The Calculation of Transport Phenomena in Electromagnetically Levitated Metal Droplets

Abstract

A mathematical representation has been developed for the electromagnetic force field, the fluid flow field, the temperature field (and for transport controlled kinetics) in a levitation melted metal droplet. The technique of mutual inductances was employed for the calculation of the electromagnetic force field, while the turbulent Navier-Stokes equations and the turbulent convective transport equations were used to represent the fluid flow field, the temperature field and the concentration field. The governing differential equations, written in spherical coordinates, were solved numerically.

The computed results were found to be in good agreement with measurements reported in the literature, regarding the lifting force and the average temperature of the specimen.

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Abbreviations

A:

vector potential (wb/m)

B:

magnetic flud density (wb/m2)

C:

concentration of diffusing species (kg/m3)

Cp:

specific heat of levitated sphere (j/kg °c)

De:

effective mass diffusivity (m/s)

F:

electromagnetically lifting force (N)

Fs:

electromagnetic stirring force (N/m3)

g:

gravitational acceleration (m/s2)

I:

coil current (A)

J:

induced eddy current (A/m2)

K:

turbulent kinetic energy (m2s2)

m:

mass of levitated sphere (kg)

p:

pressure (N/m2)

P:

total Joule heat generation (w)

r:

radial coordinate

S:

cross section area of elementary circuit (m2)

T:

temperature (°C)

µ:

magnetic permeability (H/m)

µe:

effective viscosity (kg/m.s)

αe:

effective thermal diffusivity (m2/s)

ρ:

density of levitated sphere (kg/m3)

ε:

turbulent energy dissipation (m2s3)

σ:

electric conductivity (Ohm.m)−1

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Acknowledgments

The authors wish to thank the National Aeronautics and Space Administration for support of this investigation through Grant-7645 given to the Materials Processing Center at MIT. Thanks are also due to Drs. C.W. Chang and T. Frost of the General Electric Co., Space Science Center, Valley Forge, PA for providing the program for calculating the electromagnetic force field.

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El-Kaddah, N., Szekely, J. The Calculation of Transport Phenomena in Electromagnetically Levitated Metal Droplets. MRS Online Proceedings Library 9, 191–198 (1981). https://doi.org/10.1557/PROC-9-191

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