# On the Prediction of Macrosegregation in Vacuum Arc Remelted Ingots

K. Mramor, T. Quatravaux, H. Combeau, Alain Jardy, Miha Založnik, I.

Crassous, A. Gaillac

► To cite this version:

K. Mramor, T. Quatravaux, H. Combeau, Alain Jardy, Miha Založnik, et al. On the Prediction of Macrosegregation in Vacuum Arc Remelted Ingots. Metallurgical and Materials Transactions B, 2022, 53 (5), pp.2953-2971. 10.1007/s11663-022-02578-3. hal-03874260

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Distributed under a Creative Commons Attribution 4.0 International License
1 On the prediction of macrosegregation in Vacuum Arc Remelted ingots

2 K. Mramor<sup>a</sup>, T. Quatravaux<sup>a*</sup>, H. Combeau<sup>a</sup>, A. Jardy<sup>a</sup>, M. Založnik<sup>a</sup>, I. Crassous<sup>b</sup>, A. Gaillac<sup>b</sup>

3

4 <sup>a</sup> Institut Jean Lamour – UMR 7198 CNRS/Université de Lorraine, Laboratory of Excellence

5 DAMAS, Campus ARTEM, 2 allée André Guinier, BP 50840, F-54011 NANCY, France

6 <sup>b</sup> Framatome, Components Research Center, Avenue Paul Girod, F-73403 UGINE, FRANCE

7 *corresponding author: thibault.quatravaux@univ-lorraine.fr, +33 (0)372742900

8 katarina.mramor@fs.uni-lj.si, herve.combeau@univ-lorraine.fr, alain.jardy@univ-lorraine.fr,

9 miha.zaloznik@univ-lorraine.fr, isabelle.crassous@framatome.com,

10 alexis.gaillac@framatome.com

11

12 Abstract

13 The chemical homogeneity and metallurgical structure of vacuum arc remelted (VAR)
14 zirconium ingots are directly responsible for product quality. It is therefore important to
15 understand the relationship between these properties and the operating conditions. An in-depth
16 analysis of the modelling of solidification phenomena during the VAR was carried out.

17 Such model, solves a coupled set of transient equations for heat, momentum, solute
18 transport and turbulence in an axisymmetric geometry. The remelting and cooling of a cylindrical
19 ingot are calculated for time-dependent operating parameters.

20 The solidification mechanisms implemented in the model can be applied to multi-
21 component industrial alloys such as Zircaloy-4, which provides information on the
22 macrosegregation phenomena studied in this paper.

23 The model results were validated, based on the remelting of a specially designed
24 chemically homogeneous Zircaloy-4 electrode. The results illustrate the importance of thermal
and solute convection, the importance of the permeability of the partially solid material and the weak influence of nuclei density and solute diffusion in the solid phase on the prediction of macrosegregation in Zircaloy-4 ingots.

**Keywords:** Numerical solution; Solidification; Zirconium alloys; Macrosegregation; Vacuum Arc Remelting, Secondary Metallurgy

Nomenclature

B magnetic field (T)

D diffusion coefficient of solute in the liquid or the solid phase (m².s⁻¹)

Fel electromagnetic force (N)

g gravitational acceleration (m.s⁻²)

gl volume fraction of the liquid phase

gs volume fraction of the solid phase

h specific enthalpy (J.kg⁻¹)

J melting current density (A.m⁻²)

J interfacial species transfer rate per unit volume (kg.m⁻³.s⁻¹)

Ji interfacial species transfer rate per unit volume due to diffusion (kg.m⁻³.s⁻¹)

Jri interfacial species transfer rate per unit volume due to the phase change (kg.m⁻³.s⁻¹)

k turbulent kinetic energy (m².s⁻²)

K permeability (m²)

K₀ permeability constant of the Blake-Kozeny law (m²)

N₀ nuclei density (m⁻³)

Nₗ grain density (m⁻³)
48 p pressure (Pa)

49 $\bar{u}_l$ average velocity of the liquid phase (m.s⁻¹)

50 Prt turbulent Prandtl number

51 Sct turbulent Schmidt number

52 $u'$ turbulent velocity of the liquid metal (m.s⁻¹)

53

54 Greek symbols

55 $\beta_s$ solutal expansion coefficient (wt%⁻¹)

56 $\beta_T$ thermal expansion coefficient (K⁻¹)

57 $\epsilon$ dissipation rate of turbulent kinetic energy (m².s⁻³)

58 $\lambda$ thermal conductivity (W.m⁻¹.K⁻¹)

59 $\lambda_{DAS}$ characteristic length of the dendritic structure (m)

60 $\mu$ dynamic viscosity (kg.m⁻¹.s⁻¹)

61 $\sigma$ electric conductivity (Ω⁻¹.m⁻¹)

62 $\varphi$ electric potential (V)

63 $\rho$ density (kg.m⁻³)

64 $\omega$ solute mass fraction (wt%)

65 $\Phi$ mass production rate per unit volume due to nucleation (kg.m⁻³.s⁻¹)

66 $\Gamma$ interfacial phase change rate per unit volume (kg.m⁻³.s⁻¹)

67 Subscripts

68 g grain density (m⁻³)

69 l liquid

70 m mixture
71 r radial direction

72 s solid

73 t turbulent

74 z axial direction

75 θ orthoradial direction

76 **1 Introduction**

77 The fuel assembly components, including Zirconium alloys, are one of the key elements in the
78 production process in nuclear reactors. Refinement of the metallurgical structure and
79 improvement of the cleanness of Zr alloy ingots are generally achieved by the Vacuum Arc
80 Remelting (VAR) process. The high quality and good performance of the final product are
81 ensured by this manufacturing process, which enables the regulation of the chemical
82 homogeneity

83 The VAR process, as illustrated in Fig. 1, consists in continuously remelting a
84 consumable electrode. In order to obtain a high quality structural ingot, the composition of the
85 electrode must comply with strict standards. A high power direct current, applied under vacuum,
86 provides an electric arc between the tip of the electrode and the baseplate of a water-cooled
87 copper mold. The intense heat generated by the arc melts the electrode tips that fall into the
88 crucible. The secondary ingot is gradually formed from the liquid metal that falls through the arc
89 plasma. The secondary ingot consists of a liquid metal pool above the solidified part and a mushy
90 zone where solidification takes place.
<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic representation of a VAR furnace.</figcaption></figure>

In addition, VAR is generally used in the remelting cycles of high quality special steels, as well as Nickel and Titanium alloys.

In the case of Zr or Ti remelting, the arc length is several centimeters. To prevent side-arcing due to these operating conditions, an axial magnetic field is generated by external induction coils. A periodical reversal of the coil current creates a magnetic field that can provide complex electromagnetic stirring of the melt pool. For the standard production of superior-quality Zr alloys for nuclear applications, a VAR operation cycle is applied (up to three times), where each secondary ingot feeds the next remelting as a new consumable electrode.

During solidification, chemical heterogeneities eventually appear in the mushy zone. One of the main challenges in the production of Zr alloys is to control macrosegregation in the VAR process for an optimal quality of the final ingots. Macrosegregation refers to chemical
heterogeneity at the ingot scale and is a combination of microsegregation, i.e. concentration gradient between the solid and liquid phases at the dendritic microscale, and the solute convective transport caused by the liquid metal flow within the mushy zone.

Several numerical models have already been developed to calculate the hydrodynamics of the melt pool in a VAR ingot, including the 2D SOLAR code, as described by Jardy and Ablitzer[1]. The extension of this CFD code forms the basis for the present study.

In addition, an extensive work has also been done to anticipate solidification defects in the remelting products. The SOLAR code was successfully used for simulating VAR and ESR (ElectroSlag Remelting) processes, to predict the risks of white spot and freckle occurrence in the ingot, using a Rayleigh number criterium[2]. The numerical results highlighted the positive impact of an optimized melting rate to minimize the occurrence of freckles.

Wilson and Jardy[3] used the SOLAR model to determine the influence of the alternated electromagnetic stirring sequence on the remelting of Titanium alloys. They presented in-depth numerical studies on several magnetic field stirring patterns during remelting. These numerical results were laterly validated by Venkatesh et al.[4] with an industrial campaign dedicated to the remelting of Titanium alloys. The numerical results were successfully validated, based on the measured positions of tungsten markers inside the ingots, providing cavity locations and liquid pool depth for different stirring sequences.

Besides, Chapelle et al.[5] determined the effect of electromagnetic stirring on melt pool free surface dynamics during the remelting of a Zirconium alloy. They observed the behavior of the free surface of the pool for undirectional and alternated stirring. They thus showed that the free surface deformation can affect the quality of the surface of the VAR ingot.

From all these previous studies, it is now well established that the sources of in-place (r, z) motion of the molten pool of a VAR ingot are:
128 - thermal convection (buoyancy force due to thermal gradients) which tends to favour a flow in a clockwise direction (symmetry axis being on the left),

130 - the Lorenz force resulting from the interaction between the melting current and the self-induced magnetic field that causes a counterclockwise flow,

132 - the centrifugal force resulting from the azimuthal motion caused by the interaction between the melting current and the external stirring field. The latter generates a clockwise flow in the (r,z) plane.

135 Moreover, Revil-Baudard et al.[6] identified the additional potential effect of solute-driven buoyancy force resulting from microsegregation, on macrosegregation.

137 This paper presents the recent implementation of an improved solidification modelling, based on work of Combeau et al.[7], to provide an enhanced description of macrosegregation phenomena in the ingot.

140

141

142 **2** Full-scale VAR trial

143 For this study, a 2.6 ton Zircaloy-4 (Zy4: Zr-1.3Sn-0.2Fe-0.1O-0.1Cr) VAR ingot of standard size was processed at the Framatome plant (Ugine, France). First, a dedicated electrode was entirely manufactured from recycled materials of the same grade, in order to avoid any chemical heterogeneity. It was noticeable that the mechanical strength of this electrode made of scraps was sufficient throughout the remelting process, so that the trial was carried out without any technical hitch.

145
146
147
148
149
150
151
The actual recordings of a highly non-standard melting sequence are presented in Figs. 2 and 3, in terms of melting rate and stirring coil current. During the trial, the stirring sequence was carried out in two successive steps: a strong alternating stirring was used to remelt the first 60 %
152 of the electrode, followed by a weak continuous stirring until the end of the melting process. The
153 magnitude of the alternating stirring was about 4.5 times higher than the continuous stirring one.

154 Additionally, a strong continuous stirring has been briefly applied at three moments
155 during the alternated stirring stage, in order to mark different pools in the ingot, as such
156 temporary change caused a local change in the grain structure of the solidified material. The
157 profiles of the marked melt pool are clearly visible on calculated composition maps presented in
158 figure 3.c.

<figure><img src="image_3.png" /><figcaption>159 **Fig. 2.** Melting rate of the full-scale Zy4 remelting trial.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Stirring sequence of the full-scale Zy4 remelting trial. (a) Coil current throughout the melt. (b) Detailed alternating stirring sequence. (c) Example of a computed composition map (Zr content), highlighting the melt pool markings in the remelted ingot.</figcaption></figure>

The remelted ingot was cut longitudinally along its entire length in order to perform chemical analyses and characterize the grain structure. Furthermore, chemical analyses were carried out along several radii at different heights of the ingot, ranging from 0.25 H to 0.88 H,
167 where H is the total height of the ingot. These measurements are used in section 4 to validate the
168 numerical results.

169

# 3 SOLAR model description

170

171

## 3.1 The CFD macromodel

172 As discussed in section 1, SOLAR is a 2D axisymmetric finite volume model of the VAR
173 process, with the complete set of macroscopic equations presented in Table 1. At a macroscopic
174 scale, the model (Eqs. (1 to 19)) takes into account the transport of heat and solutes coupled with
175 the turbulent flow driven by the electromagnetic stirring, as well as thermal and solute natural
176 convection.

177 The electromagnetic force **F** results from the interaction between the melting current **J**,
178 and the magnetic field **B**. The latter includes the self-induced magnetic field **B**<sub>θ</sub>e<sub>θ</sub>, and the
179 magnetic field created by the external induction coils **B**<sub>z,ext</sub>e<sub>z</sub>. The current density is in turn
180 calculated from Laplace's conservation equation (Eq. 8) and Ohm's law (Eq. 11). It is important
181 to note that the effect of EM stirring on the free surface dynamics is not considered here, in
182 accordance with the results of Chapelle et al.<sup>[5]</sup>, who show a second-order effect of EM stirring
183 on the deformation of the melt pool free surface. Thermal buoyancy and solutal buoyancy are
184 taken into account by using a conventional Boussinesq approximation (Eq. 12) to describe the
185 variations in density of the liquid phase in the gravity term of the momentum balance equation. In
186 the mushy zone, the solid phase forms a porous matrix, modelled by a Darcy term in the
187 momentum equation (Eq. 4). Permeability is estimated by the Blake-Kozeny law (Eq. 13), where
188 the Asai and Muchi formula is used to describe the permeability constant K<sub>0</sub> (Eq. 17).

189
**Table 1.** Main constitutive equations of the CFD macromodel

Averaged total mass balance

$$ \frac{\partial}{\partial t}(\rho_m) + \nabla \cdot (\rho_l g_l \overline{\mathbf{u}_l}) = 0 \qquad (1) $$

Averaged mass balance of the solid phase

$$ \frac{\partial}{\partial t}(\rho g_s) = \Gamma_s + \Phi_s, \quad \rho = \text{const} \qquad (2) $$

Averaged total heat balance

$$ \frac{\partial}{\partial t}(\rho g_s \overline{h_s} + \rho g_l \overline{h_l}) + \nabla \cdot (\rho g_l \mathbf{u}_l \overline{h_l}) = \nabla \cdot [(\lambda + g_l \lambda_t) \nabla \overline{T}] \qquad (3) $$

Averaged momentum balance for the liquid phase: fluid flow

$$ \frac{\partial}{\partial t}(\rho g_l \overline{\mathbf{u}_l}) + \nabla \cdot (\rho g_l \overline{\mathbf{u}_l} \overline{\mathbf{u}_l}) = -g_l \nabla p^* + \nabla \cdot [(\mu + \mu_t)g_l(\nabla \overline{\mathbf{u}_l} + (\nabla \overline{\mathbf{u}_l})^T)] - \frac{g_l^2 \mu}{K} \overline{\mathbf{u}_l} + g_l \overline{g} \overline{\rho_l^b} + g_l \mathbf{F}_{el} \qquad (4) $$

Averaged solute mass balance for species i in the solid phase

$$ \frac{\partial}{\partial t}(\rho g_s \overline{\omega_s^i}) = J_s^{ji} + J_s^{\Gamma i} \qquad (5) $$

Averaged solute mass balance for species i in the liquid phase

$$ \frac{\partial}{\partial t}(\rho g_l \overline{\omega_l^i}) + \nabla \cdot (\rho g_l \overline{\mathbf{u}_l} \overline{\omega_l^i}) = \nabla \cdot [(D_l^i + D_t)g_l \nabla \overline{\omega_l^i}] + J_l^{ji} + J_l^{\Gamma i} \qquad (6) $$

Averaged $k - \epsilon$ turbulence model
$$ \frac{\partial}{\partial t}(\rho g_l k) + \nabla \cdot (\rho g_l \mathbf{u}_l k) = \nabla \cdot \left[ \left( \mu + \frac{\mu_t}{\sigma_k} \right) g_l \nabla k \right] + P_k + D_k + G_k^T + G_k^S - \rho g_l \varepsilon \quad (7) $$

$$ \frac{\partial}{\partial t}(\rho g_l \varepsilon) + \nabla \cdot (\rho g_l \mathbf{u}_l \varepsilon) = \nabla \cdot \left[ \left( \mu + \frac{\mu_t}{\sigma_\varepsilon} \right) g_l \nabla \varepsilon \right] + \frac{\varepsilon}{k} \left[ c_1 P_k + c_2 D_k + c_1 c_3 (G_k^T + G_k^S) - c_2 \rho g_l \varepsilon \right] $$

Laplace conservation equation

$$ \nabla^2 \varphi = 0 \quad (8) $$

Mass balance equation at the solid-liquid interface

$$ \Gamma_s + \Gamma_l = 0 \quad (9) $$

Solute mass balance at the solid-liquid interface

$$ J_s^{j_i} + J_s^{\Gamma_i} + J_l^{j_i} + J_l^{\Gamma_i} = 0 \quad (10) $$

Ohm's law and Lorentz force

$$ \mathbf{J} = -\sigma \nabla \varphi \qquad \mathbf{F}_{el} = \mathbf{J} \wedge \mathbf{B} = -J_z B_\theta \mathbf{e}_r + J_r B_{ext,z} \mathbf{e}_\theta + J_r B_\theta \mathbf{e}_z \quad (11) $$

Liquid density in the buoyancy force

$$ \overline{\rho_l^b} = \rho_{ref} \left[ 1 - \beta_T (\overline{T} - T_{ref}) - \sum_i \beta_S^i (\overline{\omega_l^i} - \omega_{ref}^i) \right] \quad (12) $$

Permeability law

$$ K = \frac{g_l^3}{(1 - g_l^2)} K_0 \quad (13) $$

Production rate $P_k$ due to gradients of $u_l$ and dissipation rate $D_k$ due to liquid-solid interaction

$$ P_k = -\rho g_l \overline{\mathbf{u}_i' \mathbf{u}_j} \frac{\partial}{\partial x_j} (g_l \overline{u_i}) \qquad D_k = -\frac{2\mu g_l^2}{K} k \quad (14) $$

Generation rate due to thermal and solutal buoyancy
$$G_k^T = \beta_T g_l \frac{\mu_t}{Pr_t} \mathbf{g} \cdot \nabla \bar{T} \qquad G_k^S = \sum_i \beta_S^i g_l \frac{\mu_t}{Sc_t} \mathbf{g} \cdot \nabla \bar{\omega}_l^i \qquad (15)$$

Macroscopic $k-\epsilon$ model constants

$$ \begin{array}{lll} Pr_t = 0.85 & \sigma_k = 1.0 & c_\mu = 0.09 \\ Sc_t = 0.7 & \sigma_\epsilon = 1.3 & c_1 = 1.44 \\ & & c_2 = 1.92 \end{array} \qquad c_3 = \tanh \left( \frac{|\overline{\mathbf{u}_{l,z}}|}{\sqrt{\overline{\mathbf{u}_{l,\theta}}^2 + \overline{\mathbf{u}_{l,r}}^2}} \right) \qquad (16) $$

Permeability constants of the model

$$ K_0 = \frac{\lambda_{DAS}^2}{180} \qquad (17) $$

Pressure

$$ p^* = \bar{p}_l + \frac{2}{3} \rho k \qquad (18) $$

Turbulent parameters

$$ \mu_t = c_\mu \rho \frac{k^2}{\epsilon} \qquad \lambda_t = \frac{c_p \mu}{Pr_t} \qquad D_t = \frac{\mu_t}{\rho Sc_t} \qquad (19) $$

192 A macroscopic RANS two-equation $k-\epsilon$ model (Eq. 7) is adopted to describe the turbulence of the liquid flow. The transport equations for the mean turbulent kinetic energy $k$ and its corresponding mean dissipation rate $\epsilon$ include:

193

194

195

196 - turbulence production due to mean velocity gradients $P_k$ (Eq. 14),

197

198 - dissipation of turbulence due to the liquid/solid interaction in the mushy zone $D_k$ (Eq. 14),

199 - production or dissipation of turbulence caused by vertical thermal $G_k^T$ and solute content $G_k^S$ gradients (Eq. 15).
The constants used for closure (Eq. 16) correspond to the standard $k-\varepsilon$ model originally

developed by Launder and Spalding[8] for a fully liquid medium ($g_l = 1$, hence $K \to \infty$). This

approach is suitable for a wide variety of turbulent flows.

The closure of such model in the mushy zone, considered as a porous medium, is solved through

the formulation of the dissipation term $D_k$ of the turbulence model [9–12]. Precisely, the Prescott

and Incropera approach was implemented in the model (eq. 14).

The model simulates the direct radiation of the electric arc by the implementation of a heat source

as well as a current density on the top face of the secondary ingot.

All boundary conditions are discussed in the PhD theses by Hans[13], Quatravaux[14] and Revil-

Baudard[15].

The mesh used for all simulations reported in this paper is a structured one. The number of cells

must increase during the simulation to follow the ingot growth, as the material input from the

electrode melting is modeled by an advection-like term, i.e. a cell split and growth method. The

resulting mesh is scalable, both in size and number of cells, describing at each time step the ingot

growth in a continuous way. More details are available in Quatravaux’s Ph D thesis[14].

## 3.2 The solidification micromodel

The solidification micromodel is derived from the approach of Beckerman and

Viskanta[16]. They proposed a mathematical formulation for dendritic solidification in a binary

mixture. Their micro-model was based on a two-phase volume-averaged formulation and results

were compared to shadowgraph images of the experimental solidification of an ammonium

chloride solution in water. Although the numerical results were qualitatively close to
experimental measurements, the quantitative values for temperature and concentrations showed considerable differences.

Lately, an improved model was presented by Wang and Beckerman[17]. They described equiaxed dendritic solidification by considering the convection and solid phase transport. This model accounts for the nucleation, grain growth and dendrite morphology by solving a set of transport equations for solid, interdendritic, and extradendritic phases. However, the model was only validated at the laboratory scale.

A similar approach, based on decoupling physical phenomena according to the micro and macro time-scale, was presented by Zaloznik and Combeau[18] for a real industrial size ingot. The constitutive equations of the models were obtained by introducing a Representative Elementary Volume (REV), averaging the transport equations of heat, mass, and momentum over the individual (liquid, solid) phases. The process model considers mass transfer, turbulent fluid flow, heat transfer, electromagnetism and phase change using the following main assumptions:
- the local thermal equilibrium is assumed in each mesh cell, which behaves as a REV;
- the solidification shrinkage is not considered, because the densities of both phases are equal and constant, except for the buoyancy terms where the Boussinesq approximation is applied;
- the solid phase is considered rigid and fixed;
- the thermophysical properties of all alloys are constant; in particular, partition coefficients and liquidus slopes are independent of the temperature and composition.

An operator splitting scheme, described in detail in Založnik and Combeau[18], was used to define the transport and growth stages. In the first one, the convective/diffusive macroscopic transport partial differential equations were solved globally by neglecting nucleation and growth terms, while in the second one, the contributions of nucleation and growth were solved locally and initialized from the transport stage. The grain growth model was highly nonlinear and
therefore required special attention. The advantage of this approach was the decoupling of macro and micro time scales, which allows the grain growth model to be solved separately.

The microscopic terms that described the grain nucleation and growth were treated locally within each mesh cell where the interfacial chemical and thermal equilibrium were assumed to be reached: $T_s = T_l = T$ and $\omega_s^{i*} = k_p \omega_l^{i*}$. The interfacial compositions and temperatures are correlated by considering a simplified multicomponent phase diagram, where the liquidus temperature is described with a linear dependence on all alloy components. Both the liquidus slopes $m_l^i$ and partition coefficient $k_p^i$ are constant. Therefore, the solidification process starts when the enthalpy of a cell falls below the liquidus enthalpy $h_{liq}$,

$$h_{liq} = c_p \left( T_m + \sum_{i=1}^{n_{sol}} m_L^i \omega_l^i \right) + L_f \quad (20)$$

and stops when the temperature drops below a fixed temperature $T_{sol0}$ or when the solid fraction reaches $g_s = 1$ due to the primary growth as explained by Založnik and Combeau$^{[18]}$.

In the solidification interval, where the liquid and solid phases co-exist in the same REV ($0 < g_s < 1$), the solidification micromodel takes into account finite solute diffusion in both phases and considers two separate regions: the solid phase and the extra-dendritic liquid. In the mushy zone, grains are assumed to be spherical, the velocity of the extra-dendritic phase to be equal to the averaged velocity of the liquid, and the diffusivity of each solute to be constant in each phase. Grain generation is modelled by an instantaneous nucleation model, which assumes that a predefined number of grains per unit volume $N_g$ with an initial radius $R^*$ nucleate when the local temperature becomes lower than the local liquidus temperature (see Eq. 25 in Table 2). The
transfer of the interfacial solute due to the phase change (solidification or melting) is given in Eq.

21:

$$ (k_p^i - 1)\omega_l^{i*} \frac{\Gamma_s}{\rho} + \omega_l^{i*} \left( \frac{S_v D_l^n}{\delta_l} + k_p^n \frac{S_v D_s^n}{\delta_s} \right) = \omega_l^i \frac{S_v D_l^n}{\delta_l} + \omega_s^i \frac{S_v D_s^n}{\delta_s}. \quad (21) $$

During primary solidification, the diffusion flux at the solid/liquid interface is described by a simplified model: solute gradients are calculated by dividing the difference of the solute mass fraction at the interface $\omega_k^{*,i}$ and the average mass fraction $\omega_k^i$, by a diffusion length $\delta_k$. An approximation is used to estimate the boundary layer thickness in the solid phase while a stagnant film model is used to describe the boundary layer thickness in the liquid phase. This solidification micromodel, whose main equations are given in Table 2 (Eqs. 22-28), is described in detail by Tveito et al.[19]. It must be stated here that its implementation in SOLAR software represents an innovative approach over the previous versions used to simulate the VAR process.

**Table 2.** Main equations of the solidification micromodel.

Geometrical relations and surface area

$$ R^* = \left( \frac{3g_s}{4\pi N_g} \right)^{\frac{1}{3}} \qquad R_f = \left( \frac{3}{4\pi N_g} \right)^{\frac{1}{3}} \qquad S_v^* = 4\pi R^{*2} N_g \quad (22) $$

Dendrite tip kinetics

$$ \Omega_i = \frac{\omega_l^{*,i} - [<\omega_s^i>^e]^{gr}}{\omega_l^{*,i}(1 - k_p^i)} \qquad V = \frac{\partial R^*}{\partial t} \qquad \Gamma_s = \rho_s S_v^* V \quad (23) $$

Boundary layer thickness
$$ \delta_i^l = \frac{d_i}{\left( \frac{d_i}{R^*} - \frac{f(R^*, \Delta_i) + g(R_f, R^*, \Delta_i)}{d_i(R^* + d_i - (R^* - \Delta_i + d_i)\exp\left(\frac{-\Delta_i}{d_i}\right) - f(R^*, \Delta_i) + (\exp\left(\frac{-\Delta_i}{d_i-1}\right)g(R_f, R^*, \Delta_i))} \right)} \quad (24) $$

$$ \delta_i^s = \frac{R^*}{5} \qquad g(R_f, R^*, \Delta_i) = \frac{R_f^3 - (R^* + \Delta_i)^3}{3(R^* + \Delta_i)} \qquad d_i = \frac{D_l^i}{V} $$

$$ Sh_i = \frac{2}{3(1-g_s)} Sc_i^{\frac{1}{3}} Re^{n(Re)} \qquad f(R^*, \Delta_i) = \frac{1}{2}((R^* + \Delta_i)^2 - (R^*)^2) \qquad \Delta_i = \frac{2R^*}{Sh_i} $$

$$ Re = \frac{\rho_l (1 - g_s) 2R^*}{\mu_l} |\bar{u}_l| \qquad n(Re) = \frac{2Re^{0.28} + 4.65}{3(Re^{0.28} + 4.65)} \qquad Sc_i = \frac{\mu_l}{\rho_l D_l^i} $$

## Thermodynamics

$$ \omega_{s-d}^{*,i} = \omega_l^{*,i} k_p^i \qquad T_l = T_m + \sum_i m_l^i \omega_l^{*,i} \quad (25) $$

## Nucleation

$$ \Phi_s = \begin{cases} \rho \frac{\pi R^{*3}}{6} N_g \delta(t - t_0); & \text{if } T_l < T_m + \sum_i m_l^i \omega_l^{*,i} \text{ and } g_s = 0 \\ 0; & \text{else} \end{cases} \qquad \Phi_s = -\Phi_l \quad (26) $$

## Solute fluxes due to phase change

$$ J_s^{\Gamma^i} = \omega_s^{*,i} \Gamma_s \qquad J_l^{\Gamma^i} = \omega_l^{*,i} \Gamma_l \qquad \omega_s^{*,i} = k_p^i \omega_l^{*,i} \quad (27) $$

## Solute fluxes due to diffusion

$$ J_s^{j,i} = \frac{S_v^* \rho_s D_s^i}{\delta_s} (\omega_s^{*,i} - \omega_s^i) \qquad J_l^{j,i} = \frac{S_v^* \rho_l D_l^i}{\delta_l} (\omega_l^{*,i} - \omega_l^i) \quad (28) $$

# 4 Numerical results and discussion
282 industrial trial to feed all simulations presented in this section. The operating parameters were
283 entered by means of data files containing these records (see Figures 2 and 3).

284 The thermophysical properties of Zy4 used in the simulations are given in Table 3. Note
285 that the solidification properties (partition coefficients and liquidus slopes) have been extracted
286 from the binary phase diagrams for Zr-X alloys[20]. Unless otherwise specified, the values of β<sub>S</sub><sup>Sn</sup>
287 and λ<sub>DAS</sub> used in the simulations (default values) are also reported in Table 3.

288

289

<table><caption>290 **Table 3.** Thermophysical properties of Zy4 used in the simulations</caption><tbody><tr><td rowspan="4">Nominal compositions</td><td>Sn</td><td>ω<sub>0</sub><sup>Sn</sup> [wt%]</td><td>1.3</td></tr><tr><td>Fe</td><td>ω<sub>0</sub><sup>Fe</sup> [wt%]</td><td>0.2</td></tr><tr><td>O</td><td>ω<sub>0</sub><sup>O</sup> [wt%]</td><td>0.1</td></tr><tr><td>Cr</td><td>ω<sub>0</sub><sup>Cr</sup> [wt%]</td><td>0.1</td></tr><tr><td>Melting temperature of pure Zr</td><td>T<sub>m</sub> [°C]</td><td>1855</td></tr><tr><td rowspan="4">Liquidus slopes</td><td>Sn</td><td>m<sub>l</sub><sup>Sn</sup> [°C(wt%)<sup>-1</sup>]</td><td>-11</td></tr><tr><td>Fe</td><td>m<sub>L</sub><sup>Fe</sup> [°C(wt%)<sup>-1</sup>]</td><td>-57</td></tr><tr><td>O</td><td>m<sub>L</sub><sup>O</sup> [°C(wt%)<sup>-1</sup>]</td><td>147</td></tr><tr><td>Cr</td><td>m<sub>L</sub><sup>Cr</sup> [°C(wt%)<sup>-1</sup>]</td><td>-26</td></tr><tr><td rowspan="3">Partition Coefficients</td><td>Sn</td><td>k<sub>p</sub><sup>Sn</sup> [-]</td><td>0.27</td></tr><tr><td>Fe</td><td>k<sub>p</sub><sup>Fe</sup> [-]</td><td>0.25</td></tr><tr><td>O</td><td>k<sub>p</sub><sup>O</sup> [-]</td><td>2.32</td></tr></tbody></table>
<table><tbody><tr><td></td><td>Cr</td><td>k<sub>p</sub><sup>Cr</sup> [-]</td><td>0.15</td></tr><tr><td rowspan="4">Diffusion coefficients in the liquid</td><td>Sn</td><td>D<sub>l</sub><sup>Sn</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td>Fe</td><td>D<sub>l</sub><sup>Fe</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td>O</td><td>D<sub>l</sub><sup>O</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td>Cr</td><td>D<sub>l</sub><sup>Cr</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td rowspan="4">Diffusion coefficients in the solid</td><td>Sn</td><td>D<sub>s</sub><sup>Sn</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>4.9.10<sup>-11</sup></td></tr><tr><td>Fe</td><td>D<sub>s</sub><sup>Fe</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.5.10<sup>-9</sup></td></tr><tr><td>O</td><td>D<sub>s</sub><sup>O</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>3.8.10<sup>-9</sup></td></tr><tr><td>Cr</td><td>D<sub>s</sub><sup>Cr</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.3.10<sup>-9</sup></td></tr><tr><td>Temp. of end of solidification</td><td></td><td>T<sub>sol</sub> [°C]</td><td>1592</td></tr><tr><td>Latent heat</td><td></td><td>L<sub>f</sub> [J.kg<sup>-1</sup>]</td><td>2.42.10<sup>5</sup></td></tr><tr><td>Reference density</td><td></td><td>ρ<sub>0</sub> [kg.m<sup>-3</sup>]</td><td>6210</td></tr><tr><td>Dynamic viscosity</td><td></td><td>μ [Pa.s]</td><td>4.59.10<sup>-3</sup></td></tr><tr><td>Thermal conductivity</td><td></td><td>λ [W.m<sup>-1</sup>.°C<sup>-1</sup>]</td><td>36.5</td></tr><tr><td>Thermal expansion coefficient</td><td></td><td>β<sub>T</sub> [°C<sup>-1</sup>]</td><td>4.10<sup>-5</sup></td></tr><tr><td>Specific heat</td><td></td><td>c<sub>p</sub> [J.kg<sup>-1</sup>.°C<sup>-1</sup>]</td><td>435</td></tr><tr><td>Electric conductivity</td><td></td><td>σ [m<sup>-1</sup>.Ω<sup>-1</sup>]</td><td>6.8.10<sup>5</sup></td></tr><tr><td rowspan="4">Solute expansion coefficients</td><td>Sn</td><td>β<sub>s</sub><sup>Sn</sup> [(wt%)<sup>-1</sup>]</td><td>3.10<sup>-2</sup></td></tr><tr><td>Fe</td><td>β<sub>s</sub><sup>Fe</sup> [(wt%)<sup>-1</sup>]</td><td>0</td></tr><tr><td>O</td><td>β<sub>s</sub><sup>O</sup> [(wt%)<sup>-1</sup>]</td><td>0</td></tr><tr><td>Cr</td><td>β<sub>s</sub><sup>Cr</sup> [(wt%)<sup>-1</sup>]</td><td>0</td></tr><tr><td>Characteristic length of the</td><td></td><td>λ<sub>DAS</sub> [m]</td><td>134.10<sup>-6</sup></td></tr></tbody></table>
291

292 For all simulations reported in the present paper, the average cell size was approximately
293 9 mm wide and 6 mm height. Mesh refinement was applied with a geometrical reason of 0.8 on
294 the five nodes closest to the mold as well as to the free surface. The time-step was fixed to
295 0.025 s.

296 The computational time required for simulating a full melt depends on the computer. In
297 the IJL facility, the total CPU time was around 72 hours.

298 In several previous publications [2,21-23], the authors have detailed some CFD macromodel
299 results when applied to various remelting situations. In this paper, the attention will be focused on
300 the fluid flow in the liquid pool and the mushy zone, as well as the resulting macrosegregation.

301

## 4.1 Hydrodynamics in the liquid metal pool

303 The purpose of this section is to numerically study the influence of the thermal and solutal
304 buoyancy forces on the flow in the liquid bath during remelting. Mean velocity field, turbulence
305 generation are therefore investigated. Only the effect of local Sn content on solutal buoyancy was
306 considered, as the concentrations in all other alloying elements are negligible.

### 4.1.1. Assumptions on the thermal and solutal buoyancy forces

308 During solidification, a buoyancy driving force can be generated either by the thermal or
309 solutal buoyancy effects, that can be collaborating or opposing [24]. Both buoyancy effects are
310 modelled by the Boussinesq approximation (Eq. 12), therefore both the thermal and the solutal
311 expansion coefficients had to be estimated.
Unfortunately, values for the solutal and thermal expansion coefficient for Zr-4 alloy and its alloying elements are not readily available in the literature.

The thermal expansion has been estimated from values measured by Paradis and Rhim[25] on pure Zr. Moreover, other pertaining physical properties, such as reference density, come from this reference.

Figures 4 and 5 below show the evolution of Zr density and Sn density with the temperature.

<figure><img src="image_5.png" /></figure>

**Fig. 4.** Density of Zr as a function of temperature, from [26] (the vertical bar corresponds to Zr melting point).
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Density of Sn as a function of temperature, from [27] (the vertical bar corresponds to Sn melting point).</figcaption></figure>

From these figures, it is not possible to decide with a great confidence whether Sn should be considered as heavier or lighter than Zr at high temperature, around 2128 K, i.e. Zr melting temperature. Clearly, the value of the solutal expansion coefficient cannot be readily given.

In order to examine the influence of thermo-solutal convection, four simulations have been run, as shown in Table 4. In Case A, both buoyancy terms were null, which means that the flow is only driven by the forced convection generated by the electromagnetic field. In case B, thermal expansion coefficient for pure Zr has been used, while no solutal buoyancy was applied. Cases C and D take into account solutal buoyancy, with a negative (case C) and positive (case D) value of the Sn expansion coefficient. The negative coefficient $\beta_S^{Sn} = -1 \cdot 10^{-2}(wt\%)^{-1}$, defined in case C, was chosen to investigate on the consequences of cooperating buoyancy forces. On the contrary, the positive value in case D implies opposing buoyancy forces.

<caption>**Table 4.** Definition of the four simulations run to study the effect of thermo-solute convection</caption>
<table><thead><tr><th>case</th><th>β<sub>T</sub> [°C<sup>-1</sup>]</th><th>β<sub>S</sub><sup>Sn</sup>[wt%<sup>-1</sup>]</th></tr></thead><tbody><tr><td>A</td><td>0</td><td>0</td></tr><tr><td>B</td><td>4 · 10<sup>-5</sup></td><td>0</td></tr><tr><td>C</td><td>4 · 10<sup>-5</sup></td><td>-1 · 10<sup>-2</sup></td></tr><tr><td>D</td><td>4 · 10<sup>-5</sup></td><td>3 · 10<sup>-2</sup></td></tr></tbody></table>

339

## 4.1.2 Characteristics of the liquid metal pool

The metal flow in the melt pool and mushy zone has been investigated at instants corresponding to alternated or continuous stirring periods applied during the melt. In the latter case, the hydrodynamics in the (r, z) plane is quite stationary, whereas in the former one, two different moments have been considered during the stirring sequence:

- time $t_1$ is characterized by a low azimuthal velocity of the liquid, hence a low centrifugal force

- time $t_2$ is characterized by a high azimuthal velocity of the liquid, hence a high centrifugal force.

Flows during the continuous stirring stage (time $t_3$) and at time $t_2$ are similar, since the azimuthal velocity in both cases is high (see Fig. 6). However, as can be seen in Fig. 6, the transition from alternated to continuous stirring changes the depth and shape of the liquid pool as well as the size of the mushy zone. In the case of alternated stirring, the liquid pool is shallower and more parabolic in shape, whereas in the case of continuous stirring, the pool is deeper and cone-shaped, with a larger mushy zone.
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Case A – Computed pool profile and azimuthal velocity (m.s⁻¹) map in the melt pool at 3 different moments during the melt. (a) Time t₁ - alternated stirring, low azimuthal velocity. (b) Time t₂ - alternated stirring, high azimuthal velocity. (c) Time t₃ - continuous stirring, high azimuthal velocity.</figcaption></figure>

The hydrodynamic behavior at time t₁ (low azimuthal velocity) is the object of Fig. 7. The turbulence intensity is represented by the µt/µ viscosity ratio and the velocity field by velocity vectors in the upper part of the melt pool where velocities are high and by streamlines in the lower part of the melt pool and in the mushy zone, where velocities are significantly lower.

Without any buoyancy phenomena, the electromagnetic force causes two recirculation loops (Fig. 7 (a)). The interaction between the melting current and the self-induced magnetic field generates centripetal forces localized in the near-surface region, at the origin of a counter-
clockwise recirculation flow of the liquid metal in the upper part of the pool. In the mid-height zone of the pool, stirring centrifugal forces prevail, reinforcing the upper recirculation flow and creating a second clockwise loop in the lower part of the pool.

When thermal buoyancy is added (case B), it can be seen in fig. 7(b) that the two recirculation loops remain qualitatively identical. The major modification is a strong decrease in the turbulence intensity in the near-surface region, as the high vertical thermal stratification strongly dissipates turbulence (eq. 15).

Turbulence characteristics of the flow is almost identical to case B when solute buoyancy is added (cases C and D).

The collaborating solute buoyancy has a negligible impact on the “two recirculation loops” system observed for cases A and B. However, in case D, the opposing solute buoyancy leads to the apparition of a third counter-clockwise recirculation flow in the bottom of the pool and in the mushy zone. As $Sn$ partition ratio is lower than unity, solute redistribution during solidification causes a liquid enrichment in solute in the mushy zone. This enrichment causes a decrease in liquid density due to the positive solutal expansion coefficient. Finally, the resulting solutal buoyancy force, localized in the mushy zone, promotes an upward flow along the solidification front, inducing the observed third bottom loop.
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Computed flow in the melt pool and mushy zone at time $t_1$ (alternated stirring, low azimuthal velocity). Left side is the turbulence intensity and right side is the mean flow represented by vectors (pool upper part) or streamlines (pool lower part and mushy zone) and the liquid fraction map.
(a) Case A: $\beta_T = 0$, $\beta_S^{Sn} = 0$, $v_{max} = 2.03 \text{ cm.s}^{-1}$.
(b) Case B: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = 0$, $v_{max} = 1.60 \text{ cm.s}^{-1}$.
(c) Case C: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = -1$.</figcaption></figure>
388 $10^{-2}$ wt%$^{-1}$, $v_{max}$ = 1.82 cm.s$^{-1}$. (d) Case D: $\beta_T$ = $4 \cdot 10^{-5}$ °C$^{-1}$, $\beta_S^{Sn}$ = $3 \cdot 10^{-2}$ wt%$^{-1}$,
389 $v_{max}$ = 1.61 cm.s$^{-1}$.

390
391 The influence of a high azimuthal velocity on the in-plane velocity field and turbulence
392 intensity in the melt pool and mushy zone is shown in Figs. 8 and 9, for instants $t_2$ during the
393 alternated stirring and $t_3$ during the continuous stirring.

394 Compared to the previous case, azimuthal velocity generated by the stirring is 60 %
395 higher, implying an increase in centrifugal forces of 156 %. Therefore, the self-induced forces
396 become of second order and, thus, the circulation counter-clockwise loop in the near surface
397 region disappears.

398 Similar to previous simulations, the introduction of thermal buoyancy forces considerably
399 reduces the intensity of turbulence in the near-surface area (cases B, C and D).

400 A collaborative solutal buoyancy slightly decreases turbulence intensity in the bottom of
401 the pool in case C in the frame of a strong alternated stirring (Fig. 8(c)). This is due to the local
402 vertical solute stratification, in the same way as the effects of thermal buoyancy near the free
403 surface.

404 Finally, an opposite solutal buoyancy generates an additional recirculation loop in the
405 bottom of the pool for all stirring conditions (Fig. 8(d) and Fig. 9(d)).

406

407

408
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Computed flow in the melt pool and mushy zone at time $t_2$ (alternated stirring, high azimuthal velocity). Left side is the turbulence intensity and right side is the mean flow represented by vectors (pool upper part) or streamlines (pool lower part and mushy zone) and the liquid fraction map.
(a) Case A: $\beta_T = 0$, $\beta_S^{Sn} = 0$, $v_{max} = 2.53 \text{ cm.s}^{-1}$.
(b) Case B: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = 0$, $v_{max} = 2.53 \text{ cm.s}^{-1}$.
(c) Case C: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = -1$.</figcaption></figure>

409

410

411

412

413
414 10⁻² wt%⁻¹, vmax = 2.50 cm.s⁻¹. (d) Case D: βT = 4 · 10⁻⁵ °C⁻¹, βSⁿ = 3 · 10⁻² wt%⁻¹,

415 vmax = 2.51 cm.s⁻¹.

416

<figure><img src="image_10.png" /></figure>

417 **Fig. 9.** Computed flow in the melt pool and mushy zone at time t₃ (continuous stirring, high
azimuthal velocity). Left side is the turbulence intensity and right side is the mean flow
418
represented by vectors (pool upper part) or streamlines (pool lower part and mushy zone) and the liquid fraction map. (a) Case A: $β_T = 0$, $β_S^{Sn} = 0$, $v_{max} = 2.02$ cm.s⁻¹. (b) Case B: $β_T = 4 \cdot 10^{-5}$ °C⁻¹, $β_S^{Sn} = 0$, $v_{max} = 1.93$ cm.s⁻¹. (c) Case C: $β_T = 4 \cdot 10^{-5}$ °C⁻¹, $β_S^{Sn} = -1 \cdot 10^{-2}$ wt%⁻¹, $v_{max} = 1.83$ cm.s⁻¹. (d) Case D: $β_T = 4 \cdot 10^{-5}$ °C⁻¹, $β_S^{Sn} = 3 \cdot 10^{-2}$ wt%⁻¹, $v_{max} = 2.00$ cm.s⁻¹.

## 4.2. Macrosegregation

Macrosegregation behaviour of the remelted product is now numerically investigated. Particular attention is paid to the influence of the thermophysical properties defined in the macromodel, such as thermal and solutal buoyancy forces, as well as the solidification properties implemented in the micromodel, such as dendrite arm spacing and grain growth kinetics.

As computed results differed only slightly from previous calculations[28], where a simplified solidification model assumed complete diffusion of all solutes in both liquid and solid phases (i.e. lever rule), it appears that the use of a detailed solidification micromodel confirmed the trends which had been highlighted with a simplified lever-rule based model.

### 4.2.1. Influence of the thermo-solutal convection

The computed maps of Sn content in the final ingot are shown in Fig. 10. As discussed previously, the two declined Sn-depleted bands are a consequence of the strong electromagnetic stirring (see Fig. 3 (b)) that was applied to mark the melt pool profile in the ingot macrostructure. The average concentration of Sn in the liquid pool increases as the ingot grows because the
partition coefficient $k_p^{Sn}$ is lower than 1. Solidification of the last liquid pool causes a local enrichment at the top of the ingot. A strong negative segregation is predicted at the lateral surface and at the bottom of the ingot. However, results at the periphery must be handled with care because the segregation calculated there is very sensitive to mesh size.

442

<figure><img src="image_11.png" /></figure>
Fig. 10. Computed Sn content map in the remelted ingot. (a) Case A: $\beta_T = 0$, $\beta_S^{Sn} = 0$. (b) Case B: $\beta_T = 4 \cdot 10^{-5} \,^\circ C^{-1}$, $\beta_S^{Sn} = 0$. (c) Case C: $\beta_T = 4 \cdot 10^{-5} \,^\circ C^{-1}$, $\beta_S^{Sn} = -1 \cdot 10^{-2} \, \text{wt}\%^{-1}$. (d) Case D: $\beta_T = 4 \cdot 10^{-5} \,^\circ C^{-1}$, $\beta_S^{Sn} = 3 \cdot 10^{-2} \, \text{wt}\%^{-1}$.

446

In the absence of buoyancy (case A), the model predicts a Sn enrichment in the ingot
447
448 centre for the two stirring regimes used during the melt. This is the result of the clockwise flow
449 measurement cell discussed in the previous section. The liquid flows down along the
450 solidification front. As a result, the liquid enriched in Sn piles up in the mushy zone and at the
451 bottom of the melt pool, resulting in a positive segregation on the axis of symmetry. As shown in
452 Fig. 10, radial macrosegregation is weaker with alternating stirring than with continuous stirring.

The addition of thermal convection (case B) slightly increases the radial macrosegregation
453
454 for both stirring sequences. The centrifugal force is reinforced by thermal buoyancy, which
455 causes more intense circulation in the mushy zone and greater transport of enriched liquid to the
456 central axis.

The convection effects induced by the solutal buoyancy are presented in Figs. 10 (c) and
457
458 10 (d). They prevail in the central part of the mushy zone and have an impact on segregation. In
459 case C, the segregation at the central axis is notably amplified by the cooperation of solutal and
460 thermal buoyancy. The opposite effect is observed in case D, where the effects of solutal and
461 thermal buoyancy are opposed: the segregation at the central axis is considerably reduced, due to
462 the upward flow in the mushy region. A positive segregation band is therefore located at r/
463 R~0.5.

Moreover, in case D, during the continuous stirring sequence, a series of vertical channels
464
465 appear in the enriched segregation part. Fig. 11 details this effect. The Sn-enriched liquid
466 accumulated near the centerline is transported outward under the influence of solutal convection.

467 This transport of solute is effective for a liquid fraction $g_l > 0.6$. When $g_l < 0.6$, the

468 permeability of the porous media decreases rapidly as the drag forces dominate inertial and

469 buoyancy forces and macrosegregation evolution is stopped.

<figure><img src="image_12.png" /></figure>

470 **Fig. 11.** Zoom visualization of the influence of solute-driven convection at time $t_3$ - continuous

471 stirring, high azimuthal velocity. Case D - $\beta_T = 4 \cdot 10^{-5} \,^\circ\text{C}^{-1}$, $\beta_S^{Sn} = 3 \cdot 10^{-2} \,\text{wt}\%^{-1}$. (a)

472 Liquid fraction, streamlines and Sn content. (b) Liquid fraction and convective $v^1 \cdot \nabla T$ term.

473 In Fig. 12, the predicted radial Sn segregation profiles are plotted at two different heights

474 and compared to experimental measurements. Those heights are representative for the alternated stirring ($h/H = 0.25$) and continuous stirring ($h/H = 0.75$) stages. Comparison of the

475 simulation results with the experimental measurements confirms the importance of solutal convection in the mushy zone. At $h/H = 0.75$, the measured Sn concentration continuously

478 increases in the inner part of the ingot until it reaches its maximum at $r/R = 0.5$. The simulated results show the same tendency only in case D, where solutal convection is opposed to thermal
one. Among the calculations, case D, where the solute coefficient $\beta_S^{Sn} = 3 \cdot 10^{-2} (wt\%)^{-1}$, has the best fit to the measured data and was therefore chosen for subsequent simulations.

<figure><img src="image_13.png" /></figure>

**Fig. 12.** Radial profiles of Sn segregation ($\overline{\omega}^{Sn} - \omega_0^{Sn}$)/$\omega_0^{Sn}$ at 2 heights in the remelted ingot. Experimental (•) and model (-) results. Letters A, B, C, D correspond to cases listed in Table 4.

The computed profiles of the radially averaged longitudinal Sn content are shown in Fig. 13. Thermo-solutal convection has no significant effect on the overall longitudinal segregation. Although the comparison of simulated results with experimental measurements should be interpreted with caution, due to the limited number of measurements along the ingot height, the results show a fairly good agreement.
<figure><img src="image_14.png" /></figure>

Fig. 13. Radially-averaged longitudinal profiles of Sn segregation ($\overline{\omega}^{Sn} - \omega_{0}^{Sn})/\omega_{0}^{Sn}$) in the remelted ingot. Experimental (•) and model (-) results. Letters A, B, C, D correspond to cases listed in Table 4.

For a given thermo-solutal convection (case D), Fig. 14 shows the longitudinal content profiles for Sn and other alloying elements, namely Fe, O and Cr. The results for all elements are quite similar (it must be stated that the oxygen partition coefficient $k_{O}$ is larger than 1) and in fairly good agreement with measurements, except in the bottom part of the ingot. At this location, in addition to a possible effect of dendrite arm spacing (see section 4.2), it could be argued that the discrepancy between the measured chemistry and the predicted segregation is related to the motion of equiaxed grains. [7] showed that the sedimentation of free floating equiaxed grains can
501 significantly affect macrosegregation. Decantation of these grains in the bottom of the melt pool
502 during its solidification could balance the macrosegregation induced by thermo-solutal
503 convection, resulting in a flat segregation profile.

<figure><img src="image_15.png" /></figure>

504 **Fig. 14.** Radially-averaged longitudinal profiles of chemical segregation $(\overline{\omega^n} - \omega_0^n)/\omega_0^n$ in the
505 remelted ingot for elements Sn, Fe, O and Cr. Experimental (•) and model (-) results. Case D -
506 $\beta_T = 4 \cdot 10^{-5} \, ^\circ C^{-1}$, $\beta_S^{Sn} = 3 \cdot 10^{-2} \, \text{wt}\%^{-1}$.

507

### 4.2.2. Influence of the dendrite arm spacing

508 As discussed in section 3, the solidifying mushy zone is modelled as a porous medium
509 and is characterized by its local permeability which depends on the liquid volume fraction $g_l$ and
the dendrite arm spacing $\lambda_{DAS}$ (Eqs.13 and 17 in Table 1). Unfortunately, it is very difficult to obtain a true value of $\lambda_{DAS}$ from a conventional metallographic analysis due to the solid state transformation $\beta(cc) \rightarrow \alpha(hcp)$ that occurs during the cooling of Zirconium alloys. In order to assess the role of the permeability in macrosegregation prediction, three cases were tested for each of the solidification models and are presented in Table 5. The value

$\beta_S^{Sn} = 3 \cdot 10^{-2}(wt\%)^{-1}$ (corresponding to case D, see above) was adopted for all calculations.

**Table 5.** Definition of the three simulations run to study the effect of dendrite arm spacing.

<table><thead><tr><th>case</th><th>λ<sub>DAS</sub> [μm]</th><th>K<sub>0</sub>[m<sup>2</sup>]</th></tr></thead><tbody><tr><td>D'</td><td>67</td><td>2.5 · 10<sup>-11</sup></td></tr><tr><td>D</td><td>134</td><td>10<sup>-10</sup></td></tr><tr><td>D''</td><td>268</td><td>4 · 10<sup>-10</sup></td></tr></tbody></table>

The computed Sn macrosegregation maps are shown in Fig. 15. It can be observed that the intensity of macrosegregation increases with $\lambda_{DAS}$. To explain this phenomenon, the amplitude of the liquid velocity in the mushy zone during the continuous stirring stage (time $t_3$) is plotted in Fig. 16 as a function of the liquid fraction. It is obvious that a large $\lambda_{DAS}$ increases the velocity of the interdendritic liquid flow, thus promoting macrosegregation.
<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Influence of the dendrite arm spacing $\lambda_{DAS}$ on the computed Sn macrosegregation map in the remelted ingot. (a) Case D': $\lambda_{DAS} = 67$ µm. (b) Case D: $\lambda_{DAS} = 134$ µm. (c) Case D'':

$\lambda_{DAS} = 268$ µm.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 16.** Magnitude of the liquid velocity in the mushy zone as a function of liquid fraction at time t3 (continuous stirring stage).</figcaption></figure>

In Fig. 17, the simulated macrosegregation of Sn is compared to experimental measurements at two different heights ($h/H = 0.25$) and ($h/H = 0.75$). The profiles calculated for $\lambda_{DAS} = 67$ µm and $\lambda_{DAS} = 134$ µm are similar at $h/H = 0.25$. These profiles are quite flat up to $r/R = 0.6$, after which point the Sn concentration gradually decreases. Although the difference between the simulated results and the experimental measurements is not high, it should be noted that the predicted segregation in the mid-radius zone is positive while the experimental segregation is negative. The agreement is much better when $\lambda_{DAS}$ is increased to 268 µm, because the predictions for this case anticipate the negative Sn segregation values from $r/R = 0.5$ on. At $h/H = 0.75$, on the contrary, a better agreement was found when $\lambda_{DAS}$ was reduced (case D').
<figure><img src="image_18.png" /><figcaption>Fig. 17. Radial profiles of Sn segregation ($\overline{\omega}^{Sn} - \omega_{0}^{Sn}$)/$\omega_{0}^{Sn}$ at 2 heights in the remelted ingot. Experimental (•) and model results. Letters D, D' and D'' correspond to cases listed in Table 5.</figcaption></figure>

### 4.2.3. Influence of the grain growth kinetics

The effect of grain growth kinetics on macrosegregation was investigated numerically by examining four different cases listed in Table 6. The default values presented in Table 3 were used for the diffusion coefficients of alloy elements, thermal and solutal expansion coefficients, nucleat+ion density, and secondary dendrite arm spacing, unless otherwise specified. Case D, as usual, represents the case with default values. In case E, the nucleation density $N_0$ was increased to $10^{12} \text{ m}^{-3}$. In cases F and G, the diffusion coefficient of Sn in the solid phase was reduced to $4.9 \cdot 10^{-15} \text{ m}^2/\text{s}$ for a nucleation density equal to $10^{12} \text{ m}^{-3}$ and $10^{10} \text{ m}^{-3}$, respectively.

<table><caption>Table 6. Definition of the four simulations run to study the effect of grain growth kinetics</caption><thead><tr><td>case</td><td>$N_0$ [m<sup>-3</sup>]</td><td>$D_S^{Sn}$ [m<sup>2</sup>/s]</td></tr></thead></table>
<table><tr><td>D</td><td>10<sup>10</sup></td><td>4.9 · 10<sup>-11</sup></td></tr><tr><td>E</td><td>10<sup>12</sup></td><td>4.9 · 10<sup>-11</sup></td></tr><tr><td>F</td><td>10<sup>12</sup></td><td>4.9 · 10<sup>-15</sup></td></tr><tr><td>G</td><td>10<sup>10</sup></td><td>4.9 · 10<sup>-15</sup></td></tr></table>

553

554

Fig. 18 shows the final macrosegregation maps of Sn computed in all four cases.

555

Although the solidification process differs quantitatively, the comparison of the maps reveals only small differences in the calculated macrosegregation. As apparent from Fig. 18, nucleation density and solute diffusivity did not seem to play an important role in the chemistry of the remelted ingot. Nevertheless, it can be noted that the macrosegregation was slightly less intense in case E.

556

557

558

559

560

The small differences observed between these different cases can be explained by studying the value of the Fourier number in order to characterize the diffusion of solute in the solid and liquid phases in the central part of the ingot. The Fourier number associated to a given phase (solid or liquid) is defined as the ratio $o = \frac{D \cdot t_{ls}}{R_g^2}$. In this relationship, D is the solute diffusion coefficient in the considered phase, $t_s$ is the local solidification time, and $R_g$ is the final average grain radius as defined in equation (22) in Table 2. As it can be seen in this equation, the final grain radius is only dependent of the grain density which is an input of the model. If the value of the Fourier number is greater than 1 the composition profile of the alloying element can be considered as uniform in the phase, while if it is smaller than 1 the composition profile is non uniform. The values of the Fourier numbers for the liquid and solid phases are reported in Table 7 considering two local solidification times ($t_{ls}$), respectively 10 and 1260 s. The first value corresponds to the solidification time at a point close to the skin while the second one stands for a

561

562

563

564

565

566

567

568

569

570

571
location in the central part of the ingot. These points have been chosen along a line at mid-height of the ingot. They are considered to be representative of extreme values of the Fourier numbers in the solid and liquid phases.

**Table 7.** Values of the Fourier number in the liquid and solid phases for the four cases (D, E, F, G). These Fourier numbers are estimated for two local solidification times (10 s and 1260 s).

These values correspond to points located close to the skin of the ingot and for the second one to a point located at the center of the ingot. Both of them are considered at mid-height of the ingot.

<table><thead><tr><th rowspan="2"></th><th colspan="2">t<sub>ls</sub> = 10 s</th><th colspan="2">t<sub>ls</sub> = 1260 s)</th></tr><tr><th>Fo<sub>l</sub><sup>Sn</sup></th><th>Fo<sub>S</sub><sup>Sn</sup></th><th>Fo<sub>l</sub><sup>Sn</sup></th><th>Fo<sub>S</sub><sup>Sn</sup></th></tr></thead><tbody><tr><td>D</td><td>1.2</td><td>0.006</td><td>150</td><td>0.75</td></tr><tr><td>E</td><td>2.6</td><td>0.127</td><td>3270</td><td>16</td></tr><tr><td>F</td><td>2.6</td><td>10<sup>-5</sup></td><td>3270</td><td>0.0016</td></tr><tr><td>G</td><td>1.2</td><td>6 10<sup>-7</sup></td><td>150</td><td>7.5 10<sup>-5</sup></td></tr></tbody></table>

For the two points and all 4 cases, the Fourier number values in the liquid phase are greater than 1, thus the solute profile at the microscopic scale was uniform in the liquid phase during almost all the solidification. Considering the point close to the skin, the Fourier number in the solid phase is lower than 1 in all cases. This means that there is no diffusion of solute in the solid phase at the microscopic scale in all cases for the point close to the skin. The skin is thus a region for which all cases correspond to the same situation at the microscopic scale: perfect diffusion of solute in the liquid and no diffusion of solute in the solid. This is why the macrosegregation is very similar for all the cases in the region of the skin of the ingot.
Considering the point located at mid-height in the center of the ingot, for cases D, F and G, the Fourier number value in the solid phase is lower than 1 while it is greater than 1 for case E. These results show that for cases D, F and G, solidification proceeds in such a way that the solute diffuses perfectly in the liquid phase and very little in the solid phase. For case E, the diffusion of the solute is perfect in the solid and liquid phases. For all cases, the diffusion is perfect in the liquid phase, only the diffusion of the solute in the solid phase is different. It has already been shown that solute diffusion in the solid phase has very little influence on macrosegregation[29]. This can explain why cases D, E, F, G lead to similar macrosegregation.

<figure><img src="image_19.png" /></figure>
597 **Fig. 18.** Computed Sn content map in the remelted ingot, (a) case D, (b) case E, (c) case F and (d) case G. Letters D, E, F, G correspond to cases listed in Table 6.
598

599

600

## 5 Conclusions and prospects

601 A two-scale model was used to simulate the solidification of the remelting of an
602 experimental chemically homogeneous Zy-4 electrode in a production furnace. The effects of
603 thermal and solutal convection on ingot macrosegregation, as well as the influence of dendrite
604 arm spacing (i.e. the permeability of the mushy region) were studied and analyzed. In addition,
605 the influence of grain growth kinetics was investigated numerically.

606 The results show that it is necessary to take into account the thermal and solutal buoyancy
607 forces in order to obtain an accurate simulation of hydrodynamics and consequently the
608 macrosegregation in VAR ingots. Comparing the experimental measurements with the simulated
609 results, it is concluded that solutal-driven convection may promote an upward flow along the
610 solidification front in the mushy zone and thus be partially responsible for macrosegregation in
611 the central part of the ingot. Furthermore, the simulations have shown that the macrosegregation
612 predicted by the model depends on the dendrite arm spacing value used to calculate local
613 permeability in the mushy zone. However, the model is still unable to predict some of the trends
614 observed in experimental measurements.

615 In future work, the transport of free-floating grains and the flow induced by the
616 solidification shrinkage will be implemented in the SOLAR code. Such additions could
617 significantly affect the predicted hydrodynamic behaviour, resulting in macrosegregation.
618

619
619 Acknowledgments

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

623

624 Conflict of interest

625 On behalf of all authors, the corresponding author states that there is no conflict of
626 interest.

627

628 References

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668

669

670

671 **List of figures**

672 **FIG. 1.** SCHEMATIC REPRESENTATION OF A VAR FURNACE. .................................................................................................................. 5

673 **FIG. 2.** MELTING RATE OF THE FULL-SCALE ZY4 REMELTING TRIAL. .................................................................................................................. 8

674 **FIG. 3.** STIRRING SEQUENCE OF THE FULL-SCALE ZY4 REMELTING TRIAL. (A) COIL CURRENT THROUGHOUT THE MELT. (B)
675 DETAILED ALTERNATING STIRRING SEQUENCE. (C) EXAMPLE OF A COMPUTED COMPOSITION MAP (ZR CONTENT),
676 HIGHLIGHTING THE MELT POOL MARKINGS IN THE REMELTED INGOT. .................................................................................................................. 9

677 **FIG. 4.** DENSITY OF ZR AS A FUNCTION OF TEMPERATURE, FROM [26] (THE VERTICAL BAR CORRESPONDS TO ZR
678 MELTING POINT). ................................................................................................................................................................................................. 22

679 **FIG. 5.** DENSITY OF SN AS A FUNCTION OF TEMPERATURE, FROM [27] (THE VERTICAL BAR CORRESPONDS TO SN
680 MELTING POINT). ................................................................................................................................................................................................. 23

681 **FIG. 6.** CASE A – COMPUTED POOL PROFILE AND AZIMUTHAL VELOCITY (M.S⁻¹) MAP IN THE MELT POOL AT 3 DIFFERENT
682 MOMENTS DURING THE MELT. (A) TIME T₁ - ALTERNATED STIRRING, LOW AZIMUTHAL VELOCITY. (B) TIME T₂ -
683 ALTERNATED STIRRING, HIGH AZIMUTHAL VELOCITY. (C) TIME T₃ - CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY. 25

684 **FIG. 7.** COMPUTED FLOW IN THE MELT POOL AND MUSHY ZONE AT TIME T₁ (ALTERNATED STIRRING, LOW AZIMUTHAL VELOCITY).
685 LEFT SIDE IS THE TURBULENCE INTENSITY AND RIGHT SIDE IS THE MEAN FLOW REPRESENTED BY VECTORS (POOL UPPER
686 PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: BT = 0,
687 BSSN = 0, VMAX = 2.03 CM.S-1. (B) CASE B: BT = 4 · 10⁻⁵ °C − 1, BSSN = 0, VMAX = 1.60 CM.S-1. (C) CASE
688 C: BT = 4 · 10⁻⁵ °C − 1, BSSN = −1 · 10⁻² WT% − 1, VMAX = 1.82 CM.S-1. (D) CASE D: BT = 4 · 10⁻⁵ °C − 1,
689 BSSN = 3 · 10⁻² WT% − 1, VMAX = 1.61 CM.S-1. .................................................................................................................. 27

690 **FIG. 8.** COMPUTED FLOW IN THE MELT POOL AND MUSHY ZONE AT TIME T₂ (ALTERNATED STIRRING, HIGH AZIMUTHAL VELOCITY).
691 LEFT SIDE IS THE TURBULENCE INTENSITY AND RIGHT SIDE IS THE MEAN FLOW REPRESENTED BY VECTORS (POOL UPPER
692 PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: BT = 0,
693 BSSN = 0, VMAX = 2.53 CM.S-1. (B) CASE B: BT = 4 · 10⁻⁵ °C − 1, BSSN = 0, VMAX = 2.53 CM.S-1. (C) CASE
694 C: BT = 4 · 10⁻⁵ °C − 1, BSSN = −1 · 10⁻² WT% − 1, VMAX = 2.50 CM.S-1. (D) CASE D: BT = 4 · 10⁻⁵ °C − 1,
695 BSSN = 3 · 10⁻² WT% − 1, VMAX = 2.51 CM.S-1. .................................................................................................................. 29

696 **FIG. 9.** COMPUTED FLOW IN THE MELT POOL AND MUSHY ZONE AT TIME T₃ (CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY).
697 LEFT SIDE IS THE TURBULENCE INTENSITY AND RIGHT SIDE IS THE MEAN FLOW REPRESENTED BY VECTORS (POOL UPPER
698 PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: BT = 0,
699 BSSN = 0, VMAX = 2.02 CM.S-1. (B) CASE B: BT = 4 · 10⁻⁵ °C − 1, BSSN = 0, VMAX = 1.93 CM.S-1. (C) CASE
700 C: BT = 4 · 10⁻⁵ °C − 1, BSSN = −1 · 10⁻² WT% − 1, VMAX = 1.83 CM.S-1. (D) CASE D: BT = 4 · 10⁻⁵ °C − 1,
701 BSSN = 3 · 10⁻² WT% − 1, VMAX = 2.00 CM.S-1. .................................................................................................................. 30
702 **FIG. 10.** COMPUTED SN CONTENT MAP IN THE REMELTED INGOT. (A) CASE A: $bT = 0$, $bSSN = 0$. (B) CASE B: $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 0$. (C) CASE C: $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = -1 \cdot 10 - 2 \,\text{wt}\% - 1$. (D) CASE D: $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 3 \cdot 10 - 2 \,\text{wt}\% - 1$. 33

705 **FIG. 11.** ZOOM VISUALIZATION OF THE INFLUENCE OF SOLUTE-DRIVEN CONVECTION AT TIME $T_3$ - CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY. CASE D - $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 3 \cdot 10 - 2 \,\text{wt}\% - 1$. (A) LIQUID FRACTION, STREAMLINES AND SN CONTENT. (B) LIQUID FRACTION AND CONVECTIVE $v^\perp \nabla T$ TERM. 34

708 **FIG. 12.** RADIAL PROFILES OF SN SEGREGATION ($\Omega\text{SN} - \Omega\text{0SN})/\Omega\text{0SN}$ AT 2 HEIGHTS IN THE REMELTED INGOT. EXPERIMENTAL ($\bullet$) AND MODEL (-) RESULTS. LETTERS A, B, C, D CORRESPOND TO CASES LISTED IN TABLE 4. 35

710 **FIG. 13.** RADIALLY-AVERAGED LONGITUDINAL PROFILES OF SN SEGREGATION ($\Omega\text{SN} - \Omega\text{0SN})/\Omega\text{0SN}$ IN THE REMELTED INGOT. EXPERIMENTAL ($\bullet$) AND MODEL (-) RESULTS. LETTERS A, B, C, D CORRESPOND TO CASES LISTED IN TABLE 4. 36

712 **FIG. 14.** RADIALLY-AVERAGED LONGITUDINAL PROFILES OF CHEMICAL SEGREGATION ($\Omega\text{N} - \Omega\text{0N})/\Omega\text{0N}$ IN THE REMELTED INGOT FOR ELEMENTS SN, FE, O AND CR. EXPERIMENTAL ($\bullet$) AND MODEL (-) RESULTS. CASE D - $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 3 \cdot 10 - 2 \,\text{wt}\% - 1$. 37

715 **FIG. 15.** INFLUENCE OF THE DENDRITE ARM SPACING $\lambda DAS$ ON THE COMPUTED SN MACROSEGREGATION MAP IN THE REMELTED INGOT. (A) CASE D': $\lambda DAS = 67 \,\mu\text{M}$. (B) CASE D: $\lambda DAS = 134 \,\mu\text{M}$. (C) CASE D": $\lambda DAS = 268 \,\mu\text{M}$. 39

717 **FIG. 16.** MAGNITUDE OF THE LIQUID VELOCITY IN THE MUSHY ZONE AS A FUNCTION OF LIQUID FRACTION AT TIME $T_3$ (CONTINUOUS STIRRING STAGE). 40

719 **FIG. 17.** RADIAL PROFILES OF SN SEGREGATION ($\omega\text{Sn} - \omega\text{0Sn})/\omega\text{0Sn}$ AT 2 HEIGHTS IN THE REMELTED INGOT. EXPERIMENTAL ($\bullet$) AND MODEL RESULTS. LETTERS D, D' AND D'' CORRESPOND TO CASES LISTED IN TABLE 5. 41

721 **FIG. 18.** COMPUTED SN CONTENT MAP IN THE REMELTED INGOT, (A) CASE D, (B) CASE E, (C) CASE F AND (D) CASE G. LETTERS D, E, F, G CORRESPOND TO CASES LISTED IN TABLE 6. 45

723

724

725 **List of tables**

726 **TABLE 1.** MAIN CONSTITUTIVE EQUATIONS OF THE CFD MACROMODEL 11

727 **TABLE 2.** MAIN EQUATIONS OF THE SOLIDIFICATION MICROMODEL. 17

728 **TABLE 3.** THERMOPHYSICAL PROPERTIES OF ZY4 USED IN THE SIMULATIONS 19

729 **TABLE 4.** DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF THERMO-SOLUTE CONVECTION 23

730 **TABLE 5.** DEFINITION OF THE THREE SIMULATIONS RUN TO STUDY THE EFFECT OF DENDRITE ARM SPACING. 38

731 **TABLE 6.** DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF GRAIN GROWTH KINETICS 41

732 **TABLE 7.** VALUES OF THE FOURIER NUMBER IN THE LIQUID AND SOLID PHASES FOR THE FOUR CASES (D, E, F, G). THESE FOURIER NUMBERS ARE ESTIMATED FOR TWO LOCAL SOLIDIFICATION TIMES ($10 \text{ s}$ AND $1260 \text{ s}$). THESE VALUES CORRESPOND TO POINTS LOCATED CLOSE TO THE SKIN OF THE INGOT AND FOR THE SECOND ONE TO A POINT LOCATED AT THE CENTER OF THE INGOT. BOTH OF THEM ARE CONSIDERED AT MID-HEIGHT OF THE INGOT. 43

736
1 On the prediction of macrosegregation in Vacuum Arc Remelted ingots

2 K. Mramor<sup>a</sup>, T. Quatravaux<sup>a*</sup>, H. Combeau<sup>a</sup>, A. Jardy<sup>a</sup>, M. Založnik<sup>a</sup>, I. Crassous<sup>b</sup>, A. Gaillac<sup>b</sup>

3

4 <sup>a</sup> Institut Jean Lamour – UMR 7198 CNRS/Université de Lorraine, Laboratory of Excellence

5 DAMAS, Campus ARTEM, 2 allée André Guinier, BP 50840, F-54011 NANCY, France

6 <sup>b</sup> Framatome, Components Research Center, Avenue Paul Girod, F-73403 UGINE, FRANCE

7 *corresponding author: thibault.quatravaux@univ-lorraine.fr, +33 (0)372742900

8 katarina.mramor@fs.uni-lj.si, herve.combeau@univ-lorraine.fr, alain.jardy@univ-lorraine.fr,

9 miha.zaloznik@univ-lorraine.fr, isabelle.crassous@framatome.com,

10 alexis.gaillac@framatome.com

11

12 Abstract

13 The chemical homogeneity and metallurgical structure of vacuum arc remelted (VAR)
14 zirconium ingots are directly responsible for product quality. It is therefore important to
15 understand the relationship between these properties and the operating conditions. An in-depth
16 analysis of the modelling of solidification phenomena during the VAR was carried out.

17 Such model, solves a coupled set of transient equations for heat, momentum, solute
18 transport and turbulence in an axisymmetric geometry. The remelting and cooling of a cylindrical
19 ingot are calculated for time-dependent operating parameters.

20 The solidification mechanisms implemented in the model can be applied to multi-
21 component industrial alloys such as Zircaloy-4, which provides information on the
22 macrosegregation phenomena studied in this paper.

23 The model results were validated, based on the remelting of a specially designed
24 chemically homogeneous Zircaloy-4 electrode. The results illustrate the importance of thermal
and solute convection, the importance of the permeability of the partially solid material and the weak influence of nuclei density and solute diffusion in the solid phase on the prediction of macrosegregation in Zircaloy-4 ingots.

**Keywords:** Numerical solution; Solidification; Zirconium alloys; Macrosegregation; Vacuum Arc Remelting, Secondary Metallurgy

Nomenclature

B magnetic field (T)

D diffusion coefficient of solute in the liquid or the solid phase (m².s⁻¹)

Fel electromagnetic force (N)

g gravitational acceleration (m.s⁻²)

gl volume fraction of the liquid phase

gs volume fraction of the solid phase

h specific enthalpy (J.kg⁻¹)

J melting current density (A.m⁻²)

J interfacial species transfer rate per unit volume (kg.m⁻³.s⁻¹)

Ji interfacial species transfer rate per unit volume due to diffusion (kg.m⁻³.s⁻¹)

Jri interfacial species transfer rate per unit volume due to the phase change (kg.m⁻³.s⁻¹)

k turbulent kinetic energy (m².s⁻²)

K permeability (m²)

K₀ permeability constant of the Blake-Kozeny law (m²)

N₀ nuclei density (m⁻³)

Nₗ grain density (m⁻³)
48 p pressure (Pa)

49 $\bar{u}_l$ average velocity of the liquid phase (m.s⁻¹)

50 Pr<sub>t</sub> turbulent Prandtl number

51 Sc<sub>t</sub> turbulent Schmidt number

52 $\mathbf{u}'$ turbulent velocity of the liquid metal (m.s⁻¹)

53

54 *Greek symbols*

55 $\beta_S$ solutal expansion coefficient (wt%⁻¹)

56 $\beta_T$ thermal expansion coefficient (K⁻¹)

57 $\varepsilon$ dissipation rate of turbulent kinetic energy (m².s⁻³)

58 $\lambda$ thermal conductivity (W.m⁻¹.K⁻¹)

59 $\lambda_{DAS}$ characteristic length of the dendritic structure (m)

60 $\mu$ dynamic viscosity (kg.m⁻¹.s⁻¹)

61 $\sigma$ electric conductivity (Ω⁻¹.m⁻¹)

62 $\varphi$ electric potential (V)

63 $\rho$ density (kg.m⁻³)

64 $\omega$ solute mass fraction (wt%)

65 $\Phi$ mass production rate per unit volume due to nucleation (kg.m⁻³.s⁻¹)

66 $\Gamma$ interfacial phase change rate per unit volume (kg.m⁻³.s⁻¹)

67 *Subscripts*

68 g grain density (m⁻³)

69 l liquid

70 m mixture
71 r radial direction

72 s solid

73 t turbulent

74 z axial direction

75 θ orthoradial direction

76 **1 Introduction**

77 The fuel assembly components, including Zirconium alloys, are one of the key elements in the
78 production process in nuclear reactors. Refinement of the metallurgical structure and
79 improvement of the cleanness of Zr alloy ingots are generally achieved by the Vacuum Arc
80 Remelting (VAR) process. The high quality and good performance of the final product are
81 ensured by this manufacturing process, which enables the regulation of the chemical
82 homogeneity

83 The VAR process, as illustrated in Fig. 1, consists in continuously remelting a
84 consumable electrode. In order to obtain a high quality structural ingot, the composition of the
85 electrode must comply with strict standards. A high power direct current, applied under vacuum,
86 provides an electric arc between the tip of the electrode and the baseplate of a water-cooled
87 copper mold. The intense heat generated by the arc melts the electrode tips that fall into the
88 crucible. The secondary ingot is gradually formed from the liquid metal that falls through the arc
89 plasma. The secondary ingot consists of a liquid metal pool above the solidified part and a mushy
90 zone where solidification takes place.
<figure><img src="image_20.png" /><figcaption>Fig. 1. Schematic representation of a VAR furnace.</figcaption></figure>

In addition, VAR is generally used in the remelting cycles of high quality special steels, as well as Nickel and Titanium alloys.

In the case of Zr or Ti remelting, the arc length is several centimeters. To prevent side-arcing due to these operating conditions, an axial magnetic field is generated by external induction coils. A periodical reversal of the coil current creates a magnetic field that can provide complex electromagnetic stirring of the melt pool. For the standard production of superior-quality Zr alloys for nuclear applications, a VAR operation cycle is applied (up to three times), where each secondary ingot feeds the next remelting as a new consumable electrode.

During solidification, chemical heterogeneities eventually appear in the mushy zone. One of the main challenges in the production of Zr alloys is to control macrosegregation in the VAR process for an optimal quality of the final ingots. Macrosegregation refers to chemical
heterogeneity at the ingot scale and is a combination of microsegregation, i.e. concentration

gradient between the solid and liquid phases at the dendritic microscale, and the solute convective

transport caused by the liquid metal flow within the mushy zone.

Several numerical models have already been developed to calculate the hydrodynamics of

the melt pool in a VAR ingot, including the 2D SOLAR code, as described by Jardy and

Ablitzer[1]. The extension of this CFD code forms the basis for the present study.

In addition, an extensive work has also been done to anticipate solidification defects in the

remelting products. The SOLAR code was successfully used for simulating VAR and ESR

(ElectroSlag Remelting) processes, to predict the risks of white spot and freckle occurrence in the

ingot, using a Rayleigh number criterium[2]. The numerical results highlighted the positive impact

of an optimized melting rate to minimize the occurrence of freckles.

Wilson and Jardy[3] used the SOLAR model to determine the influence of the alternated

electromagnetic stirring sequence on the remelting of Titanium alloys. They presented in-depth

numerical studies on several magnetic field stirring patterns during remelting. These numerical

results were lately validated by Venkatesh et al.[4] with an industrial campaign dedicated to the

remelting of Titanium alloys. The numerical results were successfully validated, based on the

measured positions of tungsten markers inside the ingots, providing cavity locations and liquid

pool depth for different stirring sequences.

Besides, Chapelle et al.[5] determined the effect of electromagnetic stirring on melt pool

free surface dynamics during the remelting of a Zirconium alloy. They observed the behavior of

the free surface of the pool for undirectional and alternated stirring. They thus showed that the

free surface deformation can affect the quality of the surface of the VAR ingot.

From all these previous studies, it is now well established that the sources of in-place (r,

z) motion of the molten pool of a VAR ingot are:
128 - thermal convection (buoyancy force due to thermal gradients) which tends to favour a flow in a clockwise direction (symmetry axis being on the left),

130 - the Lorenz force resulting from the interaction between the melting current and the self-induced magnetic field that causes a counterclockwise flow,

132 - the centrifugal force resulting from the azimuthal motion caused by the interaction between the melting current and the external stirring field. The latter generates a clockwise flow in the (r,z) plane.

135 Moreover, Revil-Baudard et al.[6] identified the additional potential effect of solute-driven buoyancy force resulting from microsegregation, on macrosegregation.

137 This paper presents the recent implementation of an improved solidification modelling, based on work of Combeau et al.[7], to provide an enhanced description of macrosegregation phenomena in the ingot.

140

141

142 **2** Full-scale VAR trial

143 For this study, a 2.6 ton Zircaloy-4 (Zy4: Zr-1.3Sn-0.2Fe-0.1O-0.1Cr) VAR ingot of standard size was processed at the Framatome plant (Ugine, France). First, a dedicated electrode was entirely manufactured from recycled materials of the same grade, in order to avoid any chemical heterogeneity. It was noticeable that the mechanical strength of this electrode made of scraps was sufficient throughout the remelting process, so that the trial was carried out without any technical hitch.

145
146
147
148
149
150
151
The actual recordings of a highly non-standard melting sequence are presented in Figs. 2 and 3, in terms of melting rate and stirring coil current. During the trial, the stirring sequence was carried out in two successive steps: a strong alternating stirring was used to remelt the first 60 %
152 of the electrode, followed by a weak continuous stirring until the end of the melting process. The
153 magnitude of the alternating stirring was about 4.5 times higher than the continuous stirring one.

154 Additionally, a strong continuous stirring has been briefly applied at three moments
155 during the alternated stirring stage, in order to mark different pools in the ingot, as such
156 temporary change caused a local change in the grain structure of the solidified material. The
157 profiles of the marked melt pool are clearly visible on calculated composition maps presented in
158 figure 3.c.

<figure><img src="image_21.png" /><figcaption>159 **Fig. 2.** Melting rate of the full-scale Zy4 remelting trial.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. 3.** Stirring sequence of the full-scale Zy4 remelting trial. (a) Coil current throughout the melt. (b) Detailed alternating stirring sequence. (c) Example of a computed composition map (Zr content), highlighting the melt pool markings in the remelted ingot.</figcaption></figure>

The remelted ingot was cut longitudinally along its entire length in order to perform chemical analyses and characterize the grain structure. Furthermore, chemical analyses were carried out along several radii at different heights of the ingot, ranging from 0.25 H to 0.88 H,
167 where H is the total height of the ingot. These measurements are used in section 4 to validate the
168 numerical results.

169

## 3 SOLAR model description

170

171

### 3.1 The CFD macromodel

172 As discussed in section 1, SOLAR is a 2D axisymmetric finite volume model of the VAR
173 process, with the complete set of macroscopic equations presented in Table 1. At a macroscopic
174 scale, the model (Eqs. (1 to 19)) takes into account the transport of heat and solutes coupled with
175 the turbulent flow driven by the electromagnetic stirring, as well as thermal and solute natural
176 convection.

177 The electromagnetic force **F** results from the interaction between the melting current **J**,
178 and the magnetic field **B**. The latter includes the self-induced magnetic field **B**<sub>θ</sub>e<sub>θ</sub>, and the
179 magnetic field created by the external induction coils **B**<sub>z,ext</sub>e<sub>z</sub>. The current density is in turn
180 calculated from Laplace's conservation equation (Eq. 8) and Ohm's law (Eq. 11). It is important
181 to note that the effect of EM stirring on the free surface dynamics is not considered here, in
182 accordance with the results of Chapelle et al.<sup>[5]</sup>, who show a second-order effect of EM stirring
183 on the deformation of the melt pool free surface. Thermal buoyancy and solutal buoyancy are
184 taken into account by using a conventional Boussinesq approximation (Eq. 12) to describe the
185 variations in density of the liquid phase in the gravity term of the momentum balance equation. In
186 the mushy zone, the solid phase forms a porous matrix, modelled by a Darcy term in the
187 momentum equation (Eq. 4). Permeability is estimated by the Blake-Kozeny law (Eq. 13), where
188 the Asai and Muchi formula is used to describe the permeability constant K<sub>0</sub> (Eq. 17).

189
**Table 1.** Main constitutive equations of the CFD macromodel

Averaged total mass balance

$$ \frac{\partial}{\partial t}(\rho_m) + \nabla \cdot (\rho_l g_l \overline{\mathbf{u}_l}) = 0 \qquad (1) $$

Averaged mass balance of the solid phase

$$ \frac{\partial}{\partial t}(\rho g_s) = \Gamma_s + \Phi_s, \quad \rho = \text{const} \qquad (2) $$

Averaged total heat balance

$$ \frac{\partial}{\partial t}(\rho g_s \overline{h_s} + \rho g_l \overline{h_l}) + \nabla \cdot (\rho g_l \mathbf{u}_l \overline{h_l}) = \nabla \cdot [(\lambda + g_l \lambda_t) \nabla \overline{T}] \qquad (3) $$

Averaged momentum balance for the liquid phase: fluid flow

$$ \frac{\partial}{\partial t}(\rho g_l \overline{\mathbf{u}_l}) + \nabla \cdot (\rho g_l \overline{\mathbf{u}_l} \overline{\mathbf{u}_l}) = -g_l \nabla p^* + \nabla \cdot [(\mu + \mu_t)g_l(\nabla \overline{\mathbf{u}_l} + (\nabla \overline{\mathbf{u}_l})^T)] - \frac{g_l^2 \mu}{K} \overline{\mathbf{u}_l} + g_l \overline{g} \overline{\rho_l^b} + g_l \mathbf{F}_{el} \qquad (4) $$

Averaged solute mass balance for species i in the solid phase

$$ \frac{\partial}{\partial t}(\rho g_s \overline{\omega_s^i}) = J_s^{ji} + J_s^{\Gamma i} \qquad (5) $$

Averaged solute mass balance for species i in the liquid phase

$$ \frac{\partial}{\partial t}(\rho g_l \overline{\omega_l^i}) + \nabla \cdot (\rho g_l \overline{\mathbf{u}_l} \overline{\omega_l^i}) = \nabla \cdot [(D_l^i + D_t)g_l \nabla \overline{\omega_l^i}] + J_l^{ji} + J_l^{\Gamma i} \qquad (6) $$

Averaged $k - \epsilon$ turbulence model
$$ \frac{\partial}{\partial t}(\rho g_l k) + \nabla \cdot (\rho g_l \mathbf{u}_l k) = \nabla \cdot \left[ \left( \mu + \frac{\mu_t}{\sigma_k} \right) g_l \nabla k \right] + P_k + D_k + G_k^T + G_k^S - \rho g_l \varepsilon \quad (7) $$

$$ \frac{\partial}{\partial t}(\rho g_l \varepsilon) + \nabla \cdot (\rho g_l \mathbf{u}_l \varepsilon) = \nabla \cdot \left[ \left( \mu + \frac{\mu_t}{\sigma_\varepsilon} \right) g_l \nabla \varepsilon \right] + \frac{\varepsilon}{k} \left[ c_1 P_k + c_2 D_k + c_1 c_3 (G_k^T + G_k^S) - c_2 \rho g_l \varepsilon \right] $$

Laplace conservation equation

$$ \nabla^2 \varphi = 0 \quad (8) $$

Mass balance equation at the solid-liquid interface

$$ \Gamma_s + \Gamma_l = 0 \quad (9) $$

Solute mass balance at the solid-liquid interface

$$ J_s^{j_i} + J_s^{\Gamma_i} + J_l^{j_i} + J_l^{\Gamma_i} = 0 \quad (10) $$

Ohm's law and Lorentz force

$$ J = -\sigma \nabla \varphi \qquad F_{el} = J \wedge B = -J_z B_{\theta} e_r + J_r B_{ext,z} e_{\theta} + J_r B_{\theta} e_z \quad (11) $$

Liquid density in the buoyancy force

$$ \overline{\rho_l^b} = \rho_{ref} \left[ 1 - \beta_T (\overline{T} - T_{ref}) - \sum_i \beta_S^i (\overline{\omega_l^i} - \omega_{ref}^i) \right] \quad (12) $$

Permeability law

$$ K = \frac{g_l^3}{(1 - g_l^2)} K_0 \quad (13) $$

Production rate $P_k$ due to gradients of $u_l$ and dissipation rate $D_k$ due to liquid-solid interaction

$$ P_k = -\rho g_l \overline{u'_i u'_j} \frac{\partial}{\partial x_j} (g_l \overline{u_i}) \qquad D_k = -\frac{2\mu g_l^2}{K} k \quad (14) $$

Generation rate due to thermal and solutal buoyancy
$$G_k^T = \beta_T g_l \frac{\mu_t}{Pr_t} \mathbf{g} \cdot \nabla \bar{T} \qquad G_k^S = \sum_i \beta_S^i g_l \frac{\mu_t}{Sc_t} \mathbf{g} \cdot \nabla \bar{\omega}_l^i \qquad (15)$$

Macroscopic $k-\epsilon$ model constants

$$ \begin{array}{lll} Pr_t = 0.85 & \sigma_k = 1.0 & c_\mu = 0.09 \\ Sc_t = 0.7 & \sigma_\epsilon = 1.3 & c_1 = 1.44 \\ & & c_2 = 1.92 \end{array} \qquad c_3 = \tanh \left( \frac{|\overline{\mathbf{u}_{l,z}}|}{\sqrt{\overline{\mathbf{u}_{l,\theta}}^2 + \overline{\mathbf{u}_{l,r}}^2}} \right) \qquad (16) $$

Permeability constants of the model

$$ K_0 = \frac{\lambda_{DAS}^2}{180} \qquad (17) $$

Pressure

$$ p^* = \bar{p}_l + \frac{2}{3} \rho k \qquad (18) $$

Turbulent parameters

$$ \mu_t = c_\mu \rho \frac{k^2}{\epsilon} \qquad \lambda_t = \frac{c_p \mu}{Pr_t} \qquad D_t = \frac{\mu_t}{\rho Sc_t} \qquad (19) $$

A macroscopic RANS two-equation $k-\epsilon$ model (Eq. 7) is adopted to describe the turbulence of the liquid flow. The transport equations for the mean turbulent kinetic energy $k$ and its corresponding mean dissipation rate $\epsilon$ include:

- turbulence production due to mean velocity gradients $P_k$ (Eq. 14),
- dissipation of turbulence due to the liquid/solid interaction in the mushy zone $D_k$ (Eq. 14),
- production or dissipation of turbulence caused by vertical thermal $G_k^T$ and solute content $G_k^S$ gradients (Eq. 15).
The constants used for closure (Eq. 16) correspond to the standard $k-\varepsilon$ model originally

developed by Launder and Spalding[8] for a fully liquid medium ($g_l = 1$, hence $K \to \infty$). This

approach is suitable for a wide variety of turbulent flows.

The closure of such model in the mushy zone, considered as a porous medium, is solved through

the formulation of the dissipation term $D_k$ of the turbulence model [9–12]. Precisely, the Prescott

and Incropera approach was implemented in the model (eq. 14).

The model simulates the direct radiation of the electric arc by the implementation of a heat source

as well as a current density on the top face of the secondary ingot.

All boundary conditions are discussed in the PhD theses by Hans[13], Quatravaux[14] and Revil-

Baudard[15].

The mesh used for all simulations reported in this paper is a structured one. The number of cells

must increase during the simulation to follow the ingot growth, as the material input from the

electrode melting is modeled by an advection-like term, i.e. a cell split and growth method. The

resulting mesh is scalable, both in size and number of cells, describing at each time step the ingot

growth in a continuous way. More details are available in Quatravaux’s Ph D thesis[14].

## 3.2 The solidification micromodel

The solidification micromodel is derived from the approach of Beckerman and

Viskanta[16]. They proposed a mathematical formulation for dendritic solidification in a binary

mixture. Their micro-model was based on a two-phase volume-averaged formulation and results

were compared to shadowgraph images of the experimental solidification of an ammonium

chloride solution in water. Although the numerical results were qualitatively close to
experimental measurements, the quantitative values for temperature and concentrations showed considerable differences.

Lately, an improved model was presented by Wang and Beckerman[17]. They described equiaxed dendritic solidification by considering the convection and solid phase transport. This model accounts for the nucleation, grain growth and dendrite morphology by solving a set of transport equations for solid, interdendritic, and extradendritic phases. However, the model was only validated at the laboratory scale.

A similar approach, based on decoupling physical phenomena according to the micro and macro time-scale, was presented by Zaloznik and Combeau[18] for a real industrial size ingot. The constitutive equations of the models were obtained by introducing a Representative Elementary Volume (REV), averaging the transport equations of heat, mass, and momentum over the individual (liquid, solid) phases. The process model considers mass transfer, turbulent fluid flow, heat transfer, electromagnetism and phase change using the following main assumptions:
- the local thermal equilibrium is assumed in each mesh cell, which behaves as a REV;
- the solidification shrinkage is not considered, because the densities of both phases are equal and constant, except for the buoyancy terms where the Boussinesq approximation is applied;
- the solid phase is considered rigid and fixed;
- the thermophysical properties of all alloys are constant; in particular, partition coefficients and liquidus slopes are independent of the temperature and composition.

An operator splitting scheme, described in detail in Založnik and Combeau[18], was used to define the transport and growth stages. In the first one, the convective/diffusive macroscopic transport partial differential equations were solved globally by neglecting nucleation and growth terms, while in the second one, the contributions of nucleation and growth were solved locally and initialized from the transport stage. The grain growth model was highly nonlinear and
therefore required special attention. The advantage of this approach was the decoupling of macro and micro time scales, which allows the grain growth model to be solved separately.

The microscopic terms that described the grain nucleation and growth were treated locally within each mesh cell where the interfacial chemical and thermal equilibrium were assumed to be reached: $T_s = T_l = T$ and $\omega_s^{i*} = k_p \omega_l^{i*}$. The interfacial compositions and temperatures are correlated by considering a simplified multicomponent phase diagram, where the liquidus temperature is described with a linear dependence on all alloy components. Both the liquidus slopes $m_l^i$ and partition coefficient $k_p^i$ are constant. Therefore, the solidification process starts when the enthalpy of a cell falls below the liquidus enthalpy $h_{liq}$,

$$h_{liq} = c_p \left( T_m + \sum_{i=1}^{n_{sol}} m_L^i \omega_l^i \right) + L_f \quad (20)$$

and stops when the temperature drops below a fixed temperature $T_{sol0}$ or when the solid fraction reaches $g_s = 1$ due to the primary growth as explained by Založnik and Combeau$^{[18]}$.

In the solidification interval, where the liquid and solid phases co-exist in the same REV ($0 < g_s < 1$), the solidification micromodel takes into account finite solute diffusion in both phases and considers two separate regions: the solid phase and the extra-dendritic liquid. In the mushy zone, grains are assumed to be spherical, the velocity of the extra-dendritic phase to be equal to the averaged velocity of the liquid, and the diffusivity of each solute to be constant in each phase. Grain generation is modelled by an instantaneous nucleation model, which assumes that a predefined number of grains per unit volume $N_g$ with an initial radius $R^*$ nucleate when the local temperature becomes lower than the local liquidus temperature (see Eq. 25 in Table 2). The
transfer of the interfacial solute due to the phase change (solidification or melting) is given in Eq.

21:

$$ (k_p^i - 1)\omega_l^{i*} \frac{\Gamma_s}{\rho} + \omega_l^{i*} \left( \frac{S_v D_l^n}{\delta_l} + k_p^n \frac{S_v D_s^n}{\delta_s} \right) = \omega_l^i \frac{S_v D_l^n}{\delta_l} + \omega_s^i \frac{S_v D_s^n}{\delta_s}. \quad (21) $$

During primary solidification, the diffusion flux at the solid/liquid interface is described by a simplified model: solute gradients are calculated by dividing the difference of the solute mass fraction at the interface $\omega_k^{*,i}$ and the average mass fraction $\omega_k^i$, by a diffusion length $\delta_k$. An approximation is used to estimate the boundary layer thickness in the solid phase while a stagnant film model is used to describe the boundary layer thickness in the liquid phase. This solidification micromodel, whose main equations are given in Table 2 (Eqs. 22-28), is described in detail by Tveito et al.[19]. It must be stated here that its implementation in SOLAR software represents an innovative approach over the previous versions used to simulate the VAR process.

**Table 2.** Main equations of the solidification micromodel.

Geometrical relations and surface area

$$ R^* = \left( \frac{3g_s}{4\pi N_g} \right)^{\frac{1}{3}} \qquad R_f = \left( \frac{3}{4\pi N_g} \right)^{\frac{1}{3}} \qquad S_v^* = 4\pi R^{*2} N_g \quad (22) $$

Dendrite tip kinetics

$$ \Omega_i = \frac{\omega_l^{*,i} - [<\omega_s^i>^e]^{gr}}{\omega_l^{*,i}(1 - k_p^i)} \qquad V = \frac{\partial R^*}{\partial t} \qquad \Gamma_s = \rho_s S_v^* V \quad (23) $$

Boundary layer thickness
$$ \delta_i^l = \frac{d_i}{\left( \frac{d_i}{R^*} - \frac{f(R^*, \Delta_i) + g(R_f, R^*, \Delta_i)}{d_i(R^* + d_i - (R^* - \Delta_i + d_i)\exp\left(\frac{-\Delta_i}{d_i}\right) - f(R^*, \Delta_i) + (\exp\left(\frac{-\Delta_i}{d_i-1}\right)g(R_f, R^*, \Delta_i))} \right)} \quad (24) $$

$$ \delta_i^s = \frac{R^*}{5} \qquad g(R_f, R^*, \Delta_i) = \frac{R_f^3 - (R^* + \Delta_i)^3}{3(R^* + \Delta_i)} \qquad d_i = \frac{D_l^i}{V} $$

$$ Sh_i = \frac{2}{3(1 - g_s)} Sc_i^{\frac{1}{3}} Re^{n(Re)} \qquad f(R^*, \Delta_i) = \frac{1}{2}((R^* + \Delta_i)^2 - (R^*)^2) \qquad \Delta_i = \frac{2R^*}{Sh_i} $$

$$ Re = \frac{\rho_l(1 - g_s)2R^*}{\mu_l} |\bar{u}_l| \qquad n(Re) = \frac{2Re^{0.28} + 4.65}{3(Re^{0.28} + 4.65)} \qquad Sc_i = \frac{\mu_l}{\rho_l D_l^i} $$

## Thermodynamics

$$ \omega_{s-d}^{*,i} = \omega_l^{*,i} k_p^i \qquad T_l = T_m + \sum_i m_l^i \omega_l^{*,i} \quad (25) $$

## Nucleation

$$ \Phi_s = \begin{cases} \rho \frac{\pi R^*}{6} N_g \delta(t - t_0); & \text{if } T_l < T_m + \sum_i m_l^i \omega_l^{*,i} \text{ and } g_s = 0 \\ 0; & \text{else} \end{cases} \qquad \Phi_s = -\Phi_l \quad (26) $$

## Solute fluxes due to phase change

$$ J_s^{\Gamma^i} = \omega_s^{*,i} \Gamma_s \qquad J_l^{\Gamma^i} = \omega_l^{*,i} \Gamma_l \qquad \omega_s^{*,i} = k_p^i \omega_l^{*,i} \quad (27) $$

## Solute fluxes due to diffusion

$$ J_s^{j,i} = \frac{S_v^* \rho_s D_s^i}{\delta_s} (\omega_s^{*,i} - \omega_s^i) \qquad J_l^{j,i} = \frac{S_v^* \rho_l D_l^i}{\delta_l} (\omega_l^{*,i} - \omega_l^i) \quad (28) $$

# 4 Numerical results and discussion
industrial trial to feed all simulations presented in this section. The operating parameters were entered by means of data files containing these records (see Figures 2 and 3).

The thermophysical properties of Zr4 used in the simulations are given in Table 3. Note that the solidification properties (partition coefficients and liquidus slopes) have been extracted from the binary phase diagrams for Zr-X alloys[20]. Unless otherwise specified, the values of β<sub>S</sub><sup>Sn</sup> and λ<sub>DAS</sub> used in the simulations (default values) are also reported in Table 3.

<table><caption>**Table 3.** Thermophysical properties of Zr4 used in the simulations</caption><tbody><tr><td rowspan="4">Nominal compositions</td><td>Sn</td><td>ω<sub>0</sub><sup>Sn</sup> [wt%]</td><td>1.3</td></tr><tr><td>Fe</td><td>ω<sub>0</sub><sup>Fe</sup> [wt%]</td><td>0.2</td></tr><tr><td>O</td><td>ω<sub>0</sub><sup>O</sup> [wt%]</td><td>0.1</td></tr><tr><td>Cr</td><td>ω<sub>0</sub><sup>Cr</sup> [wt%]</td><td>0.1</td></tr><tr><td>Melting temperature of pure Zr</td><td>T<sub>m</sub> [°C]</td><td>1855</td></tr><tr><td rowspan="4">Liquidus slopes</td><td>Sn</td><td>m<sub>l</sub><sup>Sn</sup> [°C(wt%)<sup>-1</sup>]</td><td>-11</td></tr><tr><td>Fe</td><td>m<sub>L</sub><sup>Fe</sup> [°C(wt%)<sup>-1</sup>]</td><td>-57</td></tr><tr><td>O</td><td>m<sub>L</sub><sup>O</sup> [°C(wt%)<sup>-1</sup>]</td><td>147</td></tr><tr><td>Cr</td><td>m<sub>L</sub><sup>Cr</sup> [°C(wt%)<sup>-1</sup>]</td><td>-26</td></tr><tr><td rowspan="3">Partition Coefficients</td><td>Sn</td><td>k<sub>p</sub><sup>Sn</sup> [-]</td><td>0.27</td></tr><tr><td>Fe</td><td>k<sub>p</sub><sup>Fe</sup> [-]</td><td>0.25</td></tr><tr><td>O</td><td>k<sub>p</sub><sup>O</sup> [-]</td><td>2.32</td></tr></tbody></table>
<table><tbody><tr><td></td><td>Cr</td><td>k<sub>p</sub><sup>Cr</sup> [-]</td><td>0.15</td></tr><tr><td rowspan="4">Diffusion coefficients in the liquid</td><td>Sn</td><td>D<sub>l</sub><sup>Sn</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td>Fe</td><td>D<sub>l</sub><sup>Fe</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td>O</td><td>D<sub>l</sub><sup>O</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td>Cr</td><td>D<sub>l</sub><sup>Cr</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.10<sup>-8</sup></td></tr><tr><td rowspan="4">Diffusion coefficients in the solid</td><td>Sn</td><td>D<sub>s</sub><sup>Sn</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>4.9.10<sup>-11</sup></td></tr><tr><td>Fe</td><td>D<sub>s</sub><sup>Fe</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.5.10<sup>-9</sup></td></tr><tr><td>O</td><td>D<sub>s</sub><sup>O</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>3.8.10<sup>-9</sup></td></tr><tr><td>Cr</td><td>D<sub>s</sub><sup>Cr</sup> [m<sup>2</sup>.s<sup>-1</sup>]</td><td>1.3.10<sup>-9</sup></td></tr><tr><td>Temp. of end of solidification</td><td></td><td>T<sub>sol</sub> [°C]</td><td>1592</td></tr><tr><td>Latent heat</td><td></td><td>L<sub>f</sub> [J.kg<sup>-1</sup>]</td><td>2.42.10<sup>5</sup></td></tr><tr><td>Reference density</td><td></td><td>ρ<sub>0</sub> [kg.m<sup>-3</sup>]</td><td>6210</td></tr><tr><td>Dynamic viscosity</td><td></td><td>μ [Pa.s]</td><td>4.59.10<sup>-3</sup></td></tr><tr><td>Thermal conductivity</td><td></td><td>λ [W.m<sup>-1</sup>.°C<sup>-1</sup>]</td><td>36.5</td></tr><tr><td>Thermal expansion coefficient</td><td></td><td>β<sub>T</sub> [°C<sup>-1</sup>]</td><td>4.10<sup>-5</sup></td></tr><tr><td>Specific heat</td><td></td><td>c<sub>p</sub> [J.kg<sup>-1</sup>.°C<sup>-1</sup>]</td><td>435</td></tr><tr><td>Electric conductivity</td><td></td><td>σ [m<sup>-1</sup>.Ω<sup>-1</sup>]</td><td>6.8.10<sup>5</sup></td></tr><tr><td rowspan="4">Solute expansion coefficients</td><td>Sn</td><td>β<sub>s</sub><sup>Sn</sup> [(wt%)<sup>-1</sup>]</td><td>3.10<sup>-2</sup></td></tr><tr><td>Fe</td><td>β<sub>s</sub><sup>Fe</sup> [(wt%)<sup>-1</sup>]</td><td>0</td></tr><tr><td>O</td><td>β<sub>s</sub><sup>O</sup> [(wt%)<sup>-1</sup>]</td><td>0</td></tr><tr><td>Cr</td><td>β<sub>s</sub><sup>Cr</sup> [(wt%)<sup>-1</sup>]</td><td>0</td></tr><tr><td>Characteristic length of the</td><td></td><td>λ<sub>DAS</sub> [m]</td><td>134.10<sup>-6</sup></td></tr></tbody></table>
291

292 For all simulations reported in the present paper, the average cell size was approximately
293 9 mm wide and 6 mm height. Mesh refinement was applied with a geometrical reason of 0.8 on
294 the five nodes closest to the mold as well as to the free surface. The time-step was fixed to
295 0.025 s.

296 The computational time required for simulating a full melt depends on the computer. In
297 the IJL facility, the total CPU time was around 72 hours.

298 In several previous publications [2,21-23], the authors have detailed some CFD macromodel
299 results when applied to various remelting situations. In this paper, the attention will be focused on
300 the fluid flow in the liquid pool and the mushy zone, as well as the resulting macrosegregation.

301

## 4.1 Hydrodynamics in the liquid metal pool

302 The purpose of this section is to numerically study the influence of the thermal and solutal
303 buoyancy forces on the flow in the liquid bath during remelting. Mean velocity field, turbulence
304 generation are therefore investigated. Only the effect of local Sn content on solutal buoyancy was
305 considered, as the concentrations in all other alloying elements are negligible.

### 4.1.1. Assumptions on the thermal and solutal buoyancy forces

308 During solidification, a buoyancy driving force can be generated either by the thermal or
309 solutal buoyancy effects, that can be collaborating or opposing [24]. Both buoyancy effects are
310 modelled by the Boussinesq approximation (Eq. 12), therefore both the thermal and the solutal
311 expansion coefficients had to be estimated.
Unfortunately, values for the solutal and thermal expansion coefficient for Zr-4 alloy and its alloying elements are not readily available in the literature.

The thermal expansion has been estimated from values measured by Paradis and Rhim[25] on pure Zr. Moreover, other pertaining physical properties, such as reference density, come from this reference.

Figures 4 and 5 below show the evolution of Zr density and Sn density with the temperature.

<figure><img src="image_23.png" /></figure>

**Fig. 4.** Density of Zr as a function of temperature, from [26] (the vertical bar corresponds to Zr melting point).
<figure><img src="image_24.png" /><figcaption>**Fig. 5.** Density of Sn as a function of temperature, from [27] (the vertical bar corresponds to Sn melting point).</figcaption></figure>

From these figures, it is not possible to decide with a great confidence whether Sn should be considered as heavier or lighter than Zr at high temperature, around 2128 K, i.e. Zr melting temperature. Clearly, the value of the solutal expansion coefficient cannot be readily given.

In order to examine the influence of thermo-solutal convection, four simulations have been run, as shown in Table 4. In Case A, both buoyancy terms were null, which means that the flow is only driven by the forced convection generated by the electromagnetic field. In case B, thermal expansion coefficient for pure Zr has been used, while no solutal buoyancy was applied. Cases C and D take into account solutal buoyancy, with a negative (case C) and positive (case D) value of the Sn expansion coefficient. The negative coefficient $\beta_S^{Sn} = -1 \cdot 10^{-2}(wt\%)^{-1}$, defined in case C, was chosen to investigate on the consequences of cooperating buoyancy forces. On the contrary, the positive value in case D implies opposing buoyancy forces.

<caption>**Table 4.** Definition of the four simulations run to study the effect of thermo-solute convection</caption>
<table><thead><tr><th>case</th><th>β<sub>T</sub> [°C<sup>-1</sup>]</th><th>β<sub>S</sub><sup>Sn</sup>[wt%<sup>-1</sup>]</th></tr></thead><tbody><tr><td>A</td><td>0</td><td>0</td></tr><tr><td>B</td><td>4 · 10<sup>-5</sup></td><td>0</td></tr><tr><td>C</td><td>4 · 10<sup>-5</sup></td><td>-1 · 10<sup>-2</sup></td></tr><tr><td>D</td><td>4 · 10<sup>-5</sup></td><td>3 · 10<sup>-2</sup></td></tr></tbody></table>

339

### 4.1.2 Characteristics of the liquid metal pool

The metal flow in the melt pool and mushy zone has been investigated at instants corresponding to alternated or continuous stirring periods applied during the melt. In the latter case, the hydrodynamics in the (r, z) plane is quite stationary, whereas in the former one, two different moments have been considered during the stirring sequence:

- time $t_1$ is characterized by a low azimuthal velocity of the liquid, hence a low centrifugal force

- time $t_2$ is characterized by a high azimuthal velocity of the liquid, hence a high centrifugal force.

Flows during the continuous stirring stage (time $t_3$) and at time $t_2$ are similar, since the azimuthal velocity in both cases is high (see Fig. 6). However, as can be seen in Fig. 6, the transition from alternated to continuous stirring changes the depth and shape of the liquid pool as well as the size of the mushy zone. In the case of alternated stirring, the liquid pool is shallower and more parabolic in shape, whereas in the case of continuous stirring, the pool is deeper and cone-shaped, with a larger mushy zone.
<figure><img src="image_25.png" /><figcaption>**Fig. 6.** Case A – Computed pool profile and azimuthal velocity (m.s⁻¹) map in the melt pool at 3 different moments during the melt. (a) Time t₁ - alternated stirring, low azimuthal velocity. (b) Time t₂ - alternated stirring, high azimuthal velocity. (c) Time t₃ - continuous stirring, high azimuthal velocity.</figcaption></figure>

The hydrodynamic behavior at time t₁ (low azimuthal velocity) is the object of Fig. 7. The turbulence intensity is represented by the µt/µ viscosity ratio and the velocity field by velocity vectors in the upper part of the melt pool where velocities are high and by streamlines in the lower part of the melt pool and in the mushy zone, where velocities are significantly lower.

Without any buoyancy phenomena, the electromagnetic force causes two recirculation loops (Fig. 7 (a)). The interaction between the melting current and the self-induced magnetic field generates centripetal forces localized in the near-surface region, at the origin of a counter-
clockwise recirculation flow of the liquid metal in the upper part of the pool. In the mid-height zone of the pool, stirring centrifugal forces prevail, reinforcing the upper recirculation flow and creating a second clockwise loop in the lower part of the pool.

When thermal buoyancy is added (case B), it can be seen in fig. 7(b) that the two recirculation loops remain qualitatively identical. The major modification is a strong decrease in the turbulence intensity in the near-surface region, as the high vertical thermal stratification strongly dissipates turbulence (eq. 15).

Turbulence characteristics of the flow is almost identical to case B when solute buoyancy is added (cases C and D).

The collaborating solute buoyancy has a negligible impact on the “two recirculation loops” system observed for cases A and B. However, in case D, the opposing solute buoyancy leads to the apparition of a third counter-clockwise recirculation flow in the bottom of the pool and in the mushy zone. As $Sn$ partition ratio is lower than unity, solute redistribution during solidification causes a liquid enrichment in solute in the mushy zone. This enrichment causes a decrease in liquid density due to the positive solutal expansion coefficient. Finally, the resulting solutal buoyancy force, localized in the mushy zone, promotes an upward flow along the solidification front, inducing the observed third bottom loop.
<figure><img src="image_26.png" /><figcaption>**Fig. 7.** Computed flow in the melt pool and mushy zone at time $t_1$ (alternated stirring, low azimuthal velocity). Left side is the turbulence intensity and right side is the mean flow represented by vectors (pool upper part) or streamlines (pool lower part and mushy zone) and the liquid fraction map.
(a) Case A: $\beta_T = 0$, $\beta_S^{Sn} = 0$, $v_{max} = 2.03 \text{ cm.s}^{-1}$.
(b) Case B: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = 0$, $v_{max} = 1.60 \text{ cm.s}^{-1}$.
(c) Case C: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = -1$.</figcaption></figure>
388 10⁻² wt%⁻¹, $v_{max}$ = 1.82 cm.s⁻¹. (d) Case D: $β_T$ = 4 · 10⁻⁵ °C⁻¹, $β_S^{Sn}$ = 3 · 10⁻² wt%⁻¹,
389 $v_{max}$ = 1.61 cm.s⁻¹.

390
391 The influence of a high azimuthal velocity on the in-plane velocity field and turbulence
392 intensity in the melt pool and mushy zone is shown in Figs. 8 and 9, for instants $t_2$ during the
393 alternated stirring and $t_3$ during the continuous stirring.

394 Compared to the previous case, azimuthal velocity generated by the stirring is 60 %
395 higher, implying an increase in centrifugal forces of 156 %. Therefore, the self-induced forces
396 become of second order and, thus, the circulation counter-clockwise loop in the near surface
397 region disappears.

398 Similar to previous simulations, the introduction of thermal buoyancy forces considerably
399 reduces the intensity of turbulence in the near-surface area (cases B, C and D).

400 A collaborative solutal buoyancy slightly decreases turbulence intensity in the bottom of
401 the pool in case C in the frame of a strong alternated stirring (Fig. 8(c)). This is due to the local
402 vertical solute stratification, in the same way as the effects of thermal buoyancy near the free
403 surface.

404 Finally, an opposite solutal buoyancy generates an additional recirculation loop in the
405 bottom of the pool for all stirring conditions (Fig. 8(d) and Fig. 9(d)).

406

407

408
<figure><img src="image_27.png" /><figcaption>**Fig. 8.** Computed flow in the melt pool and mushy zone at time $t_2$ (alternated stirring, high azimuthal velocity). Left side is the turbulence intensity and right side is the mean flow represented by vectors (pool upper part) or streamlines (pool lower part and mushy zone) and the liquid fraction map.

(a) Case A: $\beta_T = 0$, $\beta_S^{Sn} = 0$, $v_{max} = 2.53 \text{ cm.s}^{-1}$.
(b) Case B: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = 0$, $v_{max} = 2.53 \text{ cm.s}^{-1}$.
(c) Case C: $\beta_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $\beta_S^{Sn} = -1$.</figcaption></figure>
414 10⁻² wt%⁻¹, $v_{max}$ = 2.50 cm.s⁻¹. (d) Case D: $β_T$ = 4 · 10⁻⁵ °C⁻¹, $β_S^{Sn}$ = 3 · 10⁻² wt%⁻¹,

415 $v_{max}$ = 2.51 cm.s⁻¹.

416

<figure><img src="image_28.png" /></figure>

417 **Fig. 9.** Computed flow in the melt pool and mushy zone at time $t_3$ (continuous stirring, high azimuthal velocity). Left side is the turbulence intensity and right side is the mean flow
418
represented by vectors (pool upper part) or streamlines (pool lower part and mushy zone) and the liquid fraction map. (a) Case A: $β_T = 0$, $β_S^{Sn} = 0$, $v_{max} = 2.02 \text{ cm.s}^{-1}$. (b) Case B: $β_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $β_S^{Sn} = 0$, $v_{max} = 1.93 \text{ cm.s}^{-1}$. (c) Case C: $β_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $β_S^{Sn} = -1 \cdot 10^{-2} \text{ wt%}^{-1}$, $v_{max} = 1.83 \text{ cm.s}^{-1}$. (d) Case D: $β_T = 4 \cdot 10^{-5} \text{ °C}^{-1}$, $β_S^{Sn} = 3 \cdot 10^{-2} \text{ wt%}^{-1}$, $v_{max} = 2.00 \text{ cm.s}^{-1}$.

## 4.2. Macrosegregation

Macrosegregation behaviour of the remelted product is now numerically investigated. Particular attention is paid to the influence of the thermophysical properties defined in the macromodel, such as thermal and solutal buoyancy forces, as well as the solidification properties implemented in the micromodel, such as dendrite arm spacing and grain growth kinetics.

As computed results differed only slightly from previous calculations[28], where a simplified solidification model assumed complete diffusion of all solutes in both liquid and solid phases (i.e. lever rule), it appears that the use of a detailed solidification micromodel confirmed the trends which had been highlighted with a simplified lever-rule based model.

### 4.2.1. Influence of the thermo-solutal convection

The computed maps of Sn content in the final ingot are shown in Fig. 10. As discussed previously, the two declined Sn-depleted bands are a consequence of the strong electromagnetic stirring (see Fig. 3 (b)) that was applied to mark the melt pool profile in the ingot macrostructure. The average concentration of Sn in the liquid pool increases as the ingot grows because the
partition coefficient $k_p^{Sn}$ is lower than 1. Solidification of the last liquid pool causes a local enrichment at the top of the ingot. A strong negative segregation is predicted at the lateral surface and at the bottom of the ingot. However, results at the periphery must be handled with care because the segregation calculated there is very sensitive to mesh size.

442

<figure><img src="image_29.png" /></figure>
Fig. 10. Computed Sn content map in the remelted ingot. (a) Case A: $\beta_T = 0$, $\beta_S^{Sn} = 0$. (b) Case B: $\beta_T = 4 \cdot 10^{-5} \,^\circ C^{-1}$, $\beta_S^{Sn} = 0$. (c) Case C: $\beta_T = 4 \cdot 10^{-5} \,^\circ C^{-1}$, $\beta_S^{Sn} = -1 \cdot 10^{-2} \, \text{wt}\%^{-1}$. (d) Case D: $\beta_T = 4 \cdot 10^{-5} \,^\circ C^{-1}$, $\beta_S^{Sn} = 3 \cdot 10^{-2} \, \text{wt}\%^{-1}$.

446

In the absence of buoyancy (case A), the model predicts a Sn enrichment in the ingot
447
448 centre for the two stirring regimes used during the melt. This is the result of the clockwise flow
449 measurement cell discussed in the previous section. The liquid flows down along the
450 solidification front. As a result, the liquid enriched in Sn piles up in the mushy zone and at the
451 bottom of the melt pool, resulting in a positive segregation on the axis of symmetry. As shown in
452 Fig. 10, radial macrosegregation is weaker with alternating stirring than with continuous stirring.

The addition of thermal convection (case B) slightly increases the radial macrosegregation
453
454 for both stirring sequences. The centrifugal force is reinforced by thermal buoyancy, which
455 causes more intense circulation in the mushy zone and greater transport of enriched liquid to the
456 central axis.

The convection effects induced by the solutal buoyancy are presented in Figs. 10 (c) and
457
458 10 (d). They prevail in the central part of the mushy zone and have an impact on segregation. In
459 case C, the segregation at the central axis is notably amplified by the cooperation of solutal and
460 thermal buoyancy. The opposite effect is observed in case D, where the effects of solutal and
461 thermal buoyancy are opposed: the segregation at the central axis is considerably reduced, due to
462 the upward flow in the mushy region. A positive segregation band is therefore located at r/
463 R~0.5.

Moreover, in case D, during the continuous stirring sequence, a series of vertical channels
464
465 appear in the enriched segregation part. Fig. 11 details this effect. The Sn-enriched liquid
466 accumulated near the centerline is transported outward under the influence of solutal convection.

467 This transport of solute is effective for a liquid fraction $g_l > 0.6$. When $g_l < 0.6$, the

468 permeability of the porous media decreases rapidly as the drag forces dominate inertial and

469 buoyancy forces and macrosegregation evolution is stopped.

<figure><img src="image_30.png" /></figure>

470 **Fig. 11.** Zoom visualization of the influence of solute-driven convection at time $t_3$ - continuous

471 stirring, high azimuthal velocity. Case D - $\beta_T = 4 \cdot 10^{-5} \,^\circ\text{C}^{-1}$, $\beta_S^{Sn} = 3 \cdot 10^{-2} \,\text{wt}\%^{-1}$. (a)

472 Liquid fraction, streamlines and Sn content. (b) Liquid fraction and convective $v^1 \cdot \nabla T$ term.

473 In Fig. 12, the predicted radial Sn segregation profiles are plotted at two different heights

474 and compared to experimental measurements. Those heights are representative for the alternated stirring ($h/H = 0.25$) and continuous stirring ($h/H = 0.75$) stages. Comparison of the

475 simulation results with the experimental measurements confirms the importance of solutal

476 convection in the mushy zone. At $h/H = 0.75$, the measured Sn concentration continuously increases in the inner part of the ingot until it reaches its maximum at $r/R = 0.5$. The simulated

479 results show the same tendency only in case D, where solutal convection is opposed to thermal
one. Among the calculations, case D, where the solute coefficient $\beta_S^{Sn} = 3 \cdot 10^{-2} (wt\%)^{-1}$, has the best fit to the measured data and was therefore chosen for subsequent simulations.

<figure><img src="image_31.png" /></figure>

**Fig. 12.** Radial profiles of Sn segregation ($\overline{\omega}^{Sn} - \omega_0^{Sn}$)/$\omega_0^{Sn}$ at 2 heights in the remelted ingot. Experimental (•) and model (-) results. Letters A, B, C, D correspond to cases listed in Table 4.

The computed profiles of the radially averaged longitudinal Sn content are shown in Fig. 13. Thermo-solutal convection has no significant effect on the overall longitudinal segregation. Although the comparison of simulated results with experimental measurements should be interpreted with caution, due to the limited number of measurements along the ingot height, the results show a fairly good agreement.
<figure><img src="image_32.png" /></figure>

Fig. 13. Radially-averaged longitudinal profiles of Sn segregation ($\overline{\omega}^{Sn} - \omega_{0}^{Sn})/\omega_{0}^{Sn}$) in the remelted ingot. Experimental (•) and model (-) results. Letters A, B, C, D correspond to cases listed in Table 4.

For a given thermo-solutal convection (case D), Fig. 14 shows the longitudinal content profiles for Sn and other alloying elements, namely Fe, O and Cr. The results for all elements are quite similar (it must be stated that the oxygen partition coefficient $k_{O}$ is larger than 1) and in fairly good agreement with measurements, except in the bottom part of the ingot. At this location, in addition to a possible effect of dendrite arm spacing (see section 4.2), it could be argued that the discrepancy between the measured chemistry and the predicted segregation is related to the motion of equiaxed grains. [7] showed that the sedimentation of free floating equiaxed grains can
501 significantly affect macrosegregation. Decantation of these grains in the bottom of the melt pool
502 during its solidification could balance the macrosegregation induced by thermo-solutal
503 convection, resulting in a flat segregation profile.

<figure><img src="image_33.png" /></figure>

504 **Fig. 14.** Radially-averaged longitudinal profiles of chemical segregation $(\overline{\omega^n} - \omega_0^n)/\omega_0^n$ in the
505 remelted ingot for elements Sn, Fe, O and Cr. Experimental (•) and model (-) results. Case D -
506 $\beta_T = 4 \cdot 10^{-5} \, ^{\circ}\text{C}^{-1}$, $\beta_S^{Sn} = 3 \cdot 10^{-2} \, \text{wt}\%^{-1}$.

507

### 4.2.2. Influence of the dendrite arm spacing

508 As discussed in section 3, the solidifying mushy zone is modelled as a porous medium
509 and is characterized by its local permeability which depends on the liquid volume fraction $g_l$ and
510
the dendrite arm spacing $\lambda_{DAS}$ (Eqs.13 and 17 in Table 1). Unfortunately, it is very difficult to obtain a true value of $\lambda_{DAS}$ from a conventional metallographic analysis due to the solid state transformation $\beta(cc) \rightarrow \alpha(hcp)$ that occurs during the cooling of Zirconium alloys. In order to assess the role of the permeability in macrosegregation prediction, three cases were tested for each of the solidification models and are presented in Table 5. The value

$\beta_S^{Sn} = 3 \cdot 10^{-2}(wt\%)^{-1}$ (corresponding to case D, see above) was adopted for all calculations.

**Table 5.** Definition of the three simulations run to study the effect of dendrite arm spacing.

<table><thead><tr><th>case</th><th>λ<sub>DAS</sub> [μm]</th><th>K<sub>0</sub>[m<sup>2</sup>]</th></tr></thead><tbody><tr><td>D'</td><td>67</td><td>2.5 · 10<sup>-11</sup></td></tr><tr><td>D</td><td>134</td><td>10<sup>-10</sup></td></tr><tr><td>D''</td><td>268</td><td>4 · 10<sup>-10</sup></td></tr></tbody></table>

The computed Sn macrosegregation maps are shown in Fig. 15. It can be observed that the intensity of macrosegregation increases with $\lambda_{DAS}$. To explain this phenomenon, the amplitude of the liquid velocity in the mushy zone during the continuous stirring stage (time $t_3$) is plotted in Fig. 16 as a function of the liquid fraction. It is obvious that a large $\lambda_{DAS}$ increases the velocity of the interdendritic liquid flow, thus promoting macrosegregation.
<figure><img src="image_34.png" /><figcaption>**Fig. 15.** Influence of the dendrite arm spacing $\lambda_{DAS}$ on the computed Sn macrosegregation map in the remelted ingot. (a) Case D': $\lambda_{DAS} = 67$ µm. (b) Case D: $\lambda_{DAS} = 134$ µm. (c) Case D'':

$\lambda_{DAS} = 268$ µm.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>**Fig. 16.** Magnitude of the liquid velocity in the mushy zone as a function of liquid fraction at time $t=3$ (continuous stirring stage).</figcaption></figure>

In Fig. 17, the simulated macrosegregation of Sn is compared to experimental measurements at two different heights ($h/H = 0.25$) and ($h/H = 0.75$). The profiles calculated for $\lambda_{DAS} = 67$ µm and $\lambda_{DAS} = 134$ µm are similar at $h/H = 0.25$. These profiles are quite flat up to $r/R = 0.6$, after which point the Sn concentration gradually decreases. Although the difference between the simulated results and the experimental measurements is not high, it should be noted that the predicted segregation in the mid-radius zone is positive while the experimental segregation is negative. The agreement is much better when $\lambda_{DAS}$ is increased to 268 µm, because the predictions for this case anticipate the negative Sn segregation values from $r/R = 0.5$ on. At $h/H = 0.75$, on the contrary, a better agreement was found when $\lambda_{DAS}$ was reduced (case D').
<figure><img src="image_36.png" /><figcaption>Fig. 17. Radial profiles of Sn segregation ($\overline{\omega}^{Sn} - \omega_{0}^{Sn}$)/$\omega_{0}^{Sn}$ at 2 heights in the remelted ingot.</figcaption></figure>

Experimental (•) and model results. Letters D, D' and D'' correspond to cases listed in Table 5.

### 4.2.3. Influence of the grain growth kinetics

The effect of grain growth kinetics on macrosegregation was investigated numerically by examining four different cases listed in Table 6. The default values presented in Table 3 were used for the diffusion coefficients of alloy elements, thermal and solutal expansion coefficients, nucleat+ion density, and secondary dendrite arm spacing, unless otherwise specified. Case D, as usual, represents the case with default values. In case E, the nucleation density $N_0$ was increased to $10^{12} \text{ m}^{-3}$. In cases F and G, the diffusion coefficient of Sn in the solid phase was reduced to $4.9 \cdot 10^{-15} \text{ m}^2/\text{s}$ for a nucleation density equal to $10^{12} \text{ m}^{-3}$ and $10^{10} \text{ m}^{-3}$, respectively.

<table><caption>**Table 6.** Definition of the four simulations run to study the effect of grain growth kinetics</caption><thead><tr><td>case</td><td>$N_0$ [m<sup>-3</sup>]</td><td>$D_S^{Sn}$ [m<sup>2</sup>/s]</td></tr></thead></table>
<table><tr><td>D</td><td>10<sup>10</sup></td><td>4.9 · 10<sup>-11</sup></td></tr><tr><td>E</td><td>10<sup>12</sup></td><td>4.9 · 10<sup>-11</sup></td></tr><tr><td>F</td><td>10<sup>12</sup></td><td>4.9 · 10<sup>-15</sup></td></tr><tr><td>G</td><td>10<sup>10</sup></td><td>4.9 · 10<sup>-15</sup></td></tr></table>

553

554

Fig. 18 shows the final macrosegregation maps of Sn computed in all four cases.

555

556

557

558

559

Although the solidification process differs quantitatively, the comparison of the maps reveals only small differences in the calculated macrosegregation. As apparent from Fig. 18, nucleation density and solute diffusivity did not seem to play an important role in the chemistry of the remelted ingot. Nevertheless, it can be noted that the macrosegregation was slightly less intense in case E.

560

561

562

563

564

565

566

567

568

569

570

571

The small differences observed between these different cases can be explained by studying the value of the Fourier number in order to characterize the diffusion of solute in the solid and liquid phases in the central part of the ingot. The Fourier number associated to a given phase (solid or liquid) is defined as the ratio $o = \frac{D \cdot t_{ls}}{R_g^2}$. In this relationship, D is the solute diffusion coefficient in the considered phase, $t_s$ is the local solidification time, and $R_g$ is the final average grain radius as defined in equation (22) in Table 2. As it can be seen in this equation, the final grain radius is only dependent of the grain density which is an input of the model. If the value of the Fourier number is greater than 1 the composition profile of the alloying element can be considered as uniform in the phase, while if it is smaller than 1 the composition profile is non uniform. The values of the Fourier numbers for the liquid and solid phases are reported in Table 7 considering two local solidification times ($t_{ls}$), respectively 10 and 1260 s. The first value corresponds to the solidification time at a point close to the skin while the second one stands for a
572 location in the central part of the ingot. These points have been chosen along a line at mid-height
573 of the ingot. They are considered to be representative of extreme values of the Fourier numbers in
574 the solid and liquid phases.

575

576 **Table 7.** Values of the Fourier number in the liquid and solid phases for the four cases (D, E, F,
577 G). These Fourier numbers are estimated for two local solidification times (10 s and 1260 s).

578 These values correspond to points located close to the skin of the ingot and for the second one to
579 a point located at the center of the ingot. Both of them are considered at mid-height of the ingot.

<table><thead><tr><th rowspan="2"></th><th colspan="2">t<sub>ls</sub> = 10 s</th><th colspan="2">t<sub>ls</sub> = 1260 s)</th></tr><tr><th>Fo<sub>l</sub><sup>Sn</sup></th><th>Fo<sub>S</sub><sup>Sn</sup></th><th>Fo<sub>l</sub><sup>Sn</sup></th><th>Fo<sub>S</sub><sup>Sn</sup></th></tr></thead><tbody><tr><td>D</td><td>1.2</td><td>0.006</td><td>150</td><td>0.75</td></tr><tr><td>E</td><td>2.6</td><td>0.127</td><td>3270</td><td>16</td></tr><tr><td>F</td><td>2.6</td><td>10<sup>-5</sup></td><td>3270</td><td>0.0016</td></tr><tr><td>G</td><td>1.2</td><td>6 10<sup>-7</sup></td><td>150</td><td>7.5 10<sup>-5</sup></td></tr></tbody></table>

580

581 For the two points and all 4 cases, the Fourier number values in the liquid phase are
582 greater than 1, thus the solute profile at the microscopic scale was uniform in the liquid phase
583 during almost all the solidification. Considering the point close to the skin, the Fourier number in
584 the solid phase is lower than 1 in all cases. This means that there is no diffusion of solute in the
585 solid phase at the microscopic scale in all cases for the point close to the skin. The skin is thus a
586 region for which all cases correspond to the same situation at the microscopic scale: perfect
587 diffusion of solute in the liquid and no diffusion of solute in the solid. This is why the
588 macrosegregation is very similar for all the cases in the region of the skin of the ingot.
Considering the point located at mid-height in the center of the ingot, for cases D, F and G, the Fourier number value in the solid phase is lower than 1 while it is greater than 1 for case E. These results show that for cases D, F and G, solidification proceeds in such a way that the solute diffuses perfectly in the liquid phase and very little in the solid phase. For case E, the diffusion of the solute is perfect in the solid and liquid phases. For all cases, the diffusion is perfect in the liquid phase, only the diffusion of the solute in the solid phase is different. It has already been shown that solute diffusion in the solid phase has very little influence on macrosegregation[29]. This can explain why cases D, E, F, G lead to similar macrosegregation.

<figure><img src="image_37.png" /></figure>
597 **Fig. 18.** Computed Sn content map in the remelted ingot, (a) case D, (b) case E, (c) case F and (d) case G. Letters D, E, F, G correspond to cases listed in Table 6.
598

599

600

**5 Conclusions and prospects**

601 A two-scale model was used to simulate the solidification of the remelting of an
602 experimental chemically homogeneous Zy-4 electrode in a production furnace. The effects of
603 thermal and solutal convection on ingot macrosegregation, as well as the influence of dendrite
604 arm spacing (i.e. the permeability of the mushy region) were studied and analyzed. In addition,
605 the influence of grain growth kinetics was investigated numerically.

606 The results show that it is necessary to take into account the thermal and solutal buoyancy
607 forces in order to obtain an accurate simulation of hydrodynamics and consequently the
608 macrosegregation in VAR ingots. Comparing the experimental measurements with the simulated
609 results, it is concluded that solutal-driven convection may promote an upward flow along the
610 solidification front in the mushy zone and thus be partially responsible for macrosegregation in
611 the central part of the ingot. Furthermore, the simulations have shown that the macrosegregation
612 predicted by the model depends on the dendrite arm spacing value used to calculate local
613 permeability in the mushy zone. However, the model is still unable to predict some of the trends
614 observed in experimental measurements.

615 In future work, the transport of free-floating grains and the flow induced by the
616 solidification shrinkage will be implemented in the SOLAR code. Such additions could
617 significantly affect the predicted hydrodynamic behaviour, resulting in macrosegregation.
618

619
619 Acknowledgments

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

623

624 Conflict of interest

625 On behalf of all authors, the corresponding author states that there is no conflict of
626 interest.

627

628 References

629 1 A. Jardy and D. Ablitzer: *Xiyou Jinshu Cailiao Yu GongchengRare Met. Mater. Eng.*, 2006, vol. 35, pp. 119-122.

630 2 A. Jardy and D. Ablitzer: *Mater. Sci. Technol.*, 2009, vol. 25, pp. 163-9.

631 3 A. Wilson and A. Jardy: in *A numerical study of the influence of stirring parameters on titanium ingot quality using SOLAR v4.2*, vol. 1, The Japan Institute of Metals, Kyoto, Japan, 2007.

632 4 V. Venkatesh, A. Wilson, M. Kamal, M. Thomas, and D. Lambert: *JOM*, 2009, vol. 61, pp. 45-50.

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634 6 M. Revil-Baudard, A. Jardy, F. Leclerc, M. Zaloznik, V. Rebeyrolle, and H. Combeau: in *CFD Modeling and Simulation in Materials Processing*, John Wiley & Sons, Ltd, 2012, pp. 107-14.

635 7 H. Combeau, M. Založnik, S. Hans, and P.E. Richy: *Metall. Mater. Trans. B*, 2009, vol. 40, pp. 289-304.

636 8 B.E. Launder and D.B. Spalding: *Comput. Methods Appl. Mech. Eng.*, 1974, vol. 3, pp. 269-89.

637 9 P.J. Prescott and F.P. Incropera: *J. Heat Transf.*, 1995, vol. 117, pp. 716-24.

638 10 B.V. Antohe and J.L. Lage: *Int. J. Heat Mass Transf.*, 1997, vol. 40, pp. 3013-24.

639 11 A. Nakayama and F. Kuwahara: *J. Fluids Eng.*, 1999, vol. 121, pp. 427-33.

640 12 M.H.J. Pedras and M.J.S. de Lemos: *Int. J. Heat Mass Transf.*, 2001, vol. 44, pp. 1081-93.

641 13 S. Hans: PhD thesis (in French), Institut National Polytechnique de Lorraine, 1995.

642 14 T. Quatravaux: PhD thesis (in French), Institut National Polytechnique de Lorraine, 2004.

643 15 M. Revil-Baudard: PhD thesis (in French), Université de Lorraine, 2012.

644 16 C. Beckermann and R. Viskanta: *Physicochem. Hydrodyn.*, 1988, vol. 10, pp. 195-213.

645 17 C.Y. Wang and C. Beckermann: *Metall. Mater. Trans. A*, 1996, vol. 27, pp. 2754-64.

646 18 M. Založnik and H. Combeau: *Comput. Mater. Sci.*, 2010, vol. 48, pp. 1-10.

647 19 K. Tveito, M. Bedel, M. Založnik, H. Combeau, and M. M'Hamdi: vol. 27, IOP Publishing, 2012, p. 012040.

648 20 T.B. Massalski, H. Okamoto, P. Subramanian, L. Kacprzak, and W.W. Scott: *Binary Alloy Phase Diagrams*, vol. 1, American society for metals Metals Park, OH, 1986.

649 21 T. Quatravaux, S. Ryberon, S. Hans, A. Jardy, B. Lusson, P.E. Richy, and D. Ablitzer: *J. Mater. Sci.*, 2004, vol. 39, pp. 7183-91.

650 22 G. Ghazal, A. Jardy, P. Chapelle, and Y. Millet: *Metall. Mater. Trans. B*, 2010, vol. 41, pp. 646-59.
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659 DOI:10.1520/JAI102992.

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661 25 P.-F. Paradis and W.-K. Rhim: *J. Mater. Res.*, 1999, vol. 14, pp. 3713–9.

662 26 T. Ishikawa and P.-F. Paradis: *J. Electron. Mater.*, 2005, vol. 34, pp. 1526–32.

663 27 M.J. Assael, A.E. Kalyva, K.D. Antoniadis, R. Michael Banish, I. Egry, J. Wu, E. Kaschnitz, and W.A.
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665 28 M. Revil-Baudard, A. Jardy, H. Combeau, F. Leclerc, and V. Rebeyrolle: *Metall. Mater. Trans. B*,
666 2014, vol. 45, pp. 51–7.

667 29 M.C. Schneider and C. Beckermann: *Int. J. Heat Mass Transf.*, 1995, vol. 38, pp. 3455–73.

671 List of figures

672 **FIG. 1.** SCHEMATIC REPRESENTATION OF A VAR FURNACE. .................................................................................................................. 5

673 **FIG. 2.** MELTING RATE OF THE FULL-SCALE ZY4 REMELTING TRIAL. .................................................................................................................. 8

674 **FIG. 3.** STIRRING SEQUENCE OF THE FULL-SCALE ZY4 REMELTING TRIAL. (A) COIL CURRENT THROUGHOUT THE MELT. (B)
675 DETAILED ALTERNATING STIRRING SEQUENCE. (C) EXAMPLE OF A COMPUTED COMPOSITION MAP (ZR CONTENT),
676 HIGHLIGHTING THE MELT POOL MARKINGS IN THE REMELTED INGOT. .................................................................................................................. 9

677 **FIG. 4.** DENSITY OF ZR AS A FUNCTION OF TEMPERATURE, FROM [26] (THE VERTICAL BAR CORRESPONDS TO ZR
678 MELTING POINT). .................................................................................................................................................................................. 22

679 **FIG. 5.** DENSITY OF SN AS A FUNCTION OF TEMPERATURE, FROM [27] (THE VERTICAL BAR CORRESPONDS TO SN
680 MELTING POINT). .................................................................................................................................................................................. 23

681 **FIG. 6.** CASE A – COMPUTED POOL PROFILE AND AZIMUTHAL VELOCITY (M.S⁻¹) MAP IN THE MELT POOL AT 3 DIFFERENT
682 MOMENTS DURING THE MELT. (A) TIME T₁ - ALTERNATED STIRRING, LOW AZIMUTHAL VELOCITY. (B) TIME T₂ -
683 ALTERNATED STIRRING, HIGH AZIMUTHAL VELOCITY. (C) TIME T₃ - CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY. 25

684 **FIG. 7.** COMPUTED FLOW IN THE MELT POOL AND MUSHY ZONE AT TIME T₁ (ALTERNATED STIRRING, LOW AZIMUTHAL VELOCITY).
685 LEFT SIDE IS THE TURBULENCE INTENSITY AND RIGHT SIDE IS THE MEAN FLOW REPRESENTED BY VECTORS (POOL UPPER
686 PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: BT = 0,
687 BSSN = 0, VMAX = 2.03 CM.S-1. (B) CASE B: BT = 4 · 10⁻⁵ °C⁻¹, BSSN = 0, VMAX = 1.60 CM.S-1. (C) CASE
688 C: BT = 4 · 10⁻⁵ °C⁻¹, BSSN = -1 · 10⁻² WT%⁻¹, VMAX = 1.82 CM.S-1. (D) CASE D: BT = 4 · 10⁻⁵ °C⁻¹,
689 BSSN = 3 · 10⁻² WT%⁻¹, VMAX = 1.61 CM.S-1. .................................................................................................................. 27

690 **FIG. 8.** COMPUTED FLOW IN THE MELT POOL AND MUSHY ZONE AT TIME T₂ (ALTERNATED STIRRING, HIGH AZIMUTHAL VELOCITY).
691 LEFT SIDE IS THE TURBULENCE INTENSITY AND RIGHT SIDE IS THE MEAN FLOW REPRESENTED BY VECTORS (POOL UPPER
692 PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: BT = 0,
693 BSSN = 0, VMAX = 2.53 CM.S-1. (B) CASE B: BT = 4 · 10⁻⁵ °C⁻¹, BSSN = 0, VMAX = 2.53 CM.S-1. (C) CASE
694 C: BT = 4 · 10⁻⁵ °C⁻¹, BSSN = -1 · 10⁻² WT%⁻¹, VMAX = 2.50 CM.S-1. (D) CASE D: BT = 4 · 10⁻⁵ °C⁻¹,
695 BSSN = 3 · 10⁻² WT%⁻¹, VMAX = 2.51 CM.S-1. .................................................................................................................. 29

696 **FIG. 9.** COMPUTED FLOW IN THE MELT POOL AND MUSHY ZONE AT TIME T₃ (CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY).
697 LEFT SIDE IS THE TURBULENCE INTENSITY AND RIGHT SIDE IS THE MEAN FLOW REPRESENTED BY VECTORS (POOL UPPER
698 PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: BT = 0,
699 BSSN = 0, VMAX = 2.02 CM.S-1. (B) CASE B: BT = 4 · 10⁻⁵ °C⁻¹, BSSN = 0, VMAX = 1.93 CM.S-1. (C) CASE
700 C: BT = 4 · 10⁻⁵ °C⁻¹, BSSN = -1 · 10⁻² WT%⁻¹, VMAX = 1.83 CM.S-1. (D) CASE D: BT = 4 · 10⁻⁵ °C⁻¹,
701 BSSN = 3 · 10⁻² WT%⁻¹, VMAX = 2.00 CM.S-1. .................................................................................................................. 30
702 **FIG. 10.** COMPUTED SN CONTENT MAP IN THE REMELTED INGOT. (A) CASE A: $bT = 0$, $bSSN = 0$. (B) CASE B: $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 0$. (C) CASE C: $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = -1 \cdot 10 - 2 \,\text{wt}\% - 1$. (D) CASE D: $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 3 \cdot 10 - 2 \,\text{wt}\% - 1$. 33

703 **FIG. 11.** ZOOM VISUALIZATION OF THE INFLUENCE OF SOLUTE-DRIVEN CONVECTION AT TIME $T_3$ - CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY. CASE D - $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 3 \cdot 10 - 2 \,\text{wt}\% - 1$. (A) LIQUID FRACTION, STREAMLINES AND SN CONTENT. (B) LIQUID FRACTION AND CONVECTIVE $v^\perp \nabla T$ TERM. 34

704 **FIG. 12.** RADIAL PROFILES OF SN SEGREGATION ($\Omega\text{SN} - \Omega\text{0SN})/\Omega\text{0SN}$ AT 2 HEIGHTS IN THE REMELTED INGOT. EXPERIMENTAL ($\bullet$) AND MODEL (-) RESULTS. LETTERS A, B, C, D CORRESPOND TO CASES LISTED IN TABLE 4. 35

705 **FIG. 13.** RADIALLY-AVERAGED LONGITUDINAL PROFILES OF SN SEGREGATION ($\Omega\text{SN} - \Omega\text{0SN})/\Omega\text{0SNIN}$ THE REMELTED INGOT. EXPERIMENTAL ($\bullet$) AND MODEL (-) RESULTS. LETTERS A, B, C, D CORRESPOND TO CASES LISTED IN TABLE 4. 36

706 **FIG. 14.** RADIALLY-AVERAGED LONGITUDINAL PROFILES OF CHEMICAL SEGREGATION ($\Omega\text{N} - \Omega\text{0N})/\Omega\text{0N}$ IN THE REMELTED INGOT FOR ELEMENTS SN, FE, O AND CR. EXPERIMENTAL ($\bullet$) AND MODEL (-) RESULTS. CASE D - $bT = 4 \cdot 10 - 5 \,^{\circ}\text{C} - 1$, $bSSN = 3 \cdot 10 - 2 \,\text{wt}\% - 1$. 37

707 **FIG. 15.** INFLUENCE OF THE DENDRITE ARM SPACING $\lambda DAS$ ON THE COMPUTED SN MACROSEGREGATION MAP IN THE REMELTED INGOT. (A) CASE D': $\lambda DAS = 67 \,\mu\text{M}$. (B) CASE D: $\lambda DAS = 134 \,\mu\text{M}$. (C) CASE D": $\lambda DAS = 268 \,\mu\text{M}$. 39

708 **FIG. 16.** MAGNITUDE OF THE LIQUID VELOCITY IN THE MUSHY ZONE AS A FUNCTION OF LIQUID FRACTION AT TIME $T_3$ (CONTINUOUS STIRRING STAGE). 40

709 **FIG. 17.** RADIAL PROFILES OF SN SEGREGATION ($\omega\text{Sn} - \omega\text{0Sn})/\omega\text{0Sn}$ AT 2 HEIGHTS IN THE REMELTED INGOT. EXPERIMENTAL ($\bullet$) AND MODEL RESULTS. LETTERS D, D' AND D'' CORRESPOND TO CASES LISTED IN TABLE 5. 41

710 **FIG. 18.** COMPUTED SN CONTENT MAP IN THE REMELTED INGOT, (A) CASE D, (B) CASE E, (C) CASE F AND (D) CASE G. LETTERS D, E, F, G CORRESPOND TO CASES LISTED IN TABLE 6. 45

711
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717
718
719
720
721
722
723
724
725
**List of tables**

726 **TABLE 1.** MAIN CONSTITUTIVE EQUATIONS OF THE CFD MACROMODEL 11

727 **TABLE 2.** MAIN EQUATIONS OF THE SOLIDIFICATION MICROMODEL. 17

728 **TABLE 3.** THERMOPHYSICAL PROPERTIES OF ZY4 USED IN THE SIMULATIONS 19

729 **TABLE 4.** DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF THERMO-SOLUTE CONVECTION 23

730 **TABLE 5.** DEFINITION OF THE THREE SIMULATIONS RUN TO STUDY THE EFFECT OF DENDRITE ARM SPACING. 38

731 **TABLE 6.** DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF GRAIN GROWTH KINETICS 41

732 **TABLE 7.** VALUES OF THE FOURIER NUMBER IN THE LIQUID AND SOLID PHASES FOR THE FOUR CASES (D, E, F, G). THESE FOURIER NUMBERS ARE ESTIMATED FOR TWO LOCAL SOLIDIFICATION TIMES (10 s AND 1260 s). THESE VALUES CORRESPOND TO POINTS LOCATED CLOSE TO THE SKIN OF THE INGOT AND FOR THE SECOND ONE TO A POINT LOCATED AT THE CENTER OF THE INGOT. BOTH OF THEM ARE CONSIDERED AT MID-HEIGHT OF THE INGOT. 43

733
734
735
736