IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010 3217 Coupled Magneto-Thermal FEM Model of Direct Heating of Ferromagnetic Bended Tubes Alexandr Aliferov , Fabrizio Dughiero , and Michele Forzan Novosibirsk Technical State University, Novosibirsk, Russia Department of Electrical Engineering, University of Padova, 35131 Padova, Italy Scalar and vector magnetic formulations have been applied to solve the current distribution in a 50 Hz direct resistance heating system of ferromagnetic tubes. The scalar formulation driven via an external circuit has been also applied to solve the Time-Harmonic EM part of the problem coupled with the thermal transient: the computed warm up curves have been compared with experimental data. Index Terms—Eddy currents, electromagnetic heating, finite element methods, resistance heating. I. INTRODUCTION D IRECT resistance heating of steel tubes is industrially achieved by supplying strong 50 Hz currents directly to the workpiece by means of suitable contact systems. The cur- rent density distribution inside a straight tube depends upon the skin effect, while for bended tube it is influenced also by the ring effect. The thermal sources for the heating are the Joule losses, which depend on the square of the current density: con- sequently the unbalanced distribution of the current density due to the ring effect produces a significant overheating in the inner part of the curved zone. In previous investigations the possibility of balance the ring effect by means of properly designed lam- inated cores has been analyzed [1]. The proposed solution has been realized in a laboratory setup and experimental measure- ments have been used to verify the reliability of the numerical models applied to this case study. Numerical models have been developed to solve the electro- magnetic problem by means of 3-D finite element solution: be- cause there is only one conductor carrying the source current, the A-AV formulation has been implemented applying Dirichlet conditions for the scalar electric potential on the edges of the conductor which means that a voltage has been applied between the extremities that are supposed to be at the same potential. The solution of the EM problem has been also implemented by means of a magnetic scalar coupled with the electric vector potential formulation. This formulation reduces substantially the computational requirements so that also the 3-D coupled electromagnetic thermal solution can be achieved in reasonable times [2]. Moreover the scalar formulation can be more effi- ciently driven by an external electrical circuit, allowing to feed a constant current in the model instead of an applied voltage [2], [3]. Some comparisons between the results obtained by means of the A-AV formulation with the ones made with are presented as well as some comparisons between the computed Manuscript received December 23, 2009; revised March 02, 2010; accepted March 11, 2010. Current version published July 21, 2010. Corresponding au- thor: M. Forzan (e-mail: Michele.forzan@unipd.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2046479 Fig. 1. Schematic of the bended tube laboratory setup. The ferromagnetic tube (‘A’ region) is surrounded by a thermal insulator (‘B’). The ‘C’ region describes the laminated yoke, placed to minimize the ring effect. temperature distribution and the experimental measurements re- sulting from some warm up processes carried out controlling the current intensity during the heating transients. II. COMPUTATION MODELS The model represents the laboratory setup built in NSTU (Novosibirsk State Technical University) and it is constituted by a ferromagnetic tube, a laminated yoke and a thermal insulator that envelops the tube (Fig. 1). Only a part of the real system has been considered, applying tangential magnetic field conditions on the boundaries, repre- sented in Fig. 2 by the ‘ ’ and ‘ ’ lines. These faces are coincident with the terminals of the tube conductor. On the plane where the tube axis is laying, normal magnetic field and tangen- tial electric field boundary conditions have been posed. In the model the entire domain can be subdivided into: , the electrical conductor, , the insulating part inside the conductor, the air region, the magnetic region where there is the lamination, considered as an electric insulator. As mentioned above, the electromagnetic solution, in partic- ular the distribution of the current density inside the tube, has been obtained by means of two different numerical formula- tions. The magnetic vector potential coupled with the scalar electric potential V formulation leads to a very accurate solu- tion for the current distribution since it can be obtained, in the 0018-9464/$26.00 © 2010 IEEE