Characterization of the thermal flux transferred by the plasma arc to the surface of the liquid bath in the Plasma Arc Melting Cold Hearth Refining process (PAMCHR)

of the liquid bath in the Plasma Arc Melting Cold Hearth Refining process (PAMCHR) Léa Décultot 1,2, Alain Jardy 1, Stéphane Hans 2, Emiliane Doridot 2, Jérôme Delfosse 3, Fabienne Ruby-Meyer4, JeanPierre Bellot1. 1. Institut Jean Lamour UMR CNRS 7198, LabEx DAMAS, Université de Lorraine, 2 allée André Guinier, Campus Artem, 54000, Nancy, France. 2. Aubert & Duval Les Ancizes – 63770, Les Ancizes Cedex BP1, France. 3. Safran Tech Rue des Jeunes Bois, Châteaufort, 78772, Magny-les-Hameaux, France. 4. MetaFensch – 109 rue de Thionville, 57270, Uckange, France.


Introduction
Hearth Refining (HR) followed by single VAR (Vacuum Arc Remelting) has been set-up as an alternative route to produced certified premium grade titanium alloys, as compared to the conventional route involving three successive VAR melts. Indeed, HR processes include a refining stage in a horizontal water-cooled crucible designed to easily eliminate Low Density Inclusions (TiN, TiO…) due to the long metal residence time that allow the dissolution of solid particles. Moreover, these processes remove readily the High Density Inclusions (WC particles…) by exploiting the density separation: the inclusions settle to the bottom of the refining crucible and are trapped in the mushy zone. HR basically stands for two different processes: Electron Beam Cold Hearth Refining (EBCHR) and Plasma Arc Melting Cold Hearth Refining (PAMCHR). The power used in the EBCHR process is generated by electron guns, which sweep rapidly the liquid surface to provide a quasi-stationary heat supply. These electron guns require a high vacuum, which favors the volatilization losses of alloy constituents with a high vapor pressure such as Al and Cr. Minimizing the volatilization is the main advantage of the PAMCHR process, which uses plasma torches as the heat source: the process is conducted under an atmosphere of inert gas, with an operating pressure lying between 0.4 bar and 1 bar. As compared to the EBCHR process, the determination of the heat distribution transfer to the bath is more complex in the PAMCHR case.
In 1997, Huang et al. [1] developed a model of the PAMCHR refining stage, assuming a Gaussian distribution of the heat transfer from the plasma arc applied to the bath surface. In 1999 Li et al. [2] developed a torch model for a reversed polarity plasma torch, and calculated a non-Gaussian distribution of the heat transfer: the predicted heat flux reached a maximum value 20 mm away from the torch center. Different studies [3][4][5] have discussed the plasma heat flux as the composition of three main distributions: a convective-diffusive contribution, an electric contribution and a radiative contribution. Nevertheless, more recently, a team from UBC [6,7] neglected the electric contribution and determined, based on experimental trials, that the heat transfer from the plasma plume could be approximated by the sum of two Gaussian inputs: one for the convective part and the second for the radiative contribution.
The present work focusses on the determination of the heat flux distribution transferred from the plasma plume to the metal bath under real conditions of the PAMCHR process. Experimental tests have been achieved in a pilot furnace, and a numerical model has been developed to simulate the trials: a comparison is carried out between experimental results and numerical ones in order to quantify the overall distribution. In addition, these trials and numerical model have been used to analyze the impact of Lorentz forces on the liquid flow.

Experimental method
A set of trials using thermocouple measurements has been achieved in a pilot furnace in Metafensch, a public research and development center. The trials mainly consisted in recording the temperature evolution during and after the melting of a Ti64 cylindrical part with a stationary plasma torch.
The dimensions of the Ti64 block are given in Figure 1a. The plasma torch power was set at 177 kW, and helium was used for the torch operation under a chamber pressure of 400 mbar. The metal part was instrumented with 11 K-type thermocouples at the depth of 40 mm (red line in Figure 1a). This depth was chosen to avoid any contact between the thermocouples and molten Ti64, which would destroy the thermocouples. The latter, numbered TC1 to TC11, were positioned along two perpendicular lines in order to assess the symmetry of temperature distribution: this layout is shown on Figure 1b. Two trials have been carried out, differing through the Ti64/copper contact ( Figure 2). In this way, electric current lines should lead to different self-induced magnetic fields, hence different Lorentz forces.

Experimental results
The total heating time was 50 s for the first trial and 54 s for the second one. In each case, the final shape of the liquid bath was characterized (Figures 3 and 4) and the evolution of the temperature measured by all thermocouples was recorded ( Figure 5). A slightly larger liquid bath radius is noticed for the second trial (see Figure 3). This increase can be caused by the longer time of heating by the plasma torch. This additional time seems to impact more the bath radius than its depth which is quite similar in both cases (see Figure 4). The evolution of the measured temperatures is in the same range of value for thermocouples positioned at the same distance from the torch plasma (see Figure 5). The small differences can be explained by: -An uncertainty in the alignment of the torch center and the center of the cylinder (for both trials).
-Again, the melting time is longer for the second trial, which increases the temperature.
However, we cannot notice any significant difference which could be attributed to a variation in the bath hydrodynamics caused by a different electrical behavior.

Figure 5. Temperature measured by the thermocouples for both trials
For this reason, an average of the two experimental results will be considered, in order to be compared to the numerical results. Thus, the final metal bath is assumed to exhibit a 15 mm depth and a 69 mm radius, for a torch heating time of 52 s. Figure

Model description
The 2D axisymmetric model uses the finite volume method. The heat diffusion equation (eq. 1) is solved to calculate the transient temperature response in the Ti64 cylinder. This equation also takes into account the melting of the material (eq. 2). Moreover, the Laplace equation (eq. 3) is solved by the model to compute the current density (eq. 4), self-induced magnetic field (eq. 5), and resulting Lorenz (or electromagnetic) forces (eq. 6).
Where: Figure 7 presents the meshing strategy: the test block is modeled in 2D (r,z) by a rectangular domain in which a regular mesh is typically composed of 2500 cells.

Top surface
The heat transfer at the top surface includes heat input from the plasma plume (φtorch (r) in eq. 7) and heat loss by thermal radiation (second term in eq. 7) toward the furnace walls. Where: The heat flux density φtorch (r) is the product of the thermal torch power t U I (where t is the efficiency of the torch) by the sum of three distributions φi(r) weighted by factors i.
The mathematical expressions of the convective-diffusive and electric contributions (see § 2) have been calculated by means of a simplified 2D modelling of the plasma jet flow on a surface. The shape of the convective-diffusive contribution (Figure 8a) was determined using the Reynolds analogy, which expresses the proportionality between the shear stress and convective heat transfer at the bath surface. The stagnation point of the shear stress under the plasma jet results in a nearly log-normal distribution for the convective-diffusive term.
Based on the thermodynamic equilibrium in the plasma plume, the electric distribution (Figure 8b) is proportional to the value of the mass flux density of helium, calculated by the 2D model of the plasma jet flow.
Furthermore, if we assume the plasma column geometry to be cylindrical, the shape of the radiative contribution ( Figure  8c) is deduced from the calculation of shape factors between the surface of the column which thermally radiates on a surface element of the bath.

Lateral and bottom surfaces
A simple Fourier condition is used to describe the heat transfer at these interfaces. The ambient temperature is Tamb = 20°C , and the value of the heat transfer coefficient is set to hwall = 10 W/m2/K.

Top surface
The top surface condition is the same for both trials. The current density reaches the bath surface through the impact of the plasma column assimilated to a cylinder with a radius rt.

Lateral and bottom surfaces (trial 2)
Lateral surface: Bottom surface: On the lateral wall, a copper crown (see Figure 1a) is placed between z = zinf and z = zsup.
Lateral surface: Bottom surface:

Results and discussion
The thermo-physical properties of Ti64 are compiled in Table 1 [8]. The density and specific heat are assumed to remain constant. However, the thermal conductivity is dependent of the temperature in the solid phase [9] and anisotropic in nature in the liquid phase: In the radial direction, it is artificially increased to take roughly into account the heat transfer by radial convection and movement induced by the blowing of the plasma at the surface. Table 1. Physical properties of Ti-6Al-4V

Heat transfer
As stated above, a comparison of the predicted bath profile at the end of melting with the experimental result (Figures 3  and 4) leads to the determination of optimal values for the efficiency t and the weighting coefficients i. With these "optimal" values (which cannot be reported here for reasons of confidentiality), the heat flux distribution at the surface of the block is shown on Figure 9: the shape of the curve is similar to the heat flux distribution proposed by Li et al. [2]. Experimental and simulated volumes of the pool match very well. However, some disparities are still apparent (see Figure   9 MATEC Web of Conferences 321, 10003 (2020) https://doi.org/10.1051/matecconf/202032110003 The 14 th World Conference on Titanium 10) between the numerically computed and experimentally recorded temperatures. The difference increases with the distance of the thermocouples from the impact of the torch. It may arise from a lack of knowledge in the boundary conditions after extinction of the plasma torch. Nevertheless, the "numerical" temperature at P1 and the shape of the bath, which are the least sensitive results to these boundary conditions, are close to experimental measurements. These results validate the φtorch(r) distribution, hence the values of i and t, which will be implemented in our future model of the PAMCHR process.

Electromagnetic stirring
The orientation and magnitude of the computed Lorentz forces are depicted for the first and second trial in Figure 11a and Figure 11b, respectively. The black line on these figures represents the liquid bath profile. Lorentz forces were similar in both trials: they are localized under the torch with a maximum value of 19 kN/m3 and decrease rapidly (2 kN/m3 at a distance of 15 mm). This similarity is responsible for the similar results observed in the paragraph Results comparison of the two trials. Unfortunately, it is not possible yet to conclude on the importance of electromagnetic stirring on the liquid bath flow. However this volume force can be compared with the gravitational force (buoyancy) by calculating the curls of these two forces. Indeed, such calculation reflects the recirculating motion created by a volume force. The ratio of these curls is presented on Figure 12. Lorentz forces are much larger than the buoyancy force just below the impact of the torch, therefore they cannot be neglected for a future computation of the overall fluid flow.

Conclusion and perspectives
Thermal results (temperature measurements and final bath shape) from trials carried out on a PAMCHR pilot furnace have been compared to numerical results of a simple 2D axisymmetric model, in order to determine the distribution of the heat flux transferred from the plasma jet to the metal surface under the conditions of the PAMCHR process. Thus, this heat flux distribution will be implemented in a thermo-hydrodynamic 3D model of the PAMCHR process developed in parallel in Nancy [9].
In addition, the model computes the Lorentz forces, to be compared to buoyancy in the liquid bath. Electromagnetic stirring is localized near the impact of the torch, however it cannot be neglected. A more comprehensive model of the trials should be achieved to take into account several phenomena such as hydrodynamics of the liquid bath, Marangoni shear stress and the blowing of the torch on the surface as well.

Nomenclature
Latin letters: Greek letters: magnetic field (T) α i weighed factors of a heat transfer contribution i