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Dimensionless Analysis and Numerical Modeling of Rebalancing Phenomena During Levitation

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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

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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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Authors and Affiliations

  1. Faculty of Metallurgical and Energy Engineering, State Key Laboratory of Complex Non-Ferrous Metal Resources Clean Utilization, Kunming University of Science and Technology, Kunming, 650093, China

    Lei Gao (PhD Candidate, Visiting Student) & Zhe Shi (Professor)

  2. Department of Materials Science and Engineering, Process Metallurgy and Modeling Group (PM2G), University of Toronto, Toronto, ON, M5S 3E4, Canada

    Lei Gao (PhD Candidate, Visiting Student), Donghui Li (Research Associate), Alexander McLean (Professor) & Kinnor Chattopadhyay (Assistant Professor)

Authors
  1. Lei Gao
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  2. Zhe Shi
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  3. Donghui Li
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  4. Alexander McLean
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  5. Kinnor Chattopadhyay
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Corresponding author

Correspondence to Kinnor Chattopadhyay.

Additional information

Manuscript submitted December 13, 2015.

Appendix

Appendix

The calculation conditions in this study are listed in Table AI as follows:

Table AI Calculation Conditions in this Study
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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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