Abstract
Electromagnetic levitation (EML) has proved to be a powerful tool for research activities in areas pertaining to materials physics and engineering. The customized EML setups in various fields, ranging from solidification to nanomaterial manufacturing, require the designing of stable levitation systems. Since the elevated droplet is opaque, the most effective way to research on EML is mathematical modeling. In the present study, a 3D model was built to investigate the rebalancing phenomenon causing instabilities during droplet melting. A mathematical model modified based on Hooke’s law (spring) was proposed to describe the levitation system. This was combined with dimensionless analysis to investigate the generation of levitation forces as it will significantly affect the behavior of the spring model.
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Abbreviations
- F :
-
Levitation force, N
- ∇ ×:
-
Curl operator
- H :
-
Magnetic field intensity vector, T
- J :
-
Total current density vector, A/m2
- Js :
-
Applied source current density vector, A/m2
- Je :
-
Induced eddy current density vector, A/m2
- Jv :
-
Velocity current density vector, A/m2
- D1 :
-
Electric flux density vector, C/m2
- t :
-
Time, s
- E :
-
Electric field intensity vector, N/C
- \( \nabla \cdot \) :
-
Curl operator
- B :
-
Magnetic flux density vector, T
- ρ 1 :
-
Electric charge density, C/m3
- N :
-
Vector of shape functions
- J* :
-
Complex conjugate of J
- G :
-
Gravity, N
- c:
-
Stiffness of the spring, N/m
- λ:
-
The static deformation of the spring, m
- Fresultant :
-
Resultant upward force, N
- ω0 :
-
Angular frequency under free vibration, Hz
- f s :
-
Frequency, Hz
- ω’ :
-
Angular frequency under underdamping, Hz
- γ :
-
Damping coefficient
- y:
-
Position of the droplet (gravity direction), m
- A:
-
Maximum amplitude, m
- α:
-
Oscillation phase
- C 0 , C :
-
Constant value
- η :
-
Dimensionless number
- I :
-
Current in the coil, A
- μ :
-
Absolute permeability, N A−2
- θ :
-
Angle
- Lei :
-
Dimensionless number
- a,b,c,d,e,f,g,e’,g’:
-
Power exponents
- ρ :
-
Electrical resistivity, Ω m
- f :
-
Angular frequency, Hz
- δ :
-
Skin depth, m
- X :
-
Horizontal distance between droplet center to coil center, m
- Y :
-
Vertical horizontal distance between droplet center to coil center, m
- D :
-
Diameter of the droplet, m
- EML:
-
Electromagnetic levitation
- Y :
-
Vertical horizontal distance between droplet center to coil center, m
- Y i , \( Y_{1i} \), \( Y_{2i} \), \( Y_{3i} \) :
-
Vertical horizontal distance between droplet center to coil center
References
E. C. Okress, D. M. Wroughton, G. Comenetz, P. H. Brace, and J. C R Kelly: J. Appl. Phys., 1952, vol. 23(5), pp. 545–52.
S. Binder, P. K. Galenko, and D. M. Herlach: J. Appl. Phys., 2014, vol. 115(053511), pp. 1–11.
A. Seidel, W. Soellner, and C. Stenzel: J. Phys. Conf. Ser., 2011, vol. 327(012015), pp. 1–11.
K. Zhou, H. P. Wang, and B. Wei: Chem. Phys. Lett., 2012, vol. 521, pp. 52–54.
Zhou K, Wang HP, Wei B (2013) Philos. Mag. Lett. 93(3): 138–41
A. Kermanpur, B. N. Rizi, M. Vaghayenegar, and H. G. Yazdabadi: Mater. Lett., 2009, vol. 63(5), pp. 575–77.
J. S. Luo, K. Li, X. B. Li, Y. J. Shu, and Y. J. Tang: J. Alloys Compd., 2014, vol. 615, pp. 333–37.
J. Siwka: ISIJ Int., 2008, vol. 48(4), pp. 385–94.
J. Siwka and A. Hutny: Metalurgija, 2009, vol. 48(1), pp. 23–27.
A. McLean: Metall. Mater. Trans. B, 2006, vol. 37, pp. 319–32.
Zuliani DJ, McLean A (1979) Can. Metall. Q. 18: 323–31
S. R. Berry, R. W. Hyers, L. M. Racz, and B. Abedian: Int. J. Thermophys., 2005, vol. 26(5), pp. 1565–81.
V. Bojarevics and R. W. Hyers: JOM., 2012, vol 64, pp. 1089–96.
L. Feng and W. Shi: Metall. Mater. Trans. B, 2015, vol. 46(4), pp. 1895–1901.
J. Lee, X. Xiao, D. M. Matson, and R. W. Hyers: Metall. Mater. Trans. B, 2014, vol. 46, pp. 199–207.
S. Spitans, A. Jakovičs, E. Baake, and B. Nacke: Magnetohydrodynamics, 2011, vol. 47(4), pp. 461–73.
L. Gao, Z. Shi, D. Li, Y. Yang, G. Zhang, A. McLean, and K. Chattopadhyay: Metall. Mater. Trans. B, 2015, published online.DOI: 10.1007/s11663-015-0457-0.
Matula R.A. (1979) J. Phys. Chem. 8(4):1147
L. Gao, K. Chattopadhyay, G. Zhang, Y. Yang, Z. Shi, and A. McLean: in Conf. Metall., Canadian Institute of Mining, Metallurgy and Petroleum, Toronto, 2015, pp. 1–11.
Acknowledgements
The thanks are to the SimuTech Group, ANSYS Inc. for their support toward the mathematical modeling research performed in this study.
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Manuscript submitted December 13, 2015.
Appendix
Appendix
The calculation conditions in this study are listed in Table AI as follows:
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Gao, L., Shi, Z., Li, D. et al. Dimensionless Analysis and Numerical Modeling of Rebalancing Phenomena During Levitation. Metall Mater Trans B 47, 1905–1915 (2016). https://doi.org/10.1007/s11663-016-0608-y
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DOI: https://doi.org/10.1007/s11663-016-0608-y