Stability of Liquid Metal Interface Affected by a High-Frequency Magnetic Field

Transcription

1 International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, March 4-6, 3 Stability of Liquid Metal Interface Affected by a High-Frequency Magnetic Field J-U Mohring, Ch Karcher, D Schulze Abstract The stability of a free liquid metal surface influenced by an alternating magnetic field is investigated both experimentally and analytically The experimental set-up consists of an annulus filled with liquid metal The outer wall of the annulus is made of quartz glass to allow visualization using a high-speed camera system The magnetic field is generated by ring-like inductor located at some distance above the free surface of the liquid metal The inductor is supplied by an alternating current of variable frequency between 5 khz and 5 khz The alternating magnetic field induces Lorentz forces within the liquid and generates an electromagnetic pressure acting on the free surface Two different types of instabilities in the are detected At lower inductor currents we observe sinusoidal surface waves, while at higher currents, above a critical value, an electromagnetic pinch is generated During the experiments we measure the critical current and the amplitude of the surface waves for various frequencies A simple analytical model has been developed in order to predict the instability of a plane liquid metal surface affected by an alternating electromagnetic field The hydrodynamic equations are simplified by applying the Hele-Shaw approximation for a thin annulus gap The electromagnetic equations are simplified by using the skin depth approximation Here we exploit the fact that for high frequencies the penetration depth of the magnetic field is small compared to the liquid layer height Within this assumption, the electromagnetic pressure on the free surface can be calculated by using the method of mirror charges Linear stability analysis shows that the plane surface becomes unstable to perturbations of wave number k whenever the effect of the electromagnetic pressure exceeds the stabilizing effects of surface tension and gravity The critical inductor current predicted by the model is in good agreement with the critical value of the instability observed in experiment Introduction The control of metal surfaces by electromagnetic forces is used in many metallurgical processes, including electromagnetic levitation, cold crucible technologies, and electromagnetic slit sealing In the latter application, a thin slit between a rotating roll and a refractory must be sealed to prevent leakage of the liquid steel This problem was investigated numerically by Belkessam [1] In a D quasi-steady simulation the position of the free surface was calculated in dependence of various electrical and geometrical parameters However, in application the problem is more complicated The stability of the free surface along the slit is the most important problem and its control is crucial for success Analytical studies by Fautrelle and Sneyd [] have demonstrated that an instability of a flat conducting free surface sets in when the applied alternating magnetic field exceeds a certain critical value B c 55

2 Within this context we investigate the stability of a liquid metal interface in a thin gap affected by an alternating magnetic field In Section 1 we describe the experimental set-up and show some results in Section To support the experimental results, in Section 3 we present a simple linear stability analysis of a flat liquid metal interface in a Hele-Shaw cell A comparison of experimental observations and analytical predictions is provided in Section 4 1 Experimental set-up The apparatus used in the experimental investigations is shown schematically in Fig 1 It consists of a water-cooled bottom made of aluminum covered by a thin sheet of copper Aluminum was chosen because of its low specific weight and high thermal conductivity The copper disc has a thickness of only 1 mm It is used because of its relatively high chemical resistance against the aggressive liquid metal The outer wall of the annulus is formed by a circular ring of transparent fused silica of thickness 1 mm and an inner diameter of 15 mm The inner wall of the annulus is realized using a PVC cylinder Hence, the width of the annulus can be varied by using cylinders of various diameters The gap between the inner and outer wall is filled with GaInSn in such a composition that its melting temperature is as low as 19 C (Galinstan) To prevent the free liquid metal surface from oxidazing, it is covered with a thin layer of diluted hydrochloric acid The electromagnetic field is generated by a water-cooled copper inductor consisting of two layers, each of which has 5 windings The distance L between the liquid metal surface and the lower site of the inductor can be adjusted by a special device sustaining the aluminum bottom The liquid metal interface is observed from the side by a high-speed camera system taking up to 185 pictures per second characteristic dimensions parameter mm annulus width 5 annulus radius 7-75 metal height 5 gap between inductor and 13 metal Fig 1 Sketch of the experimental apparatus The inductor is fed by a high-frequency electric current The eddy currents induced in the liquid metal lead to an overall rise in temperature, thereby changing material properties like electrical conductivity and surface tension These unwelcome Joule heat losses can only partly be removed through the water-cooled bottom, limiting each experimental run to only a few seconds On the other hand, due to interactions of the eddy currents and the generated magnetic field, Lorentz forces F L are induced in the liquid metal They act mainly in direction of gravity and excite waves of various wavelengths at the free surface If the applied inductor current exceeds a critical value I c, an electromagnetic pinch is observed The measurements aim to visualize the surface waves and the pinch and to determine I c in dependence of the frequency of the applied inductor current 56

3 Observation of the instabilities in the metal The used power source for the inductor is a parallel resonant circuit generator with 3 frequency ranges The frequency is adjustable in discrete stages depending on the ohmicinductive burden Measurements were made between 5 khz and 1 khz and between khz and 5 khz We detect two different types of instabilities of the liquid metal surface Fig Surface instability in the liquid metal at khz Fig shows a part of the annulus observed from outside with the high speed camera system If the inductor current I is small surface waves are generated along the slit Their amplitude increases upun increasing the current The wavelength depends on the frequency of the exciting current Such surface waves are generated in a subcritical range, ie within the range I I c The measured values for I c are compared with the theoretical values in Sec 5 (Fig 5) Fig 3 Pinch channel instability in the liquid metal at khz Above the critical current, the waves lose their sinusoidal form and become irregularly eventually resulting in the formation of a pinch channel (Fig 3) This channel is filled with the covering hydrochloric acid Such a thin channel can only be observed at frequencies of about khz The pinch starts at the surface and moves down to the bottom without touching it Upon approaching the bottom, the metal flow bends the pinch channel and the equilibrium of static and electromagnetic pressure prevents touching the bottom The pinch channel survives several seconds because the forces on the channel walls act repulsively (proximity effect) 3 Theoretical analysis 31 Governing Equations To support the experimental observations we perform a linear stability analysis of a plane liquid metal interface in a Hele-Shaw cell subject to a high-frequency magnetic field To simplify the analysis we consider a Cartesian geometry as shown in Figure 4 The 57

4 magnetic field is generated by an inductor located at distance z = L above the unperturbed interface The inductor is fed by an alternating current Icosωt pointing in y-direction Hence, the induced magnetic vector potential is given by A = A(x,z) e y In the analysis we assume that A vanishes at the interface at z = h(y,t) This assumption corresponds to the case where the skin depth δ = /( σωµ ) is very small compared to the liquid metal height H, ie that S the electrical conductivity σ and the frequency ω are sufficiently high (Moreau [3]) L z I cos ϖt x z,w y,v h(y,t) I cos ϖt p a, p M n p H d g Liquid metal ρ,υ,σ Fig 4 Sketch of the problem used in the analytical modeling Furthermore, we apply the usual Hele-Shaw approximation (Schlichting [4]) that the velocity field v = (u,v,w) within the liquid metal can be represented by x x u =, v = v( y, z, t) 1, w (,, ) 1 = w y z t d d and that porous friction υ / d is dominant Within these assumptions the coupled hydrodynamic and electrodynamic equations read as follows: v w v 1 p w 1 p + =, = υ v, = υ w g, y t ρ y d t ρ d x + A = µ I cosωt δ ( z L) δ ( x) Here, g represent gravity and δ is the Dirac function These equations have to be solved with respect to the following boundary conditions: h h w( z = H ) =, A v t y, ( z = h) =, w( z = h) = +, Additionally, at the interface at z = h the following pressure condition holds: p M ( 1+ ( h / ) ) 3/ h + pa p = S y y 58

5 Here, S is surface tension and p a, p M represent the ambient pressure and the magnetic pressure Once the magnetic vector potential A is known, p M can be calculated using the relations B z= h p M = µ, where B = A, and denotes time averaging 3 Linear Stability Analysis We now introduce the perturbation expansion Φ = Φ + ε Φ1, Φ = ( v, w, p, A, h, pm ), into the governing equations and respective boundary conditions and collect terms of equal power in ε For the leading order we obtain the basic state given by µ v = w = h = ; A = I cosωt ln π ( L z) ( L + z) + x + x ; p M µ I ( z = ) = π L Here we applied the classical mirror method (Jackson [5]) to solve for the vector potential and thus the magnetic pressure In the next order we find that the problem is described by two Laplace equations according to w1 y w1 + A1 A1 h1 A =, + = ; w1( z = H ) =, w1( z = ) =, A1 ( z = ) = h1 x t We solve the above equations by introducing normal modes of the form w 1 cosky and using the skin depth approximation [3] for the magnetic vector potential A 1 Inserting the solution into the pressure balance at the interface we obtain after some lengthy but straightforward algebra an evolution equation for interfacial perturbations of wavelength k We find ρ 1+ exp k 1 exp ( kh ) ( kh ) t υ + d µ I h1 + ρg h1 = Sk h1 + h1 t 4π δ SL It is obvious that the most dangerous modes are characterized by k =, as these modes are not stabilized by surface tension In this case the criterion for the onset of neutral / t = reads as instability ( ) I c = π µ L ρgδ S 4 Hence, we obtain the typical scaling laws I ω 1/ C and I C L for the critical electrical current Physically, the above relation may be interpreted as follows: Instability sets in when the induced magnetic pressure p M B / µ at the interface equals the hydrostatic pressure evaluated with the skin depth, ie p ρgδ 4 Comparison Fig 5 shows the results of the critical inductor current in dependence of the applied current frequency The measured values correspond with the case where the amplitude of the surface waves is equal to electromagnetic skin depth of a given frequency, while the values predicted by theory refer to the onset of a long-wavelength instability of an initially flat interface There is a remarkably good qualitative agreement between the data and the S 59

6 current / A predictions as both curves show the typical dependence 1/ 4 I ω However, the C quantitative comparison shows that the critical inductor currents I c predicted by theory are lower than the values measured in the experiments We explain this deviation with the simplification made in the analytical model when calculating the magnetic pressure at the free surface In theory the inductor is modeled by just one winding, while the real inductor consists of 1 windings arranged in two layers It is obvious that the real inductor induces a weaker magnetic field and, therefore, a weaker magnetic pressure on the surface of the liquid metal A careful comparison of the magnetic induction B of the real and the theoretical inductor yields a constant factor of 1,44 The corrected line now shows a good quantitative agreement to the measurements Conclusions Ic in A (measurement) Ic in A (theory) Ic in A (theory corrected) frequency / Hz Fig 5 Comparison of measured and calculated critical currents We have investigated both experimentally and analytically the stability of a free liquid metal interface subject to an applied magnetic pressure The magnetic pressure induced by an inductor fed by an alternating electrical current In the experiments we observe that an electromagnetic pinch sets in when the feeding current exceeds a certain critical value A simple linear stability analysis based on Hele-Shaw and skin-depth approximation predicts 1/ 4 that the critical current scales as I C ω The theoretical predictions and experimental findings are in good qualitative and quantitative agreement References [1] O Belkessam: Lagestabilisierung von Metallschmelzen mit freier Oberfläche im elektromagnetischen Wechselfeld, PhD dissertation, Ilmenau Technical University, 1998 [] Y Fautrelle, A Sneyd: Instability of a plane conducting free surface submitted to an alternating magnetic field, J Fluid Mech (1998), vol 375, pp [3] R Moreau: Magnetohydrodynamics, Kluver, Dortrecht (199) [4] H Schlichting: Boundary-Layer Theory, McGraw-Hill, New York (1979) [5] J D Jackson: Classical Electrodynamics, J Wiley & Sons, New York (1999) Authors Dipl-Ing Mohring, Jens-Uwe Dr-Ing Karcher, Christian Prof Dr-Ing habil Schulze, Dietmar Department for Department for Electroheat Mechanical Engineering Department for Electroheat Ilmenau Technical University Ilmenau Technical University Ilmenau Technical University PO Box 1565 PO Box 1565 PO Box 1565 D Ilmenau D Ilmenau D Ilmenau Germany Germany Germany jens-uwemohring@tu-ilmenaude christiankarcher@tu-ilmenaude dietmarschulze@tu-ilmenaude 6