AC & DC Magnetic Levitation and Semi-Levitation Modelling

International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, March 24-26, 2003 AC & DC Magnetic Levitation and Semi-Levitation Modelling V. Bojarevics, K. Pericleous Abstract This work presents computational analysis of levitated liquid thermal and flow fields with free surface oscillations in AC and DC magnetic fields. The volume electromagnetic force distribution is continuously updated with the shape and position change. The oscillation frequency spectra are analysed for droplets levitated against gravity in AC and DC magnetic fields at various combinations. For larger volume liquid metal confinement and melting the semi-levitation induction skull melting process is simulated with the same numerical model. Applications are aimed at pure electromagnetic material processing techniques and the material properties measurements in uncontaminated conditions. Introduction Magnetic levitation is a well known technique for treating liquid metals without or with an absolute minimum contact at crucible solid walls [1-4]. AC magnetic field can be used to support, stir, heat, melt and evaporate electrically conducting liquid at high temperatures. DC magnetic field can be added to this in order to stabilize [5] the liquid motion, improve the heat-mass transfer characteristics or even to levitate a poorly conducting liquid material [6]. These techniques allow for reactive high temperature materials treatment, the material properties measurements [7] and energy saving processes development. Typically the material is solid at the beginning of the process, then it heats up owing to the induced currents Joule heating, gradually melts and assumes the shape dictated by the balance of forces acting on the liquid volume. The numerical modelling for the magnetic semi-levitation involving coupling of the electromagnetic field, liquid shape change, fluid velocities and the temperature field during the time evolution was introduced in [8].The generated fluid flow easily becomes turbulent and the 2-equation k-omega time dependent turbulence model [9] is used to predict the efficient viscosity and turbulent heat transport. The simulation model is validated against measurements of velocity and turbulent kinetic energy in the cylindrical liquid metal experiment [9] and measurements of temperature and heat losses in the commercial size cold-crucible [10]. The same model can be applied for small levitated droplets to test skin-layer approximation accuracy [11] and to predict surface oscillations for ideal fluid, laminar viscous and with dynamically varying effective turbulent viscosity. 1. Mathematical Model The present modelling approach is based on the turbulent momentum and heat transfer equations for an incompressible fluid: 1 T 1 t v + ( v ) v = ρ p + ( ν e ( v + v )) + ρ f + g, (1) v = 0, (2) 99

* 1 2 C p ( tt + v T ) = ( C pα e T ) + ρ J / σ, (3) where v is the velocity vector, p - the pressure, ρ - the density, ν e = ν T + ν (summ of turbulent and laminar viscosity) is the effective viscosity which is variable in time and position, f is the electromagnetic force, g - the gravity vector, T - the temperature, α e = α T + α (summ of turbulent and laminar) is the effective thermal diffusivity, C p - the specific heat, C * p - the solid fraction modified specific heat function which accounts for latent 2 heat effects (see [8] for details), and J / σ is the Joule heat. The momentum equation (1) contains the nonlinear term in the convective (in difference to the rotational) form which, according to our tests, gives greater stability for the long time development problems. The boundary conditions used for the fluid flow problem are: at the free surface normal stress is compensated by the surface tension only, tangential stress is zero, and the kinematic condition which states that the new interface location moves with material fluid particles. The no-slip condition is applied for the velocity at solid walls (if any). For the temperature boundary conditions we adopt the radiation and the effective heat transfer at solid walls. The respective expressions in co-ordinate representation are given in [8]. The temperature boundary conditions depend on the local turbulent thermal diffusion coefficient α e, which is proportional to the effective turbulent viscosity ν e determined from the numerical k-ω turbulence model [9]. The k-ω model used is a low Re number version which resolves the flow from laminar to developed turbulent states, and therefore is considered suitable for turbulent flow evolution simulations. The ω variable is related to the reciprocal turbulent time scale (frequency of vorticity fluctuations) and the k variable is the turbulence kinetic energy per unit mass. In the present work we apply the k-ω model within the pseudospectral framework [9,11]. The computation follows in detail the time development of the turbulent characteristics determined by the coupled non-linear transport equations accounting for a continuous generation and destruction of the turbulent energy. For the Navier-Stokes and the heat transfer equations (1-3), and also the k-ω model equations, the pseudo-spectral spatial representation and the implicit time stepping with iterative linearisation for the non-linear terms is used [9]. The electromagnetic force distribution is highly sensitive to the shape of liquid metal free surface and is recalculated at every time step. The computational procedure for the electromagnetic field is implemented with the same grid as the fluid dynamic equations, which ensures a high resolution within the surface boundary layer because of the dense grid in this region. The electromagnetic force f is computed by the previously tested [8, 9] accurate integral equation based algorithm. In the AC case the time-average force in the fluid is concentrated in the skin layer and it is variable along the layer. This means that, however small the skin layer depth is, the force is always rotational and drives the fluid flow. 2. Modelling Results 2.1. AC and DC field Levitation in Normal Gravity For the tests of AC magnetic levitation under normal gravity conditions we chose the levitation coil configuration from the surface tension measurement experimental setup [12]. It is shown in Figure 1, where the additional DC coil is our modification to the original setup. Electric current of a fixed effective magnitude 195 A flows in the positive azimuthal direction in the bottom four turns of the coil and in the negative direction at the top two turns. The coil current frequency is f ν = 450 khz, therefore one can expect a very small penetration depth for the electromagnetic field in a well conducting material like liquid aluminium with the 100

electrical conductivity 6 1 σ = 3.85 10 ( Ω m) used for the tests. Indeed, the skin layer depth 1/ 2 3 according to the classical expression for a flat surface δ = ( µ 0σ 2πf ν ) = 0.27 10 m. For the levitated droplet of initial diameter 8 10 3 m, the penetration depth, which is numerically computed and analytically confirmed for the conducting sphere [8], is apparently larger, and it is quite non-uniform because of the coil configuration (Figure 2). From the Figure 2 we can see that the magnetic field behaves in a similar manner, being tangential over most of the external boundary (as expected from the asymptotic theories), yet there is a significant penetration into the material, and at the bottom part the field is entering normally into the droplet. There is also a closed loop of the secondary magnetic field lines enclosing the induced current in the metal. An external DC magnetic field can be added using a coaxial coil surrounding the AC coil as shown in Figure 1. Passing 200 A DC current in the 8 turn external coil creates an almost uniform additional DC magnetic field inside the droplet. T 960 959 959 959 958 958 957 957 Liquid levitation DC coil 0.01 0.005 0 Z (m) -0.005 AC coil -0.01-0.015-0.02 Figure 1. Perspective view of the AC levitation coil similar to the experimental [12] (a sectioned half of the full arrangement) and the additional DC coil. The computed typical velocity and the temperature field are shown at the end of 4 seconds physical time simulation. Magnetic field and electric current t=4. s 0.02 T J(A/m**2) 0.004 0.003 0.002 0.001 z 0-0.001-0.002-0.003-0.004 8.40E+06 2.64E+06-3.13E+06-8.89E+06-1.47E+07-2.04E+07-2.62E+07-3.20E+07-3.77E+07-4.35E+07-4.93E+07-5.50E+07 Figure 2. The typical instantaneous induced AC electric current (level lines) and the magnetic field distribution (arrows) in the section of the deformed levitated Al droplet. 101

Let us consider the oscillations generated in the droplet, which is assumed initially of spherical shape and in molten quiescent state at an initial temperature of 700 o C for liquid aluminium. The droplet is positioned relative to the coil so that the initial total electromagnetic force balances the weight of the droplet. However, there is an initial transient time during which the droplet assumes the shape imposed by the force balance, and an intense fluid flow develops. The initial oscillations are damped significantly after some time, and a new non-decaying quasi-stationary oscillation pattern is established in 3-4 seconds time (Figure 3a). The final oscillation pattern is of a slightly distorted sinusoidal shape. The centre of mass for the droplet is subject to vertical oscillating motion affecting the flow and the surface oscillations. The final oscillation pattern does not show damping, which suggests a net energy transfer from the external field to the droplet mechanical motion. It is quite instructive to consider the numerically calculated Fourier power spectra for the simulated oscillations (Figure 3c). For the liquid aluminium properties the computed dominant frequency is 35.40 Hz, corresponding closely to the ideal fluid nearly spherical droplet Rayleigh capillary oscillation frequency 35.36 Hz of the l=2 mode [7,11,12]. The computed secondary frequency is 63.78 Hz resulting from nonlinear interaction in presence of the varying electromagnetic field, which is still very close to the l=3 mode Rayleigh frequency 68.48 Hz. There is also the translational motion frequency at 8.76 Hz for the electromagnetically excited droplet. The exact mechanism of the translational and normal mode oscillation interaction needs a further analysis. a) Amplitude (m) 0.0044 Al droplet oscillations d=8 mm, I=195 A, γ =0.94 0.004 R top b) Amplitude (m) 0.0044 0 1 2 3 4 t(s) Al droplet oscillations d=8 mm, I=195 A, γ =0.94 Idc=200 A 0.004 Idc=200 A c) Power 10-9 10-10 0 2 4 t(s) F ourier power spectra 10-8 R top Idc=20 0 A γ =0.94 50 100 150 f(h z) Figure 3. Oscillations of the levitated liquid Al droplet in normal gravity: a) by means of AC field only, b) with the additional presence of DC magnetic field, c) the computed Fourier power spectra. 102

When the DC magnetic field is added to the aluminium droplet oscillation in the presence of the same AC coil, the droplet stability is greatly enhanced, and the resulting oscillation amplitude is significantly reduced after a 2 second damping interval (Figure 3b). However for the case of 200 A DC in the coil, the oscillation still reaches a quasi-stationary state, and the power spectra in Figure 3c exhibit essentially the same frequencies as without the DC field. The remaining oscillation can be completely suppressed when 500 A current is supplied in the DC coil. The oscillations generated depend on the material properties of the liquid metal. When the surface tension coefficient in the numerical experiment is increased by a factor of two, the motion of the droplet is dominated by the centre of mass translational vertical oscillation. Apart from the oscillation mode interaction, there is a considerable influence also of the intense circulation flow (see Figure 1) consisting of two vortices, the intensity of which changes with the oscillation phase. The lower, smaller vortex is particularly affected by the bottom oscillation. The turbulent viscosity is mainly generated in this bottom part and then transported to the rest of the volume. The maximum magnitude for the time dependent turbulent viscosity is about 15-20 times the laminar value, and it greatly enhances overall flow stability by limiting the velocity magnitude to below 0.3-0.4 m/s. Attempts to simulate the flow with only the laminar viscosity failed when the flow velocities started to increase in a persistent continuous way. 2.2 Semi-Levitation in Cold Crucible (Induction Scull Melting ISM) In the ISM process the coil is positioned behind the water cooled sectioned copper walls (see Figure 4). If the initial charge is of cylindrical shape, there is usually a gap at the sidewall. After the melting starts the temperature is highest in the surface film zone and the heat is slowly penetrating the rather thick cylindrical charge in the radial direction. At the following few seconds the gravity driven flow is directed mainly downwards, waving and filling the bottom gap. After the initial stage, when the bottom is filled, the largest Joule heating concentration is shifted to the bottom part of the side skin layer, which is now closer to the source coil and the segmented wall induced electric currents. Subsequently the melting front progresses slowly radially inwards. The heat conduction coefficient within the solid part is small compared to the turbulent flow region, and the phase change from solid to liquid consumes significant energy. The bottom water-cooled wall takes away the heat efficiently when the bottom gap is filled. Therefore the optimisation for the coil and crucible design are of prime importance if the resulting superheat must be increased. At the final stage shown in Figure 4 the arrangement of the bottom made of a special Ti alloy permits rather good semilevitation containing the melt away from the side-walls. The electric current and the heating are induced in the side layer, from where the intense turbulent mixing distributes the heat. The wall adherent mushy zone damps the turbulence efficiently. No significant destruction of the turbulence occurs at the top free surface with the zero stress boundary condition. The flow and the temperature distribution changes dramatically when the electric current is switched off in preparation to pouring of the molten metal into a casting mould. See more detailed comparisons to experimental measurements in [10]. Conclusion The presented model calculations permit to analyse flow patterns, free surface behaviour and to estimate the energy consumption of the magnetically levitated and confined liquid metal. A detailed time history and the clear physical insight to the driving forces and the sequence of events are available from the simulations. The modelling procedure can be extended to other generally unsteady flows with the moving melting fronts and free surfaces. 103

t=718 s z 4 coil turns TIAl melting: coil current 6700 A and Ti bottom segmented crucible wall copper base 0.5 m/s Z Y T 1606 1602 1598 1594 1590 1586 1582 1578 1574 1570 1566 1562 1558 1554 1550 1546 1542 1538 X x Figure 4. Computed final quasi-steady stage of the ISM semi-levitation with AC coil similar to the experimental [10]. y References [1] E. Okress et al.: J. Appl. Phys., Vol. 23, 1952, pp. 545-552. [2] A. D. Sneyd and H. K. Moffatt: J. Fluid Mech., Vol. 117, 1982, pp. 45-70. [3] J. Szekely and E. Schwartz: Proc. Int. Symp. On Electromagnetic Processing of Materials, Nagoya, 1994, pp. 9-14. [4] C. H. Winstead, P. C. Gazzerro and J. F. Hoburg: Metall. Materials Trans., Vol. 29B, 1998, pp. 275-281. [5] J. Priede, G. Gerbeth, A. Mikelsons and Y. Gelfgat: Proc. 3 rd Int. Symp. on Electromagnetic Processing of Materials, Nagoya, 2000, pp. 352-357. [6] P. Gillon: Proc. 3 rd Int. Symp. on Electromagnetic Processing of Materials, Nagoya, 2000, pp. 635-640. [7] I. Egry, A. Diefenbach, W. Dreier and J. Piller: Int. J. Thermophys., Vol. 22, 2001, pp. 569-578. [8] V. Bojarevics, K. Pericleous and M. Cross: Metall. Materials Trans., Vol. 31B, 2000, pp. 179-189. [9] A. Bojarevics, V. Bojarevics, J. Gelfgat and K. Pericleous: Magnetohydrodynamics, Vol. 35, 1999, pp. 258-277. [10] V. Bojarevics et al.: Proc. 5 th Int. Conf. on Fundamental and Applied MHD, Ramatuelle, 2002, Vol. 2, pp. 77-82. [11] V.Bojarevics, K.Pericleous: Magnetohydrodynamics, Vol. 37, 2001, pp. 93-102. [12] R. F. Brooks and A. P. Day: Int. J. Thermophys., Vol. 20, 1999, pp. 1041-1050. Authors Dr. Bojarevics, Valdis Prof. Pericleous, Koulis School of Computing and Mathematics School of Computing and Mathematics University of Greenwich University of Greenwich 30 Park Row 30 Park Row London SE10 9LS, UK London SE10 9LS, UK E-mail: v.bojarevics@gre.ac.uk E-mail: k.pericleous@gre.ac.uk 104