International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, September 16-19, 2014 Frequency dependence of an alternating magnetic field driven flow A. Cramer, V. Galindo Abstract The flow induced by a single-phase alternating magnetic field (AMF) is studied mainly numerically, and by experiments at mains frequency. For validation, the well-known dependence of the characteristic velocity υc on the magnetic induction B in the turbulent case when the mean velocity scales with the shear velocity, υc ∝ B, is reproduced experimentally and in the simulations. Ultrasonic flow mapping reveals that the eddies in the well believed flow structure of two toroidal vortices one on top of another do not only simply oscillate, but rather the topology of the flow changes slowly on a large scale. Besides such turbulence characteristics, it will be shown that the global flow structure depends strongly also on the frequency f of the AMF. An investigation of the change of υc with f suggests a quantitative difference of flows in an AMF compared to rotating and travelling magnetic fields. Introduction The flow created by an imposition of an alternating magnetic field (AMF) on electrically high conducting liquids receives attention in the field of metallurgy since quite a long time. In 1976, Tarapore & Evans studied the flow in induction furnaces [1]. They conducted fluid velocity measurements in an inductively stirred mercury pool of radius R = 0.1445 m at a fixed frequency f of 3 kHz. The shielding parameter S = µσωR2, which measures the crowding out of an electromagnetic field by its own eddy currents from the interior of an electrical conductor, was 514. In the definition of S, ω = 2πf, µ is the permeability, and σ the electrical conductivity. For such a high S » 1, the field does not penetrate the conductor; it is expelled to a thin skin at the surface. The measurements in [1] served the validation of the numerical code, the flow structure was calculated. It consists of two toroidal vortices one on top of another, is symmetric in the axial direction, and the rolling direction of the vortices is from the rim to the centre at half the height of the fluid cylinder. Taberlet & Fautrelle (1985) varied the frequency in a similar configuration in such a way that S was 3.9, 30, 166, and 372 [2]. Except for S = 30, where a small third counterrotating toroidal vortex appeared in the upper corner around the rim, the two vortices described in [1] were observed. Opposing to there, they were not of equal size. For S = 3.9 and 30 the lower one was much larger than the upper one, and vice versa for S = 166 and 372. While stating that the flow structure is determined by the particular distribution of the Lorentz force, which distribution significantly changes with frequency, a perspicuous explanation for the intermediate appearance of a third vortex and the reversal of the size of the eddies was not provided. Regarding the dependence of the characteristic velocity υc on S, the maximum υcmax(S) was found for S = 30. υc for S = 3.9 is much smaller than those for S = 166 and 372, i.e., the decrease towards higher S is significantly flatter than the increase for smaller S. Recently, studying flows in inductively processed melts gained stimulus from the field of solar silicon block casting. A new crystallisation furnace is proposed in [3–5], in which the traditional heating by resistance is replaced by induction heating. It is not only that the flow driven by a single-phase alternating magnetic field obviously influences the shape of the solidification front, thereby leading to higher efficiency photovoltaic material. Based on the furnace described in [3–5], a project within the Framework Programme 7 of the European Commission was yet launched to re-use silicon saw dust accruing during slicing the block casts to wafers [6]. The challenge in this project is cleaning the silicon melt from polluting silicon oxide and silicon carbide particles. Means for such cleaning is an electromagnetic separation process (ESP), in which the non-conducting particles are moved to the vertical crucible wall by Leenov-Kolin force (LKF) where they are accumulated in a dirty skin during solidification. As the LKF is applied via an AMF, one is once more concerned with the flow driven by this type of magnetic field. The influence of flow on the ESP is discussed controversially in the literature; a too strong flow inhibits ESP, whereas some flow is needed to move the particles from the bulk of the melt to the crucible walls where they can be captured by the LKF. It is however not only the vigour, but also the topology and the turbulence characteristics of the flow affecting the effectiveness of the ESP. Two successful model experiments on ESP are reported in [7, 8]. Despite the high frequency mentioned in the title, S was in a moderate range owing to the small scale of the experiments. Besides reporting on turbulence characteristics, which were not subject in [7, 8], the present paper revisits the parameter range of the experiment in [8] done on silicon. The manuscript is organised as follows. Sections 1. and 2. describe the experiment and the numerical procedure, respectively. Findings are reported and discussed in Section 3. Besides a summarising interpretation, the Conclusions outline future work planned on flows driven by single-phase alternating magnetic fields. 1. Experimental setup All measurements have been carried out in the cylindrical container of 20 mm in radius and 60 mm in height depicted to the left in Fig. 1. Although the magnetic system allows for a height covering more than that of the melt volume as it was the case in [1, 2, 7, 8], only the central part of the segmented coil was used. Since limitations were only imposed by the maximum available power of the current source, which could be fed to that one segment, the choice of the small height of the coil provided the maximum possible vigour of stirring. This is due to the particularity of an AMF that the magnitude of the rotational part of the Lorentz force depends not only on the strength of Fig. 1. Drawn to true scale sketch of the the magnetic field but on its gradient as well. fluid container, the magnetic system, and The velocity measurements were done by the measuring unit. ultrasound Doppler velocimetry (UDV). A transducer installed at the top with the ultrasonic beam directed downward measures the vertical velocity component υz. Traversing the transducer allowed mapping mean properties of the flow field. For similar measurements with a description of the principles of UDV see [9]. 2. Numerical procedure For the numerical simulations of the induced electric current j and magnetic induction B in electrically conducting regions, the finite element code OPERA 3d (Cobham plc., [10]) was used. The computational grid was refined near the walls in order to resolve the skin layers and has a total number of 2 million finite elements. The flow in the volume containing the melt was simulated numerically by means of the open source library OpenFOAM [11] solving the Navier-Stokes equation together with the incompressibility condition ∇ ⋅ υ = 0 and including an electromagnetic force density term averaged over one period T = 1/f: ρ ⋅ ρ, p, and η are the density, the pressure, and the dynamic viscosity. The boundary conditions for the flow field is the no-slip condition υ = 0 at the solid container wall. For the melt surface, either υ = 0 or the conditions for a stress-free non-deformable surface υn = 0 and ∂υt / ∂z = 0 (the subscripts n and t denote the normal and the tangential component, respectively) are applied depending on whether the melt flow is to be calculated in an open or an enclosed container. A computational grid with 650000 volume elements was used. The discretisation scheme for the convective term was of second order. η Fig. 2. Dependence of velocity on induction. 3. Results and discussion 3.1. Characteristic velocity vs. induction Numerical calculation and measurement of the dependence of υ on B was done mainly for validation purposes. Since the experimental setup allowed measuring υz over the height of the container, the maximum υzmax(z) along the ultrasonic beam was taken as a representative for υc. The computations were done with a stationary solver where possible. Higher values of B required transient calculation and using a turbulence model, which was the simple k–ε one. Fig. 2 shows that the well-known linear relation for υc continues to be true for υz measured on the cylinder axis: υzmax(z) ∝ B. The numerically obtained results are consistent between the stationary and transient calculations. Agreement can be stated between the computations and the experiments. 3.2. Turbulence It is reported in the numerical work [12] that the time Fig. 3. Relative (top) dependence of an AMF driven flow in configurations similar to the and absolute stanpresent one consists in an oscillation of the mean flow eddies (c.f. Fig. 1 dard deviation of υ. there). This means, however, that the mean flow structure of two vortices one on top of another is preserved. Further findings were that these oscillations show a sharp peak in the spectrum at quite low frequencies, and that the flow is most unstable in between the vortices adjacent to the rim. The low frequency peak was reproduced in the present work at about 0.1 Hz; a spectrum is not shown due to space limitations. As the turbulence degree, in its strict definition, is meaningless in a recirculating flow, Fig. 3 plots a modification thereof calculated from the local standard deviation sd and the local time averaged velocity (upper panel). The lower panel shows sd for comparison—note that only a quarter of a central section through the cell is plotted. A free-hand drawn rough course of streamlines should serve as a guide to the eye. Both mappings of statistical properties largely support the findings in [12], except that the region of strong fluctuations extends farther in both directions, vertically towards the eye of the vortex and radially towards half the radius. Fig. 4. Equidistant in time series of vertical sections of υz(z) at r = 15 mm. For pictorially better comprehendible presentation, abscissas and ordinates are exchanged. The series of υz(z) in Fig. 4 evinces that one is not concerned with mere oscillations of the mean flow eddies. If it were so, the shape should be an “S” as, e.g., in the second panel, with changing amplitudes and the root moving up and down. Whereas, the third panel exhibits only an upper vortex and the existence of two or even three roots (panels 1 and 5) makes evident that eddies are created and destroyed. That is to say, the well believed flow structure exists only as a long-term average and is probably never realised at any instant in time. 3. 3. Dependence of velocity on frequency It is well known that, keeping B constant, a maximum of υc is observed for an f where S is in the range from 5 to 10 in the case of rotating (RMF) and travelling magnetic fields (TMF). Among their coarsely spaced S, Taberlet & Fautrelle found υcmax(S) at S = 30 [2]. This is noticeable since shielding, i.e. expelling an AC field from a conductor thereby diminishing the Lorentz force, depends on f, only. These authors published the dependence of υc on S in a joint work with a numerical group [13], based on the experiments in [2], one year later. υcmax(S) was Fig. 5. Dependence of the maximum velocity found at S about 80 in the calculations and, on f. The upper abscissa translates the surprisingly, a second local maximum at frequency to the shielding parameter S. higher S appeared. The interest in the present work is in the range of S for which successful ESP experiments were done. Figure 5 plots the numerically obtained maxima of |υ| for the conditions in [8]. A Levenberg-Marquardt fit suggests that |υ|max(S) > 40. The difference for 10 and 11 kHz is noticeable; both solutions, albeit the velocity fluctuates distinctly, are converged with respect to the total kinetic energy. Such a significant difference may be attributed to a change of the global flow structure. This aspect will be returned to in the next section, where it is discussed in conjunction with the obvious result that υcmax(S) for an AMF is distinctly higher than those for an RMF and a TMF. 3.4. Lorentz force and flow structure It is well known that the flow structure evoked by an RMF neither depends significantly on f, nor on the axial distribution of the Lorentz force FL. The movement of an electrically conducting fluid follows the azimuthal FL, accompanied by a secondary poloidal flow due to Ekman pumping. Shielding affects essentially the vigour of the flow. From own experience, it turns out that the same holds for the axial forcing in a TMF. That is to say, the more complex spatially-temporarily alternating fields drive a comparably simple flow. To imagine why a simple singlephase, i.e. not moving in space, AMF may lead to rather complex flow patterns, an axially infinite configuration is considered. The fluid stays at rest because ∇×FL = 0. Said Fig. 6. Lorentz force distribution for 50 Hz (left) and in words, the AMF produces only for 11 kHz (right). Plotted is the Lorentz force density pressure. In any finite system, it is the divided by the density, so the dimension of F is m/s2. spatial variation of FL which determines the global flow structure. Iso-plots of |FL| for the system investigated in [8] are shown in Fig. 6. 50 Hz and 11 kHz correspond to S = 0.19 and 42.7, respectively. The change of |FL| with S is remarkable, albeit not in the focus here. The interest is in the topology of the flow, which is to be seen in Fig. 7. Also at mains frequency, |FL| is maximal in the corners. Despite this, a considerable central portion of the height where |FL| is sufficiently high drives the fluid inwards since FL is made mainly of the radial component. The fluid consequently moves opposing to the Lorentz force at the top and at the bottom, the well-believed double-torus is observed. Shielding obviously does not only expel B simply radially outwards. The magnetic field is also crowded out in the axial direction, with the consequence that FL loses dominance in the same central-axial region that determined the flow at mains frequency. Driven all but by the strong force in the corners, four vortices one on top of another establish. That the flow Fig. 7. Plot of velocity vectors for the same appears asymmetric in the vertical direction is a matter of the plot depicting the flow at a frequencies as in Fig. 6. certain instance in time; also the lower of the two central vortices is the larger one at times. Loss of axial symmetry is to be seen at the different height of the eyes of the vortices along the circumference. This may explain the significant difference in |υ|max(S) between 10 and 11 kHz. The flow is still almost axisymmetric at 10 kHz. While increasing f merely by 10 %, the asymmetry with respect to the cylinder axis becomes significant. Conclusion and perspective Being subject of investigation for decades, the flow driven by an AMF is still far from being completely understood. Even restricting to the case where the axial extension of the coil is much larger than that of the fluid volume, which quasi generic configuration will hardly be realisable for silicon recycling, both dependences of the characteristic velocity and flow topology on f are not precisely know. Besides more parametric numerical studies on different geometries aiming at suitable ones for recycling purposes via separation, experiments are badly needed, the latter of which should be capable to visualise the flow structure. Mapping of turbulent flow by means of ultrasonic Doppler anemometry is readily available [14]. Acknowledgement This work was financially supported by the European Commission in the Framework Programme 7 ENV-2013 under the grant agreement number 603-718. References [1] Tarapore, E.D., Evans, J.W.: Fluid velocities in induction melting furnaces: Part I. Theory and laboratory experiments. Metall. Trans. B, Vol. 7, 1976, pp. 343–351. 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Authors Dr. Cramer, Andreas Institute of Fluid Dynamics Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstraße 400 D-01328 Dresden, Germany E-mail: a.cramer@hzdr Dr. Galindo, Vladimir Institute of Fluid Dynamics Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstraße 400 D-01328 Dresden, Germany E-mail: v.galindo@hzdr.de