Finite Element Analysis of the Transverse Flux Induction Heating of Moving Magnetic Steel Sheets

Transcription

1 The 8 th International Symposium on ADVANCED TOPICS IN ELECTRICAL ENGINEERING The Faculty of Electrical Engineering, U.P.B., Bucharest, May 23-24, 2013 Finite Element Analysis of the Transverse Flux Induction Heating of Moving Magnetic Steel Sheets Virgiliu FIRETEANU 1, Philippe ROEHR 2 1 POLITEHNICA University of Bucharest, EPM_NM Laboratory, 313 Splaiul Independentei, Bucharest, Romania 2 Fives Celes, 89bis rue Principale, Lautenbach, France virgiliu.fireteanu@upb.ro, philippe.roehr@fivesgroup.com Abstract- This paper deals with the study of transverse flux induction heating of magnetic steel sheets in motion based on 3D finite element models. The computations take into account the magnetic non-linearity and the dependence on temperature of the sheet properties. A non-magnetic model of the magnetic non-linear sheets electromagnetically and thermally thin is studied. Keywords: induction heating, transverse flux, magnetic steel steel sheets, finite element analysis x z O y I. INTRODUCTION There is a very important research experience in the last forty years [1 17] in the domain of transverse flux induction heating of metallic sheets in motion. Many authors confirm that the design and optimization of the transverse flux devices and processes are not possible without finite element models. The interest for continuous transverse flux heating of the magnetic steel sheets is confirmed by the development in the last period of industrial testing facilities [18]. Unlike the similar processing of nonferrous sheets, the heating range can be larger, from the ambient temperature until around thousand degrees Celsius. The variation of sheet properties during such a heating is very important. Consequently, it is necessary to take into account the nonlinear behavior of the magnetic steels, the phase transformations at the Curie point and the temperature dependence of the electromagnetic and thermal properties in the evaluation of the heating parameters. The finite elements models and the investigations in this paper confirm such a possibility. II. DESCRIPTION OF A TRANSVERSE FLUX INDUCTION HEATING DEVICE The studied transverse flux device, Fig. 1, contains two identical non-conductive linear magnetic cores, characterized by the relative permeability 1000, length 1175 mm, width 600 mm and thickness 235 mm. Each of the four coils of the inductor, is supplied with the total current 18.9 ka with the frequency f = 1000 Hz. The steel sheet having the thickness 2a and the width 2b = 1000 mm is symmetrically placed with respect the symmetry planes xoy and xoz of the inductor, Fig. 1. The sheet moves with the speed v = 1 m/s along the Ox coordinate. Fig. 1. The main regions of the finite element computation domain. The computation domain of the electromagnetic field contains only the right-upper quarter of the geometry in Fig. 1. The region sheet of the computation domain has the length 3375 mm. The position of the sheet represented in Fig. 1 figure corresponds to the end of the interval of step-bystep in time domain computation. The carbon steel taking into account in this paper is characterised by a non-linear B(H) curve that depends on the temperature θ. The numerical applications consider the B(H,θ) dependencies in Fig. 2, with the saturation magnetic flux density 1.9 T, the initial value of the relative magnetic permeability 1000 and the Curie temperature 760 C. Fig. 2. The nonlinearity temperature dependence B(H,θ) of carbon steel.

2 The curves in Fig. 3 are the dependences on temperature of the carbon steel resistivityρ, thermal conductivity λ and heat capacity γc. III. Fig. 3. Temperature dependences ρ(θ), λ(θ), γc(θ) of the carbon steel. DIFFERENTIAL MODEL OF THE MAGNETOHARMONIC TRANSIENT THERMAL TRANSLATING MOTION COUPLING Magnetic non-linearity and the equivalent harmonic electromagnetic fields. The study of the electromagnetic field generated in computation domains with magnetic linear behavior by harmonic current or voltage sources is the object of a frequency domain analysis. The real time dependent quantities are replaced by its complexe images. In the magnetic non-linear regions, where the magnetic flux density B has a harmonically variation in time, the intensity of the magnetic field H has no a similar harmonic variation in time. Vice versa, in case of a harmonically time variation of H, the magnetic flux density B it is not harmonically in time. Consequently, a non-harmonics electromagnetic field can be the object of a step-by-step in time domain computation. Such a computation requires very high computation times. An equivalent harmonic electromagnetic field can be defined in case of computation domains including magnetic non-linear regions. In such cases the real non-linear B(H) dependence is replaced by an equivalent curve B(H e ), if B has a harmonic variation in time, by an equivalent curve B e (H) if H has a harmonic variation in time, or by a combination B e (H e ) of the two previous equivalences [19]. Differential model of the harmonic electromagnetic field. The frequency domain analysis of the harmonic electromagnetic field of pulsation ω = 2πf in the transverse flux devices concerns the evaluation in the solid conductor regions with the resistivity ρ and magnetic permeability μ of the couple of the two complex unknowns, the electric vector potential T, which defines the density of the induced currents J 2 = curlt, and the scalar magnetic potential Φ. These unknowns satisfy the equations: curl[ρcurlt] + jωμ(t - gradφ) = 0, (1) divt = 0, div[μ(t - gradφ)] = 0 In the magnetic cores, which are nonconductive and nosource regions with high magnetic permeability, the unknown Φ that defines the magnetic flux density B = - μgradφ, satisfies the equation:. div[μgradφ)] = 0 (2) In the non-conductive and non-magnetic regions, which are source regions of stranded coil type, in the insulations and in the surrounding air, the magnetic field strength is expressed by the formula H = H 0 gradφ r. The component H 0 of this field, generated in the free space by the current density J 1 in the inductor coils, is given by the Biot-Savart formula: 1 J = 1 x r H dv (3) 0 4π 3 V r The second component of the magnetic field strength in the non-conductive and non-magnetic regions is computed through the reduced scalar magnetic potential Φ r, which satisfies the equation: div(μ 0 gradφ r ) = div(μ 0 H 0 ). (4) The TΦ-Φ-Φ r model of the electromagnetic field, respectively the scalar formulation of the electromagnetic field is the most convenient for numerical computations in transverse flux devices. Time dependent thermal field. The time dependent field θ(r, t) of the sheet temperature heated by the induced power density p = ρj 2 2 satisfies the differential equation: γc θ/ t = p + div(λgradθ) (5) The initial field of the sheet temperature is θ(r, 0) = 20 C. The computation of the transient temperature

3 field in the sheet takes into account the thermal losses by convection and radiation on the sheet surface. Coupling harmonic electromagnetic field transient thermal field - sheet translating motion. The interest in transverse flux heating is represented by the steady state operation parameters of the heating device, respectively by the steady state temperature of the sheet with respect the inductor. The study of the sheet heating starts from an initial geometry of the computation domain and an initial field of the sheet temperature. The end of this study corresponds to the moment when the steady state of the sheet temperature in the inductor coordinate system is reached. The step-by-step in time domain computation of heating in transverse flux based on the coupling between the harmonic electromagnetic field and the transient temperature field of the sheet in motion, updates each time step the geometry of the computation domain and the sheet properties. IV. ANALYSIS OF THE RESULTS The results in the first line of Table 1 correspond to the sheet thickness 2a = 0.5 mm, the others to 4.0 mm. P is the active power in the sheet and Q is the reactive power of the system. The sheet temperatures at the exit from the inductor, Table 1, are as follows: θ 1 in the central point y = 0, Fig. 1, θ 2 at the coordinate y = b/2 and θ 3 on the sheet border y = b. v [m/s] TABLE I COMPUTATION DATA AND MAIN RESULTS P Q θ 1 θ 2 [MW] [MVAr] [ C] [ C] θ 3 [ C] Thickness 2a = 0.5 mm, speed v = 0.8 m/s. The time variation of the power induced in the sheet, Fig. 4, shows that the electromagnetic and thermal steady states of sheet heating with respect the inductor is globally reached after 1.25 s, respectively after 1 m of sheet motion. Nevertheless, for a better image of the moveable sheet temperature, the limit t f = 2.2 s of computation time was considered. The minimum value of the penetration depth in the sheet steel, δ min = mm, corresponds to the value 1000 of the relative magnetic permeability and to the minimum value μωm of the resistivity. Thus, during the sheet heating the maximum value of the ratio 2a/δ is (2a/δ) max = first time step last time step - steady state Fig. 5. Maps of the induced current density mm sheet thickness. first time step steady state Fig. 6. Maps of the magnetic flux density mm sheet thickness. first time step Fig. 4. Time variation of the active power in the sheet region. The figures 5, 6 and 7 show the induced current density, the magnetic flux density and the sheet temperature after the first time step t = Δt = s and at the study end t f. These three quantities have practically unchanged values on the sheet thickness. There where the temperature is under the Curie point, the magnetic flux density, Fig. 6, has values close to the saturation. steady state Fig. 7. Maps of the sheet temperature mm sheet thickness. In the part of the sheet placed in the inductor air gap, there where the magnetic field is non-negligible, the value of the penetration depth is higher, because the relative magnetic permeability is under The maximum of the penetration

4 depth is reached there where the temperature has the highest value, over the Curie point. During the heating transient, the value of the penetration depth depends also on the temperature, whose value influences the resistivity. Since the increase of the temperature determines the increase of the steel resistivity, the penetration depth increases when the temperature increases and the ratio 2a/δ decreases. The maps of the two sheet properties in Fig. 8 reflect the correlation between the dependences of steel properties in Figs. 2 and 3 and the steady state temperature in Fig. 7. Only a small part to the sheet lateral border is heated over the Curie, Fig. 11, the most part of the sheet rests magnetic after heating. The permeability of this part of the sheet is higher or much higher than one and the resistivity has a lower value than in the application with 0.5 mm sheet thickness. As consequence, the penetration depth of the electromagnetic field in this part of the sheet it is not eight times or more higher than in the previous application and the ratio 2a/δ is higher than This explains the differences between the two maps of the current density in Fig. 10. relative magnetic permeability plane z = 0.00 mm heat capacity Fig. 8. Steady state properties of the sheet with 0.5 mm thickness. The transversal profile of the sheet temperature at the end of heating, Fig. 9, corresponds to a path placed 72.5 mm distance from the inductor border. Taking into account the extreme values θ_max = 1121 C, θ_min = 811 C and the temperature in the middle of the sheet, θ_y0 = 1086 C, the gap of the transversal non-uniformity of the sheet heating is Gap_θ = (θ _max θ_min)/θ_y0 = 28.5 %. plane z = 0.25 mm Fig. 10. Steady state current density in the sheet mm sheet thickness. plane z = 0.00 mm Transversal path after the sheet exit from inductor Fig. 9. Transversal profile of the temperature mm sheet thickness. Thickness 2a = 4.0 mm, speed v = 0.1 m/s. The maximum value of the ratio between the sheet thickness and the minimum value of the penetration depth of the electromagnetic field, (2a/δ) max = 19.52, is now eight time higher than in the previous application. There are non-negligible differences between the steady state maps of the current density in the symmetry plane z = 0.00 mm of the sheet and on the sheet surface z = 2.00 mm, Fig. 10. Nevertheless, the corresponding steady state maps of the temperature, Fig. 11, are similar. plane z = 2.00 mm Fig. 11. Maps of sheet temperature 4.00 mm sheet thickness. While there are differences between the values of the current density across the sheet thickness, the transversal profile of sheet heating, Fig. 12, in this new application is practically the same in the symmetry plane z = 0.00 mm and on the sheet surface z = 2.00 mm. The good temperature uniformity on the sheet thickness is the effect of the thermal transfer phenomenon. The new sheet of 4.00 mm thickness is a thermally thin sheet. The transversal profile in Fig. 12 shows that the sheet is heated over the Curie point only on a stripe of 67 mm width

5 to the lateral sides. The extremes of sheet temperature are θ_max = 1201 C, θ_min = 305 C and the temperature in the middle of the sheet is θ_y0 = 365 C. The gap of the transversal non-uniformity of heating for 4.00 mm sheet thickness, Gap_θ = %, has a very high value. magnetic sheets transverse flux induction heating. It rest to see the limits of the good concordance between the results for the real magnetic non-linear model of the sheet and the results for the non-magnetic sheet model. Transversal path after the sheet exit from inductor Fig. 12. Transversal profile of the 4.00 mm sheet temperature. magnetic non-linear model V. STUDY OF THE TRANSVERSE FLUX HEATING OF MAGNETIC SHEETS USING A NON-MAGNETIC SHEET MODEL The very important computation time required by the numerical solution of the transverse flux heating of moving sheets taking into account the temperature dependent magnetic non-linearity of the sheet steel is first of all associated with the treatment of the magnetic non-linearity. Consequently, it is interesting to compare the results obtained with the real model B(H, θ) in Fig. 2, characterizing the temperature dependent magnetic non-linearity, with the results for a non-magnetic model B = μ 0 H of the sheet. (a) Thickness 2a = 0.5 mm, v = 0.8 m/s. The two maps of the induced current density in Fig. 13 and the maps of the sheet steady state temperature, Fig. 14, are practically identical. The transversal profile of the sheet temperature after heating for the non-magnetic sheet model is also practically identical with those in Fig. 9. non-magnetic model Fig. 14. Maps of sheet temperature mm sheet thickness. (b) Thickness 2a = 4.0 mm, v = 0.1 m/s. This application considers a sheet with the thickness eight times higher and the speed eight times lower than the previous one. The two maps of the current density in Figs. 15-1, 15-2show nonnegligible differences between the two groups of two maps, the first group for the magnetic non-linear model and the second for the non-magnetic sheet model. The two maps of the sheet temperature in Fig. 16 are different.. plane z = 0.00 mm, magnetic non-linear model magnetic non-linear model non-magnetic model Fig. 13. Current density maps mm sheet thickness. The computation time for the non-magnetic sheet model is much lower than for the real temperature dependent magnetic non-linear model. Consequently, the non-magnetic model represents a very attractive option in the computation of plane z = 2.00 mm magnetic non-linear model Fig Current density maps for the magnetic non-linear model mm sheet thickness, The transversal profile of the sheet temperature after heating for the non-magnetic sheet model, Fig. 17, is different from the profile in Fig. 12, which corresponds to the magnetic non-linear sheet. In this case θ_max = 1048 C, θ_min = 162 C and θ_y0 = 243 C.

6 between the finite element numerical solution and the stepby-step computation in time domain usually requires important computation times. If in the cold state the magnetic nonlinear sheets are electromagnetically thin, the computation of transverse flux heating with a non-magnetic sheet model, which is much more efficient, gives satisfactory enough results. plane z = 0.00 mm plane z = 2.00 mm Fig Current density maps for the non-magnetic sheet model 4.0 mm sheet thickness plane z = 0.00 mm, magnetic non-linear model plane z = 0.00 mm, non-magnetic model Fig. 16. Temperature maps mm sheet thickness. Transversal path after the sheet exit from inductor Fig. 17. Temperature profile for the non-magnetic model mm sheet thickness. The comparison of the results for the sheet thickness 0.5 mm and 4.0 mm of suggests precautions in the use of the non-magnetic sheet model of in the computation of transverse flux heating of magnetic sheets. The results for the non-magnetic sheet model become more and more approximates when the ratio between the sheet thickness and the penetration depth of the electromagnetic field increases. CONCLUSIONS It is essential to consider the magnetic non-linearity and the temperature dependence of properties in the study the transverse flux induction heating of moveable magnetic nonlinear sheets when the heating starts at ambient temperature and stops over the Curie point. The coupling REFERENCES [1]. W. B. Jackson, Transverse flux induction heating of flat metal products, Proc. of 7 th UIE Congress, Warsaw, Poland, [2] W. Andree, D. Schulze and Z. Wang, 3D eddy current computation in the transverse flux induction heating equipment, IEEE Trans. on Magnetics, Vol. 30, pp , Sept [3] F. Dughiero, M. Forzan and S. Lupi, 3D solution of electromagnetic and thermal coupled field problems in the continuous transverse flux heating of metal strips,, IEEE Trans. on Magnetics, Vol. 33, pp , Mar [4] T. Tudorache, V. Fireteanu and J.C. Bourhis, 3D numerical modeling of new structures for transverse flux heating of metallic sheets, Proc. of ISH-98 Symposium, Padua, Italy, [5] Z. Wang, W. Huang, W. Jia, Q. Zhao, Y. Wang, W. Yan, D. Schulze, G. Martin and U. Luedtke, 3D multifields FEM computation of transverse flux induction heating for moving strips, IEEE Trans. on Magnetics, Vol. 35, pp , May [6] V. Fireteanu and T. Tudorache, Electromagnetic forces in transverse flux induction heating,, IEEE Trans. on Magnetics, Vol. 36, pp , July [7] J. Nerg and J. Partanen, Numerical solution of 2D and 3D induction heating problems with nonlinear material properties taken into account,, IEEE Trans. on Magnetics, Vol. 36, pp , Sept [8] Y. Neau, B. Paya, T. Tudorache and V. Fireteanu, Numerical evaluation and experimental validation of eddy currents and electromagnetic forces in transverse flux induction heating of magnetic steel sheets, Proc. of EPM 2000 Symposium, pp , Nagoya, Japan, [9] V. Fireteanu, A. Geri, T. Tudorache and G.M. Veca, Transverse flux induction heating: Comparison between numerical models and experimental validation, Proc. of HIS-01 Symposium, Padua, Italy, [10] X. G. Yang, Y. H. Wang and W. Yan, Eddy current and temperature field computation in transverse flux induction heating equipment for galvanizing line, IEEE Trans. on Magnetics, Vol. 37, pp , Sept [11] C. Monzel and G. Henneberger, Temperature solver for highly nonlinear ferromagnetic materials for thin moving sheets in transversal flux induction heating,, IEEE Trans. on Magnetics, Vol. 38, pp , Mar [12] V. Fireteanu and T. Tudorache, Numerical simulations of continuous induction heating of magnetic billets and sheets, COMPEL Revue, Vol. 22, pp 68-78, [13] A. Nikanorov, H. Schulze, B. Nacke, M. Zlobina, S. Galunin and Y Blinov, Non linear effects in transverse flux heating systems, Proc. of HES-04 Seminar, Padua, Italy, [14] V. Fireteanu and T. Tudorache, Couplings of electromagnetic field formulations in finite element analysis of transverse flux induction heating systems, Proc. of ISEF-2005 Symposium, pp , Baiona, Spain, [15] T. Tudorache and V. Fireteanu, Magneto-thermal-motion coupling in transverse flux heating, COMPEL Revue, Vol. 27, pp , [16] V. Fireteanu, M. Popa and P. Taras, Magnetoharmonic - transient thermal - translating motion finite element analysis of scanning induction heating of sheets in transversal flux, Proc. of ISEF-2009 Symposium, Arras, France, [17] V. Fireteanu, M. Popa, S. Pasca and P. Taras, Transversal flux scanning induction heating of magnetic nonlinear steel sheets with temperature dependent properties, Proc. of EPM-2012 Conference, Beijing, China, [18] EcoTransFlux project FIVES CELES [19] FLUX V11.1 Users Guide CEDRAT Group, France.