K Mramor

T Quatravaux
email: thibault.quatravaux@univ-lorraine.fr

H Combeau
email: herve.combeau@univ-lorraine.fr

A Jardy

M Založnik
email: miha.zaloznik@univ-lorraine.fr

I Crassous

A Gaillac
email: alexis.gaillac@framatome.com

On the prediction of macrosegregation in Vacuum Arc Remelted ingots

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

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

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

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

The model results were validated, based on the remelting of a specially designed 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.

D

diffusion coefficient of solute in the liquid or the solid phase (m 2 .s -1 )

F el electromagnetic force (N) g gravitational acceleration (m.s -2 ) g l volume fraction of the liquid phase g s 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 )

J ji interfacial species transfer rate per unit volume due to diffusion (kg.m-3 .s -1 )

J Гi interfacial species transfer rate per unit volume due to the phase change (kg.m -3 .s The VAR process, as illustrated in Fig. 1, consists in continuously remelting a consumable electrode. In order to obtain a high quality structural ingot, the composition of the electrode must comply with strict standards. A high power direct current, applied under vacuum, provides an electric arc between the tip of the electrode and the baseplate of a water-cooled copper mold. The intense heat generated by the arc melts the electrode tips that fall into the crucible. The secondary ingot is gradually formed from the liquid metal that falls through the arc plasma. The secondary ingot consists of a liquid metal pool above the solidified part and a mushy zone where solidification takes place. 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 sidearcing 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:

-thermal convection (buoyancy force due to thermal gradients) which tends to favour a flow in a clockwise direction (symmetry axis being on the left),

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

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

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.

Full-scale VAR trial

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.

The actual recordings of a highly non-standard melting sequence are presented in Figs. 2 and3, 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 % of the electrode, followed by a weak continuous stirring until the end of the melting process. The magnitude of the alternating stirring was about 4.5 times higher than the continuous stirring one.

Additionally, a strong continuous stirring has been briefly applied at three moments 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, where H is the total height of the ingot. These measurements are used in section 4 to validate the numerical results.

SOLAR model description

The CFD macromodel

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

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

Laplace conservation equation

∇ 𝜑 = 0 (8)
Mass balance equation at the solid-liquid interface

𝛤 + 𝛤 = 0 (9)
Solute mass balance at the solid-liquid interface

𝐽 + 𝐽 + 𝐽 + 𝐽 = 0 (10)
Ohm's law and Lorentz force 𝑱 = -𝜎𝛻𝜑 𝑭 𝒆𝒍 = 𝑱 ∧ 𝑩 = -𝐽 𝐵 𝒆 𝒓 + 𝐽 𝐵 , 𝒆 𝜽 + 𝐽 𝐵 𝒆 𝒛 (11) Liquid density in the buoyancy force

𝜌 = 𝜌 [1 -𝛽 𝑇 -𝑇 - 𝛽 (𝜔 -𝜔 ) ] (12)
Permeability law

𝐾 = 𝑔 (1 -𝑔 ) 𝐾 (13)
Production rate 𝑃 due to gradients of 𝑢 and dissipation rate 𝐷 due to liquid-solid interaction

𝑃 = -𝜌𝑔 𝒖 𝒊 𝒖 𝒋 𝜕 𝜕𝑥 (𝑔 𝑢 ) 𝐷 = - 2𝜇𝑔 𝐾 𝑘 (14)
Generation rate due to thermal and solutal buoyancy

𝐺 = 𝛽 𝑔 𝜇 𝑃𝑟 𝒈 ⋅ 𝛻𝑇 𝐺 = 𝛽 𝑔 𝜇 𝑆𝑐 𝒈 ⋅ 𝛻𝜔 (15)
Permeability constants of the model

𝐾 = 𝜆 180 ( 1 7 )
Pressure

𝑝 * = 𝑝 + 2 3 𝜌𝑘 (18)
Turbulent parameters

𝜇 = 𝑐 𝜌 𝑘 𝜀 𝜆 = 𝑐 𝜇 𝑃𝑟 𝐷 = 𝜇 𝜌𝑆𝑐 (19)
A macroscopic RANS two-equation 𝑘 -𝜀 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 𝜀 include: -turbulence production due to mean velocity gradients 𝑃 (Eq. 14), -dissipation of turbulence due to the liquid/solid interaction in the mushy zone 𝐷 (Eq. 14), -production or dissipation of turbulence caused by vertical thermal 𝐺 and solute content 𝐺 gradients (Eq. 15).

The constants used for closure (Eq. 16) correspond to the standard 𝑘 -𝜀 model originally developed by Launder and Spalding [8] for a fully liquid medium (𝑔 = 1, hence 𝐾 → ∞). 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][10][11][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] .

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 = 𝑇 = 𝑇 and 𝜔 * = 𝑘 𝜔 * . 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 and partition coefficient 𝑘 are constant. Therefore, the solidification process starts when the enthalpy of a cell falls below the liquidus enthalpy ℎ , (20) and stops when the temperature drops below a fixed temperature 𝑇 or when the solid fraction reaches 𝑔 = 1 due to the primary growth as explained by Založnik and Combeau [18] .

f i l n i i L m p liq L m T c h sol + + = = ω 1
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 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:

( )

s n s v i s l n l v i l s n s v n p l n l v i l s i l i p D S D S D S k D S k δ ω δ ω δ δ ω ρ ω + = + + Γ - 1 . (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 𝜔 * , and the average mass fraction 𝜔 , by a diffusion length 𝛿 . 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.

4

Numerical results and discussion

The full-scale melt of a Zy4 electrode (see Section 2) was simulated using the model fully described in Section 3. A complete set of process operating parameters (melting rate, arc voltage, melting current and current in the external induction coils) was previously recorded during the 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 and3).

The thermophysical properties of Zy4 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 and 𝜆 used in the simulations (default values) are also reported in Table 3. For all simulations reported in the present paper, the average cell size was approximately 9 mm wide and 6 mm height. Mesh refinement was applied with a geometrical reason of 0.8 on the five nodes closest to the mold as well as to the free surface. The time-step was fixed to 0.025 s.

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

In several previous publications [2,[21][22][23] , the authors have detailed some CFD macromodel results when applied to various remelting situations. In this paper, the attention will be focused on the fluid flow in the liquid pool and the mushy zone, as well as the resulting macrosegregation.

Hydrodynamics in the liquid metal pool

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

Assumptions on the thermal and solutal buoyancy forces

During solidification, a buoyancy driving force can be generated either by the thermal or solutal buoyancy effects, that can be collaborating or opposing [24] . Both buoyancy effects are modelled by the Boussinesq approximation (Eq. 12), therefore both the thermal and the solutal expansion coefficients had to be estimated.

Unfortunately, values for the solutal and thermal expansion coefficient for Zy-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. Fig. 4. Density of Zr as a function of temperature, from [26] (the vertical bar corresponds to Zr melting point).

Fig. 5. Density of Sn as a function of temperature, from [27] (the vertical bar corresponds to Sn melting point).

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 𝛽 = -1 ⋅ 10 (𝑤𝑡%) , 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. 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. 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. The influence of a high azimuthal velocity on the in-plane velocity field and turbulence intensity in the melt pool and mushy zone is shown in Figs. 8 and9, for instants t 2 during the alternated stirring and t 3 during the continuous stirring.

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

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

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

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

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.

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 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. This transport of solute is effective for a liquid fraction 𝑔 > 0.6. When 𝑔 < 0.6, the permeability of the porous media decreases rapidly as the drag forces dominate inertial and buoyancy forces and macrosegregation evolution is stopped. 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 significantly affect macrosegregation. Decantation of these grains in the bottom of the melt pool during its solidification could balance the macrosegregation induced by thermo-solutal convection, resulting in a flat segregation profile.

Influence of the dendrite arm spacing

As discussed in section 3, the solidifying mushy zone is modelled as a porous medium and is characterized by its local permeability which depends on the liquid volume fraction 𝑔 and the dendrite arm spacing 𝜆 (Eqs.13 and 17 in Table 1). Unfortunately, it is very difficult to obtain a true value of 𝜆 from a conventional metallographic analysis due to the solid state transformation 𝛽(cc) → 𝛼(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 𝛽 = 3 ⋅ 10 (𝑤𝑡%) (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.

case In Fig. 17 5.

𝜆 [µm] 𝐾 [m ] D' 67

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 was increased to 10 m -3 . In cases F and G, the diffusion coefficient of Sn in the solid phase was reduced to 4.9 ⋅ 10 m 2 /s for a nucleation density equal to 10 m -3 and 10 m -3 , respectively. Fig. 18 shows the final macrosegregation maps of Sn computed in all four cases.

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.

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 =

• . 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 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. 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. case G. Letters D, E, F, G correspond to cases listed in Table 6.

Conclusions and prospects

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

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

In future work, the transport of free-floating grains and the flow induced by the solidification shrinkage will be implemented in the SOLAR code. Such additions could significantly affect the predicted hydrodynamic behaviour, resulting in macrosegregation. The VAR process, as illustrated in Fig. 1, consists in continuously remelting a consumable electrode. In order to obtain a high quality structural ingot, the composition of the electrode must comply with strict standards. A high power direct current, applied under vacuum, provides an electric arc between the tip of the electrode and the baseplate of a water-cooled copper mold. The intense heat generated by the arc melts the electrode tips that fall into the crucible. The secondary ingot is gradually formed from the liquid metal that falls through the arc plasma. The secondary ingot consists of a liquid metal pool above the solidified part and a mushy zone where solidification takes place. In addition, VAR is generally used in the remelting cycles of high quality special steels, as well as Nickel and Titanium alloys.

List of figures

In the case of Zr or Ti remelting, the arc length is several centimeters. To prevent sidearcing 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:

-thermal convection (buoyancy force due to thermal gradients) which tends to favour a flow in a clockwise direction (symmetry axis being on the left),

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

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

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.

Full-scale VAR trial

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.

The actual recordings of a highly non-standard melting sequence are presented in Figs. 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, where H is the total height of the ingot. These measurements are used in section 4 to validate the numerical results.

SOLAR model description

The CFD macromodel

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

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

Laplace conservation equation

∇ 𝜑 = 0 (8)
Mass balance equation at the solid-liquid interface

𝛤 + 𝛤 = 0 (9)
Solute mass balance at the solid-liquid interface

𝐽 + 𝐽 + 𝐽 + 𝐽 = 0 (10)
Ohm's law and Lorentz force 𝑱 = -𝜎𝛻𝜑 𝑭 𝒆𝒍 = 𝑱 ∧ 𝑩 = -𝐽 𝐵 𝒆 𝒓 + 𝐽 𝐵 , 𝒆 𝜽 + 𝐽 𝐵 𝒆 𝒛 (11) Liquid density in the buoyancy force

𝜌 = 𝜌 [1 -𝛽 𝑇 -𝑇 - 𝛽 (𝜔 -𝜔 ) ] (12)
Permeability law

𝐾 = 𝑔 (1 -𝑔 ) 𝐾 (
Permeability constants of the model

𝐾 = 𝜆 180 ( 1 7 )
Pressure

𝑝 * = 𝑝 + 2 3 𝜌𝑘 (18)
Turbulent parameters

𝜇 = 𝑐 𝜌 𝑘 𝜀 𝜆 = 𝑐 𝜇 𝑃𝑟 𝐷 = 𝜇 𝜌𝑆𝑐 (19)
A macroscopic RANS two-equation 𝑘 -𝜀 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 𝜀 include: -turbulence production due to mean velocity gradients 𝑃 (Eq. 14), -dissipation of turbulence due to the liquid/solid interaction in the mushy zone 𝐷 (Eq. 14), -production or dissipation of turbulence caused by vertical thermal 𝐺 and solute content 𝐺 gradients (Eq. 15).

The constants used for closure (Eq. 16) correspond to the standard 𝑘 -𝜀 model originally developed by Launder and Spalding [8] for a fully liquid medium (𝑔 = 1, hence 𝐾 → ∞). 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][10][11][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] .

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 = 𝑇 = 𝑇 and 𝜔 * = 𝑘 𝜔 * . 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 and partition coefficient 𝑘 are constant. Therefore, the solidification process starts when the enthalpy of a cell falls below the liquidus enthalpy ℎ , (20) and stops when the temperature drops below a fixed temperature 𝑇 or when the solid fraction reaches 𝑔 = 1 due to the primary growth as explained by Založnik and Combeau [18] .

f i l n i i L m p liq L m T c h sol + + = = ω 1
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 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:

( )

s n s v i s l n l v i l s n s v n p l n l v i l s i l i p D S D S D S k D S k δ ω δ ω δ δ ω ρ ω + = + + Γ - * * 1 . (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 𝜔 * , and the average mass fraction 𝜔 , by a diffusion length 𝛿 . 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 * = 3𝑔 4𝜋𝑁 𝑅 = 3 4𝜋𝑁 𝑆 * = 4𝜋𝑅 * 𝑁 (22) Dendrite tip kinetics Ω = 𝜔 * , -< 𝜔 > 𝜔 * , 1 -𝑘 𝑉 = 𝜕𝑅 * 𝜕𝑡 Γ = 𝜌 𝑆 * 𝑉 (23)
Boundary layer thickness

𝛿 = 𝑑 * - ( * , ) ( , * , ) ( * ( * ) ) ( * , ) ( ) ( , * , ) (24)
Solute fluxes due to diffusion

𝐽 , = 𝑆 * 𝜌 𝐷 𝛿 (𝜔 * , -𝜔 ) 𝐽 , = 𝑆 * 𝜌 𝐷 𝛿 (𝜔 * , -𝜔 ) (28)
4

Numerical results and discussion

The full-scale melt of a Zy4 electrode (see Section 2) was simulated using the model fully described in Section 3. A complete set of process operating parameters (melting rate, arc voltage, melting current and current in the external induction coils) was previously recorded during the 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 and3).

The thermophysical properties of Zy4 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 and 𝜆 used in the simulations (default values) are also reported in Table 3.

F e 𝑚 [°C(wt%) -1 ] -57 O 𝑚 [°C(wt%) -1 ] 147 C r 𝑚 [°C(wt%) -1 ] -26 Partition Coefficients Sn k [-] 0.27 F e k [-] 0.25 O k [-] 2.32 C r k [-] 0.15 Diffusion coefficients in the liquid Sn D [m 2 .s -1 ] 1•10 -8 F e D [m 2 .s -1 ] 1•10 -8 O D [m 2 .s -1 ] 1•10 -8 C r D [m 2 .s -1 ] 1•10 -8 Diffusion coefficients in the solid Sn D [m 2 .s -1 ] 4.9•10 -11 F e D [m 2 .s -1 ] 1.5•10 -9 O D [m 2 .s -1 ] 3.8•10 -9 C r D [m 2 .s -1 ] 1.3•10 -9
Temp. of end of solidification

T [°C] 1592 Latent heat L [J.kg -1 ] 2.42•10 5 Reference density ρ [kg.m -3 ] 6210 Dynamic viscosity μ [Pa.s] 4.59•10 -3 Thermal conductivity λ [W.m -1 .°C -1 ] 36.5 Thermal expansion coefficient 𝛽 [°C -1 ] 4•10 -5 Specific heat c [J.kg -1 .°C -1 ] 435 Electric conductivity σ [m -1 .Ω -1 ] 6.8•10 5 Solute expansion coefficients Sn β [(wt%) -1 ] 3•10 -2 F e β [(wt%) -1 ] 0 O β [(wt%) -1 ] 0 C r β [(wt%) -1 ] 0 Characteristic length of the λ [m] 134•10 -6
dendritic structures For all simulations reported in the present paper, the average cell size was approximately 9 mm wide and 6 mm height. Mesh refinement was applied with a geometrical reason of 0.8 on the five nodes closest to the mold as well as to the free surface. The time-step was fixed to 0.025 s.

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

In several previous publications [2,[21][22][23] , the authors have detailed some CFD macromodel results when applied to various remelting situations. In this paper, the attention will be focused on the fluid flow in the liquid pool and the mushy zone, as well as the resulting macrosegregation.

Hydrodynamics in the liquid metal pool

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

Assumptions on the thermal and solutal buoyancy forces

During solidification, a buoyancy driving force can be generated either by the thermal or solutal buoyancy effects, that can be collaborating or opposing [24] . Both buoyancy effects are modelled by the Boussinesq approximation (Eq. 12), therefore both the thermal and the solutal expansion coefficients had to be estimated. Fig. 4. Density of Zr as a function of temperature, from [26] (the vertical bar corresponds to Zr melting point).

Fig. 5. Density of Sn as a function of temperature, from [27] (the vertical bar corresponds to Sn melting point).

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 𝛽 = -1 ⋅ 10 (𝑤𝑡%) , 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.

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. 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. The influence of a high azimuthal velocity on the in-plane velocity field and turbulence intensity in the melt pool and mushy zone is shown in Figs. 8 and9, for instants t 2 during the alternated stirring and t 3 during the continuous stirring.

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

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

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

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

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.

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 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. This transport of solute is effective for a liquid fraction 𝑔 > 0.6. When 𝑔 < 0.6, the permeability of the porous media decreases rapidly as the drag forces dominate inertial and buoyancy forces and macrosegregation evolution is stopped. 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 significantly affect macrosegregation. Decantation of these grains in the bottom of the melt pool during its solidification could balance the macrosegregation induced by thermo-solutal convection, resulting in a flat segregation profile.

Influence of the dendrite arm spacing

As discussed in section 3, the solidifying mushy zone is modelled as a porous medium and is characterized by its local permeability which depends on the liquid volume fraction 𝑔 and the dendrite arm spacing 𝜆 (Eqs.13 and 17 in Table 1). Unfortunately, it is very difficult to obtain a true value of 𝜆 from a conventional metallographic analysis due to the solid state transformation 𝛽(cc) → 𝛼(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 𝛽 = 3 ⋅ 10 (𝑤𝑡%) (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.

case 𝜆 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 𝜆 = 67 µm and 𝜆 = 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 𝜆 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 𝜆 was reduced (case D'). 5.

[µm] 𝐾 [m ] D' 67

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 was increased to 10 m -3 . In cases F and G, the diffusion coefficient of Sn in the solid phase was reduced to 4.9 ⋅ 10 m 2 /s for a nucleation density equal to 10 m -3 and 10 m -3 , respectively. Fig. 18 shows the final macrosegregation maps of Sn computed in all four cases.

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.

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 =

• . 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 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. 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. case G. Letters D, E, F, G correspond to cases listed in Table 6.

Conclusions and prospects

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

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

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

List of figures

  the liquid metal (m.s -1 ) Greek symbols β S solutal expansion coefficient (wt% -1 ) β T thermal expansion coefficient (K -1 ) ε dissipation rate of turbulent kinetic energy (m 2 .s -3 ) λ thermal conductivity (W.m -1 .K -1 ) λ DAS characteristic length of the dendritic structure (m) μ dynamic viscosity (kg.m -1 .s -per unit volume due to nucleation (kg.m -3 .s -1 ) Г interfacial phase change rate per unit volume (kg.m -3 .s -1 ) The fuel assembly components, including Zirconium alloys, are one of the key elements in the production process in nuclear reactors. Refinement of the metallurgical structure and improvement of the cleanness of Zr alloy ingots are generally achieved by the Vacuum Arc Remelting (VAR) process. The high quality and good performance of the final product are ensured by this manufacturing process, which enables the regulation of the chemical homogeneity

Fig. 1 .

 1 Fig. 1. Schematic representation of a VAR furnace.

  figure 3.c.

Fig. 2 .

 2 Fig. 2. Melting rate of the full-scale Zy4 remelting trial.

Fig. 3 .

 3 Fig. 3. Stirring sequence of the full-scale Zy4 remelting trial. (a) Coil current throughout the

Macroscopic

  |𝒖 𝒍 , 𝒛 | 𝒖 𝒍 , 𝜽 + 𝒖 𝒍 , 𝒓

Figures 4 and 5

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

Fig. 6 .

 6 Fig. 6. Case A -Computed pool profile and azimuthal velocity (m.s -1 ) map in the melt pool at 3

Fig. 7 .

 7 Fig. 7. Computed flow in the melt pool and mushy zone at time t 1 (alternated stirring, low

Fig. 8 .Fig. 9 .

 89 Fig. 8. Computed flow in the melt pool and mushy zone at time t 2 (alternated stirring, high

Fig. 10 .Fig. 10 ,𝑅~0. 5 .

 10105 Fig. 10. Computed Sn content map in the remelted ingot. (a) Case A: β = 0, β = 0. (b) Case

Fig. 11 .

 11 Fig. 11. Zoom visualization of the influence of solute-driven convection at time t 3 -continuous

Fig. 12 .

 12 Fig. 12. Radial profiles of Sn segregation (ω -ω )/ω at 2 heights in the remelted ingot.

Fig. 13 .

 13 Fig. 13. Radially-averaged longitudinal profiles of Sn segregation (ω -ω )/ω in the

Fig. 14 .

 14 Fig. 14. Radially-averaged longitudinal profiles of chemical segregation (ω -ω )/ω in the

Fig. 15 .

 15 Fig. 15. Influence of the dendrite arm spacing 𝜆 on the computed Sn macrosegregation map in

Fig. 16 .

 16 Fig. 16. Magnitude of the liquid velocity in the mushy zone as a function of liquid fraction at

  , 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 𝜆 = 67 µm and 𝜆 = 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 𝜆 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 𝜆 was reduced (case D').

Fig. 17 .

 17 Fig. 17. Radial profiles of Sn segregation (𝜔 -𝜔 )/𝜔 at 2 heights in the remelted ingot.

Fig. 18 .

 18 Fig. 18. Computed Sn content map in the remelted ingot, (a) case D, (b) case E, (c) case F and (d)

9 FIG. 4 . 22 FIG. 5 . 23 FIG. 6 . 25 FIG. 7 . 27 FIG. 8 . 29 FIG. 9 .FIG. 10 .FIG. 11 .FIG. 12 .

 94225236257278299101112 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. ............................................................................................... 9FIG. 4. DENSITY OF ZR AS A FUNCTION OF TEMPERATURE, FROM[26] (THE VERTICAL BAR CORRESPONDS TO ZR MELTING POINT). .................................................................................................................................................................................... 22FIG. 5. DENSITY OF SN AS A FUNCTION OF TEMPERATURE, FROM[27] (THE VERTICAL BAR CORRESPONDS TO SN MELTING POINT). .................................................................................................................................................................................... 23 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 T 1 -ALTERNATED STIRRING, LOW AZIMUTHAL VELOCITY. (B) TIME T 2 -ALTERNATED STIRRING, HIGH AZIMUTHAL VELOCITY. (C) TIME T 3 -CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY. . 25 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: ΒT = 0, ΒSSN = 0 , VMAX = 2.03 CM.S-1. (B) CASE B: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 0, VMAX = 1.60 CM.S-1. (C) CASE C: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = -1 ⋅ 10 -2 WT% -1 , VMAX = 1.82 CM.S-1. (D) CASE D: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 3 ⋅ 10 -2 WT% -1 , VMAX = 1.61 CM.S-1. ............................................................................................... 27 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: ΒT = 0, ΒSSN = 0, VMAX = 2.53 CM.S-1. (B) CASE B: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 0, VMAX = 2.53 CM.S-1. (C) CASE C: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = -1 ⋅ 10 -2 WT% -1, VMAX = 2.50 CM.S-1. (D) CASE D: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 3 ⋅ 10 -2 WT% -1, VMAX = 2.51 CM.S-1. ................................................................................................ 29 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 REPRESENTED BY VECTORS (POOL UPPER PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: ΒT = 0, ΒSSN = 0, VMAX = 2.02 CM.S-1. (B) CASE B: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 0, VMAX = 1.93 CM.S-1. (C) CASE C: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = -1 ⋅ 10 -2 WT% -1, VMAX = 1.83 CM.S-1. (D) CASE D: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 3 ⋅ 10 -2 WT% -1, VMAX = 2.00 CM.S-1. ................................................................................................ 30

  per unit volume due to nucleation (kg.m -3 .s -1 ) Г interfacial phase change rate per unit volume (kg.m -3 .s -1 ) The fuel assembly components, including Zirconium alloys, are one of the key elements in the production process in nuclear reactors. Refinement of the metallurgical structure and improvement of the cleanness of Zr alloy ingots are generally achieved by the Vacuum Arc Remelting (VAR) process. The high quality and good performance of the final product are ensured by this manufacturing process, which enables the regulation of the chemical homogeneity

Fig. 1 .

 1 Fig. 1. Schematic representation of a VAR furnace.

2 and 3 ,

 23 figure 3.c.

Fig. 2 .

 2 Fig. 2. Melting rate of the full-scale Zy4 remelting trial.

Fig. 3 .

 3 Fig. 3. Stirring sequence of the full-scale Zy4 remelting trial. (a) Coil current throughout the

  = 𝜔 * , 𝑘 𝑇 = 𝑇 + 𝑚 𝜔 * ,

  𝑡 -𝑡 ); if 𝑇 < 𝑇 + 𝑚 𝜔 * , and 𝑔 = to phase change 𝐽 = 𝜔 * , Γ 𝐽 = 𝜔 * , Γ 𝜔 * , = 𝑘 𝜔 * ,

Figures 4 and 5

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

Fig. 6 .

 6 Fig. 6. Case A -Computed pool profile and azimuthal velocity (m.s -1 ) map in the melt pool at 3

Fig. 7 .

 7 Fig. 7. Computed flow in the melt pool and mushy zone at time t 1 (alternated stirring, low

Fig. 8 .Fig. 9 .

 89 Fig. 8. Computed flow in the melt pool and mushy zone at time t 2 (alternated stirring, high

Fig. 10 .Fig. 10 ,𝑅~0. 5 .

 10105 Fig. 10. Computed Sn content map in the remelted ingot. (a) Case A: β = 0, β = 0. (b) Case

Fig. 11 .

 11 Fig. 11. Zoom visualization of the influence of solute-driven convection at time t 3 -continuous

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Radial profiles of Sn segregation (ω -ω )/ω at 2 heights in the remelted ingot.

Fig. 13 .

 13 Fig. 13. Radially-averaged longitudinal profiles of Sn segregation (ω -ω )/ω in the

Fig. 14 .

 14 Fig. 14. Radially-averaged longitudinal profiles of chemical segregation (ω -ω )/ω in the

Fig. 15 .

 15 Fig. 15. Influence of the dendrite arm spacing 𝜆 on the computed Sn macrosegregation map in

Fig. 16 .

 16 Fig. 16. Magnitude of the liquid velocity in the mushy zone as a function of liquid fraction at

Fig. 17 .

 17 Fig. 17. Radial profiles of Sn segregation (𝜔 -𝜔 )/𝜔 at 2 heights in the remelted ingot.

Fig. 18 .

 18 Fig. 18. Computed Sn content map in the remelted ingot, (a) case D, (b) case E, (c) case F and (d)

FIG. 1 . 5 FIG. 2 . 8 FIG. 3 . 9 FIG. 4 . 22 FIG. 5 . 23 FIG. 6 . 25 FIG. 7 . 27 FIG. 8 . 29 FIG. 9 .FIG. 10 .FIG. 11 .FIG. 12 .FIG. 13 .FIG. 14 .FIG. 15 .

 1528394225236257278299101112131415 FIG. 1. SCHEMATIC REPRESENTATION OF A VAR FURNACE. .......................................................................................................................... 5 FIG. 2. MELTING RATE OF THE FULL-SCALE ZY4 REMELTING TRIAL. ........................................................................................................... 8 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. ............................................................................................... 9FIG. 4. DENSITY OF ZR AS A FUNCTION OF TEMPERATURE, FROM[26] (THE VERTICAL BAR CORRESPONDS TO ZR MELTING POINT). .................................................................................................................................................................................... 22FIG. 5. DENSITY OF SN AS A FUNCTION OF TEMPERATURE, FROM[27] (THE VERTICAL BAR CORRESPONDS TO SN MELTING POINT). .................................................................................................................................................................................... 23 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 T 1 -ALTERNATED STIRRING, LOW AZIMUTHAL VELOCITY. (B) TIME T 2 -ALTERNATED STIRRING, HIGH AZIMUTHAL VELOCITY. (C) TIME T 3 -CONTINUOUS STIRRING, HIGH AZIMUTHAL VELOCITY. . 25 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: ΒT = 0, ΒSSN = 0 , VMAX = 2.03 CM.S-1. (B) CASE B: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 0, VMAX = 1.60 CM.S-1. (C) CASE C: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = -1 ⋅ 10 -2 WT% -1 , VMAX = 1.82 CM.S-1. (D) CASE D: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 3 ⋅ 10 -2 WT% -1 , VMAX = 1.61 CM.S-1. ............................................................................................... 27 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: ΒT = 0, ΒSSN = 0, VMAX = 2.53 CM.S-1. (B) CASE B: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 0, VMAX = 2.53 CM.S-1. (C) CASE C: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = -1 ⋅ 10 -2 WT% -1, VMAX = 2.50 CM.S-1. (D) CASE D: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 3 ⋅ 10 -2 WT% -1, VMAX = 2.51 CM.S-1. ................................................................................................ 29 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 REPRESENTED BY VECTORS (POOL UPPER PART) OR STREAMLINES (POOL LOWER PART AND MUSHY ZONE) AND THE LIQUID FRACTION MAP. (A) CASE A: ΒT = 0, ΒSSN = 0, VMAX = 2.02 CM.S-1. (B) CASE B: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 0, VMAX = 1.93 CM.S-1. (C) CASE C: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = -1 ⋅ 10 -2 WT% -1, VMAX = 1.83 CM.S-1. (D) CASE D: ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 3 ⋅ 10 -2 WT% -1, VMAX = 2.00 CM.S-1. ................................................................................................ 30

Table 1 .

 1 Main constitutive equations of the CFD macromodel

	191
	Averaged total mass balance
	𝜕 𝜕𝑡	(𝜌 ) + 𝛻 ⋅ (𝜌 𝑔 𝒖 𝒍 ) = 0

𝒍 ℎ = 𝛻 ⋅ [(𝜆 + 𝑔 𝜆 )𝛻𝑇 ] (3) Averaged momentum balance for the liquid phase: fluid flow 𝜕 𝜕𝑡 (𝜌𝑔 𝒖 𝒍 ) + 𝛻 ⋅ (𝜌𝑔 𝒖 𝒍 𝒖 𝒍 ) = -𝑔 𝛻𝑝 * + 𝛻 ⋅ [(𝜇 + 𝜇 )𝑔 (𝛻𝒖 𝒍 + (𝛻𝒖 𝒍 ) )] -

Table 2 .

 2 Main equations of the solidification micromodel.

	Geometrical relations and surface area
	𝑅 * =	3𝑔 4𝜋𝑁	𝑅 =	4𝜋𝑁 3	𝑆 * = 4𝜋𝑅 * 𝑁 (22)
	Dendrite tip kinetics
	Ω =	𝜔 * , -< 𝜔 > 𝜔 * , 1 -𝑘	𝑉 =	𝜕𝑅 * 𝜕𝑡	Γ = 𝜌 𝑆 * 𝑉	(23)
	Boundary layer thickness

Table 3 .

 3 Thermophysical properties of Zy4 used in the simulations

	Nominal compositions	Sn 𝜔	[wt%]	1.3
		F e 𝜔	[wt%]	0.2
		O	𝜔	[wt%]	0.1
		C r 𝜔	[wt%]	0.1
	Melting temperature of pure Zr		T	[°C]	1855
	Liquidus slopes	Sn 𝑚	[°C(wt%) -1 ]	-11
		F e 𝑚	[°C(wt%) -1 ]	-57
		O	𝑚	[°C(wt%) -1 ]	147
		C r 𝑚	[°C(wt%) -1 ]	-26
	Partition Coefficients	Sn k	[-]	0.27
		F e k	[-]	0.25
		O	k	[-]	2.32

Table 4 .

 4 Definition of the four simulations run to study the effect of thermo-solute convection

	case 𝛽 [°𝐶 ]	𝛽 [𝑤𝑡% ]
	A	0	0
	B	4 ⋅ 10	0
	C	4 ⋅ 10	-1 ⋅ 10
	D	4 ⋅ 10	3 ⋅ 10
	4.1

.2 Characteristics of the liquid metal pool

Table 6 .

 6 Definition of the four simulations run to study the effect of grain growth kinetics

	case 𝑁 [m ]	𝐷 [m /𝑠]

Table 7 .

 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.

			t ls = 10 s	t ls = 1260 s)
		𝐹𝑜	𝐹𝑜	𝐹𝑜	𝐹𝑜
	D	1.2	0.006	150	0.75
	E	2.6	0.127	3270
	F	2.6	10 -5	3270	0.0016
	G	1.2	6 10 -7	150	7.5 10 -5

TABLE 4 .

 4 ................................................ 35 FIG. 13. RADIALLY-AVERAGED LONGITUDINAL PROFILES OF SN SEGREGATION (ΩSN -Ω0SN)/Ω0SNIN THE REMELTED INGOT. EXPERIMENTAL (•) AND MODEL (-) RESULTS. LETTERS A, B, C, D CORRESPOND TO CASES LISTED IN TABLE 4. ................... 36 FIG. 14. RADIALLY-AVERAGED LONGITUDINAL PROFILES OF CHEMICAL SEGREGATION (ΩN -Ω0N)/Ω0N IN THE REMELTED INGOT FOR ELEMENTS SN, FE, O AND CR. EXPERIMENTAL (•) AND MODEL (-) RESULTS. CASE D -ΒT = 4 ⋅ 10 -5 °C -1, ΒSSN = 3 ⋅ 10 -2 WT% -1. ........................................................................................................................................................... 37 FIG. 15. INFLUENCE OF THE DENDRITE ARM SPACING 𝜆𝐷𝐴𝑆 ON THE COMPUTED SN MACROSEGREGATION MAP IN THE REMELTED INGOT. (A) CASE D': 𝜆𝐷𝐴𝑆 = 67 µM. (B) CASE D: 𝜆𝐷𝐴𝑆 = 134 µM. (C) CASE D'': 𝜆𝐷𝐴𝑆 = 268 µM. ............................. 39 FIG. 16. MAGNITUDE OF THE LIQUID VELOCITY IN THE MUSHY ZONE AS A FUNCTION OF LIQUID FRACTION AT TIME T3 RADIAL PROFILES OF SN SEGREGATION (𝜔𝑆𝑛 -𝜔0𝑆𝑛)/𝜔0𝑆𝑛 AT 2 HEIGHTS IN THE REMELTED INGOT.

(CONTINUOUS STIRRING STAGE). ...................................................................................................................................................... 40 FIG. 17. EXPERIMENTAL (•) AND MODEL RESULTS. LETTERS D, D' AND D'' CORRESPOND TO CASES LISTED IN TABLE 5. .................. 41 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

List of tables

TABLE 1 .

 1 MAIN CONSTITUTIVE EQUATIONS OF THE CFD MACROMODEL 1 1

TABLE 2 .

 2 MAIN EQUATIONS OF THE SOLIDIFICATION MICROMODEL. 17

TABLE 3 .

 3 THERMOPHYSICAL PROPERTIES OF ZY4 USED IN THE SIMULATIONS 1 9

TABLE 4 .

 4 DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF THERMO-SOLUTE CONVECTION 2 3

TABLE 5 .

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

TABLE 6 .

 6 DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF GRAIN GROWTH KINETICS 4 1

TABLE 7 .

 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.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 (m 2 .s -1 )
	F el	electromagnetic force (N)
	g	gravitational acceleration (m.s -2 )
	g l	volume fraction of the liquid phase
	g s	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 )
	J ji	interfacial species transfer rate per unit volume due to diffusion (kg.m-3 .s -1 )
	J Гi	interfacial species transfer rate per unit volume due to the phase change (kg.m -3 .s -1 )
	k	turbulent kinetic energy (m 2 .s -2 )
	K	permeability (m 2 )
	K 0	permeability constant of the Blake-Kozeny law (m 2 )
	N 0	nuclei density (m -3 )
	N g	grain density (m -3 )

Table 1 .

 1 Main constitutive equations of the CFD macromodel

	191
	Averaged total mass balance
	𝜕 𝜕𝑡	(𝜌 ) + 𝛻 ⋅ (𝜌 𝑔 𝒖 𝒍 ) = 0

  13) Production rate 𝑃 due to gradients of 𝑢 and dissipation rate 𝐷 due to liquid-solid

	𝐺 = 𝛽 𝑔	𝜇 𝑃𝑟	𝒈 ⋅ 𝛻𝑇	𝐺 =	𝛽 𝑔	𝜇 𝑆𝑐	𝒈 ⋅ 𝛻𝜔	(15)
	Macroscopic 𝑘 -𝜀 model constants
	𝑃𝑟 = 0.85				𝜎 = 1.0	𝑐 = 0.09	𝑐 = tanh	|𝒖 𝒍 , 𝒛 |
	𝑆𝑐 = 0.7				𝜎 = 1.3	𝑐 = 1.44			𝒖 𝒍 , 𝜽 + 𝒖 𝒍 , 𝒓
						𝑐 = 1.92
	interaction
	𝑃 = -𝜌𝑔 𝒖 𝒊 𝒖 𝒋	𝜕 𝜕𝑥	(𝑔 𝑢 )	𝐷 = -	2𝜇𝑔 𝐾	𝑘	(14)
	Generation rate due to thermal and solutal buoyancy

Table 3 .

 3 Thermophysical properties of Zy4 used in the simulations

	Nominal compositions	Sn 𝜔	[wt%]	1.3
		F e 𝜔	[wt%]	0.2
		O	𝜔	[wt%]	0.1
		C r 𝜔	[wt%]	0.1
	Melting temperature of pure Zr		T	[°C]	1855
	Liquidus slopes	Sn 𝑚	[°C(wt%) -1 ]	-11

Table 4 .

 4 Definition of the four simulations run to study the effect of thermo-solute convection

	case 𝛽 [°𝐶 ]	𝛽 [𝑤𝑡% ]
	A	0	0
	B	4 ⋅ 10	0
	C	4 ⋅ 10	-1 ⋅ 10
	D	4 ⋅ 10	3 ⋅ 10

Table 6 .

 6 Definition of the four simulations run to study the effect of grain growth kinetics

	case 𝑁 [m ]	𝐷 [m /𝑠]

Table 7 .

 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.

			t ls = 10 s	t ls = 1260 s)
		𝐹𝑜	𝐹𝑜	𝐹𝑜	𝐹𝑜
	D	1.2	0.006	150	0.75
	E	2.6	0.127	3270
	F	2.6	10 -5	3270	0.0016
	G	1.2	6 10 -7	150	7.5 10 -5

  ..... 39 FIG. 16. MAGNITUDE OF THE LIQUID VELOCITY IN THE MUSHY ZONE AS A FUNCTION OF LIQUID FRACTION AT TIME T3 (CONTINUOUS STIRRING STAGE). ...................................................................................................................................................... 40 FIG. 17. RADIAL PROFILES OF SN SEGREGATION (𝜔𝑆𝑛 -𝜔0𝑆𝑛)/𝜔0𝑆𝑛 AT 2 HEIGHTS IN THE REMELTED INGOT. EXPERIMENTAL (•) AND MODEL RESULTS. LETTERS D, D' AND D'' CORRESPOND TO CASES LISTED IN TABLE 5. .................. 41 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

	List of tables

TABLE 1 .

 1 MAIN CONSTITUTIVE EQUATIONS OF THE CFD MACROMODEL 1 1

TABLE 2 .

 2 MAIN EQUATIONS OF THE SOLIDIFICATION MICROMODEL. 17

TABLE 3 .

 3 THERMOPHYSICAL PROPERTIES OF ZY4 USED IN THE SIMULATIONS 1 9

TABLE 4 .

 4 DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF THERMO-SOLUTE CONVECTION 2 3

TABLE 5 .

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

TABLE 6 .

 6 DEFINITION OF THE FOUR SIMULATIONS RUN TO STUDY THE EFFECT OF GRAIN GROWTH KINETICS 4 1

TABLE 7 .

 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.

Unfortunately, values for the solutal and thermal expansion coefficient for Zy-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.

wt% , v max = 1.82 cm.s -1 . (d) Case D: β = 4 ⋅ 10 °C , β = 3 ⋅ 10 wt% ,

Acknowledgments

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

Acknowledgments

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

Conflict of interest

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

Conflict of interest

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