Karcher, Christian

Minchenya, Vitaly

Digitale Bibliothek Thüringen

   Information Technology and Electrical  Engineering - Devices and Systems, Materials  and Technologies for the Future        Faculty of Electrical Engineering and Information Technology    Startseite / Index:  http://www.db-thueringen.de/servlets/DocumentServlet?id=14089 54. IWK Internationales Wissenschaftliches Kolloquium International Scientific Colloquium 07 - 10 September 2009 PROCEEDINGS Impressum  Herausgeber: Der Rektor der Technischen Universität llmenau  Univ.-Prof. Dr. rer. nat. habil. Dr. h. c. Prof. h. c. Peter Scharff  Redaktion: Referat Marketing   Andrea Schneider   Fakultät für Elektrotechnik und Informationstechnik  Univ.-Prof. Dr.-Ing. Frank Berger  Redaktionsschluss: 17. August 2009  Technische Realisierung (USB-Flash-Ausgabe):  Institut für Medientechnik an der TU Ilmenau  Dipl.-Ing. Christian Weigel  Dipl.-Ing. Helge Drumm  Technische Realisierung (Online-Ausgabe):  Universitätsbibliothek Ilmenau    Postfach 10 05 65  98684 Ilmenau  Verlag:   Verlag ISLE, Betriebsstätte des ISLE e.V.  Werner-von-Siemens-Str. 16  98693 llmenau   © Technische Universität llmenau (Thür.) 2009   Diese Publikationen und alle in ihr enthaltenen Beiträge und Abbildungen sind urheberrechtlich geschützt.   ISBN (USB-Flash-Ausgabe): 978-3-938843-45-1 ISBN (Druckausgabe der Kurzfassungen): 978-3-938843-44-4  Startseite / Index: http://www.db-thueringen.de/servlets/DocumentServlet?id=14089  SUPPRESSION OF ELECTROMAGNETICALLY INDUCED INTABILITIES  OF THE LIQUID METAL DISK  Vitaly Minchenya, Christian Karcher   Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology P.O. Box 100565, D-98684 Ilmenau, Germany  ABSTRACT  In this work we study a method to suppress the electromagnetically induced free surface instabilities of the free surface subject to a high-frequency magnetic field. The object of our current investigation is a liquid metal disk (GaInSn) exposed to a high-frequency magnetic field (with frequencies in a range of 10*n kHz – 100*n kHz). This simplified configuration can be realized by putting the liquid metal within the thin gap between two horizontal plates, so that gravitational effects can be omitted. This way a 3D liquid metal drop is reduced to 2D model problem. We consider a local inhomogenity of the high-frequency magnetic field as an instrument for the suppression of the instable oscillation modes of the disk.   Index Terms – Liquid metal, Magnetic field, Instability, Stabilization  1. INTRODUCTION Instabilities of the capillary surfaces of liquid metal subject to high frequency magnetic field can take place in many technological processes like Levitation physical vapor deposition and Laser beam evaporation with electromagnetic shaping [1, 2]. In the current investigation we study a simplified model of such capillary surface - a 2D liquid metal disk of GaInSn (eutectic metal alloy which is liquid at room temperature). This simplified configuration can be realized by putting the liquid metal within the thin gap between two horizontal plates, so that gravitational effects can be omitted. Moreover, we reduce the order of the instable geometries to mode 2 (ellipsoid). The results of the previous investigations [3] allow us to choose the range of the experiment parameters like liquid metal volumes and the excitation magnetic field frequency 31 kHz to reach exactly the 2nd mode of the instability oscillations at the appropriate eigenfrequency.  2. MECHANICAL MODEL Let us represent the local moving part of the peripheral area of the disk as a harmonic oscillator. Considering the moving volume of the fluid as given in Fig. 1 and taking into account the continuity of the fluid motion, we can assume for the first order approach that the flow is two-dimensional and a mean velocity of the flow can be considered for the given moment of time.  Moreover, there are two halves of the oscillating disk in which the flow has opposite directions. Hence, the symmetrical problem can be formulated in which the impulse in one direction is defined by the half of the disk mass (in case of the 2nd mode of oscillations).  Representation of the moving disk contour as a harmonic oscillator can be given as follows  kxdtxdm −=222,   (1)  where m – effective mass of the disk, x – contour displacement, k – effective elasticity of the oscillating disk.   Fig. 1 Geometry of the instable form of the disk  (2nd mode of oscillation)  From which the effective elasticity can be found in a form:  22mk ω= .   (2)  here mk2=ω  is eigenfrequency of the disk. Considering a potential energy of the disk at maximum displacement x from the equilibrium position  © 2009 - 54th Internationales Wissenschaftliches Kolloquium2max 21 kxE p = ,   (3)  one can estimate a maximum velocity maxu of the disc during the instable azimuthal displacement:  mEu p max2= .   (4)  Using units’ analogy from Bernoulli equation, one can finally express the local pressure in the fluid on the disc periphery as follows:  2ρυ=p .   (5)  3. ASSESSMENT OF THE ACTUATOR Using the scaling equation for the magnetic energy density we can estimate the minimal magnetic induction B in a vicinity of the disk surface, which is necessary to counteract the instable deformation. The equation is given as follows:       1022 ABAPp M µ== , or ApAB /210µ= .      (6)  Here A1 – unit area, A – effective area of the free surface of the liquid metal disk, B – amplitude of the magnetic field, 0µ  - magnetic constant. To obtain the necessary local increase of the magnetic induction in a vicinity of the disk periphery an actuator can be used in a form of the coil fed with AC current.  The schematic diagram for the calculation is given in Fig. 2.    Fig. 2: Schematic representation of the electromagnetic actuator for the calculation of the axial component of magnetic induction.  The axial component of the magnetic induction is given as follows:   ∫ −++ −=πϕϕϕπµ02/322 )cos21(cos12dRRXRrNiBx   (7)  The next parameters of typical actuator are used for the calculation using the equation (7):  Parameter name  Parameter symbol/equation Windings number N  Peak Current i  Mean radius of the actuator r  Absolute magnetic permeability rµµµ 0=  Axial coordinate of the view point x  Radial coordinate of the view point 1r  View point axial aspect ratio 1/ rxX =  View point radial aspect ratio 1/ rrR =   4. CONCLUSIONS In this work we present an example of analytical calculation of the AC actuator for the counteraction to the instable azimuthal motion of the liquid metal disk subject to high-frequency magnetic field. We start with the mechanical model of the disk on the basis of the equation of harmonic oscillator. Further the local magnetic pressure and the necessary local magnetic induction was evaluated. For an actuator in a form of cylindrical coil the magnetic induction near the liquid metal disk periphery can obtained on the basis of Biot-Savart law.   5. ACKNOWLEDGEMENT  Authors gratefully acknowledge the financial support of Deutsche Forschungsgemeinschaft (DFG) within the framework of the project KA 1660/3-1.   6. REFERENCES  [1] T. Felici, On the surface stability of liquid conductors in electromagnetic shaping, J. Fluid. Mech. 302, 1-28.  [2] J. Priede, G. Gerbeth, Spin-up instability of electromagnetically levitated spherical bodies, IEEE Transactions on magnetics 36, No. 1, 349-353.  [3] V. Kocourek, Ch. Karcher, M. Conrath and D. Schilze (2006) Stability of liquid metal drops affected by a high frequency magnetic field, Physical Review E 74, 026303. 

Suppression of electromagnetically induced instabilities of the liquid metal disk

https://www.db-thueringen.de/servlets/MCRFileNodeServlet/dbt_derivate_00019768/54_IWK_2009_7_1_02.pdf

Suppression of electromagnetically induced instabilities of the liquid metal disk

Abstract

Similar works

Full text

Available Versions

Digitale Bibliothek Thüringen