Силовыe и энергетические воздействия магнитного поля на проводящую среду. Модели и эксперименты
DOI:
https://doi.org/10.17072/1994-3598-2020-2-10-37Ключевые слова:
постоянное магнитное поле, бегущее магнитное поле, вращающееся магнитное поле, переменное маг-нитное поле, магнитная гидродинамика, индукционный ток, электродинамическая сила, индукционный нагрев, число Гартмана, параметр диффузии магнитного поляАннотация
Выполнен обзор существующих методов и технологий воздействия на проводящие среды магнитными полями, которые могут быть приложены принципиально тремя способами: постоянное, бегущее (и его вариант – вращающееся) и переменное (гармонически изменяющееся). Проанализированы общие уравнения электромагнетизма применительно к движущейся проводящей среде, приведены важные для описания безразмерные критерии. Для каждого вида приложения магнитного поля рассмотрены математические модели, уравнения моделей записаны настолько подробно, насколько этого достаточно для расчёта соответствующего воздействия. По каждому виду поля приведены данные натурных и вычислительных экспериментов. Основное внимание уделено переменному магнитному полю, остальные виды представлены более кратко, но достаточно для начала движения в интересующем направлении. В заключении обзора перечислены работы, полезные при верификации математических моделей. Для удобства работы с англоязычными поисковыми и наукометрическими базами для наиболее важных терминов приведён их английский вариант.Библиографические ссылки
Hussain Z. et al. Instability of three dimensional electrically conducting fluid of magnetohydrodynamics Couette flow. AIP Adv. AIP Publishing, LLC, 2019. Vol. 9, № 10.
Yang Z. et al. Instability of magnetohydrodynamic flow of Hartmann layers between parallel plates. AIP Adv. 2019. Vol. 9, № 5.
Pavlinov A. et al. Eddy current flowmeter for sodium flow. IOP Conf. Ser. Mater. Sci. Eng. 2017. Vol. 208, № 1.
Vatazhin A.B., Lyubimov G.A., Regirer S.A. Magnitogidrodinamicheskie techeniya v kanalah. Moskva: Nauka, 1970. 672 p. (In Russian)
Lielpeter Ya.Ya. ZHidkometallicheskie indukcionnye MGD-mashiny. Riga: Zanatne, 1969. 246 p. (In Russian)
Gel'fgat YU.M., Lielausis O.A., SHCHerbinin E.V. ZHidkij metall pod dejstviem elektromagnitnyh sil. Riga: Zanatne, 1976. 248 p. (In Russian)
Tsaplin A.I. Dinamika cirkulyacii zhidkogo yadra kristallizuyushchegosya nepreryvnogo slitka v begushchem pole induktora. Magnitnaya gidrodinamika. 1986. № 1. pp. 127–131. (In Russian)
Tsaplin A.I., Grachyov A.B. Eksperimental'no-raschetnoe modelirovanie elektromagnitnogo peremeshivaniya zhidkogo yadra slitka. Magnitnaya gidrodinamika. 1987. № 2. pp. 103. (In Russian)
Zeleneckij A.B., Hripchenko S.YU., Tsaplin A.I. Modelirovanie kristallizacii metalla v ploskom sloe pri elektromagnitnom peremeshivanii. Magnitnaya gidrodinamika. 1992. № 1. pp. 96–100. (In Russian)
Tsaplin A.I., Rogatchikov YU.M. Modelirovanie vozdejstviya rolikovogo nepreryvnogo peremeshivatelya v mashinah nepreryvnogo lit'ya zagotovok. Magnitnaya gidrodinamika. 1993. № 2. pp. 105–112. (In Russian)
Tsaplin A.I. Teplofizika vneshnih vozdejstvij pri kristallizacii stal'nyh slitkov na mashinah nepreryvnogo lit'ya. Ekaterinburg, Izd-vo UrO RAN, 1995, 238 s., 1995. 238 p. (In Russian)
Tsaplin A.I., Nikulin I.L., Nechaev V.N. Modelling of electromagnetic actions in sponge titanium production. Magnetohydrodynamics. 2015. Vol. 51, № 4. pp. 749–755.
Khripchenko S. et al. Laboratory model of the aluminum furnace with MHD stirring induced by a rod-like inductor generating a travelling magnetic field. Magnetohydrodynamics. 2017.
Lekomtsev S. V., Khripchenko S.Y. Evaluation of the Temperature Regime of the Rods of the Inductor of an MHD Stirrer for Possible Use in an Industrial Aluminum Furnace. J. Appl. Mech. Tech. Phys. 2018.
Khripchenko S. et al. Numerical and experimental modelling of various mhd induction pumps. Magnetohydrodynamics. 2010.
Denisov S. et al. The MHD travelling magnetic field pump for liquid magnesium. Magnetohydrodynamics. 2013.
Kirko I.M., Kirko G.E. Magnitnaya gidrodinamika. Sovremennoe videnie problem. M.- Izhevsk: NIC “Regulyarnaya i haoticheskaya dinamika”, Izhevskij institut komp'yuternyh issledovanij, 2009. 632 p. (In Russian)
Siraev R.R., Khripchenko S.Y. Liquid metal exposed to rotating and travelling magnetic fields in crucibles with circular and square cross-sections. Magnetohydrodynamics. 2018.
Denisov S. et al. The effect of traveling and rotating magnetic fields on the structure of aluminum alloy during its crystallization in a cylindrical crucible. Magnetohydrodynamics. 2014. Vol. 50, № 4.
Khripchenko S.Y. et al. Influence of the position of the MHD stirrer relative to the crucible on the driven liquid metal flow. Magnetohydrodynamics. 2018.
Khlybov O.A., Lyubimova T.P. Effect of rotating magnetic field on mass transfer during directional solidification of semiconductors. Magnetohydrodynamics. 2016.
Tir L.L.; Gubchenko A. P.; Indukcionnye plavil'nye pechi dlya processov povyshennoj tochnosti i chistoty. Moskva: Energoatomizdat, 1988. P. 120. (In Russian)
Tarapore E.D., Evans J.W. Fluid Velocities in Induction Melting Furnaces : Part I . Theory and Laboratory Experiments. Metall. Trans. B. 1976. Vol. 7, № September.
Tarapore E.D., Evans J.W., Langfeldt J. Fluid velocities in induction melting furnaces: Part II. large scale measurements and predictions. Metall. Trans. B. 1977. Vol. 8, № 1. pp. 179–184.
Musaeva D. et al. Numerical simulation of the melt flow in an induction crucible furnace driven by a Lorentz force pulsed at low frequency. Magnetohydrodynamics. 2015. Vol. 51, № 4. pp. 771–783.
Musaeva D. et al. Analysis of the Almgsi-alloy structure formed under the influence of low-frequency pulsed Lorentz force. Magnetohydrodynamics. 2017. Vol. 53, № 1. pp. 526–536.
Nikulin I.L. Numerical simulation of melt flow control by controlling averaged electromagnetic forces generated in high frequency magnetic field. Magnetohydrodynamics. 2016. Vol. 52, № 4. pp. 527–533.
Floymayr J. Coreless induction furnace and method of melting and stirring metals in this furnace: pat. patent US 3472941 USA. 1969. Vol. 2, № 12. pp. 1–3.
Frizen V.E., Smolianov I.A., Sarapulov S.F. Induction crucible furnace with reactive power non- symmetrical compensation of inductor sections. 2018 20th Int. Symp. Electr. Appar. Technol. IEEE, 2018. P. 1–4.
Hermann R. et al. Influence of growth parameters and melt convection on the solid-liquid interface during RF-floating zone crystal growth of intermetallic compounds. J. Cryst. Growth. 2001. Vol. 223, № 4. pp. 577–587.
Hermann R. et al. Magnetic field controlled FZ single crystal growth of intermetallic compounds. J. Cryst. Growth. 2005. Vol. 275, № 1–2. pp. 1533–1538.
Hermann R., Gerbeth G., Priede J. Magnetic field controlled floating-zone single crystal growth of intermetallic compounds. Eur. Phys. J. Spec. Top. 2013. Vol. 220, № 1. pp. 227–241.
Bioul F., Dupret F. Free surface shear flows induced by Marangoni and alternating electromagnetic forces. J. Non-Equilibrium Thermodyn. 2005. Vol. 30, № 3. pp. 205–213.
Huang M.S., Huang Y.L. Effect of multi-layered induction coils on efficiency and uniformity of surface heating. Int. J. Heat Mass Transf. Elsevier Ltd, 2010. Vol. 53, № 11–12. pp. 2414–2423.
Shih S.Y. et al. Nonplanar mold surface heating using external inductive coil and magnetic shielding materials. Int. Commun. Heat Mass Transf. 2016. Vol. 71.
Shih S.Y., Nian S.C., Huang M.S. Comparison between single- and multiple-zone induction heating of largely curved mold surfaces. Int. Commun. Heat Mass Transf. 2016. Vol. 75.
Zhou J. et al. Effect of laser surface melting with alternating magnetic field on wear and corrosion resistance of magnesium alloy. Surf. Coatings Technol. Elsevier B.V., 2017. Vol. 309, № June. pp. 212–219.
Li C. et al. Chemical segregation and coarsening of γ′ precipitates in Ni-based superalloy during heat treatment in alternating magnetic field. J. Alloys Compd. 2017. Vol. 720, № May. pp. 272–276.
Hömberg D. et al. Simulation of multi-frequency-induction-hardening including phase transitions and mechanical effects. Finite Elem. Anal. Des. Elsevier, 2016. Vol. 121. pp. 86–100.
Krauter N. et al. Multi-frequency inductive system for magnesium level detection in a titanium reduction reactor. IOP Conf. Ser. Mater. Sci. Eng. 2018. Vol. 424, № 1. pp. 1–5.
Looney R., Priede J. Concept of a next-generation electromagnetic phase-shift flowmeter for liquid metals. Flow Meas. Instrum. 2019. Vol. 65, № November 2018. pp. 128–135.
Zhanwei S., Xinghus Z. A, phi-Omega method for 3-D eddy current analysis. Appl. Math. Mech. 1998. Vol. 19, № 11. pp. 1017–1023.
Cramer A., Galindo V., Zennaro M. Frequency dependence of an alternating magnetic field driven flow. Magnetohydrodynamics. 2015. Vol. 51, № 1. pp. 133–147.
Nikulin I.L., Perminov A. V., Tsaplin A.I. Mathematical model of conducting fluid convection in a non-uniform alternating magnetic field. Magnetohydrodynamics. 2013. Vol. 49, № 1–2. pp. 203–209.
Chen R. et al. Dimensionless parameters controlling fluid flow in electromagnetic cold crucible. J. Mater. Process. Technol. 2018. Vol. 255, № December 2017. pp. 242–251.
Gellert M. et al. Nonaxisymmetric Mhd Instabilities of Chandrasekhar States in Taylor-Couette Geometry. Astrophys. J. IOP Publishing, 2016. Vol. 823, № 2. p. 99.
Hripchenko S.Yu. Elektrovihrevye techeniya v kanalah MGD-ustrojstv. Ekatterinburg: UrO RAN, 2009. 261 p. (In Russian)
Tagawa T. Linear stability of parallel flow of liquid metal in a rectangular duct driven by a constant pressure gradient under the influence of a uniform magnetic field. IOP Conf. Ser. Mater. Sci. Eng. 2018. Vol. 424, № 1.
Sposito G., Ciofalo M. One-dimensional mixed MHD convection. Int. J. Heat Mass Transf. 2006. Vol. 49, № 17–18. pp. 2939–2949.
Lyubimova T.P., Skuridin R. V., Faizrakhmanova I.S. Effect of a magnetic field on the hysteresis transitions during floating-zone crystal growth. Tech. Phys. Lett. 2007. Vol. 33, № 9. pp. 744–747.
Demin V.A., Makarov D. V. Vliyanie vrashchayushchegosya magnitnogo polya na rasplav v cilindricheskoj zhidkoj zone. Vestnik permskogo universiteta. Fizika. 2004. № 1. pp. 106–111. (In Russian)
Jacoutot L. et al. Numerical modeling of coupled phenomena in a mechanically stirred molten-glass bath heated by induction. Chem. Eng. Sci. 2008. Vol. 63, № 9. pp. 2391–2401.
Tavakoli M.H. et al. Numerical study of induction heating in melt growth systems-Frequency selection. J. Cryst. Growth. Elsevier, 2010. Vol. 312, № 21. pp. 3198–3203.
Tavakoli M.H., Karbaschi H., Samavat F. Influence of workpiece height on the induction heating process. Math. Comput. Model. Elsevier Ltd, 2011. Vol. 54, № 1–2. pp. 50–58.
Khodamoradi H., Tavakoli M.H., Mohammadi K. Influence of crucible and coil geometry on the induction heating process in Czochralski crystal growth system. J. Cryst. Growth. Elsevier, 2015. Vol. 421. pp. 66–74.
Honarmandnia M., Tavakoli M.H., Sadeghi H. Global simulation of an RF Czochralski furnace during different stages of germanium single crystal growth. CrystEngComm. 2016. Vol. 18, № 21. pp. 3942–3948.
Honarmandnia M., Tavakoli M.H., Sadeghi H. Global simulation of an RF Czochralski furnace during different stages of germanium single crystal growth, part II: to investigate the effect of the crucible’s relative position against the RF coil on the isotherms, flow fields and thermo-elastic stresses. CrystEngComm. 2017. Vol. 19, № 3. pp. 576–583.
Kranjc M. et al. Numerical analysis and thermographic investigation of induction heating. Int. J. Heat Mass Transf. Elsevier Ltd, 2010. Vol. 53, № 17–18. pp. 3585–3591.
Jankowski T.A. et al. Approximate analytical solution for induction heating of solid cylinders. Elsevier Inc., 2016. Vol. 40. P. 2770–2782.
Kennedy M.W. et al. Empirical Verification of a Short-Coil Correction Factor. IEEE Trans. Ind. Electron. IEEE, 2014. Vol. 61, № 5. pp. 2573–2583.
Nayfeh A.H. Perturbation Methods. C. John Wiley, 1973, 446 p.
Davis E.J. Conduction and Induction Heating. The Institution of Engineering and Technology, 1990. 417 p.
Ango A. Matematika dlya elektro- i radioinzhenerov. M.: Izdatel'stvo “Nauka,” 1967. 780 p. (In Russian)
Nikulin I.L., Perminov A.V. Matematicheskaya model' processov teplomassoperenosa i diffuzii magnitnogo polya v indukcionnoj pechi. IFZH. 2016. Vol. 89, № 2. (In Russian)
Mistrangelo C., Bühler L. Magneto-convective instabilities in horizontal cavities. Phys. Fluids. 2016. Vol. 28, № 2.
Hudoba A., Molokov S. The effect of the Prandtl number on magnetoconvection in a horizontal fluid layer. Int. J. Heat Mass Transf. Elsevier Ltd, 2018. Vol. 116. P. 1292–1303.
Tagawa T., Ozoe H. Enhancement of heat transfer rate by application of a static magnetic field during natural convection of liquid metal in a cube. J. Heat Transfer. 1997. Vol. 119, № 2. pp. 265–271.
Tagawa T., Ozoe H. Enhanced heat transfer rate measured for natural convection in liquid gallium in a cubical enclosure under a static magnetic field. J. Heat Transfer. 1998. Vol. 120, № 4. pp. 1027–1032.
Tagawa T., Ozoe H. The natural convection of liquid metal in a cubical enclosure with various electro-conductivities of the wall under the magnetic field. Int. J. Heat Mass Transf. 1998. Vol. 41, № 13. P. 1917–1928.
Barannikov V.A., Zimin V.D. Neustojchivost' pokoya neizotermicheskoj provodyashchej zhidkosti v shcheli ferromagnitnogo massiva pri protekanii elektricheskogo toka. Magnitnaya gidrodinamika. 1982. № 2. pp. 117–122. (In Russian)
Burnysheva A. V., Lyubimova T.P. Oscillatory instability of advective flow in a horizontal cylinder in the presence of a rotating magnetic field. Fluid Dyn. 2012.
Bermúdez A. et al. Numerical simulation of a thermo-electromagneto-hydrodynamic problem in an induction heating furnace. Appl. Numer. Math. Elsevier B.V., 2009. Vol. 59, № 9. pp. 2082–2104.
Bermúdez A. et al. Numerical analysis of a finite-element method for the axisymmetric eddy current model of an induction furnace. IMA J. Numer. Anal. 2010. Vol. 30, № 3. pp. 654–676.
Bermúdez a., Muñoz-Sola R., Vázquez R. Analysis of two stationary magnetohydrodynamics systems of equations including Joule heating. J. Math. Anal. Appl. Elsevier Inc., 2010. Vol. 368, № 2. pp. 444–468.
Bermúdez A. et al. A thermo-electrical problem with a nonlocal radiation boundary condition. Math. Comput. Model. Elsevier Ltd, 2011. Vol. 53, № 1–2. pp. 63–80.
Bermúdez a. et al. Finite element approximation of nonlinear transient magnetic problems involving periodic potential drop excitations. Comput. Math. with Appl. Elsevier Ltd, 2013. Vol. 65, № 8. pp. 1200–1219.
Bermúdez A. et al. Numerical solution of a transient nonlinear axisymmetric eddy current model with nonlocal boundary conditions. Math. Model. Methods Appl. Sci. 2013. Vol. 23, № 13. pp. 2495–2521.
Bermúdez A. et al. Numerical analysis of a transient non-linear axisymmetric eddy current model. Comput. Math. with Appl. Elsevier Ltd, 2015. Vol. 70, № 8. pp. 1984–2005.
Burke P. Induction heating and melting systems having improved induction coils: pat. patent US 4874916 USA. 1989.
Beckstein P., Galindo V., Vukčević V. Efficient solution of 3D electromagnetic eddy-current problems within the finite volume framework of OpenFOAM. J. Comput. Phys. 2017. Vol. 344, № December 2016. pp. 623–646.
Perminov A. V., Nikulin I.L. Mathematical Model of the Processes of Heat and Mass Transfer and Diffusion of the Magnetic Field in an Induction Furnace. J. Eng. Phys. Thermophys. 2016. Vol. 89, № 2. pp. 397–409.
Nikulin I.L., Perminov A.V. Numerical investigation of electromagnetic effects and averaged metal melt flows generated by high-frequency magnetic field. Magnetohydrodynamics. 2016. Vol. 52, № 1.
Luozzo N. Di, Fontana M., Arcondo B. Modelling of induction heating of carbon steel tubes : Mathematical analysis , numerical simulation and validation. J. Alloys Compd. Elsevier B.V., 2012. Vol. 536. P. S564–S568.
Jakubovi L. et al. Optimization of the induction heating process in order to achieve uniform surface temperature. Procedia Eng. 2016. Vol. 136. P. 125–131.
Munakata T., Someya S., Tanasawa I. Three-dimensional CZ silicon melt flow under induction heating. J. Cryst. Growth. 2005. Vol. 275, № 1–2. pp. 1565–1569.
Fishman O. et al. Induction furnace with improved efficiency coil system: pat. patent US 6542535 B2 USA. patent, 2003. Vol. 2, № 12.
Bioul F., Dupret F. Application of Asymptotic Expansions to Model Two-Dimensional Induction Heating Systems . Part I : Calculation of Electromagnetic Field Distribution. IEEE Trans. Magn. 2005. Vol. 41, № 9. pp. 2496–2505.
Bioul F., Dupret F. Application of asymptotic expansions to model two-dimensional induction heating systems. Part II: Calculation of equivalent surface stresses and heat flux. IEEE Trans. Magn. 2005. Vol. 41, № 9. P. 2508–2514.
Lu L. et al. Numerical study of titanium melting by high frequency inductive heating. Int. J. Heat Mass Transf. Elsevier Ltd, 2017. Vol. 108. pp. 2021–2028.
Galpin J.M.., Fautrelle Y. Liquid-metal flows Induced By Low-Frequency Alternating Magnetic Fields. J. Fluid Mech. 1992. Vol. 239. pp. 383–408.
Kirpo M. et al. Analysis of Experimental and Simulation Data for Liquid Metal Flow in a Cylindrical Container. International Scientific Colloquium: Modelling for Material Processing. 2006.
Ščepanskis M. et al. Solid inclusions in an electromagnetically induced recirculated turbulent flow: Simulation and experiment. Int. J. Multiph. Flow. 2014. Vol. 64. pp. 19–27.
Nikulin I.L. Analisys of possibilities of melt surface cleaning by controlling AMF frequency and distribution. Magnetohydrodynamics. 2017. Vol. 53, № 3. pp. 537–546.
Nikulin I.L. Mathematical modelling of amf geometry and frequency impacts on volume and surface melt flows at induction melting. Magnetohydrodynamics. 2016. Vol. 52, № 4. pp. 513–526.
Nikulin I.L. Numerical simulation of melt flow control by controlling averaged electromagnetic forces generated in high frequency magnetic field. Magnetohydrodynamics. 2016. Vol. 52, № 4.
Musaeva D. et al. Analysis of the Almgsi-alloy structure formed under the influence of low-frequency pulsed Lorentz force. Magnetohydrodynamics. 2017. Vol. 53, № 2. pp. 245–254.
Umbrashko A. et al. Modeling of the turbulent flow in induction furnaces. Metall. Mater. Trans. B Process Metall. Mater. Process. Sci. 2006. Vol. 37, № 5. pp. 831–838.
Ščepanskis M. et al. Analysis of the oscillating behaviour of solid inclusions in induction crucible furnaces. Magnetohydrodynamics. 2012. Vol. 48, № 4. pp. 677–686.
Nikulin I.L., Perminov A. V. International Journal of Heat and Mass Transfer Mathematical modelling of frequency and force impacts on averaged metal flows in alternating magnetic field. Int. J. Heat Mass Transf. 2019. Vol. 128. pp. 1026–1032.
Yang J.R. et al. Thermal characteristics of induction heating in cold crucible used for directional solidification. Appl. Therm. Eng. Elsevier Ltd, 2013. Vol. 59, № 1–2. pp. 69–76.
Yang Y. et al. Experimental and numerical investigation on mass transfer induced by electromagnetic field in cold crucible used for directional solidification. Int. J. Heat Mass Transf. Elsevier Ltd, 2017. Vol. 114. pp. 297–306.
Chen R. et al. Numerical Research on Magnetic Field, Temperature Field and Flow Field During Melting and Directionally Solidifying TiAl Alloys by Electromagnetic Cold Crucible. Metall. Mater. Trans. B Process Metall. Mater. Process. Sci. Springer US, 2017. Vol. 48, № 6. pp. 3345–3358.
Yang Y. et al. Numerical analysis for electromagnetic field influence on heat transfer behaviors in cold crucible used for directional solidification. Int. J. Heat Mass Transf. Elsevier Ltd, 2018. Vol. 122. pp. 1128–1137.
Abbas M., Bossis G. Separation of two attractive ferromagnetic ellipsoidal particles by hydrodynamic interactions under alternating magnetic field. Phys. Rev. E. 2017. Vol. 95, № 6. pp. 1–11.
Nikulin I.L. Analysis of AMF impact on oxide scab rupture and surface cleaning in induction melting technology. Magnetohydrodynamics. 2019. Vol. 55, № 1–2. pp. 141–148.
Buliński P. et al. Effect of turbulence modelling in numerical analysis of melting process in an induction furnace. Arch. Metall. Mater. 2015. Vol. 60, № 3A. pp. 1575–1579.
Buliński P. et al. Numerical and experimental investigation of heat transfer process in electromagnetically driven flow within a vacuum induction furnace. Appl. Therm. Eng. 2017. Vol. 124. pp. 1003–1013.
Courtessole C., Etay J. Flows and mass transfers in two superimposed liquid layers in an induction furnace. Int. J. Heat Mass Transf. Elsevier Ltd, 2013. Vol. 65. pp. 893–906.
Teimurazov A.S., Frick P. G. Numerical study of molten magnesium convection in a titanium reduction apparatus. J. Appl. Mech. Tech. Phys. 2016. Vol. 57, № 7. pp. 1264–1275.
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