Induction furnaces that ensure contact less control of electromagnetic

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1 A new approach for coupled simulation of liquid metal flow, free surface dynamics and electromagnetic field in induction furnaces by Sergejs Spitans, Egbert Baake, Andris Jakovics Induction furnaces that ensure contact less control of electromagnetic (EM) stirring, molten metal free surface and temperature are widely applied in metallurgical industry. Requirements for the free surface shape and behavior are defined by the different tasks of particular technological processes. Induction furnaces that ensure contact less control of electromagnetic (EM) stirring, molten metal free surface and temperature are widely applied in metallurgical industry. Requirements for the free surface shape and behavior are defined by the different tasks of particular technological processes. For example, overheating temperatures for metal evaporation and coating applications can be obtained in case of EM levitation since there is no contact between the free surface that should be stabilized and the crucible [1]. Meanwhile, the dynamics of free surface might be complicated by the interaction between the free surface shape, EM field and flow, as well as notably unsteady due to the switch between the furnace power regimes, mean flow instability and turbulence. Since the control of free surface is significant for EM processing of metallic materials, the numerical models that consider free surface dynamics are in high demand. Advanced multiphysical processes like energy and mass transfer, crystallization and homogenization of alloying particles are calculated nowadays in 3D with fixed hydrostatic steady free surface shape and précised Large Eddy Simulation (LES) turbulence description [2]. Free surface dynamics of EM levitated melt, flow and energy transfer in 2D consideration, as well as crystallization processes with free surface behavior in EM induction furnaces were successfully simulated using simplified two-parameter turbulence models [3]. The first results of 3D numerical calculation of liquid droplet dynamics in a high DC magnetic field were published recently [4]. However, at the present moment there is no approach developed for 3D calculation of multiphysical processes in EM induction equipment with consideration of free surface dynamics and application of LES description for turbulent flow. The previous investigations revealed that in case of Induction Crucible Furnace (ICF) with two characteristic mean flow vortexes only the LES model gives comparable results to experimental measurements [2]. In this article a new general approach for coupled 3D simulation of liquid metal flow, free surface dynamics and EM field in induction furnaces of various designs is presented [5]. Furthermore, the implemented model is adjusted for the case of EM levitation and can be used with précised LES turbulence description [6]. ASSUMPTIONS OF THE NUMERICAL MODEL Due to harmonic nature of EM field and induced eddy currents, the Lorentz force f Lor can be decomposed into a steady and harmonic part that oscillates with double frequency f = f + f cos 2ωt+ ϕ Lor Lor Lor ( ) where ω is an angular frequency of harmonic EM field and φ is a phase. Because of much greater inertia times of melt in comparison to the alternate EM field timescale (ω/(2π) > 50 Hz), only the steady part of the Lorentz force is taken into account. Let us consider the non-dimensional frequency ŵ that shows (1) 79

Research & Development the relation between the induced and external EM field, and magnetic Reynolds number Re m that gives the relation between EM field that is generated by the flow and external EM

2 Research & Development the relation between the induced and external EM field, and magnetic Reynolds number Re m that gives the relation between EM field that is generated by the flow and external EM field 2 00 m 00 0 ˆω = ωσμ r and Re = σμ r v where σ is electric conductivity, μ 0 permeability of vacuum, r 0 and v 0 characteristic length and velocity. Combining ŵ and Re m it can be shown that in typical case of induction furnace EM field generated on account of the flow is insignificant in comparison to induced EM field. Assuming no free charge in the system and neglecting displacement currents (no EM wave radiation) the reduced Maxwell equation system in addition with reduced Ohms law (no EM field generation by the flow) is used for harmonic analysis with finite element method in ANSYS Classic and the Lorentz force distribution in melt at particular free surface shape is obtained. In the hydrodynamic (HD) part of calculation the Navier-Stokes equation for incompressible fluid is solved with finite volume method in ANSYS Fluent. In typical cases of ICF the Reynolds number Table 1: Externally coupled EM and HD problems for numerical calculation of free surface dynamics of molten metal (2) (3) Re = r v ρ/ η> 10 3 (4) 0 0 indicates on fully developed turbulent flow (ρ and η stand for density and dynamic viscosity of fluid), thus k-ω SST or LES turbulence model is used additionally. Volume of Fluid (VOF) numerical technique is applied for calculation of two-phase flow dynamics. In VOF technique the phase distribution is represented with a scalar volume fraction field F(x i, y i, z i, t). In particular case, F = 1 when mesh element contains primary phase (melt) and F = 0 when element contains secondary phase (air). Accordingly, when phase surface crosses element - 0 < F < 1. For phase dynamics the transport equation is solved F/ t+ v F= 0 (5) and free surface is reconstructed as isosurface of F = 0.5. Volume density of surface tension force is calculated as fγ = γ nnδ γ ( ) (6) where γ is surface tension coefficient, is free surface normal and δ γ is Delta function that ensures that surface tension force is located only at free surface. TECHNICAL IMPLEMENTATION OF NUMERICAL MODEL Calculation of free surface dynamics of EM induced metal flow is arranged by means of ANSYS Classic for EM calculation, ANSYS Fluent for two-phase flow calculation, ANSYS CFX for post-processing and their external coupler a batch file (Table 1). Initial free surface shape of molten metal, as well as every instant shape obtained with HD calculation, is written into a file. This file contains free surface keypoint (KP) numbers, KP coordinates and series of KP number sequences that indicate the order of free surface KP connection for definition of elementary polygons. Transferring free surface KPs and elementary polygons from CFX-Post to ANSYS Classic a self written filtering procedure is performed in order to avoid generation of degenerate surface polygons that have great edge length ratios and cause problems in ANSYS Classic volume mesher. Hence, filtered free surface consisting of elementary triangular non-degenerate areas is obtained and the finite element mesh for EM problem is constructed. Then the distribution of harmonic EM field is calculated for the fixed free surface shape. The coordinates of alloy mesh element centroids, as well as the values of Lorentz force density components, are retrieved and written into a file. In the beginning of the transient HD calculation the Lorentz force density is interpolated on the Fluent finite volume mesh and 80 heat processing

3 Fig. 1: Geometry of IFCC with sectioned crucible [7] (a) and mesh for EM calculation (b). As well as measured [7] ( ) and calculated ( ) steady free surface shapes, Lorentz force distributions (on the left) and steady flow patterns (on the right) for different initial fillings h ( ) and power regimes (c) h = 46 %, I ef = 3154 A; (d) h = 87 %, I ef = 3789 A; (e) h = 65 %, I ef = 1929 A; (f) h = 65 %, I ef = 2956 A; (g) h = 65 %, I ef = 3566 A used as mechanical momentum source in two-phase flow equations. Then the calculation of unsteady flow is performed for sufficiently small time interval for which the slight change of free surface shape can be considered insignificant for the Lorentz force distribution. Unphysical air acceleration due to inevitably diffused interface in VOF formulation is damped by regular air velocity reinitialization in a small distance from the free surface of the melt. Such technical trick allowed to ensure a stable calculation of free surface dynamics for the case of highly pronounced EM skin-effect. By the end of HD calculation a new transient free surface state is obtained and written into a file. The recalculation of the Lorentz force distribution upon the new free surface shape is performed further and the repeat of the whole calculation loop ensures fully automatic free surface dynamics computation in 2D or 3D consideration. MODEL CAPABILITIES Molten metal meniscus shape in Induction Furnace with Cold Crucible In Induction Furnace with Cold Crucible (IFCC), which is widely used for melting reactive metals for high purity castings, the melt is confined by EM field and abutted only upon the skull at the bottom of water-cooled crucible. Experimental measurements of aluminum melt free surface shape in industrial IFCC [7] were used for validation of the model. The furnace consisted of copper crucible wall divided on 26 sections with a short circuit ring in the lower part, unsectioned copper bottom and copper inductor with five turns (Fig. 1, a). On account of the symmetry of setup the EM calculation was performed only for one section considering azimuthal inhomogeneity of EM field due to the sectioned crucible (Fig. 1, b). Air gap of 1 mm was ensured between the melt, the bottom and the crucible walls due to the great electrical resistivity of the skull that appears in the contact regions between the melt and water-cooled crucible. Meanwhile, the HD calculation was performed on a mesh with one element resolution of section in azimuthal direction that ensured azimuthal averaging of the Lorentz force for the 2D axisymmetric approximation. The comparison between the measured [7] and calculated 81

Research & Development free surface shapes of molten aluminum in IFCC at different initial fillings and power regimes revealed a fine correlation between the model prediction and experiment (Fig.

4 Research & Development free surface shapes of molten aluminum in IFCC at different initial fillings and power regimes revealed a fine correlation between the model prediction and experiment (Fig. 1, c-g) and approved accuracy of developed numerical approach. Free surface dynamics of melt in Induction Crucible Furnace The oscillation period of molten metal free surface in axisymmetric ICF is estimated analytically [8]. In this small amplitude approximation the Lorentz force is considered radial and constant and the typical oscillation period T is fully dependent on the crucible geometry (1) ( ) ( ) theor 2 12 / 12 / T = πr λ g tan λ h / r (7) where r 0 is crucible radius, h 0 is initial filling and λ 1 = 3.83 first zero of the Bessel function J 1. In order to verify the free surface dynamics predicted by our model an industrial type ICF adopted from [8] with crucible that is already filled with molten aluminum is considered and it is assumed that at initial time moment of t = 0 s the furnace instantly reaches its operating state. 2D transient calculation results for Lorentz force density distribution, developing flow pattern and free surface shapes at particular time moments are shown in Fig. 2 (video_1 - QR Code). The dynamics of free surface profile (Fig. 3) sketches free surface regular oscillations and proves that the discrepancy between the numerically obtained oscillation period T calc = 0.68 s and analytical approximation (7) T theor = s is less than 1%. Moreover, the calculated instant free surface states are in good qualitative agreement with calculation from [8] (Fig. 3). Parameter studies for conventional EM levitation For EM processing of metallic materials at great temperatures and high purity a contactless method of EM levitation melting is known to be appropriate since older times. For instance, the behavior and conditions of EM levitated molten aluminum sample (m = 21.5 g) have been investigated experimentally [9]. The laboratory-scale EM levitation furnace consisted of two coils that were fed with counter oriented alternate currents (Fig. 4, a). A numerical simulation of particular experiment in 2D axisymmetric consideration has already been performed by V. Bojarevics et. al. using a self written software [3]. The results appeared to be in a good agreement with experimentally observed spinning top shape and indicated on a fully turbulent two torroidal vortex flow structure (Fig. 4, b). Particular levitation experiment was numerically reproduced by our model in 2D axisymmetric (Fig. 4, c) and full 3D (Fig. 4, a) consideration (video_2 - QR Code). The comparison of literature data for the flow pattern, Lorentz force and droplet shape revealed a good correlation with our simulations. Using the developed 2D numerical model of E. C. Okress et. al. levitation experimental setup [9] (Fig. 4, a) a series of Youtube Videolink Video 1 Spitans, Baake, Jakovics Youtube Videolink Video 2 Spitans, Baake, Jakovics Fig. 2: 2D calculation results for Lorentz force density (on the left), flow pattern (on the right) and free surface dynamics at different time moments in big industrial ICF 82 heat processing

steady state free surface calculations were performed in order to illustrate the effect of parameter change on the levitated drop (Fig. 5). Experimental values of surface tension γ = 0.

The parameter studies performed clearly illustrate that for the case of conventional EM levitation in axisymmetric vertical EM field, the Lorentz force singularity is obtained on the symmetry axis.

Pursuing the interest of scaling-up the levitated charge it is reasonable to consider EM levitation in a horizontal field.

Pesteanu experimental measurements and his 2D steady simulation results of aluminum melt levitation in a single frequency EM levitation melting device [10].

5 steady state free surface calculations were performed in order to illustrate the effect of parameter change on the levitated drop (Fig. 5). Experimental values of surface tension γ = 0.94 N/m, melt density ρ = 2,300 kg/m 3, AC frequency f = 9.8 khz and inductor effective current Ief = 0.6 ka were used as a reference case. The parameter studies performed clearly illustrate that for the case of conventional EM levitation in axisymmetric vertical EM field, the Lorentz force singularity is obtained on the symmetry axis. The melt outflow and leakage can be hindered in this lowest point on the axis of a levitated sample only by the melt surface tension and therefore, the charge weight is limited. Pursuing the interest of scaling-up the levitated charge it is reasonable to consider EM levitation in a horizontal field. EM levitation in a horizontal single-frequency field The next step of developed model verification is based on O. Pesteanu experimental measurements and his 2D steady simulation results of aluminum melt levitation in a single frequency EM levitation melting device [10]. This EM levitation furnace consists of ferrite yoke and copper inductor with 16 turns. Quartz tube is placed in the air gap between yoke ends and inductor coils in order to prevent undesirable contact between the melt and furnace parts. With out of magnetic material teeth the position of EM levitated sample is unstable and it is pushed towards the quartz tube wall (Fig. 6, a). Because of that four magnetic teeth from FLUX- TROL are introduced for redistribution of magnetic field and stabilization of EM levitating sample (Fig. 6, b). In the beginning of the calculation liquid aluminum drop was given a spherical shape and zero velocity. It was placed a few millimeters above its experimentally observed position. The qualitative comparison between experiment photo and picture of numerical model reveals a good agreement for the droplet shape at a steady state (Fig. 7). The comparison between our 3D calculation results, experimental Fig. 3: 2D calculation results for molten aluminum free surface dynamics in big industrial ICF, as well as typical meniscus shape comparison to calculation [8] measurements of droplet positions and 2D steady calculation of O. Pesteanu are presented in Fig. 8. It can be noticed that the droplet shapes obtained with both models are in a good agreement with experiment. Alternate current in inductor generates alternate magnetic field that due to the great magnetic permeability of ferrite is mainly concentrated in the yoke. In the air gap region magnetic field lines spread and due to the skin effect (δ EM = 1.55 mm) flow around electrically conductive aluminium drop from one yoke end to another. In the regions where magnetic field lines are separating at the surface of the drop the minimum of Lorentz force is expected due to the small magnetic field component parallel to free surface. Meanwhile maximum of Lorentz force is expected at the bottom of the drop due to the greater field intensity and dominating field component along free surface. The following Lorentz force distribution (Fig. 8) contributes to the stretching of the drop along magnetic field lines. In some time curvature radius of droplet free surface where a) b) c) Fig. 4: Geometry of E. C. Okress levitation melting furnace in a 3D model (a) and comparison between V. Bojarevics (b) and simulation of ETP (c) 83

Research & Development Fig. 5: Steady state free surface shape, Lorentz force (0-0.

Okress experimental setup for different values of (a) - surface tension η, (b) - inductor effective current I ef a) Fig.

modified setup with introduced magnetic material teeth (in green) and stabilized molten aluminum droplet (b) a) Fig.

field line separation takes place becomes small enough and growing contribution of the surface tension effects will stop the droplet stretching.

field lines. In the meantime, the length of the droplet is limited due to the diameter of the quartz tube, distance between inductor coils or yoke ends.

6 Research & Development Fig. 5: Steady state free surface shape, Lorentz force ( MN/m 3, on the left) and flow pattern (0-22 cm/s, on the right) calculated for EM levitation of molten sample in E. C. Okress experimental setup for different values of (a) - surface tension η, (b) - inductor effective current I ef a) Fig. 6: EM levitation of solid aluminum cylinder that touches quartz tube walls in a single-frequency EM levitation melting setup without additional magnetic material teeth (a) and numerical model of modified setup with introduced magnetic material teeth (in green) and stabilized molten aluminum droplet (b) a) Fig. 7: Qualitative comparison between experimentally observed [11] (a) and numerically predicted (b) free surface shape of EM levitating drop in the single frequency EM levitation device b) b) magnetic field line separation takes place becomes small enough and growing contribution of the surface tension effects will stop the droplet stretching. For the greater volumes of EM levitated droplets the particular single-frequency horizontal EM field configuration will contribute to the droplet shapes that are significantly stretched along EM field lines. In the meantime, the length of the droplet is limited due to the diameter of the quartz tube, distance between inductor coils or yoke ends. In order to increase the EM levitated droplet weight, it is considered to install additional orthogonal horizontal EM field [11]. 3D numerical simulation of droplet (m = 30 g) flow and free surface dynamics in two-frequency levitation melting setup has been performed and a good agreement with experiment has been obtained [12]. Using the developed numerical approach the novel levitation melting setup that meets conditions for stable levitation of 1 kg of aluminum melt is being designed. CONCLUSIONS The new general approach for coupled 3D simulation of liquid metal flow, free surface dynamics and EM field is developed. It is adjusted for the case of EM levitation and can be used with LES turbulence approach. In the next step it is planned to supplement the model with calculation of energy transfer and crystallization. The comparison of our calculation results to experimental measurements and results of other models for the steady state free surface in induction furnaces and EM levitation melting setup, as well as comparison of free surface oscillation period to analytical estimation, revealed a good correlation and approved accuracy of our model. The new method for drip- and leakage-free EM levitation melting of metallic samples with greater weights and stabilized positions proposed by O. Pesteanu et. al. has been validated by our numerical model. Using the developed approach it is planned to tailor the design of the novel levitation melting setup and configuration of EM field in order to meet the conditions for stable EM levitation of industrial-scale molten metal charge and reproduce it in the laboratory experiment. ACKNOWLEDGEMENTS Current research was performed with financial support of the ESF project of the University of Latvia, contract No. 2009/0138/1DP/ /09/IPIA/ VIAA/004. The authors wish to thank the German Research Association (DFG) for supporting this study under the Grant No. BA 3565/ heat processing

7 The authors would like to acknowledge the great scientist Prof. Ovidiu Pesteanu (* ) for development of the novel technology of EM levitation in horizontal field, his contribution and support in this research and Dr. Valdis Bojarevics for kindly sharing his simulation data of E. C. Okress et. al. experiment. LITERATURE [1] Baptiste, L. et al. (2007): Electromagnetic levitation: A new technology for high rate physical vapour deposition of coatings onto metal strips. Surface & Coatings Technology, Vol. 202, [2] Kirpo, M. Modelling of turbulence properties and particle transport in recirculated flows. Ph.D. Thesis, University of Latvia, Riga, [3] Bojarevics, V., Harding, R., Pericleous, K., Wickins, M. (2004): The development and experimental validation of a numerical model of an induction skull melting furnace. Metallurgical and Materials Transactions B, Vol. 35, Fig. 8: 3D calculation results for the Lorentz force, flow pattern and free surface shape in comparison to O. Pesteanu 2D calculation and his experimental measurements [4] Easter, S., Bojarevics, V., Pericleous, K. (2011): Numerical modelling of liquid droplet dynamics in microgravity. Journal of Physics: Conference Series, [5] Spitans S., Jakovics A., Baake E., Nacke B. (2013): Numerical Modelling of Free Surface Dynamics of Melt in an Alternate Electromagnetic Field. Part I. Implementation and verification of model. Metallurgical and Materials Transactions B, Vol. 44 (3), [6] Spitans S., Jakovics A., Baake E., Nacke B. (2012): Numerical modelling of free surface dynamics of melt in an alternate electromagnetic field. Journal of iron and steel research international, Vol. 19, Suppl. 1/1, [7] Westphal, E. (1996): Elektromagnetisches und thermisches Verhalten des Kaltwand-Induktions-Tiegelofens. Dr-Ing. Dissertation, Dusseldorf, 21(210) [8] Hegewaldt, F., Buligins, L., Jakowitsch, A. (1993): Transient bath surface bulging at energization of an induction-type crucible furnace. Elektrowärme International, Vol. 1, [9] Okress, E. C., Wroughton, D. M., Comenetz, G., Brace, P. H., Kelly, J. C. R. (1952): Electromagnetic Levitation of Solid and Molten Metals. Journal of Applied Physics, Vol. 23, [10] Pesteanu, O., Baake, E. (2011): The multicell VOF method for free surface simu-lation of MHD flows. Part I: Mathematical model and Part II: Experimental verifications and results. ISIJ International, Vol. 51(5), [11] Pesteanu, O., Baake, E. (2012): New Method and Devices for Electromagnetic Drip and Leakage-Free Levitation Melting. ISIJ International, Vol. 52(5), [12] Baake, E., Spitans, S., Jakovics, A. (2013): New technology for electromagnetic levitation melting of metals. In the Proceeding of the International Conference on Heating by Electromagnetic Sources, Padua, Italy, (addendum) 1-8 AUTHORS Prof. Dr.-Ing. Egbert Baake Institute for Electrotechnology Leibniz Universität Hannover, Germany Tel.: +49 / (0)511/ baake@etp.uni-hannover.de Prof. Dr. Phys. Andris Jakovics Laboratory for Mathematical Modelling of Environmental and Technological Processes University of Latvia, Latvia Tel.: +371 / (0) andris.jakovics@lu.lv MSc. Phys. Sergejs Spitans Institute for Electrotechnology Leibniz Universität Hannover, Germany Tel.: +49 / (0)511/ spitans@etp.uni-hannover.de 85

Modelling of Levitation Melting using a Fixed Mesh Method

International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, October 27-29, 2008 Modelling of Levitation Melting using a Fixed Mesh Method D. Hectors, E. Toorman, K. Van Reusel