# Influence of the Mold Current on the Electroslag Remelting Process

Mathilde Hugo, Bernard Dussoubs, Alain Jardy, Jessica Escaffre, Henri Poisson

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Mathilde Hugo, Bernard Dussoubs, Alain Jardy, Jessica Escaffre, Henri Poisson. Influence of the Mold Current on the Electroslag Remelting Process. Metallurgical and Materials Transactions B, 2016, 47 (4), pp.2607-2622. 10.1007/s11663-016-0694-x. hal-05170195

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MATHILDE HUGO, BERNARD DUSSOUBS, ALAIN JARDY, JESSICA ESCAFFRE, and HENRI POISSON

The electroslag remelting process is widely used to produce high value-added alloys. The use of numerical simulation has proven to be a valuable way to improve its understanding. In collaboration with Aubert & Duval, the Institute Jean Lamour has developed a numerical transient model of the process. The consumable electrode is remelted within a mold assumed to be electrically insulated by the solidified slag skin. However, this assumption has been challenged by some recent studies: the solidified slag skin may actually allow a part of the melting current to reach the mold. In this paper, the evolution of our model, in order to take into account this possibility, is presented and discussed. Numerical results are compared with experimental data, while several sensitivity studies show the influence of some slag properties and operating parameters on the quality of the ingot. Even, a weakly conductive solidified slag skin at the inner surface of the mold may be responsible for a non-negligible amount of current circulating between the slag and crucible, which in turn modifies the fluid flow and heat transfer in the slag and ingot liquid pool. The fraction of current concerned depends mainly on the electrical conductivities of both the liquid and solidified slag.

DOI: 10.1007/s11663-016-0694-x

© The Minerals, Metals & Materials Society and ASM International 2016

# I. INTRODUCTION

CRITICAL applications such as aeronautics, energy, and tooling require high-quality and defect-free metal, which cannot be fully achieved after primary melting. One way to achieve this quality is the use of remelting processes such as electroslag remelting (ESR) which allows a good control of the alloy chemistry and solidification structure of the ingot.[1] The materials concerned encompass high value-added metallic alloys such as special steels or nickel-based superalloys.

Remelting is caused by means of an electric current, usually an alternative one, which flows from the primary ingot—acting as a consumable electrode—whose lower part, as shown in Figure 1(a), is immersed in a highly resistive calcium fluoride-based slag, to a water-cooled crucible, thus generating Joule heating in the slag. The electrode melts by contact, forming metal droplets which fall down through the slag to the liquid pool of the secondary ingot. Because of the water cooling, the

latter solidifies continuously while growing, and its contraction leads to the building of an air gap between the solidified ingot and crucible.

It must be noticed that the slag is also in contact with the water-cooled mold, which results in the formation of a layer of solidified slag at the slag/mold interface, as shown in Figure 1(b). As the secondary ingot rises, this layer is partially remelted and/or crushed between the metal and crucible, resulting in a slag skin which acts as a thermal insulator and provides ESR ingots with a smooth lateral surface. After the end of melting, complete solidification, and removal of the mold, this slag skin may remain in the form of a few-millimeter-thick slag crust around the ingot. The thickness of this crust, at different heights, can be measured to compare model predictions with the actual observation.

Besides, chemical reactions occur at the electrode/slag, droplets/slag, and pool/slag interfaces, thus influencing the final ingot chemistry. Non-metallic inclusions are dissolved in the liquid slag.[1]

The strategic importance of the remelted products makes it essential to acquire a detailed understanding of the process. Actual remelting experiments are very costly; so their repeated use cannot be considered as a suitable solution to get a better insight into the process and help for its optimization. Apart from being easy to use, mathematical modeling is a valuable tool to enhance fundamental understanding, since it allows linkage of operating parameters, such as the fill ratio, electrode immersion depth, slag composition and properties, ingot diameter, or cooling conditions, to local solidification conditions, hence the ingot final quality.

<footnote>MATHILDE HUGO, formerly Graduate Student with the Institut Jean Lamour - UMR 7198 CNRS/Université de Lorraine (Laboratory of Excellence DAMAS), Parc de Saurupt, CS 50840, 54011, Nancy Cedex, France, is now Engineer with Sopra Banking Software, Tour Manhattan, 5 place de l'Iris, 92095, La Défense Cedex, France.
BERNARD DUSSOUBS, Research Engineer, and ALAIN JARDY, Senior Research Fellow, are with the Institut Jean Lamour - UMR 7198 CNRS/Université de Lorraine, France (Laboratory of Excellence DAMAS). Contact e-mail: alain.jardy@univ-lorraine.fr
JESSICA ESCAFFRE and HENRI POISSON, Research Engineers, are with Aubert & Duval, BP1, 63770, Les Ancizes, France.
Manuscript submitted January 20, 2015.
Article published online June 1, 2016.</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1—(a) Schematic representation of an Electroslag Remelting furnace.[2] (b) Growth of the solidified slag skin.[1]</figcaption></figure>

Therefore, several models of the ESR process have been developed during the last 40 years. The first ones were purely analytical and mainly focused on heat transfer and Joule heating in the slag bath,[3] but the complexity of the ESR process rapidly led to the use of computer simulation and the development of numerical models.

In 1977, Dilawari and Szekely introduced a model coupling electromagnetic phenomena, fluid motion, and heat transfer in steady-state conditions, for an ESR operation using a DC current,[4] which was later adapted to AC current[5] and improved.[6]

The liquid pool profile, originally used as input data, was a result of the simulation in Choudhary's work in 1980, allowing the observation of the consequences of the operating parameters on the shape and depth of the melted bath.[7,8]

In 1984, with Jardy's two-part model,[9,10] the transient ingot solidification was determined, based on the slag temperature field and fluid motion computed in steady-state conditions.

More recently, several studies have been presented, aiming to simulate the whole process in a transient way, in particular by Kharicha et al.,[11,12] Kelkar et al.,[13] and Rückert and Pfeifer[2,14]

Over the years, the influence of operating parameters and thermophysical properties of the materials on the heat transfer, fluid flow, and final quality of the ingot has been addressed[15,16] and numerical studies have challenged some of the assumptions used in the majority

of models, such as the flatness of the immersed extremity of the electrode.[17,18]

Among those models, a simulation software of the ESR process named SOLECS (which stands for SOLar-type ESR Complete Simulation) was developed at Institut Jean Lamour (IJL) in Nancy, in close collaboration with Aubert & Duval.[19] This 2D axisymmetric transient model, similar in nature to the SOLAR code used for the simulation of Vacuum Arc Remelting,[20,21] uses a finite volume method and a fully implicit time scheme to account for electromagnetic, fluid flow, heat transfer, and phase change in a coupled way.[22]

In the paper by Weber et al.,[19] it was stated that “we assume that the solidified slag skin insulates electrically the slag and ingot from the mold. This assumption is particularly questionable and needs to be confirmed. Indeed, in some cases, the model predicts a discontinuous solid skin surrounding the slag cap, implying a possible electrical contact between liquid slag and mold.” While this strong assumption is made classically in the literature devoted to ESR simulation,[23–25] it was sometimes claimed that a certain amount of current is able to flow in the copper crucible.[26,27] This phenomenon could modify the thermohydrodynamic behavior of the slag and liquid pool, hence influencing the solidification process. Therefore, the goal of our study is to quantify this phenomenon and determine the impact of the mold current on the phenomena which take place during electroslag remelting, including ingot solidification.
In this paper, the whole simulation model of the ESR process, as well as the modifications of the electromagnetic part which were necessary to take into account the possibility of mold current, are described. A study aiming to determine the impact of the electrical con-

## 1. Fluid flow and turbulence

The mean fluid flow is solved using Reynolds averaged Navier–Stokes (RANS) and continuity equations that can be written as

$$ egin{cases} 
\rho \frac{\partial u_r}{\partial t} + \rho u_r \frac{\partial u_r}{\partial r} + \rho v_z \frac{\partial u_r}{\partial z} = -\frac{\partial P}{\partial r} + \frac{\partial}{\partial r} \left( (\mu + \mu_t) \frac{1}{r} \frac{\partial (ru_r)}{\partial r} \right) + \frac{\partial}{\partial z} \left( (\mu + \mu_t) \frac{\partial u_r}{\partial z} \right) - \frac{\mu}{K} u_r + \hat{F}_r \\ 
\rho \frac{\partial v_z}{\partial t} + \rho u_r \frac{\partial v_z}{\partial r} + \rho v_z \frac{\partial v_z}{\partial z} = -\frac{\partial P}{\partial z} + \frac{1}{r} \frac{\partial}{\partial r} \left( (\mu + \mu_t) r \frac{\partial v_z}{\partial r} \right) + \frac{\partial}{\partial z} \left( (\mu + \mu_t) \frac{\partial v_z}{\partial z} \right) + \rho g_z - \frac{\mu}{K} v_z + \hat{F}_z, & [1] \\ 
\frac{1}{r} \frac{\partial}{\partial r} (ru_r) + \frac{\partial v_z}{\partial z} = 0 
\end{cases} $$

ductivity and thickness of the solidified slag skin on the electromagnetic phenomena presented, for a simplified geometry, is then detailed. Finally, results of numerical simulations accounting for mold current or assuming perfect insulation during an actual full-scale remelting are compared, and the effect of this phenomenon on the remelted ingot is discussed.

## II. THE MATHEMATICAL MODELS

### A. Principle of the Overall Thermohydrodynamic Model

In the present study, the furnace geometry is supposed to be 2D axisymmetric. The electrode–slag and slag–metal interfaces are assumed to be flat: the influence of metal droplets on the fluid flow is not considered, as it is in some models,[2,11,14] in which a volume-of-fluid (VOF) method is generally used to compute the geometry of the

where P is the pressure and with

$$ \hat{F}_r = -\frac{\mu_0}{2} \mathrm{Re}(\bar{J}_z \hat{H}) = -\frac{\mu_0}{2} (\mathrm{Re}(\hat{J}_z) \mathrm{Re}(\hat{H}) + \mathrm{Im}(\hat{J}_z) \mathrm{Im}(\hat{H})) \quad [2] $$

$$ \hat{F}_z = \frac{\mu_0}{2} \mathrm{Re}(\bar{J}_r \hat{H}) = \frac{\mu_0}{2} (\mathrm{Re}(\hat{J}_r) \mathrm{Re}(\hat{H}) + \mathrm{Im}(\hat{J}_r) \mathrm{Im}(\hat{H})). \quad [3] $$

In their papers, Jardy et al.,[10] and Kelkar et al.[28,29] have highlighted the possibility for the fluid motion to be turbulent in the slag bath and liquid metal pool. This is taken into consideration in the model by computing the turbulent viscosity using a k-ε model.[19,22]

$$ egin{cases} 
\rho \frac{\partial k}{\partial t} + \rho \left( u_r \frac{\partial k}{\partial r} + v_z \frac{\partial k}{\partial z} \right) = \frac{1}{r} \frac{\partial}{\partial r} \left\{ \left( \mu + \frac{\mu_t}{\sigma_k} \right) r \frac{\partial k}{\partial r} \right\} + \frac{\partial}{\partial z} \left\{ \left( \mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial z} \right\} \\ 
+ P_k + G_k - \rho \varepsilon - \frac{2\mu}{K} k \\ 
\rho \frac{\partial \varepsilon}{\partial t} + \rho \left( u_r \frac{\partial \varepsilon}{\partial r} + v_z \frac{\partial \varepsilon}{\partial z} \right) = \frac{1}{r} \frac{\partial}{\partial r} \left\{ \left( \mu + \frac{\mu_t}{\sigma_\varepsilon} \right) r \frac{\partial \varepsilon}{\partial r} \right\} + \frac{\partial}{\partial z} \left\{ \left( \mu + \frac{\mu_t}{\sigma_\varepsilon} \right) \frac{\partial \varepsilon}{\partial z} \right\} \\ 
+ \left( c_1 (P_k + c_{\varepsilon3} G_k) - c_2 \rho \varepsilon - c_3 \frac{2\mu}{K} k \right) \frac{\varepsilon}{k} 
\end{cases} 
\quad [4] $$

inter interfaces. The immersion depth of the electrode, which is a model input parameter, remains constant during the whole simulation. The transient-state numerical model accounts for electromagnetic phenomena and coupled heat and momentum transfers, in order to simulate the continuous growth of the electroslag remelted ingot and the solidification of both metal and slag. To achieve that goal, Maxwell equations as well as Navier–Stokes, turbulence, heat transfer, and solidification equations are solved.

with

$$ \mu_t = \rho c_\mu \frac{k^2}{\varepsilon}, \quad [5] $$

where $c_1 = 1.44$, $c_2 = 1.92$, $c_3 = 1$, $\sigma_k = 1$, $\sigma_\varepsilon = 1.3$ are constant,[19] $\mu_t$ is the turbulent dynamic viscosity, and $P_k$ and $G_k$ are the creation terms, respectively, depending on fluid motion and temperature.[19–22]

In order to account for the influence of solidification on the hydrodynamic behavior of the liquid, the model classically assumes that the mushy zone of the remelted
ingot can be assimilated to a porous medium whose permeability is estimated by the Kozeny–Carman relation:

$$K = K_0 \frac{g_l^3}{(1 - g_l^2)}, \quad [6]$$

where $K_0$ depends on the size of the solidification structure, i.e., the dendrite arm spacing.

The solidification of slag is, in a first approximation, treated in a similar way

From that representation, source terms $-\frac{\mu}{K}u_r$ and $-\frac{\mu}{K}v_z$ in Eq. [1] as well as $-\frac{2\mu}{K}k$ in Eq. [4] allow for the decay of mean motion and turbulence during solidification to be fully accounted for, as stated by Prescott and Incropera[30] and Antohe and Lage.[31] This feature is strictly identical to the one previously used by the present authors.[19]

## 2. Heat transfer and solidification

The temperature distribution is governed by the energy conservation equation:

$$\rho \frac{\partial H}{\partial t} + \rho C_p \left[ u_r \frac{\partial T}{\partial r} + v_z \frac{\partial T}{\partial z} \right] = \frac{1}{r} \frac{\partial}{\partial r} \left( (\lambda + \lambda_t) r \frac{\partial T}{\partial r} \right) + \frac{\partial}{\partial z} \left( (\lambda + \lambda_t) \frac{\partial T}{\partial z} \right) + Q, \quad [7]$$

where $Q$ is the volumetric power dissipated by Joule effect.

The turbulent conductivity $\lambda_t$ is calculated explicitly from the turbulent viscosity $\mu_t$, assuming that the turbulent Prandtl number keeps a constant value $Pr_T = \frac{C_p \mu_t}{\lambda_t} = 1$.

An explicit enthalpy formulation is used to take into account the solidification of the metal and slag:

$$H = C_p T + q_l g_l. \quad [8]$$

At each time step, Eq. [7] is solved by assuming a given liquid fraction. A new temperature–liquid fraction couple of values are then deduced, depending on both the solidification path and enthalpy–temperature relation. This approach is repeated until convergence is reached. The use of such a method requires a precise knowledge of the solidification path $gl = f(T)$ for a given alloy.

Solving the heat transfer equation in the slag bath enables prediction of slag solidification and the subsequent formation of a solidified slag layer, whose thickness is calculated at each location at the slag/mold interface.

## 3. Boundary conditions

a. Fluid flow and turbulence. A no-slip condition is applied at all liquid/solid interfaces and a zero shear stress condition is imposed at the slag-free surface.

The slag–liquid pool interface is assumed to remain flat: the radial velocity is continuous, while the axial velocity is assumed to be equal to zero.

Wall functions are employed to compute the turbulent kinetic energy, dissipation rate, and heat flux on liquid/solid interfaces and, at the free surface, a zero normal gradient is imposed for both $k$ and $\epsilon$, meaning

that surface tension effects are neglected. Finally, at the slag–pool interface, $k$ and $\epsilon$ are continuous variables.

b. Heat transfer. The present model does not include the calculation of heat transfer inside the electrode. To take into account the melting process in a very simplified way, the temperature of the electrode tip in contact with the molten slag is assumed constant and equal to the liquidus temperature of the alloy. Moreover, the model assumes that the immersion depth of the electrode remains constant during the whole melting process.

At the slag-free surface, heat loss due to thermal radiation is considered.

At the baseplate and on the lateral ingot/mold and slag/mold interfaces, a heat transfer coefficient is determined using an electrical analogy. The air gap and solidified slag skin are treated as additional thermal resistances.

The temperature distribution in the copper mold is calculated with a finite volume approach which simulates radial and axial conduction in a transient way, while the computation of heat transfer between the mold and cooling water accounts for the water flow and, if necessary, the nucleated boiling mechanism.

Illustrations of typical results obtained with this transient model are presented in the final section of this article.

## B. Details: Electromagnetism

In the ESR process, the consumable electrode is melted because of the resistive heating generated by an electric current flowing through the slag. In most cases, AC is used at a frequency of 50 Hz. In that case, the magnetic and electric fields in the system are dependent on each other, and hence the phenomena are governed by the four Maxwell's equations:

$$\text{Gauss's law for electricity: } \operatorname{div} \vec{D} = 0 \quad [9]$$

$$\text{Gauss's law for magnetism: } \operatorname{div} \vec{B} = 0 \quad [10]$$

$$\text{Faraday's law of induction: } \overrightarrow{\operatorname{rot}} \vec{E} = - \frac{\partial \vec{B}}{\partial t} = - \frac{\partial (\mu \vec{H})}{\partial t} \quad [11]$$

$$\text{Ampere's circuital law: } \overrightarrow{\operatorname{rot}} \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t} = \sigma \vec{E} + \frac{\partial (\varepsilon \vec{E})}{\partial t}. \quad [12]$$

The system formed by these equations can be reduced to a single one under some assumptions[4]:

* magnetization is negligible;

* the magnetic Reynolds number remains very low, which means that the fluid flow does not affect the electromagnetic field. In such conditions, Ohm's law $\vec{J} = \sigma(\vec{E} + \vec{v} \times \vec{B})$ can be simplified to $\vec{J} = \sigma\vec{E}$;
* current displacement is much lower than the electric conduction, which is valid if the electrical conductivity is not too small.

Moreover, we use the phasor notation in order to account for the periodic time dependence. In a 2D axisymmetric geometry, as no external field is applied, the magnetic field has only one component in the azimuthal direction, which varies with time as

$$H = H_{\max} \cos(\omega t + \varphi) = H_{\max} \cos(2\pi ft + \varphi), \quad [13]$$

where $H_{\max}$ is the maximum value of the magnetic field, $f$ is the frequency of the alternating current, and $\varphi$ is the local phase angle. Both $H_{\max}$ and $\varphi$ are dependent on the position. The phasor of the magnetic field is defined by

$$H = \hat{H} e^{j\omega t} \text{ with } j^2 = -1 \text{ and } \hat{H} = H_{\max} e^{j\varphi}. \quad [14]$$

Therefore, the system of Maxwell's equations reduces to a single one, with the stream function $r\hat{H}$ as the main variable and incorporating the electrical conductivity as a transfer coefficient. All interfacial conditions at the boundaries between adjacent media are automatically accounted for. The resulting equation for the phasor of the azimuthal component of the magnetic stream function is

$$\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{1}{r\sigma} \frac{\partial(r\hat{H})}{\partial r} \right) + \frac{\partial}{\partial z} \left( \frac{1}{r\sigma} \frac{\partial(r\hat{H})}{\partial z} \right) - \frac{1}{r^2\sigma} \frac{\partial(r\hat{H})}{\partial r} - j\omega\mu_0\hat{H} = 0. \quad [15]$$

This formulation is strictly identical to the one used, for instance, by Kharicha et al. [23] It is solved within the whole calculation domain, i.e., the electrode, slag, ingot (liquid and solid), and mold.

The real and imaginary parts of Eq. [15] are coupled through their source term.

### 1. Boundary conditions

The calculation domain for electromagnetic phenomena includes the electrode, slag, ingot, and crucible.

On top of the electrode and at the bottom of the ingot and mold, the model assumes that there is no radial current. Consequently, the boundary condition is expressed by

$$\hat{J}_r = 0 \Leftrightarrow \frac{\partial(\hat{H})}{\partial z} = 0. \quad [16]$$

At the lateral edge of the non-immersed electrode, the inner wall of the mold in contact with air, and the external wall of the crucible, the generalized Ampere theorem is applied:

$$\hat{H} = \frac{I}{\sqrt{2\pi R}}, \quad [17]$$

where R is the radius of the wall.

This equation is also applied to determine the value of the magnetic field at the surface of the slag bath. In this

particular case, R is the radial distance from the axis of symmetry.

### C. Computational Technique

The overall model is based on the solution of all partial differential equations (PDEs), along with their required boundary conditions. PDEs are discretized using a finite volume method with a fully implicit time scheme. [32] Pressure-velocity coupling is treated with the semi-implicit method for pressure-linked equations—consistent (SIMPLEC) algorithm, as proposed by Van Doormaal and Raithby. [33]

Fluid flow and heat transfer equations are solved for each time step in a coupled way. On the contrary, the calculation of the electromagnetic phenomena occurs in case of a change in the electric current or an evolution of the mesh, due to the growth of the ingot and the melting of the electrode.

The ESR model is written in FORTRAN 77. Its computational time, for a full remelting simulation, depends on the operating parameters and is typically 48 to 72 hours on a PC, when mold current is not considered. When this phenomenon is taken into consideration, the computational time can be dramatically increased.

### D. Main Grid and EM Grid

The growth of the ingot and melting of the electrode are simulated through a mesh-splitting method. In the original SOLECS model, the mold was assumed to be perfectly insulated from the melting current. An identical mesh was used whatever the equation to be solved.

If the current is allowed to flow from the slag to the crucible, the necessity to mesh the solidified slag skin, in order to simulate its effects on the current distribution, implies the use of an extremely refined mesh at the slag/mold interface.

Consequently, to reduce the overall simulation time, we chose to develop a specific grid for the electromagnetic phenomena. This grid—referred below as the EM grid—is built around a minimum number of control volumes imposed within the solidified slag skin, and covers the electrode, slag, growing ingot, and crucible.

Whenever the electromagnetic phenomena are calculated, the EM grid is generated and interpolation subroutines determine the values of liquid fraction and temperature for every node. After the electromagnetic phenomena are solved, values of Joule heating and Lorentz forces are interpolated back into the main grid and used to solve the thermohydrodynamic equations.

### E. Working Method

To be able to simulate the mold current and its influence on the ESR process, a step-by-step method has been used. At first, we developed an independent model of the electromagnetic phenomena occurring during an electroslag remelting, for the simplified geometry presented in Figure 2. With this model, we aimed to validate the possibility of mold current and its intensity,
<figure><img src="image_4.png" /><figcaption>Fig. 2—Simplified geometry for the independent electromagnetic model.</figcaption></figure>

according to the value of the electrical resistance of the solidified slag layer. Results are presented in the next section of this article.

It has to be noted that this model considers a uniform solidified slag skin thickness along the inner wall of the crucible. However, during an actual electroslag remelting, the value of this thickness varies with time and location, depending on the fluid flow and slag solidification. To take into account these phenomena as well as the consequence of mold current on the whole ESR process, the second step was to implement the detailed electromagnetic approach in the transient model. The thickness of the solidified slag skin is then determined according to the temperature distribution within the slag cap, and the transient geometry of the system corresponds to the one of an actual experimental remelting. Results obtained are presented in Section IV of this article.

### III. RESULTS OF THE INDEPENDENT ELECTROMAGNETIC MODEL

#### A. Input Data and Main Results

To reach the water-cooled crucible and baseplate, the melting current supplied to the electrode and liquid slag can either flow in the ingot pool or directly through the solidified slag skin. The resulting current distribution depends on the electrical resistance of that medium, hence the solid slag conductivity and skin thickness. In this section, we only address the computation of electromagnetic phenomena for the simplified geometry depicted in Figure 2. The thickness of the solidified slag skin is assumed to be uniform. The main input data are gathered in Table I. The electrical conductivity of the metal is assumed to be identical in the electrode and ingot, regardless of the actual liquid or solid state. The

<table><caption>Table I. Parameters Used in the Simulations</caption><tr><td>Melting current (maximum value)</td><td>10 kA</td></tr><tr><td>AC frequency</td><td>50 Hz</td></tr><tr><td>Electrode radius</td><td>26 cm</td></tr><tr><td>Mold external radius</td><td>30 cm</td></tr><tr><td>Mold thickness</td><td>2.5 cm</td></tr><tr><td>Height of the slag cap</td><td>15 cm</td></tr><tr><td>Electrode immersion depth</td><td>1 cm</td></tr><tr><td>Thickness of the solidified slag skin</td><td>1/2/4/6 mm</td></tr><tr><td>Electrical conductivity of the metal</td><td>10<sup>6</sup> S m<sup>-1</sup></td></tr><tr><td>Electrical conductivity of the liquid slag</td><td>400 S m<sup>-1</sup></td></tr><tr><td>Electrical conductivity of the solid slag</td><td>10<sup>-3</sup> to 400 S m<sup>-1</sup></td></tr></table>

effects of the variations of two parameters (electrical conductivity and thickness of the solid slag layer) on the current distribution and resulting Joule heating were studied. The electrical conductivity was allowed to vary in the range from 10<sup>-3</sup> to 400 S m<sup>-1</sup>, where 10<sup>-3</sup> S m<sup>-1</sup> corresponds to a full insulation, while 400 S m<sup>-1</sup> is the (assumed) conductivity of the liquid slag. The thickness of the solidified slag skin was set to 1, 2, 4, or 6 mm.

In the literature, the computed current distribution is most often represented by visualizing the magnitude of the current density phasor (i.e., the maximum value for each component of the current density), which classically leads to the observation of an important skin effect in the electrode and ingot.[4,19,24] Indeed, it is well known that the current distribution is related to the value of the skin depth within the different materials: if the latter is larger than the actual dimension of the domain, the current distribution is homogeneous, e.g., in the liquid slag.

However, within this study, we chose to represent the instantaneous current distribution at a given moment in the alternating period $t = 0$. In addition to the visualization of the skin effect, such a representation also highlights the local variation in the phase angle caused by the variation in the induced magnetic field in the metallic conductors, as shown by Li *et al.*[25]

The first step of the study consists in confirming that part of the current is likely to flow through the solidified slag layer and directly enter the mold. Figure 3 presents the computed results obtained either when the solid layer behaves as a perfect insulator (such behavior is reached as soon as the electrical conductivity is less than or equal to 10<sup>-3</sup> S m<sup>-1</sup>) or when the electrical conductivity is set to 15 S m<sup>-1</sup>, a value that lies in the range from 0.1 to 50 S m<sup>-1</sup>, recently proposed by Kharicha *et al.*[26] The solidified slag skin is supposed to be 4 mm thick.

Clearly, when the electrical conductivity of the solid slag is set to 15 S m<sup>-1</sup>, part of the current flows through the skin to reach the mold. The solidified slag layer does not act as a perfect electrical insulator and this modifies the current distribution in the system.

This conclusion is biased by the assumption made on the value of the electrical conductivity of the solid slag. However, as will be shown further in Figure 8, this statement remains valid as soon as the conductivity of the solid slag is assumed to be greater than 0.1 S m<sup>-1</sup>.
<figure><img src="image_5.png" /><figcaption>Fig. 3—Current density distribution (A m⁻²) computed for two values of the electrical conductivity of the solidified slag skin (thickness 4 mm): 10⁻³ S m⁻¹ (a) and 15 S m⁻¹ (b).</figcaption></figure>

It must be stated that in this study, we voluntarily used a wrong value for the electrical conductivity of the copper mold (10⁶ S m⁻¹ instead of 6.10⁷ S m⁻¹). Indeed, the model predicts that, in the latter case, important eddy currents take place within the crucible during the remelting, and their visualization prevents us from observing the melting current which flows from the slag.[³⁴] To avoid this phenomenon, we attribute to copper a lower electrical conductivity, for which the skin depth is larger than the mold width, which results in negligible eddy currents. All the results obtained remain unchanged in the electrode, slag, and ingot, while the melting current flowing in the mold may be observed.[³⁵]

## B. Influence of the Electrical Conductivity of the Solid Slag

This parametric study aims to compute the impact of a variation in the conductivity—in the range from 10⁻³ to 400 S m⁻¹—on the current distribution, i.e., to observe the behavior of the melting current when the slag/mold electrical contact varies from a total insulation to perfect contact. Figures 4 and 5 present, respectively, the current density distribution and the resistive heating computed if the conductivity of the 4-mm-thick slag skin is set to 10⁻³ (a), 1 (b), 15 (c), or 400 S m⁻¹ (d).

If the solid slag layer acts as a perfect insulator, the current distribution is homogeneous within the liquid slag (Figure 4(a)), leading to a homogeneous Joule heating (Figure 5(a)). Moreover, the amount of current inside the insulating layer and the resulting Joule effect are negligible.

As the resistivity of the solidified slag decreases, more current flows through the solid skin: the remaining current in the liquid slag generates a lower Joule heating. On the contrary, an important Joule heating can be noticed in the solid phase, which could lead to a partial remelting of the skin. It can be noted that in the absence of any partially insulating skin, the majority of the current would be attracted toward the mold (Figure 4(d)). As a result, resistive heating in the slag (Figure 5(d)) would very probably be sufficient for neither the slag to remain in the liquid state nor the electrode to be melted.

## C. Influence of the Thickness of the Slag Skin

As stated above, the thickness of the solidified slag layer is here assumed to be uniform along the height of the crucible. In the next chapter, coupling the electromagnetic model to the transient thermohydrodynamic calculations will allow us accounting for the calculated variation of that thickness with the axial position.

Based on industrial observations, we tested several values for this parameter: 1, 2, 4, and 6 mm. It must be noted that a slag skin thickness of 6 mm lies in the upper part of the range of measured values.

Figures 6 and 7 present a comparison of the computed current distribution and resistive heating for all cases. The electrical conductivity of the solid slag was set to 15 S m⁻¹.

As it can be deduced from the observation of Figures 6 and 7, the model predicts that even a thick
<figure><img src="image_6.png" /><figcaption>Fig. 4—Current density distribution (A m⁻²) computed with different values for the electrical conductivity of the solidified slag (thickness 4 mm): 10⁻³ (a), 1 (b), 15 (c), and 400 (d) S m⁻¹.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5—Computed Joule heating (W m⁻³) for different values of the electrical conductivity of the solidified slag (thickness 4 mm): 10⁻³ (a), 1 (b), 15 (c), and 400 (d) S m⁻¹.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 6—Computed current density distributions (A m⁻²) for different values of the skin thickness (mm): 1 (a), 2 (b), 4 (c), and 6 (d)—σ = 15 S m.</figcaption></figure>

slag skin (6 mm) cannot insulate completely the crucible as it was assumed until recently. Obviously, increasing the skin thickness has the same consequences as decreasing the electrical conductivity, since in both cases the electrical resistance of the solid layer is increased:

* the amount of current “lost” in the mold is lower;

* the total Joule heating in the liquid slag resulting from the current distribution is higher;

* on the contrary, the current within the skin, hence the Joule effect in the solid phase, is reduced.
<figure><img src="image_9.png" /><figcaption>Fig. 7—Joule heating (W m⁻³) for different values of the skin thickness – σ = 15 S m⁻¹.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8—Evolution of the total resistive heating in the slag, according to the electrical conductivity and the thickness of the solidified slag skin.</figcaption></figure>

## D. Conclusion

Figure 8 summarizes the main effects discussed in the previous sections: it represents the evolution of the total Joule heating generated according to both parameters. Indeed, in order to represent the influence of the equivalent electrical conductance of the solidified slag skin on the total Joule heating, it must be recalled that while the order of magnitude of the skin thickness is well known (a few mm), there is a much larger uncertainty concerning its conductivity. Therefore, the influence of the two parameters is better evidenced if they are studied independently.

The electrical conductivity of the solid slag appears to be a crucial parameter of the process. When its assumed value is lower than 0.1 S m⁻¹, the mold could be

considered as being fully insulated from the melting current. However, for higher values, the fraction of current flowing through the solidified slag skin and reaching the mold increased dramatically. This observation emphasizes the necessity to have access to actual measurements.

The thickness of the solidified slag layer also influences the current distribution in the system. However, in the range of values tested, the impact of this factor remains of secondary importance although not negligible.

The current distribution during an electroslag remelting operation is a major issue to achieve the efficiency and the final quality of ingot required. The 2-D axisymmetric steady-state model predicts that a part of the melting current may flow directly from the slag within the mold. This amount depends on the electrical resistance of the solidified slag skin, hence its electrical conductivity and thickness.

Our results confirm some previous claims in the literature[27,36] and raise new questions regarding the consequences of such a loss of current on the process efficiency.

As discussed above, in this section, the model considered a uniform layer thickness along the slag/crucible interface. However, this parameter is prone to vary from a negligible value to few millimeters during a real remelting. To take into account this variation, as well as to assess the influence of the electrical current distribution on the ingot solidification, the next step of our study consists in a full coupling of the model with a numerical simulation of the whole ESR process. Results presented in the following section are compared to those obtained without mold current and with experimental data.
<figure><img src="image_11.png" /><figcaption>Fig. 9—Current and melt rate profiles of the full-scale remelting.</figcaption></figure>

<table><caption>**Table II.** Main "Electromagnetic" Parameters of the Simulation</caption><tr><td>AC frequency</td><td>50 Hz</td></tr><tr><td>Electrode diameter</td><td>432 mm</td></tr><tr><td>Fill ratio</td><td>0.664</td></tr><tr><td>Electrode immersion depth</td><td>1 cm</td></tr><tr><td>Electrical conductivity of the metal</td><td>10<sup>6</sup> S m<sup>-1</sup></td></tr><tr><td>Electrical conductivity of the liquid slag</td><td>400 S m<sup>-1</sup></td></tr><tr><td>Electrical conductivity of the solid slag</td><td>1 S m<sup>-1</sup></td></tr></table>

## IV. RESULTS OF THE OVERALL COUPLED MODEL

In this section, we do not assume anymore the thickness of the solidified slag skin to be an input of the model. On the contrary, it is calculated explicitly, since all phenomena (i.e., electromagnetic, thermal, and hydrodynamics) are coupled. The computation of the slag temperature field enables the determination of its solidified part. In the following, we compare the results obtained with and without consideration of the mold current, for a steel remelting specifically designed and achieved at Aubert&Duval's plant in Les Ancizes.

This process can be decomposed into three parts: 150 minutes with a 18 kA AC current and a melt rate of 10 kg minute<sup>-1</sup>, 150 minutes with a 16 kA current and a melt rate of 6 kg minute<sup>-1</sup>, and after a brutal increase of the immersion depth and the disconnection of the regulation system, 100 minutes at 19 kA and 8.5 kg minute<sup>-1</sup>, as shown in Figure 9.

The main parameters of the simulation are presented in Tables II and III. As in the previous section, the electrical conductivities of the liquid and solid metal are assumed to be equal.

The full remelting has been simulated using the transient model, able to predict at any time the temperature, fluid flow, turbulence, current distribution,

<table><caption>**Table III.** Other Thermophysical Properties</caption><thead><tr><th>Property (Units)</th><th>Metal</th><th>Slag</th></tr></thead><tbody><tr><td>Viscosity (kg m<sup>-1</sup> s<sup>-1</sup>)</td><td>4.2 &times; 10<sup>-3</sup></td><td>2 &times; 10<sup>-2</sup></td></tr><tr><td>Density (kg m<sup>-3</sup>)</td><td>6982</td><td>2611</td></tr><tr><td>Specific heat (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>804</td><td>1260</td></tr><tr><td>Thermal conductivity (W m<sup>-1</sup> K<sup>-1</sup>)</td><td>30</td><td>34</td></tr><tr><td>Emissivity of the liquid</td><td>1 &times; 10<sup>-2</sup></td><td>8.7 &times; 10<sup>-1</sup></td></tr><tr><td>Volumetric thermal expansion coefficient of the liquid (K<sup>-1</sup>)</td><td>1.1 &times; 10<sup>-4</sup></td><td>1.9 &times; 10<sup>-4</sup></td></tr><tr><td>Latent heat of solidification (J kg<sup>-1</sup>)</td><td>1.87 &times; 10<sup>5</sup></td><td>4.25 &times; 10<sup>5</sup></td></tr><tr><td>Liquidus temperature [K (°C)]</td><td>1638.15 (1365)</td><td>1748.15 (1475)</td></tr><tr><td>Solidus temperature [K (°C)]</td><td>1399.15 (1126)</td><td>1633.15 (1360)</td></tr></tbody></table>

and solidification conditions. In the following, results are represented either as values evolving with time or with snapshots at specific moments of the remelting.

### A. Electromagnetic Phenomena

Figure 10 presents a comparison of the Joule effect generated within the slag, with and without mold current, at two different times of the remelting (50 and 90 pct of the total remelting time).

When current is allowed to flow through the solidified slag skin, the Joule heating within the liquid slag is less homogeneous, as it is reduced along the slag/mold interface. Due to the progressive “loss” of current along this wall, the area affected by this phenomenon increases with the depth.

Consequently, the total power generated within the system by Joule effect is lower when the mold is not fully insulated, as seen in Figure 11.
<table><caption>Fig. 10—Volumetric power generated by Joule effect within the slag bath (W m⁻³) with and without mold current.</caption><thead><tr><th></th><th>time</th><th>no mold current</th><th>mold current</th></tr></thead><tbody><tr><td rowspan="2">Joule heating (W.m<sup>-3</sup>)<br>1E+06 5E+06 9E+06 4E+07 8E+07 1E+09</td><td>50 %</td><td></td><td></td></tr><tr><td>90 %</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Fig. 11—Comparison of the power generated with and without mold current with the experimental data.</figcaption></figure>

The partial electrical insulation of the mold has a non-negligible impact on the power generated in the system. If the crucible is assumed to be fully insulated, the model overestimates the power generated during the remelting.

If not, the resistive heating is underestimated during the first period and slightly overestimated during the second one. This tends to show that the value used for the electrical conductivity of the solidified slag is not accurately known: the computed fraction of current “lost” to the mold is too high during the first part of the remelting and too low during the second one. Experimental measurements of this property are therefore required.

During the third period of the remelting, the regulation system was disconnected. The immersion depth is not known anymore, which explains the variations in the power generated and the inaccuracy of the simulation results.

## B. Fluid Motion

Within the slag bath, fluid motion results from the combined action of buoyancy and Lorentz forces. The

existence of a current circulating from the slag to the mold modifies the current distribution, hence the computed Lorentz forces and resulting fluid motion. Figures 12 and 13, respectively, present the map of stream function and velocity field when mold current is taken into account or not.

On the right part of each figure, vortices orientated in the clockwise direction are induced by buoyancy. In the absence of electromagnetic forces, the downward motion of the “colder” slag near the edge and the upward motion near the axis of symmetry are the consequences of the lateral cooling. In contrast, Lorentz force tends to induce a recirculating motion in the counter-clockwise direction.

As we can see, in both cases, the fluid flow is mainly driven by Lorentz forces near the axis of symmetry and by thermal convection along the mold wall, due to the contact with the water-cooled mold, and under the tip of the immersed electrode.

When the mold is not fully insulated, the intensity of the fluid flow decreases with the reduction of the current density in the slag. However, due to the reduction of both thermal convection (through the reduction of the Joule heating) and Lorentz forces, the motion distribution is not affected.

In the liquid pool, the fluid motion is low. Due to the high electrical conductivity of the metal, Joule heating and Lorentz forces are reduced and the fluid flow is mainly driven by buoyancy.

## C. Heat Transfer and Solidification

Figure 14 presents the computed temperature map in the slag bath and liquid pool of the solidifying ingot.

Mold current leads to a lower temperature of the slag bath, due to the decrease of the power generated by Joule heating (see above). As a result, the thickness of the solidified slag layer increases at the slag/ingot/mold interface.

The partial electrical insulation of the mold has a major influence on the computed thickness of the slag skin surrounding the ingot: while the ingot rises, the amount of slag trapped and crushed against the mold is more important, leading to a thicker slag skin as it can be seen in Figure 15.

It must be stated at this point that the measurement of the solidified slag skin thickness has been performed
<figure><img src="image_13.png" /><figcaption>Fig. 12—Stream function (kg s⁻¹) in the melt pool and slag bath with and without mold current at 50 and 90 pct of the remelting time, also showing iso-liquid fraction gI = 0.01, gI = 0.5, and gI = 0.99.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 13—Velocity field in the slag bath with and without mold current at 50 and 90 pct of the remelting time.</figcaption></figure>

experimentally after complete solidification, by measuring the width of the slag crust remaining around the ingot.

If the insulation is assumed to be perfect, the model predicts a slag skin thickness lower than 1 mm, smaller

than the experimental values measured after the removal of the mold, due to the overestimation of the power generated within the slag bath.

Otherwise, the power generated inside the slag is reduced. The resulting thickness of the solidified slag
<figure><img src="image_15.png" /><figcaption>Fig. 14—Temperature map in the slag bath and the ingot pool with and without mold current at 50 and 90 pct of the total remelting time, also showing iso-liquid fraction $g/l = 0.01$, $g/l = 0.5$, and $g/l = 0.99$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15—Experimental and calculated thickness of the slag skin surrounding the ingot.</figcaption></figure>

increases and the values calculated by the model are similar to experimental data.

As it was deduced from the evolution of the total Joule power (Figure 11), the thickness of the solidified slag skin increases during the second period of the remelting, when the melt current and melting rate are lowered.

It has to be noted that in the presence of mold current, this computed thickness presents noticeable variations with the height when the current and melt rate are reduced, while it is relatively uniform when the melt rate is high. These results confirm the experimental observation: the aspect of the ingot skin varies from smooth to rough when the melt rate decreases.

The slag skin acts like a thermal insulator from the water-cooled mold. A variation of its thickness consequently influences the heat loss, hence the depth and shape of the liquid pool profile. Figure 16 shows the evolution of the computed centreline liquid pool depth during the remelting, with and without mold current.

Predictions of both models ($g/l = 0.5$) are compatible with the measurements based on the hypothesis of grain perpendicularity to the solidification front, except when the immersion depth is voluntarily increased (between 400 and 450 minutes). Indeed, this modification of one of the main operating parameters is not taken into account in the simulation.

The solidification of the ingot is mainly a consequence of the lateral heat loss. As the thermal resistance caused by the solid slag skin is proportional to its thickness, computed local solidification time increases when the mold is not assumed to be fully insulated as shown in Figure 17. As it is generally claimed that the occurrence of macrosegregation defects is higher if the solidification time is increased, this means that the resulting quality of
<figure><img src="image_17.png" /><figcaption>Fig. 16—Evolution of the depth of the liquid pool during the remelting with and without mold current.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 17—Local solidification time in the secondary ingot (s) with and without mold current.</figcaption></figure>

the final ingot is not as good as it was determined by the first version of the model assuming no mold current.[19,22]

the Joule heating and its consequences on the ingot solidification cannot be neglected. Indeed, results high-light that even a weakly conductive solidified slag skin at the inner surface of the mold can be responsible for a non-negligible amount of current circulating between the slag and mold.

The use of the fully coupled transient model indicated that the existence of a mold current reduces the Joule heating and fluid flow inside the slag bath, leading to a lower slag temperature and thicker solidified slag skin around the ingot. When mold current is accounted for, results are consistent with experimental measurements of the slag skin thickness after solidification and removal of the mold. This shows the validity of the results and the necessity to integrate this phenomenon in the model in order to be able to predict accurately the solidification conditions of the remelted material.

## ACKNOWLEDGMENTS

As a part of the OPERAS (Optimizing Processes based on Electrode Remelting with Arc or Slag) Project, this work has been supported by the French National Research Agency (ANR-08-MAPR-0006-04).

## V. CONCLUSIONS

The electrical insulation of the mold during an ESR operation depends on the thickness and electrical conductivity of the solidified slag layer. Whereas the thickness can be deduced from the computation of heat transfer and fluid flow, the electrical conductivity has to be measured experimentally and the lack of accurate data has been highlighted in the present work.

Kharicha et al.[24] assumed its value to lie in the range from 0.1 to 50 S m⁻¹, for which the study realized with the electromagnetic part of the overall model confirms the existence of a mold current. The resulting decrease of

## NOMENCLATURE

### SYMBOLS

<table><tr><td>B&#773;</td><td>Magnetic induction (T)</td></tr><tr><td>C<sub>p</sub></td><td>Specific heat (J kg<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>D&#773;</td><td>Electric displacement field (C m<sup>-2</sup>)</td></tr><tr><td>E&#773;</td><td>Electric field (V m<sup>-1</sup>)</td></tr><tr><td>f</td><td>Frequency (Hz)</td></tr><tr><td>F&#773;(&Fcirc;r, &Fcirc;z)</td><td>Phasor of the Lorentz forces (N m<sup>-3</sup>)</td></tr><tr><td>g</td><td>Gravity (m s<sup>-2</sup>)</td></tr></table>
<table><tr><td>g<sub>l</sub></td><td>Liquid fraction</td></tr><tr><td>H&#773;</td><td>Magnetic field (A m<sup>-1</sup>)</td></tr><tr><td>H&#773;</td><td>Phasor of the magnetic field (A m<sup>-1</sup>)</td></tr><tr><td>I</td><td>Current (A)</td></tr><tr><td>Im ()</td><td>Imaginary part of a complex number</td></tr><tr><td>J&#773;(J<sub>r</sub>, J&#773;z)</td><td>Phasor of the current density (A m<sup>-2</sup>)</td></tr><tr><td>P</td><td>Total power (W)</td></tr><tr><td>Q</td><td>Volumetric Joule power (W m<sup>-3</sup>)</td></tr><tr><td>q<sub>l</sub></td><td>Latent heat of solidification (J kg<sup>-1</sup>)</td></tr><tr><td>r</td><td>Radial coordinate</td></tr><tr><td>R</td><td>Radial distance from the axis (m)</td></tr><tr><td>Re ()</td><td>Real part of a complex number</td></tr><tr><td>t</td><td>Time (s)</td></tr><tr><td>T</td><td>Temperature (K)</td></tr><tr><td>u<sub>r</sub></td><td>Radial component of velocity (m s<sup>-1</sup>)</td></tr><tr><td>v<sub>z</sub></td><td>Axial component of velocity (m s<sup>-1</sup>)</td></tr><tr><td>z</td><td>Axial coordinate</td></tr></table>

## GREEK SYMBOLS

<table><tr><td>β<sub>T</sub></td><td>Thermal expansion coefficient (K<sup>-1</sup>)</td></tr><tr><td>λ</td><td>Thermal conductivity (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>λ<sub>t</sub></td><td>Turbulent thermal conductivity (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr><tr><td>μ</td><td>Dynamic viscosity (kg m<sup>-1</sup> s<sup>-1</sup>)</td></tr><tr><td>μ<sub>t</sub></td><td>Turbulent viscosity (kg m<sup>-1</sup> s<sup>-1</sup>)</td></tr><tr><td>ρ</td><td>Density (kg m<sup>-3</sup>)</td></tr><tr><td>σ</td><td>Electrical conductivity (S m<sup>-1</sup>)</td></tr><tr><td>φ</td><td>Phase shift (rad)</td></tr><tr><td>ω</td><td>Pulsation (rad s<sup>-1</sup>)</td></tr></table>

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