3D Numerical Simulation of the Var Consumable Electrode Melting Process

TL;DR: In this paper, a 3D numerical model was set up to simulate the formation and dynamics of the liquid metal film under the consumable electrode during VAR process, which was developed using the open source computational fluid dynamics (CFD) software OpenFOAM.

Abstract: A 3D numerical model was set up to simulate the formation and dynamics of the liquid metal film under the consumable electrode during VAR process. In the present paper, the implementation of this model is described. It was developed using the open source computational fluid dynamics (CFD) software OpenFOAM. The model solves coupled momentum and energy equations combined with a volume of fluid (VOF) method to track the liquid metal free surface. The melting of the electrode material is modeled with an enthalpy-porosity approach. The electric power supplied by the arc is supposed to be uniformly distributed over the surface of the electrode tip. For a given electric arc power, the model enables to quantitatively predict the dripping rate and hence the overall melt rate. Besides the thermal behavior of the electrode, simulation results illustrate the dynamics of the liquid film and the transfer mechanisms of the liquid metal during VAR melts performed with short and long interelectrode gaps.

Summary

I. Introduction

• Vacuum arc remelting (VAR) is a secondary remelting process used to improve cleanliness as well as chemical and mechanical homogeneity of metal ingots.
• The arc is created between the electrode and the base plate of a water-cooled copper crucible at the beginning of the melt, then between the electrode and the secondary ingot forming in the crucible.
• At any instant, the ingot is composed of three zones: the fully solidified metal, the liquid pool fed by metal drops and an intermediate mushy zone.

II. Numerical model

• The formation and dynamics of the liquid film under the consumable electrode during the VAR process is simulated with a multiphase CFD approach.
• The developed model is based on the following assumptions.
• (2) As a first step towards a complete description of the liquid film behavior, magnetohydrodynamic effects produced by the arc current (i.e. electromagnetic forces acting on the liquid metal) are not taken into account.
• (4) All thermophysical properties are considered to be independent of temperature and identical in the solid and liquid phases.
• (5) As explained later in section II.A, when computing the flow, a specific procedure was introduced to eliminate non-physical spurious velocities generated near the interface in the gas region.

A. Governing equations

• Fluid flow and behavior of the free surface.
• This term compresses the interface by minimizing the numerical diffusion of the volume fraction while ensuring its boundedness.
• To the best of the authors knowledge, existing attempts to model melting (e.g. [34,35]) are all based on the assumption of the existence of a mushy region and treats this region using the classical enthalpy-porosity technique originally developed to deal with solidification.
• In the fully liquid cells (γ = 0), the source term is zero and the "classical" Navier-Stokes equation is solved.
• Since the solid phase is kept at rest in their simulations, this velocity reduces here to the fluid velocity.

Heat transfer with phase change

• The second term on the right hand side of Eq. 10 accounts for the evolution of the latent heat during phase change.
• The liquid-solid interface is not tracked explicitly.
• The third term on the right hand side of Eq. 10 is the thermal power provided by the electric arc to the metal.
• The radiative heat fluxes at the electrode base can be neglected, given the small difference between the electrode and ingot diameters and the similarity of the metal temperatures at the electrode base and on the ingot top.
• The energy flux density radiated by a surface element of the electrode lateral wall may thus simply be expressed as: EQUATION.

B. Boundary conditions

• Figure 2 shows a vertical cross-section of the 3D computational domain and illustrates the boundary conditions used for the simulation.
• The red lines represent the surface of application of the arc power while the purple lines represent the surface associated to radiative losses.
• In order to get a realistic simulation of the formation of molten metal bridges (i.e. drip-short) between the electrode and the ingot, the domain includes a liquid region under the electrode representing a part of the molten metal pool present at the ingot top.
• Turbulent kinetic energy and its dissipation rate:.
• The normal gradients are set to zero at all boundaries.

C. Computational details

• The pressure-velocity coupling is handled by the PIMPLE algorithm (combination of the PISO [24] and SIMPLE [25] algorithms).
• The convective terms in the conservation equations are discretized with a second order VanLeer scheme, while the diffusion terms are central differenced.
• The transport equation of the metal volume fraction is solved using the OpenFOAM solver called InterFoam, which is based on the MULES limited interface compression method.
• Temporal integration is done using a first order implicit Euler method.

III. Application of the model

• The numerical model described above was applied to simulate two different melt 300 configurations.
• The first one is the melt of a small diameter Ti-6Al-4V electrode with a long 301 interelectrode gap performed experimentally by Chapelle et al. [6] , while the second one is [26].
• A. Small diameter electrode with a long gap Fig. 3 shows a vertical cut of the computational domain with the initial metal represented in yellow.
• The electrode diameter is 160 mm, its height is 150 mm and the interelectrode gap is 66 mm.

Gas

• The electrode and the metal bath at the ingot top are considered to receive respectively 60% and 40 % of this power [11] .
• The liquid bath is at the liquidus temperature of the alloy.
• The computational grid is a uniform structured mesh consisting of around 3.5 million cells.

Liquid film behavior

• At the initial state, the electrode is isothermal.
• As soon as the temperature of the tip reaches the solidus temperature, the electrode starts to melt and a distorted liquid film is formed.
• The drop remains attached to the liquid film by a filament which gradually becomes thinner.
• The simulated behavior remains close to the experimental visualizations.
• Protuberances tend to form near the center of the electrode and migrate to the periphery of the electrode forming approximately concentric circles.

Fluid flow inside a protuberance

• The fluid flow is fed by the drainage of the liquid film and is governed by buoyancy forces.
• In the protuberance, the turbulence level is moderate, with a maximum value of the ratio of the turbulent to molecular viscosities of about 180.
• In the liquid film, the viscosity ratio is much lower, which corresponds to a flow regime remaining laminar.

Heat transfer in the electrode

• Due to the consumption of the electrode, the power supplied by the electric arc affects mainly a small region at the electrode tip, usually referred to as the "heat affected zone".
• Accordingly, figure 7 .b shows a sharp exponential decay in the axial direction of the temperature from the base of the electrode toward the upper part of the electrode.
• Also, the temperature distribution over the surface of the liquid film (excluding the drops) is relatively homogeneous, which is partly related to the uniform arc heat flux distribution considered here.
• On the other hand, a much higher superheat is calculated inside the protuberance, which reaches about 400 °C at the tip of the protuberance.
• The predicted value of the superheat of the metal at the free surface of the liquid film is consistent with values reported in the literature for different materials (namely about 100 °C for IN718 [10] and between 150 °C and 200 °C for a zirconium alloy [12] ).

B. Large diameter electrode with a short gap

• Simulation results are presented for an industrial VAR melt of a maraging steel electrode.
• In order to reduce further the computational time, an irregular mesh was used.
• The mesh is refined inside the gap and in its vicinity and gets progressively coarser moving further away from this area.

Predicted melt rate

• The time evolution of the electrode melt rate calculated (with a sampling rate of 10 s) from the evolution of the electrode total mass (including both solid and liquid regions) is plotted in Fig. 12 .
• The electrode begins effectively to melt after a pre-heating stage of about 7 min.
• The variation of the melt rate is characterized by a sharp increase until reaching a quasi-stationary regime.
• In the current model, it should be noted that the process starts at full power, contrary to the real process in which the electric current is gradually increased.
• This value is in good agreement with that monitored during the actual melt which quantitatively validates the present model.

Conclusion

• This work couples the enthalpy-porosity approach to simulate the melting process and the volume of fluid method to model the deformation of the liquid film formed under the electrode.
• The model was applied to simulate the melt of a small diameter Ti-6Al-4V electrode with a long interelectrode gap and that of a large diameter maraging steel electrode with a short interelectrode gap.
• The model was shown to be able to reproduce two different modes of electrode consumption in accordance with experimental observations.
• The metal temperature is larger inside the protuberance than that in the liquid film, which may be caused by the flow structure of the metal in the protuberance.

Figures

Figure 7: (a) Temperature field of the outer surfaces of the electrode, (b) Temperature profile 436 along a vertical line cutting through a protuberance. Results obtained during the quasi-437 stationary regime at t=22.2 s. 438

Figure 2: Cross-section of the computational domain. 281

Figure 6: (a) Velocity vectors (3D) inside a protuberance. (b) Turbulent to molecular viscosity 401 ratio in a vertical cross-section of a protuberance. 402

Figure 10: Temporal evolution of the protuberance distribution at the tip of the electrode. 490

Figure 4: Depth of protuberances formed from the liquid film. 365

Figure 5: Formation and detachment of a drop observed experimentally [6] (top row) 376 and predicted by the numerical model (bottom row). 377

Figure 1: Schematic representation of the vacuum arc remelting process. 50

Figure 11: (a) Temperature field of the outer surfaces of the electrode, (b) Temperature 534 profile along the electrode central axis. Results obtained at t = 662 s 535

Figure 9: (a) Example of a calculated capillary bridge (drip-short). (b) Velocity vectors (3D) in 483 the bridge.(c) Turbulent to molecular viscosity ratio in a vertical cross-section of the bridge. 484

Figure 3: Vertical cross-section of the computational domain used for the simulation of 341 the Ti-6Al-4V electrode. 342

Figure 8: Vertical cross-section of the computational domain used for the simulation of the 459 maraging steel electrode. 460

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Submitted on 19 Oct 2021
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3D Numerical Simulation of the Var Consumable
Electrode Melting Process
Rayan Bhar, A. Jardy, P. Chapelle, Vincent Descotes
To cite this version:
Rayan Bhar, A. Jardy, P. Chapelle, Vincent Descotes. 3D Numerical Simulation of the Var Consum-
able Electrode Melting Process. Metallurgical and Materials Transactions B, Springer Verlag, 2020,
51 (6), pp.2492-2503. �10.1007/s11663-020-01966-x�. �hal-03093861�

1
3D NUMERICAL SIMULATION OF THE VAR CONSUMABLE
1
ELECTRODE MELTING PROCESS
2
3
R. Bhar
1,2
, A. Jardy
1
, P. Chapelle
1*
and V. Descotes
2
4
5
6
1
Institut Jean Lamour – UMR CNRS 7198, LabEx DAMAS, Université de Lorraine, 2 allée
7
André Guinier, Campus Artem, 54011 Nancy Cedex, France
8
2
Aperam Alloys Imphy – Avenue Jean Jaurès, BP-1, 58160 Imphy, France
9
10
*e-mail: pierre.chapelle@univ-lorraine.fr
11
12
Keywords: VAR process, consumable electrode, dripping, liquid metal film, melt rate
13
14
Abstract
15
16
A 3D numerical model was set-up to simulate the formation and dynamics of the liquid metal
17
film under the consumable electrode during VAR process. In the present paper the
18
implementation of this model is described. It was developed using the open source
19
computational fluid dynamics (CFD) software OpenFOAM. The model solves coupled
20
momentum and energy equations combined with a volume-of-fluid (VOF) method to track the
21
liquid metal free surface. The melting of the electrode material is modelled with an enthalpy-
22
porosity approach. The electric power supplied by the arc is supposed to be uniformly
23
distributed over the surface of the electrode tip. For a given electric arc power, the model
24
enable to quantitatively predict the dripping rate, hence the overall melt rate. Besides the
25
thermal behavior of the electrode, simulation results illustrate the dynamics of the liquid film
26
and the transfer mechanisms of the liquid metal during VAR melts performed with short and
27
long interelectrode gaps.
28
29
Nomenclature
30
Symbol
Description Unit
α
m
Metal volume fraction
[-]
β
Dilatation coefficient [K
-
1
]
γ
Solid volume fraction [-]
ε
Turbulent kinetic energy dissipation rate [ m
2
.s
-
3
]
λ
2
Secondary dendrite arm spacing [m]
µ
Dynamic viscosity [Pa.s
-
1
]
µ
t
Turbulent dynamic viscosity [Pa.s
-
1
]
ρ
Density [kg.m
-
3
]
σ
Surface tension [N.m
-
1
]
σ
ST
Stefan Boltzmann constant [W.m
-
2
.K
-
4
]
Cp
Specific heat [J.K
-
1
.kg
-
1
]
f
σ
Volumetric surface tension force [N.m
-
3
]
h
Total enthalpy [J.m
-
3
]
k
Turbulent kinetic energy [m². s
-
2
]

2
k
c
Curvature [m
-
1
]
k
Thermal conductivity [W.m
-
1
.K
-
1
]
k
t
Turbulent thermal conductivity [W.m
-
1
.K
-
1
]
L
Latent heat of melting [J.kg
-
1
]
P
arc
Power delivered by the arc to the electrode
P
rad
Power radiated from the electrode lateral wall
P
Pressure
T
Temperature [K]
T
sol
Solidus temperature [K]
T
liq
Liquidus temperature [K]
U
Velocity vector [m.s
-
1
]
m
Metal
VOF
Volume Of Fluid
CSF
Continuum Surface Force
CFL
Courant-Friedrich-Lewy
MULES
Multidimensional universal limiter with
explicit solution
31
32
I. Introduction
33
34
Vacuum arc remelting (VAR) is a secondary remelting process used to improve cleanliness
35
as well as chemical and mechanical homogeneity of metal ingots. VAR was the first remelting
36
process to be used commercially for superalloy processing. It is also typically the final stage
37
in the melting cycle of reactive metals such as titanium and zirconium alloys.
[1]
38
The process consists of melting a consumable electrode under vacuum (see Fig. 1). The
39
heat source is a DC electric arc of low voltage and high current. The arc is created between
40
the electrode (cathode) and the base plate of a water-cooled copper crucible at the beginning
41
of the melt, then between the electrode and the secondary ingot (anode) forming in the
42
crucible. The melting of the tip of the electrode generates a liquid metal film under the
43
electrode, from which metal drops are produced that fall under the action of gravity into the
44
crucible and progressively solidify to form the secondary ingot. At any instant, the ingot is
45
composed of three zones: the fully solidified metal, the liquid pool fed by metal drops and an
46
intermediate mushy zone.
47
48

3
49
Figure 1: Schematic representation of the vacuum arc remelting process.
50
The quality of the produced ingots strongly depends on the operating conditions of
51
remelting. Among them, the melt rate and the interelectrode gap play a key role, since they
52
have significant effects on heat transfer conditions at the free surface of the liquid pool, which
53
have important implications on the ingot structure and chemical homogeneity
[2]
.
54
The VAR process has been investigated previously with both experimental and numerical
55
approaches. On the experimental side, some studies were devoted to establish various
56
correlations between the operating parameters
[3,4]
, whereas some research work focused on
57
the electric arc behavior and metal transfer mechanisms in the interelectrode region, which
58
were observed using high speed video cameras in specifically instrumented VAR furnaces
59
[5,6]
. Modelling of the VAR process is a difficult task, because the process involves a wide
60
range of coupled complex physical and chemical phenomena, such as fluid flow, heat and
61
mass transfer, solidification (macro and microsegregation), electromagnetic forces… In the
62
literature, most modelling works deal with the development of Computational Fluid
63
Dynamics (CFD) models of the ingot growth and solidification. The majority of authors
64
considers a 2D axisymmetric geometry of the ingot and solves the conservation equations of
65
mass, momentum and energy, accounting for turbulence phenomena and electromagnetic
66
forces in the liquid pool as well as the solidification of the metal. Examples of such models
67
are the SOLAR code
[7]
and the MeltFlow-VAR code
[8]
. More recently, a multiscale 3D
68
numerical model of VAR was developed by Pericleous et al.
[9]
, which deals also with the
69
ingot behavior. Contrary to the ingot, the consumable electrode has received relatively little
70
attention, with very few modelling works reported, all restricted to thermal phenomena in the
71
electrode. Numerical studies on the formation and dynamics of the liquid film under the
72
electrode are in particular missing up to now. Bertram and Zanner
[10]
described a transient
73
and one-dimensional model of the heat transfer in the electrode, which was applied to study
74
the effect of the melting current and the gap length on the electrode melting. A similar model,
75
including an explicit account of radiative losses from the lateral walls of the electrode, was
76
presented by Jardy et al.
[11]
. Lately, a step forward was made by El Mir et al.
[12]
and Jardy et
77
al.
[13]
, who reported an unsteady model of heat transfer in the electrode, considering
78
respectively 2D and 3D geometries. Besides the electrode melt rate, these latter models enable
79
to predict, contrary to previous studies, the evolution of the shape of the electrode tip
80
throughout the melt.
81
82

4
The present work focuses on the interelectrode gap of the VAR process. The aim is to
83
numerically study the formation and deformation of the liquid film under the electrode and
84
the transfer mechanisms of the liquid metal in the interelectrode gap. A further objective is to
85
predict the melt rate of the electrode for a given electric arc power. For this purpose, a 3D
86
model describing the melting of the consumable electrode and the dynamics of the liquid film
87
formed at the electrode tip was developed using the CFD open source software OpenFOAM.
88
The model considers fuid flow under turbulent regime, heat transfer with phase change and
89
the deformation of the free surface of the liquid film. The model is concerned with both large
90
interelectrode gaps, for which the metal transfer results from the formation of molten metal
91
drops from the electrode and their detachment before contacting the ingot, and short
92
interelectrode gaps, for which the metal transfer involves the formation of intermittent molten
93
metal bridges (drip-shorts) between the electrode and the ingot. In section 2, the model is
94
described, including physical and mathematical issues, constitutive equations, boundary
95
conditions and the numerical procedure. In section 3, examples of model results detailing the
96
computed dynamics of the liquid film and thermal behavior of the electrode during the VAR
97
melt of a small-scale electrode and a fully-scale one are presented. Finally, conclusions of the
98
present study are drawn in section 4.
99
100
II. Numerical model
101
102
The formation and dynamics of the liquid film under the consumable electrode during the
103
VAR process is simulated with a multiphase CFD approach. The metal phase change is
104
accounted for using the enthalpy-porosity method
[14]
and the shape and position of the free
105
surface of the liquid film are calculated using the volume of fluid (VOF) interface capturing
106
method
[15]
.
107
108
The developed model is based on the following assumptions.
109
(1) The heat flux provided by the arc to the electrode is considered to be uniformly
110
distributed at the base of the electrode. The influence on the arc heat flux distribution of the
111
motion of individual cathode spots and of the possible existence of a relatively slow ensemble
112
motion of the arc (see e.g.
[16]
[17]
) is not examined in this study.
113
(2) As a first step towards a complete description of the liquid film behavior,
114
magnetohydrodynamic effects produced by the arc current (i.e. electromagnetic forces acting
115
on the liquid metal) are not taken into account.
116
(3) In a VAR furnace, the liquid metal film is exposed to a low pressure arc plasma. The
117
present model does not deal with the description of this complex latter phase, which is
118
represented here as a neutral gas phase.
119
(4) All thermophysical properties are considered to be independent of temperature and
120
identical in the solid and liquid phases. The metal density is made temperature dependent only
121
in the buoyancy term in the momentum equation.
122
(5) As explained later in section II.A, when computing the flow, a specific procedure was
123
introduced to eliminate non-physical spurious velocities generated near the interface in the
124
gas region. This procedure is applied at the end of each time step after calculating the flow in
125
both the liquid and gas regions. It is based on a simple filtering scheme designed to set to zero
126

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