Free Surface Controlled by Magnetic Fields

Transcription

1 Review Free Surface Controlled by Magnetic Fields ISIJ International, Vol. 43 (2003), No. 6, pp Yves FAUTRELLE, Damien PERRIER and Jacqueline ETAY INPG/CNRS/EPM-Madylam, ENSHMG BP 95, St Martin d Hères Cedex, France. Yves.Fautrelle@inpg.fr (Received on November 20, 2002; accepted in final form on February 18, 2003 ) In metallurgy it is needed to master the shape or the motion of the liquid metal free surface. The present paper reviews some effects of a magnetic field on the behaviour of liquid metal free surfaces. First, the various effects associated with A.C. fields are presented, static dome shaping, dome oscillation, instability, emulsion. Then, the case of a DC magnetic field is presented. Such field is generally stabilising. Nevertheless, in particular case, its application may promote low frequency oscillations. Finally, the effects of magnetic field exhibiting more than one frequency are presented. All these cases are illustrated by photographies produced in the EPM-Madylam laboratory. KEY WORDS: electromagnetic processing; liquid metals; free surface motion; instabilities; free surface control. 1. Introduction When a liquid metal is submitted to a magnetic field, electromagnetic forces, called Lorentz or Laplace forces, can be created in the metal due to the interaction between the induced electric currents and the applied magnetic field. Those electric currents can be generated by three types of mechanisms: (i) application of an external voltage by the use of electrodes, (ii) time variation of the applied magnetic field, (iii) or by the motion of the liquid across the magnetic field lines. When the magnetic field is pulsating and according to its frequency f (which vanishes in the case of DC magnetic field), the electromagnetic forces generate various effects both on the bulk motion and at the free surface of the liquid metal. 1,2) DC magnetic fields generally exert a damping effect both on the bulk flow and on the free surface instabilities. However, that effect is not isotropic and depends on the orientation of the magnetic field with respect to the velocity. 2) In the present paper, we shall focus on some significant features of the behaviour of free surfaces submitted to electromagnetic fields, pointing out new or unexpected results. 2. Static Deformation in AC Magnetic Fields When the electric current is alternating, the electromagnetic forces comprise both a mean steady part and an alternating one. In the middle frequency range because of the fluid inertia, the alternating part of the electromagnetic forces has no significant effect on the liquid metal free surface provided that the magnetic field frequency is much larger than the free surface natural frequencies. However, the mean part of the electromagnetic forces is responsible for various phenomena. It generates a significant bulk fluid flow. 1 5) Furthermore, the mean Lorentz forces lead to free surface static deformations thanks to the well-known repulsion effect. The electromagnetic forces are able to levitate totally or partially a liquid metal blob located in a coil. 6) In classical single-phase induction furnaces, the dome formation has been widely shown both experimentally and numerically. 7) Its typical height h m evolves as the square of the applied magnetic field according to the following relation: B hm (1) 2µρg m, r, g and B 0 being respectively the permeability of the vacuum, the liquid metal density, the gravity and the typical magnetic field strength. Figure 1(a) shows a typical dome-shaped free surface obtained in a cold crucible induction furnace. 8) However, the dome shape may not always remain axisymmetric even in axisymmetric configurations, especially for large magnetic field amplitude. For example, for shallow liquid metal pools other types of static deformation may be obtained. 9) Highly non-symmetric steady shapes may be observed when the magnetic field amplitude increases. Figure 1(b) illustrates such phenomenon on a gallium layer in a 10 khz- AC magnetic field. In such a particular geometry, the force balance is such that both the path of the induced electric current as well as the interface area between the liquid metal and the surrounding atmosphere tend to increase. 3. Free Surface Motion under AC Magnetic Field The free surface behaviour (shape, oscillations, instabilities) varies with the value of the magnetic field amplitude and the applied frequency. The magnetic field strength is involved through the non-dimensional interaction parameter N such that ISIJ

2 Fig. 1. Photographs of the static deformation of a liquid metal free surface under the effect of a AC magnetic field, (a) classical stable dome shape on an aluminium pool for f 7.5 khz, 8) (b) static shape of a shallow gallium pool for f 10 khz. 9) Fig. 2. Top view of forced waves on the free surface of a mercury pool located inside a single phase coil, 10) the oscillation frequency is 2 f, (a) circular tank of radius 100 mm, (b) rectangular tank of dimension 120 mm. B N σ 0 2 2πρ f...(2) s and r being respectively the electrical conductivity of the liquid metal and its density. The parameter N may be considered as the ratio between the electromagnetic force and the liquid inertia. Regarding the frequency parameter, it is characterised by the screen parameter R w, i.e. the ratio between the typical length L of the considered pool and the thickness d of the electromagnetic skin depth R ω L 2 δ...(3) Various types of phenomena may be observed according the values of those two parameters Low Frequency Single-phase Magnetic Field It has been shown that very low frequency AC magnetic fields could generate motions at the free surface of a liquid metal when the natural frequencies of the interface are of the same order as the magnetic field frequency. 10,12) Moreover if the screen parameter is weak (in practice R w 1), the oscillating part of the electromagnetic forces is dominant and is responsible for two types of actions. 13) Firstly, it generates forced standing free surface waves on the free surface. Such waves have the same degree of symmetry as that of the force. For example, in an axisymmetric geometry, the 2 wave pattern is axisymmetric as well. Such types of forced waves are illustrated in Fig. 2. The second effect appears for higher amplitude magnetic fields, the interaction parameter N being of the order of one. The free surface becomes unstable and non-symmetric waves appear on the free surface. Such instabilities, which are illustrated in Fig. 3, mainly come from parametric resonance effects due to the alternating part of the Lorentz forces. 13) Note that, now, contrarily to the forced wave case, the free surface oscillates at the applied magnetic field frequency. For large magnetic field values, the free surface becomes highly agitated and liquid metal ejection may be observed (Fig. 3(a)). Coupling of electromagnetic free surface oscillations with gas stirring may also be achieved. It is observed that the presence of the magnetic field widens the gas plume coming from the nozzle, although the electromagnetic interface deformations are mainly localised near the pool side. Such an experiment is illustrated in Fig ) Large gas bubbles are formed beneath the mercury water interface. Then they cross the interface by entraining a thin mercury layer which breaks up in the water producing small mercury droplets. It must be emphasised that the pool aspect ratio is an important parameter in the present case too. For example, let us consider a flat liquid metal drop on a substrate located in a low-frequency vertical AC magnetic field. A strongenough magnetic field intensity may trigger free surface instabilities, 11) which consist mainly of horizontal motions 2003 ISIJ 802

3 Fig. 3. Free surface instabilities of a mercury pool located inside a single phase coil, 10,12) the oscillation frequency is f, (a) circular tank, (b) rectangular tank. viewed from above. Fig. 4. Electromagnetic interface oscillations of a mercury pool coupled with gas stirring; the 180 mm-diameter pool is located inside a single-phase coil and covered by a water layer; gas is injected by a nozzle located in the centre of the bottom wall; (a) scheme of the apparatus, (b) side view of the agitated mercury water interface; the large gas bubbles are covered with a thin mercury film which breaks up into droplets. 14) Fig. 5. Plane liquid metal layer viewed from above 11) ; the various pictures show the regime according to both the magnetic field amplitude and the applied frequency, the typical drop diameter is 60 mm, (a) mercury drop f 2.10 Hz, (b) mercury drop f 10 Hz, (c) gallium drop f 6.2 Hz. (Fig. 5). For large magnetic field amplitude, this motion may become highly turbulent (Fig. 5(b)) and even leads to the formation of an emulsion as illustrated in Fig. 5(c) Middle and High Frequency Magnetic Field However, free surface instabilities may also exist even if the magnetic field frequency is not very low. This has been shown analytically in the case of a liquid metal semi-infinite pool submitted to a parallel single-frequency AC magnetic field. 15,16) Increasing the frequency suppresses the instability of surface waves perpendicular to the applied magnetic field; however, increasing the magnetic field strength may enhance it. 16) That effect is illustrated in Fig. 6. It may be seen from Fig. 6 that the higher the magnetic strength, the easier the ability of the surface of the pool to be destabilised. Thus, when a medium-frequency magnetic field is used, for example to shape a liquid metal free surface or to levitate the liquid blob, the free surface may be subject to electromagnetic instabilities according to the frequency. This was for example, observed in cold crucible configurations for quasi-levitation. 8) This is illustrated by Fig. 7 where the shape of the free surface looks like daisy flower petals. 17) Note that the free surface is strongly unsteady, and the petals are rapidly moving ISIJ

4 3.3. Two-frequency Magnetic Field In some metallurgical refining processes, it may be interesting to generate free surface oscillations to enhance mass transfers. In continuous casting for example, free surface oscillations may replace mould vibrations, which are usually devoted to improve the quality of the cast metals by reducing the oscillation marks. 18,19) Single-frequency devices might be used, but multi-frequency magnetic fields are preferred, since they may simultaneously achieve a free surface control of the meniscus as well as a bulk stirring. The electric current comprises two frequencies, namely: ( i ) a very low frequency f 1 to generate resonant free surface oscillations, (ii) a middle frequency f 2 (generally f 2 f 1 ) to heat, to shape and to stir the liquid bulk. The amplitude of the middle frequency electric current is modulated by a periodic modulation function m(t) whose frequency f 1 is much lower than f 2. It is easy to show that the electromagnetic forces are also modulated, and we may write 16) : F m 2 (t)f 0,...(4) F 0 being the electromagnetic forces in the single frequency case when f 1 is zero. Fig. 6. Effect of a parallel AC magnetic field on a liquid metal free surface 16) ; maximum growth rate s of the wave perturbations versus the magnetic field pulsation w 2p f for three values of the non dimensional magnetic field strength M B 0 2 /(2mrgd); instability occurs when the growth rate is positive; instability disappears for the highest frequencies, but increasing the magnetic field strength M widens the unstable region. According to the shape of the modulation function m(t), it may possible to weight the various effects. The function m 2 (t) may be split into a time average part and an oscillating part. The time-average part of m 2 (t) controls the bulk stirring, whereas the r.m.s.-value of the oscillating part will act on the amplitude of the surface motion. It is also noteworthy to observe from Eq. (4) that the modulation of the forces actually contains two frequencies, namely f 1 and 2 f 1. Note that all the other quantities, i.e., the Joule power dissipated in the bath, are modulated in the same way. Thus, the time average of those quantities is lower than in the single frequency case, and we may expect a slight overall decrease of the efficiency of the other functions. Figure 8 comes from measurements performed on gallium by means of a contact probe. 20) Note that the height of the peak can be as high as the radius of the pool even for moderate magnetic field amplitudes. 4. Selective Damping of Free Surface Agitation by DC Magnetic Fields It is admitted that a DC magnetic field exerts a stabilising action on free surface waves. 21,22) This effect is related to the fact that as soon as the motion of the free surface crosses the magnetic field lines, electric currents are induced in the liquid metal. Accordingly, electromagnetic forces are developed in the metal acting against the generating motion. The free surface of inductively heated liquid metal is often unstable. Those fluctuations often come from the turbulent electromagnetic stirring as well as electromagnetic instabilities as discussed previously. In some metallurgical applications, it is desirable to have a quiet free surface. One way to solve that problem is to properly choose the magnetic field frequency. However, it may be more secure to superimpose a DC magnetic field to the AC component of the initial field. Recent experiments show that the free surface may be efficiently stabilised by the superimposition of a DC magnetic field. 23,24) That situation may be considered as an asymptotic two-frequency system, for which one of the frequencies is zero. We want to focus here on a case of importance where this stabilizing effect may be not strong enough to suppress all the fluctuations. 25) Figure 9 illustrates such a phenomenon. The experiments are performed on a 1/3-scale mercury Fig. 7. Daisy instabilities of a dome in a semi-levitation cold crucible 17) ; the liquid is a nickel-base alloy; the pool diameter is 60 mm and the electric current frequency is 30 khz; (a) scheme of the apparatus, (b) top view of the liquid metal pool ISIJ 804

5 Fig. 8. Maximum amplitudes (in mm) of the free surface oscillations for various modulation frequencies in an induction furnace, 20) the pool radius is 42 mm; the surface mode (0, n) identified by video recording are noted on the corresponding resonance peaks. The typical magnetic field strength is B mt, the modulation coefficient a depends on the frequency, for example a ( f Hz) 0.43 and a ( f 1 5 Hz) Fig. 9. Photographs of the free surface of mercury without and with a DC field in a cold model of a steel continuous caster. 25) model of a steel continuous caster where a horizontal transverse DC magnetic field is applied. In the absence of any magnetic field, the free surface is slightly disturbed, but various ripples are observed. However, in the presence of a DC magnetic field the free surface is quiet everywhere, but exhibits a single travelling wave. Depending on the experimental conditions the amplitude of the wave ranges between 2 and 5 mm and its celerity is 0.5 cm s 1. At the front of the wave, the shape of the free surface is strongly disturbed indicating high vorticity. This phenomenon has some similarity with the classical hydraulic jump observed in open channel flows. From a Fourier transform of the continuous signal registered by a resistive probe (see for example the right-hand-side of Fig. 9), the obtained spectra reveal two main features: in the absence of magnetic field we observe one main peak at the frequency 1.27 Hz which is the lowest natural frequency of the rectangular tank. the value of the frequency associated with this peak is lowered by the presence of the magnetic field. On the presented example the value of the main peak is 0.17 Hz. The mechanism of this phenomenon is related on one hand to the bidimensionalisation effect of the magnetic field which concentrates the kinetic energy in the large scale of ISIJ

6 the flow, on the other hand to the feedback effect of the upper recirculating flow on the flapping frequency of the feeding jets. 5. Conclusions The application of AC or DC magnetic fields is an elegant way to fulfil various metallurgical functions, such as heating, bulk and surface stirring, free surface control. However, electromagnetic devices may have some drawbacks due to secondary effects such as bulk turbulence, free surface instabilities. Thanks to the progress of the electric power sources, a new generation of electromagnetic devices is emerging. Those devices use complex electric currents, which are able either to correct some of the drawbacks or to superimpose various functions. Those devices involved multi-frequency electric power sources. The possibility to modulate one frequency with another one offers new expectations and likely new ideas. Acknowledgements Part of the presented work results from collaborations between the Japan Research Centre of Metal, USINOR, the Centre d Etudes Atomiques (VALRHO) and the EPM- Madylam laboratory. The authors are grateful to their interlocutors for financial support and stimulating discussions. REFERENCES 1) Y. Fautrelle: Proc. of the Int. Symp. on Electromagnetic Processing of Materials EPM 2000, ISIJ, Tokyo, (2000), ) R. Moreau: Magnetohydrodynamics, Kluwer Academic Pub., Dordrecht, (1990), ) S. Yoshitsuru: Kawasaki Steel Corp. Patent No. JP , (1989). 4) K.-H. Spitzer, G. Reiter and K. Schwerdtfeger: Proc. of the Int. Symp. on Electromagnetic Processing of Materials EPM 94, ISIJ, Tokyo, (1994), ) E. Taberlet and Y. Fautrelle: J. Fluid Mech., 159 (1985), ) A.D. Sneyd and H.K. Moffatt: J. Fluid Mech., 117 (1982), 45. 7) J. N. Barbier, Y. Fautrelle, J. W. Evans and P. Cremer: J. Méc. Theor. Appliquée, 1 (1982), No. 3, ) I. Leclerq: PhD Dissertation, Institut National Polytechnique de Grenoble, (1989). 9) M. Mekroum: Master Report, Institut National Polytechnique de Grenoble, (2001). 10) J. M. Galpin and Y. Fautrelle: J. Fluid Mech., 239 (1992), ) S. Daugan, Y. Fautrelle and J. Etay: Fluid Flow Phenomena in Metals Processing, TMS Publication, Warrendale, PA, (1999), ) F. Debray and Y. Fautrelle: Exp. Fluids, 16 (1994), ) J. M. Galpin, Y. Fautrelle and A. Sneyd: J. Fluid Mech., 239 (1992), ) F. Debray: PhD. Dissertation, Institut National Polytechnique de Grenoble, (1994). 15) Deepak and J. W. Evans: J. Fluid Mech., 287 (1995), ) Y. Fautrelle and A. Sneyd: J. Fluid Mech., 375 (1998), ) Y. Boussant-Roux: PhD Dissertation, Institut National Polytechnique de Grenoble, (1990). 18) T. Li, K. Sassa and S. Asai: Proc. of the Int. Symp. on Electromagnetic Processing of Materials EPM 94, ISIJ, Tokyo, (1994), ) E. Takeuchi and K. Miyazawa: Proc. of the Int. Symp. on Electromagnetic Processing of Materials EPM 2000, ISIJ, Tokyo, (2000), ) D. Perrier, Y. Fautrelle, J. Etay, R. Piccinato and R. Boen: Proc. of the 4th PAMIR International Conf. on MagnetoHydroDynamic at Dawn of Third Millennium, A. Alemany, (2000), ) P. Rivat, J. Etay and M. Garnier: Eur. J. Fluid Mech., 10 (1991), No. 5, ) C. Kazemi and J. Etay: CRASc, t.320, Série Iib, Elsevier, Paris, (1995), ) J. M. Galpin, J. Y. Lamant, P. Gardin, M. Anderhuber, J. Etay and Y. Delannoy: CAMP-ISIJ, 14 (2001), ) A. Cramer, A. Boyarevics, G. Gerbeth and Y. Gelfgat: Fluid Flow Phenomena in Metals Processing, TMS Publication, Warrendale, PA, (1999), ) J. Etay and Y. Delannoy: Proc. of the 5th PAMIR Int. Conf. on Fundamental and Applied MHD, A. Alemany, (2002), II ISIJ 806