Sergejs Spitansa,b,∗, Egbert Baakea, Bernard Nackea and Andris Jakovicsb
aInstitute of Electrotechnology, Leibniz University of Hannover, Hannover, Germany
bLaboratory for Mathematical Modelling of Environmental and Technological Processes, University of
Latvia, Riga, Latvia
Abstract.On account of ANSYS Classic, ANSYS Fluent and ANSYS CFX-Post external coupling a new approach for joined simulation of liquid metal flow, free surface dynamics and electromagnetic (EM) field in induction furnaces is developed. The model is adjusted for the case of EM levitation and extended on 3D consideration with application of standardk-ω Shear Stress Transport (SST) or précised Large Eddy Simulation (LES) turbulence description.
Calculated steady state free surface shapes of molten metal are compared to other models and experimental measurements in traditional and EM levitation induction furnaces. Calculated free surface dynamics of melt is compared to analytical estimation of free surface oscillation period.
Parameter studies performed in ICF and conventional EM levitation setup briefly illustrate capabilities of the model and demon-strate the influence of current, frequency, surface tension and viscosity on free surface dynamics and steady shape of the melt in 2D approximation. Finally, full 3D calculation of free surface dynamics in ICF usingk-ω SST and LES turbulence models is performed and the impact of turbulence model on meniscus is discussed.
Keywords: Numerical simulation, electromagnetic induction furnace, electromagnetic levitation, two-phase turbulent flow, free surface
1. Introduction
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.
∗Corresponding author: Sergejs Spitans, Institute of Electrotechnology, Leibniz University of Hannover, Wilhelm-Busch Str.
4, D-30167 Hannover, Germany. Tel.: +49 511 762 5116; Fax: +49 511 762 3275; E-mail: [email protected]. 1383-5416/14/$27.50 c 2014 – IOS Press and the authors. All rights reserved
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 [2,3].
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 simu-lated using simplified two-parameter turbulence models [4].
The first results of 3D numerical calculation of liquid droplet dynamics in a high DC magnetic field were published recently [5].
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 Cru-cible 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 [6]. Furthermore, the im-plemented model is adjusted for the case of EM levitation and can be used with précised LES turbulence description.
2. Formulation of the model
2.1. Assumptions of numerical model
Due to harmonic nature of EM field and induced eddy currents, the Lorentz force fLorcan be decom-posed into a steady and harmonic part that oscillates with double frequency
fLor =fLor+ ˜fLor· cos (2ωt + ϕ) , (1) 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 ˆω that shows the relation between the induced and external EM field, and magnetic Reynolds number Remthat gives the relation between EM field that is generated by the flow and external EM field
ˆω = ωσμ0r20 and Rem = σμ0r0v0. (2)
where σ is electric conductivity, μ0 – permeability of vacuum, r0 and v0 – characteristic length and velocity.
Combiningˆω and Remit can be shown that in typical case of induction furnace
ˆω/Rem = ωr0/v0 1, ω/ (2π) > 50 Hz, r0≈ 1 m, v0 ≈ 1 m/s (3) EM field generated on account of the flow is insignificant in comparison to induced EM field.
Fig. 1. Assumptions of developed model.
Assuming no free charge in the system and neglecting displacement currents (no EM wave radiation) the reduced Maxwell equation system (Fig. 1) in addition with reduced Ohms law (no EM field gener-ation 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
Re = r0vρ/η > 103 (4)
indicates on fully developed turbulent flow (ρ and η stand for density and dynamic viscosity of fluid), thus k-ω Shear Stress Transport (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(xi, yi, zi, 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 (∇ · n) nδγ (6)
where γ is surface tension coefficient, n is free surface normal and δγis Delta function that ensures that surface tension force is located only at free surface.
Table 1
Externally coupled EM and HD problems for numerical calculation of free surface dynamics of molten metal
2.2. Technical implementation of numerical model
Calculation of free surface dynamics of EM induced metal flow is arranged by means of ANSYS
Clas-sic 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 coordi-nates 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 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 melt free surface. Such technical trick allowed to ensure a stable calculation of free surface dynamics for the case of highly pronounced EM skin-effect.
(a) (b) (c) (d)
Fig. 2. Lorentz force (left), steady flow (right) and meniscus shape obtained by our 2D model in comparison to (a) – mea-surements [2] (circles – on the left) and calculation [9] (squares – on the right) in big ICF and (b), (c), (d) – experimental measurements of meniscus heights (ticks) above initial filling (horizontal line) in small ICF at three different power regimes.
By the end of HD calculation a new transient free surface state is obtained and written into a file. The recalculation of 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.
3. Verification of the model
3.1. Steady meniscus shape in Induction Crucible Furnace
The first verification of developed numerical approach is performed by comparison between experi-mental measurements of a steady state meniscus shape in big laboratory scale ICF [2], our 2D calculation results and results of 2D steady simulation of O. Pesteanu [10] (Fig. 2(a)).
Distribution of the Lorentz force (0–0.1 MN/m3), steady flow (0–25 cm/s) with characteristic two torroidal vortices and calculated meniscus shape in comparison to experiment [2] and 2D calculation [10] proves a good agreement.
The weak point of this verification is insignificant meniscus height in comparison to initial filling. Imprecision in geometry or material properties might lead to insensible deviations of meniscus shape in respect to initial filling that in the scale of meniscus height can reach several times.
Hence, experimental measurements of meniscus heights above initial filling (by E. Baake) in a small laboratory scale ICF were used for the next verification of the model. Due to the smaller furnace size and greater inductor turn linear density it was possible to achieve greater power densities that ensured prominent melt squeeze by the EM field and greater meniscus heights in respect to initial filling.
Calculated steady flow pattern (0–55 cm/s) and Lorentz force density distribution (0–2.5 MN/m3), as well as meniscus shape in comparison to the experimentally measured meniscus heights above initial flat free surface, are shown in Figs 2(b)–(d). Measurements are marked by the ticks, while the ticks vertical lines correspond for the experiment errors due to the turbulence and mean flow instability that caused free surface fluctuations.
The calculated meniscus is in good agreement with experiment and obtained discrepancies are below the measurement error. Meniscus heights are comparable to initial filling, however, these experiments are
unable to show the agreement between the real and calculated shape of meniscus. Therefore, experimen-tal measurements of highly pronounced meniscus shapes were used for the further model verification.
3.2. Steady 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 a 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. 3(a)).
On account of the symmetry of setup the EM calculation was performed only for one section consid-ering azimuthal inhomogeneity of EM field due to the sectioned crucible (Fig. 3(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 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 (Figs 3(c)–(g)) and approved accuracy of developed numerical approach.
3.3. 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
Ttheor= 2πr1/20 (λ1· g)−1/2· tan (λ1· h0/r0) , (7)
where r0 is crucible radius, h0is initial filling and λ1 = 3.83 – first zero of the Bessel function J1. 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. 4.
The dynamics of free surface profile (Fig. 5) sketches free surface regular oscillations and proves that the discrepancy between the numerically obtained oscillation period Tcalc = 0.68 s and analytical approximation Eq. (7) Ttheor= 0.676 s is less than 1%.
Moreover, the calculated instant free surface states are in good qualitative agreement with calculation from [8] (Fig. 5).
4. Capabilities of the model
4.1. Steady state droplet shape in conventional EM levitation
Fig. 3. Geometry of IFCC with sectioned crucible [6] (a) and mesh for EM calculation (b). As well as measured (points) [6] and calculated (bold line) steady free surface shapes, Lorentz force (on the left) and steady flow (on the right) for different initial fillings (dashed line) and power regimes (c)–(g).
fLor, kN/m3v, cm/s
t = 0.4 s t = 0.8 s t = 3.1 s t = 4.3 s
Fig. 4. 2D calculation results for Lorentz force density (left), flow pattern (right) and free surface dynamics at different time moments in big industrial ICF.
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 experimen-tally [9]. The laboratory-scale EM levitation furnace consisted of two coils that were fed with counter oriented alternate currents (Fig. 6(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 [4]. The results appeared to be in a good
Fig. 5. 2D calculation results for free surface profile dynamics and meniscus shape comparison to calculation [7].
(a) (b) (c)
Fig. 6. Geometry of Okress levitation melting furnace in our 3D model (a) and comparison between V. Bojarevics (b) and our simulation (c).
agreement with experimentally observed “spinning top” shape and indicated on a fully turbulent two torroidal vortex flow structure (Fig. 6(b)).
Particular levitation experiment was numerically reproduced by our model in 2D axisymmetric (Fig. 6(c)) and full 3D (Fig. 6(a)) consideration. 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 Okress et al. levitation experimental setup [9] (Fig. 6(a)) a series of steady state free surface calculations were performed in order to illustrate the effect of parameter change on the levitated drop. Experimental values of surface tension γ = 0.94 N/m, melt density ρ = 2300 kg/m3, AC frequency f = 9.6 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 (Fig. 7). 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.
The developed numerical model was used for determination of levitating aluminium droplet steady state regime in a horizontal single- and two-frequency field of a novel EM levitation melting setup. Ob-tained numerical results revealed specific conditions that must be considered in case of EM levitation in
Fig. 7. Steady state free surface shape, Lorentz force (0–0.35 MN/m3, on the left) and flow pattern (0–22 cm/s, on the right) calculated for EM levitation of molten sample in Okress experimental setup for different values of (a) – surface tensionη, (b) – inductor effective currentIef.
horizontal orthogonal EM field and appeared to be in a good agreement with experimental measurements and other models [12].
4.2. Parameter studies of 2D free surface dynamics in Induction Crucible Furnace
It is assumed that the crucible of big ICF [2] is filled with melt (zero initial velocity and flat free surface) and at t= 0 s inductor is instantly fed with AC current.
Greater Lorentz force density, flow velocity, free surface oscillation amplitude and period, as well as a quasi steady state meniscus height, is obtained for greater current values (Fig. 8(a)).
Greater AC frequency leads to a greater initial perturbation, because greater Lorentz force densities are acting on smaller masses due to a smaller skin-depth, and highly pronounced nonlinearity of free surface behavior that cannot be described with simplified analytical estimations (Fig. 8(b)). For the case of smaller frequency (greater EM field penetration depths) the steady state is achieved faster.
Meanwhile, free surface oscillation period at a small current value (small oscillation amplitude) and small AC frequency (reciprocal interaction between melt shape and EM field is less significant) appears to be in a good agreement with simplified analytical estimation (Ttheor= 0.39 s) from [8].
For the case of greater dynamic viscosity (quasi hydrostatic approximation) the less intensive steady flow and lower meniscus are obtained faster (Fig. 8(c)).
4.3. Calculation of 3D free surface dynamics in Induction Crucible Furnace
It is assumed that cylindrical crucible of big ICF [2] is filled with melt (zero velocity and flat free surface) and at t= 0 s inductor is instantly fed with AC current of Ief = 5 kA.
0.09 0.105 0.12 0.135 0.15 0.165 0.18 0.195 0 0.5 1 1.5 2 2.5 [t , s] [z , m] f = 50 Hz f = 385 Hz f = 4000 Hz 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 0.5 1 1.5 2 2.5 [t , s] [z , m] η = 100 Pa.s η = 1 Pa.s 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 0.5 1 1.5 2 2.5 [t , s] [z , m] Ief = 5 kA Ief = 4 kA Ief = 3 kA (a) (b) (c)
Fig. 8. Molten metal free surface oscillations on the symmetry axis of big ICF for the case of different (a) – effective currents Ief, (b) – current frequenciesf, (b) – dynamic viscosities η and all the rest of parameters held constant.
Fig. 9. Free surface point (r = 0 m) oscillations along z axis (axis of furnace) obtained with 2D and 3D models in big ICF with Ief= 5 kA.
The first 2 s of flow calculated with 2D and 3D models (Fig. 9) indicate on a similar behavior of free surface point oscillations on the symmetry axis. Moreover, the period of these damping free surface oscillations (T1 = 0.4 s) is in good agreement with analytical estimation (Ttheor = 0.39 s) from [8] according to Eq. (7).
The next 4 s correspond to the flow transition to a steady regime and development of turbulence and from t= 6 s the fully developed turbulent flow (0–0.6 m/s) starts to oscillate with periodical realloca-tion of upper and lower torroidal vortex. Free surface shape, EM field and flow reciprocal interacrealloca-tion contributes to meniscus regular staggering with constant period of T2 = 2.5 s (Fig. 9).
Higher meniscus height and maximal velocity values, as well as similar low frequency quasi steady state free surface oscillations with period of T = 2 s, are obtained also for the case of greater inductor current Ief = 7 kA (Fig. 10). For this free surface calculation, the same as for the previous one, Lorentz force was recalculated upon new free surface shape every 50 ms. The calculation was performed for 10 s of flow and took 3 months of computational time on 4 kernel machine (3.4 GHz, 8 Gb RAM).
4.4. Calculation of 3D free surface dynamics of melt in Induction Crucible Furnace with LES turbulence description
t = 10.2 s t = 12.3 s t = 13.9 s t = 15.4 s t = 17.4 s t = 19.8 s Fig. 10. Instantaneous flow patterns (0–0.75 m/s) and free surface shapes obtained with 3D transient calculation at different time moments for the case of big ICF operating atIef= 7 kA.
(a) (b) (c)
Fig. 11. Comparison between experiment (a) and numerical results using LES (b) andk-ω SST (c) turbulence models of a free surface shape at a quasi steady state in ICF.
(Fig. 11(c)) the smaller flow structures resolved with LES on account of dynamic pressure contribute to meniscus perturbations on a smaller spatial scale (Fig. 11(b)) and, as expected, are in better qualita-tive agreement with experimentally observed meniscus (Fig. 11(a)). Comparison of instantaneous flow patterns obtained with LES and k-ω SST models points on the difference between small scale turbulent structures resolved with LES and smooth Reynolds-averaged flow obtained with k-ω SST turbulence model.
5. 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 k-ω SST 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.
Profound numerical investigation of parameter influence in case of 2D EM levitation and 2D ICF, as well as 3D LES calculations of free surface dynamics in ICF with a more pronounced meniscus, form the further plans of research.
Using the developed approach it is planned to tailor the design of the novel levitation melting setup [11] 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.
Acknowledgments
Current research was performed with financial support of the ESF project of the University of Latvia, contract No. 2009/0138/1DP/1.1.2.1.2/09/IPIA/VIAA/004. The authors wish to thank the German Re-search Association (DFG) for supporting this study under the Grant No. BA 3565/3-1.
The authors would like to acknowledge the great scientist prof. Ovidiu Pesteanu (*1945–†2012) for his contribution and support in this research and Dr. Valdis Bojarevics for kindly sharing his simulation data of E.C. Okress et al. experiment.
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