Loïc Rondot

Vincent Mazauric

Yves Delannoy

Gérard Meunier

Lo'ic Rondot

Yves Delannoy2

Gerard Meunier

Electromagnetic model in finite volumes for magnetohydrodynamic breaking application

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• either a FE-CFD code, which involves time-consuming calculations as soon as non-linearity occurs;

• or an hybrid method combining a FV and a FE, thereby sacrificing the high level of integration and the accuracy achieved with a single mesh (2).

Thus, the search for a conunon, effective and integrated model calls for a single nwnerical method for the two phenomena. Tn this work, we opted for an electromagnetic model based on the FV method.

The FV method is based on local conservation, i.e. the dissipative term matches the variation over time in any elementary volwne and the flux exchanged at it boundary. For example, the local power balance is represented by:

H• d,B +�= � ( I) un�y--;1rm 1��:i, dis����1tive
Thanks to the Poynting's theorem, the conservation of the electromagnetic power density enforces the Faraday's law provided that the Max\.vell-Amperc equation is checked [3 J.

Hence, the continuity laws to consider to solve a magnetodynamic problem are as follow:

• the conservation of the current density;

• the conservation of the magnetic flux density;

• the conservation of the power density.

II. MODELING

The case shown at figure I represents a metal slab excited by a timc-hannonic magnetic field H. The aim is to validate with the CFD Fluent® code, the calculation of the induced field in the slab only using a ID model of a material with a uniform magnetic penneability. The solution of U1e Maxwell Faraday equation is derived from the local conservation of power density (I). ----------; ----------;----------;---------

H () t-

Ill. RESULTS

As expected in the classical skin depth effect, the excitation oscillating around the slab yields an evanescent field in the conductor (figure 2). The agreement with 1 D analytical solution is lower than 0. 7% mainly located in the skin depth.

[

-Magnetic flCld • • •• • Relative error I E 50E<02 +--�--- � ,, � .!! OOE.00 &i c i' ::E -6.0E-<>2 •-. .. •••••••••••• 0,,.,,, 0,6% 0,5'4 0,•'4 0,1'4 -10E'°3 �-----------���-----< o.� • •
relative position (xldelta) Fig. 2. Induction in a metal slab (continuous) and error introduced by an FV calculation in comparison to the analytical solution (dashed). (3) V. Mazauric, "From thcnuostatistics to Ma,,.wcll's equations: A variational approach of electromagnetism," IEEE Transactions on Magnetics, vol. 40, pp. 945 -948, 2004.

IV. CONCLUSION

  models. Whereas conventional Finite Volwne (FV) Computational Fluid Dynamics codes enforce local conservation of mass, momcntwn and energy [1]; low frequency electromagnetic sofhvare minimizes a global energy fw1ctional within the finite elements method (FEM). Hence, magnetohydrodynamics problems are currently resolved by using:

1 Fig. l .

 1l Fig. l. Diagram of the slab along with the applied magnetic licld and its FV mesh where the conservation laws are expressed.

  Faraday's law of induction may be derived from the pO\.\'Cr density conservation. Its implementation within a 30 finite volwne electromagnetic software is under progress and will allow: • to model within a strong coupling both fluid and electrical aspects of the plasma physics, • to consider parallel computation of the conservation equations on the finite volwnes. Subsequent improvements and productivity in the design of breaking products (MV and L V circuit-breaker) are expected. V. REFERENCES [I) P. Chevrier, M. Barrault, and C. Ficvet, "Hydrodynamic model for electrical arc modcling," IEEE Transactions on Power Delive1y, vol. 11. pp. 1824 -1829. 1996. f2] Ian J. Lyttle "Use of l\1PCCI to Perform Multidisciplinary Analyses for Electrical Distribution E quipment", American Institute of Aeronautics and Astronautics, paper 0122, 2005.