D geometry.

Figure 2: 2D axisymmetric model of the casting process.

As the melt cools down in the mold it solidifies. The phase transition releases latent heat, which the model includes. For metal alloys, the transition is often spread out over a temperature range. However, using the ALE approach to model the phase transition, a sharp interface is assumed between the two phases, and the latent heat of phase change is released at the corresponding boundary.

This example models the casting process with a transient study until it reaches a stationary state. The Heat Transfer in Fluids interface combined with the Laminar Flow interface are used.

Model Definition

The transient heat transport is described by the equation:

where k, ρ, Cp, and Q denote thermal conductivity, density, specific heat, and heating power per unit volume (heat source term), respectively.

As the melt cools down in the mold, it solidifies. During the phase transition, a significant amount of latent heat is released. The total amount of heat released per unit mass of alloy during the transition is given by the change in enthalpy, ΔH. In addition, the specific heat capacity, Cp, also changes considerably during the transition.

In this example, the boundary condition is used to model the phase change interface. This feature uses the Stefan Condition, which derives the normal interface velocity from the incoming heat fluxes, the melting latent heat and the solid density. To allow this interface to move in the geometry according to the calculated normal velocity, this feature is used along with a interface.

This example models the laminar flow by describing the fluid velocity, u, and the pressure, p, according to the equations

where ρ is the density (in this case constant), μ is the viscosity, and κ is the dilatational viscosity (here assumed to be zero).

Table 1 reviews the material properties in this model.

Table 1: Material Properties.
Property	Symbol	Melt	Solid
Density	ρ (kg/m3)	8500	8500
Heat capacity at constant pressure	Cp (J/(kg·K))	530	380
Thermal conductivity	k (W/(m·K))	200	200
Dynamic viscosity	μ (N·s/m2)	0.0434	-

Furthermore, the melting temperature, Tm, and enthalpy of phase change, ΔH, are set to 1356 K and 205 kJ/kg, respectively.

Results and Discussion

Figure 3: Velocity field with streamlines near the inlet part of the process.

In Figure 3, velocity streamlines are plot along with the phase change interface that delimits the fluid outlet. This interface stretches out toward the center of the rod because of poorer cooling in that area. With the modeled casting rate, the rod is fully solidified before leaving the mold (the first section after the die). This means that the process engineers can increase the casting rate without running into problems, thus increasing the production rate.

To help determine how to optimize process cooling, Figure 4 plots the conductive heat flux. It shows that the conductive heat flux is very large in the mold zone. This is a consequence of the heat released during the phase transition, which is cooled by the water-cooling jacket of the mold. An interesting phenomenon of the process is the peak of conductive heat flux appearing in the center of the flow at the transition zone.

Figure 4: The cooling viewed as conductive heat flux in the domains (top), and through the outer boundary (the cooling zones) after the die (bottom).

Furthermore, by plotting the conductive heat flux at the outer boundary for the process as in the lower plot in Figure 4, you can see that a majority of the process cooling occurs in the mold. More interestingly, the heat flux varies along the mold wall length. This information can help in optimizing the cooling of the mold (that is, the cooling rate and choice of cooling method).

This method allows a coarser mesh compared to the Continuous Casting — Apparent Heat Capacity Method model and by consequence a faster calculation. It provides also transient results hence the ability to compute the response of the system with time varying input (typically the casting velocity).

Reference

1. V.R. Voller and C. Prakash, “A fixed grid numerical modeling methodology for convection — diffusion mushy region phase-change problems,” Int.J.Heat Mass Transfer, vol. 30, pp. 1709–1719, 1987.

Heat_Transfer_Module/Thermal_Processing/continuous_casting_ale

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file continuous_casting_ale_parameters.txt.

Definitions

Piecewise 1 (pw1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 0.01.

Find the subsection. Click

Browse to the model’s Application Libraries folder and double-click the file continuous_casting_ale_pw1.txt.

Locate the section. In the text field, type m.

In the text field, type W/m^2/K.

Piecewise 2 (pw2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 0.01.

Find the subsection. Click

Browse to the model’s Application Libraries folder and double-click the file continuous_casting_ale_pw2.txt.

Locate the section. In the text field, type m.

In the text field, type 1.

Since the boundary edges will be translated due to the deformed geometry sliding conditions, add variables to impose spacially fixed boundary condition coefficients.

Variables 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
h_rod	pw1(Z)	W/(m²·K)	Heat transfer coefficient along the rod
eps_rod	pw2(Z)		Surface emissivity along the rod

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.065.

In the text field, type 0.1.

Locate the section. In the text field, type -0.1.

In the toolbar, click

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.0625.

In the text field, type 0.025.

Locate the section. In the text field, type -0.125.

In the toolbar, click

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.11575.

In the text field, type 1.4075.

Locate the section. In the text field, type -1.5725.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	0.04

Clear the check box.

Select the check box.

In the toolbar, click

Click the

button in the toolbar.

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


r (m)	z (m)
0	-0.125
0	-0.165
0.11575	-0.165
0.0625	-0.125
0	-0.125

In the toolbar, click

Click the

button in the toolbar.

This completes the geometry modeling stage.

Materials

Now, add the following two materials to the model, labeled and . The solid metal alloy is used in the feature for the solid phase, while the liquid metal alloy is used for the liquid phase. The liquid metal alloy also defines fluid properties used in the interface.

Solid Metal Alloy

In the toolbar, click

In the window for , type Solid Metal Alloy in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Dynamic viscosity	mu	0.0434	Pa·s	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	200	W/(m·K)	Basic
Density	rho	8500	kg/m³	Basic
Heat capacity at constant pressure	Cp	Cp_s	J/(kg·K)	Basic

Liquid Metal Alloy

In the toolbar, click

In the window for , type Liquid Metal Alloy in the text field.

Select Domains 2–5 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Dynamic viscosity	mu	0.0434	Pa·s	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	200	W/(m·K)	Basic
Density	rho	8500	kg/m³	Basic
Heat capacity at constant pressure	Cp	Cp_l	J/(kg·K)	Basic