1. Heating of Ferromagnetic Materials up to Curie Temperature by Induction Method Ing. Dušan MEDVEĎ, PhD. Pernink, 26. May 2009 TECHNICAL UNIVERSITY OF KOŠICE FACULTY OF ELECTRIC ENGINEERING AND INFORMATICS Department of Electric Power Engineering

2. Contents General formulation of induction heating process Description of physical properties of ferromagnetic materials -Relative permeability and methods for its determination Mathematical model of induction heating -Modeling of electromagnetic field -Modeling of thermal field Solution of coupled problem of electromagnetic and thermal field Conclusion

3. General formulation of induction heating process Induction heating is electric heating of ferromagnetic metal material in the alternating electromagnetic field. The source of electromagnetic field is every conductor, that is flowed by alternating current. Fig. 1 Scheme of cylindrical induction heating device

4. Mathematical description of induction heating (1) (2) (3) (4) (5)

5. Basic types of algorithms for coupled problem solving Algorithm for solving of supposed problem as deeply coupled problem Algorithm for solving of supposed problem as quasi coupled problem Algorithm for solving of supposed problem as weekly coupled problem

6. Description of physical properties of ferromagnetic materials Magnetic materials – expressed by significant spontaneous magnetism Strong magnetism – in the material structure there are not mutually compensated magnetic moments of atoms Structure types of magnetic moment arrangement – ferromagnetic, antiferromagnetic, ferrimagnetic, metamagnetic Magnetic arrangement exists always up to critical temperature, for ferromagnetic material up to Curie temperature C, for ferrimagnetic and antiferromagnetic up to Neel temperature N. Above this temperature the materials become paramagnetic and their magnetic susceptibility  decrease by increased temperature.

7. Basic properties of ferromagnetics The uniform feature of ferromagnetic material is the occurrence of the non-zero resulting magnetic moment. Fig. 2 Magnetisation and hysteresis curve of ferromagnetics Fig. 3 Typical processes during the magnetisation of ferromagnetic

8. Change of ferromagnetic material properties by induction heating process Magnetic permeability  Initial permeability  p Magnetic susceptibility  Electric resistivity  (eventually electric conductivity  ) Heat capacity c Coefficient of thermal conductivity Magnetic dipole moment m etc.

9. Relative permeability and methods for its determining Relative permeability  r is the non-unit quantity, that characterizes the magnetic properties of material. It is defined by magnetic permeability , permeability of vacuum  0 and magnetic susceptibility  m according to: (6) where:  0 permeability of vacuum;  0 = 4. .10 -7 T.m.A -1  1,26.10 -6 H.m -1 ;  m magnetic susceptibility (relative susceptibility) [  ] (7)

10. Determining of  r by Vasiljev method This method includes the dependence of relative permeability on temperature and also on external magnetic field  r = f(H, ): (8) where:  r20 relative permeability  r at temperature 20 °C for given magnetic intensity H  ( )correction function dependent on temp. (graphically) Fig. 4 Example of correction function  ( ) for steel 40H [  ][  ]

11. Approximation of BH-curve It is possible to evaluate the necessary values from the measured values for average magnetization curve (arithmetic average) and consequently these values will serve for approximation by useful curve, for example goniometric function arctg(x): (9) where:B S saturation magnetic flux density; constant for given material; [T] H S saturation magnetic intensity; constant for given material; [A.m -1 ]; Hinput magnetic intensity, i.e. independent quantity; [A.m -1 ] [  ][  ]

12. Approximation of correction function  ( ) Correction function  ( ) can be simplified for example by hyperbola: (10) where: temperature, at which is determined  r ( ); [°C] cconstant, dependent on curve inclination  ( ); [°C] C temperature of magnetic change, Curie temp.; [°C] Fig. 5 Graph of correction function  ( ) approximated by hyperbola according to (10) for various values of constant c [  ][  ]

13. Correction function  ( ) can be simplified for example by quarter- ellipse: (11) where: temperature, at which is determined  r ( ); [°C] Fig. 6 Graph of correction function  ( ) approximated by quarter- ellipse according to (11) for various values of half-axes a and b badjacent half-axis of ellipse; [  ] a main half-axis of ellipse; [  ]  ( )correction function; [  ] C temperature of magnetic change, Curie temp.; [°C] Correction function  ( ): (12) [  ]

14. Correction function  ( ) can be simplified also by exponential: (13) were:cconstant dependent on curve inclination [°C -1 ] Fig. 7 Graph of correction function  ( ) approximated by exponential according to (13) for various values of constant c Similarly as in previous case, the given expression (13) can be consider in the temperature range < C. [  ]

15. The final expression of relative permeability dependence on and H approximated by various curves -approximation of  r ( ) by quarter-ellipse by (11) a  r20 (H) by (9): -approximation of  r ( ) by hyperbola by (10) a  r20 (H) by (9): -approximation of  r ( ) by exponential by (13) a  r20 (H) by (9): (14) (15) (16) [  ][  ] [  ][  ] [  ][  ]

16. Mathematical model of induction heating Induction heating is the complex of electromagnetic, thermal and metallurgical processes.

17. Mathematical model of electromagnetic field (Ampere law)(17) (Faraday law)(18) (Gauss law)(19) (21) (20)

18. Modification of Maxwell equations (21) (22) (23) (24) Modification of expression (22) for 2D rectangular axis system: Modification of expression (22) for polar axis system:

19. Mathematical model of thermal field (25) Modification of expression (25) for 2D rectangular axis system: Modification of expression (25) for polar axis system: (26) (27) (28)

20. Numerical method of mathematical modeling of induction heating Numerical methods -Utilizing of multiphysical modeling programs: ANSYS, COMSOL FEMLAB, FLUX3D, Opera-3D, Celia, ABAQUS, etc. -Advantages, disadvantages, application Finite differences method Finite element method

21. Finite difference method (FDM) The calculation algorithm consists of substitution of all partial differential equations of electromagnetic and thermal field by difference equations, which values are derived from the closest neighboring nodes of investigated area. This method allows to approximate the partial differential equations point by point. (30) (29)

22. Solution of 1D thermal field [W.m -3 ] (31) 1-dimensional thermal field inside of planar board: where the magnitude of current density J e changes in dependence on penetration depth of electromagnetic wave from the source to charge according to expression (32) where q e means the internal source – induced specific power per volume unit, which is determined from the current density dissipation in electromagnetic field. (33)

23. Solution of 1D thermal field (34) 1-dimensional thermal field in cylindrical charge: (36) where a is coefficient of thermal diffusivity [m 2.s -1 ] (35) The value of internal source q e can be determined from the current density dissipation in electromagnetic field and electric conductivity where: [W.m -3 ]

24. Solution of 1D thermal field Boundary and marginal conditions in cylindrical charge: temperature on the cylinder surface by radius r 2, i.e. in the node n: temperature in center of cylindrical charge (r = 0), i.e. in the node 1: stability condition of solution  : (37) (38) (39)

25. Finite element method (FEM) Finite element method is some modification of finite difference method, but the operation are executed in the particular elements (not in nodes). Fig. 8 Discretization of area in finite number of triangular elements Fig. 9 Numbering of particular triangular elements

26. Solution of thermal field by FEM (42) (41) (40) Solution of equation (40) by finite weight residues by integration and respecting boundary conditions: where: Combination of finite element method in spatial domain and forward finite differences method in time domain can be the solution of equation (41) as following: (43)

27. Solution of electromagnetic field by FEM (46) (45) (44) The energetic functional corresponding to equation (44) for 2- dimensional electromagnetic field: Derived quantities of electromagnetic field: (47) (48) (49)

28. Determination of thermal characteristics 1-dimensional coupled problem in planar board with respecting of material properties change of charge (  r = f( ),  = f( ), c = const., = const.), 1-dimensional coupled problem in planar board without respecting of material properties change of charge (  r = 1,  = const., c = const., = const.), 1-dimensional coupled problem in cylindrical charge with respecting of material properties change of charge (  r = f( ),  = f( ), c = const., = const.), 1-dimensional coupled problem in cylindrical charge without respecting of material properties change of charge (  r = 1,  = const., c = const., = const.), 1-dimensional coupled problem in cylindrical chargevsádzke with respecting of electric conductivity change by temperature (  r = 1,  = f( ), c = konšt., = konšt.), 2-dimensional coupled problem in cylindrical charge without respecting of material properties change of charge (  r = 1,  = konšt., c = konšt., = konšt.). Solution of the following problem types:

29. Calculation of 1-dimensional coupled problem in planar board with respecting of material properties change (  r = f( ),  = f( ), c = const., = const.) parameters of charge:wall board thickness d 2 = 10 cm, (d = d 2 /2 = 5 cm) volume weight density  m = 7700 kg.m -3, thermal conductivity coefficient = 14,88 W.m -1.K -1, heat capacity c = 510 J.kg -1.K -1, initial charge temperature 0 = 20 °C, parameters of inductor:current in inductor I 1 = 2050 A, number of inductor turns per 1 m of length N 11 = 49, current frequency in inductor f = 50 Hz, parameters for boundary conditions: external temperature (surrounding) pr = 20 °C, heat transfer coefficient  = 150 W.m -2.K -1, parameters of calculation step: number of charge divisions: n = 50, calculation time step (stability condition must be valid): selected time step  t = 0,05 s heating time t k = 540 s

30. Fig. 10 Dependence of temperature arrangement on heating time in particular locations of charge (  r = f( ),  = f( ), c = const., = const.)

31. Fig. 11 Dependence of current density dissipation on heating time in particular places in the charge (  r = f( ),  = f( ), c = const., = const.)

32. Fig. 12 Dependence of current density dissipation in particular locations in charge (  r = f( ),  = f( ), c = const., = const.)

33. Fig. 13 Dependence of temperature in particular charge locations (  r = f( ),  = f( ), c = const., = const.)

34. Fig. 14 Dependence of internal source dissipation in particular charge locations (  r = f( ),  = f( ), c = const., = const.)

35. Fig. 15 Dependence of relative permeability on heating time in particular charge locations (  r = f( ),  = f( ), c = const., = const.)

36. Fig. 16 Dependence of relative permeability in particular locations of charge (  r = f( ),  = f( ), c = const., = const.)

37. Fig. 17 Dependence of electric conductivity on heating time in particular locations of charge (  r = f( ),  = f( ), c = const., = const.)

38. Fig. 18 Dependence of electric conductivity in particular location of charge (  r = f( ),  = f( ), c = const., = const.)

39. Fig. 18 Dependence of current density dissipation on temperature at particular selected times (  r = f( ),  = f( ), c = const., = const.)

40. Fig. 19 Dependence of relative permeability on charge temperature in particular heating times (  r = f( ),  = f( ), c = const., = const.)

41. Fig. 20 Dependence of electric conductivity on charge temperature in particular heating times (  r = f( ),  = f( ), c = const., = const.)

42. Calculation of 1-dimensional coupled problem in cylindrical charge with respecting material parameters change (  r = f( ),  = f( ), c = const., = const.) parameters of inductor:current in inductor I 1 = 2050 A, number of inductor turns per 1 m of length N 11 = 49, current frequency in inductor f = 50 Hz, parameters of charge:charge radius r 2 = 5 cm, volume weight density  m = 7700 kg.m -3, thermal conductivity coefficient = 14,88 W.m -1.K -1, heat capacity c = 510 J.kg -1.K -1, initial charge temperature 0 = 20 °C, parameters for boundary conditions: spatial temperature (surrounding) pr = 20 °C, heat transfer coefficient  = 150 W.m -2.K -1, parameters of calculating step: number of charge divisions: n = 50, time calculation step (stability condition must be valid): selected time step  t = 0,05 s heating time t k = 640 s

43. Fig. 21 Dependence of temperature on heating time in particular locations of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

44. Fig. 22 Dependence of current density dissipation on heating time in particular locations of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

45. Fig. 23 Dependence of current density dissipation in particular locations of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

46. Fig. 24 Dependence of temperature arrangement in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

47. Fig. 25 Dependence of internal source dissipation on heating time in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

48. Fig. 26 Dependence of relative permeability on heating time in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

49. Fig. 27 Dependence of internal source dissipation in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

50. Fig. 28 Dependence of relative permeability in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

51. Fig. 29 Dependence of electric conductivity on heating time in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

52. Fig. 30 Dependence of electric conductivity in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

53. Fig. 31 Dependence of internal source dissipation in particular places of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

54. Fig. 32 Dependence of current density on relative permeability in particular locations of cylindrical charge (  r = f( ),  = f( ), c = const., = const.)

55. Calculation of 2-dimensional coupled problem in cylindrical charge without respecting material properties change (  r = 1,  = const., c = const., = const.) parameters of inductor:current in inductor I 1 = 2050 A, number of inductor turns per 1 m of length N 11 = 49, current frequency in inductor f = 1000 Hz, parameters of charge:charge radius r 2 = 5 cm, relative permeability  r = 1, electric conductivity  = 1,38  10 6 S.m -1, volume weight density  m = 7700 kg.m -3, thermal conduction coefficient = 14,88 W.m -1.K -1, heat capacity c = 510 J.kg -1.K -1, initial charge temperature 0 = 20 °C, parameters for boundary conditions: spatial temperature (surrounding) pr = 20 °C, heat transfer coefficient  = 150 W.m -2.K -1, parameters of calculation step: number of charge divisions: n = 110, calculation time step:  t = 1 s heating time t k = 600 s

56. Fig. 33 Nodes network of charge division

57. Fig. 34 Dependence of temperature on heating time in particular selected places of cylindrical charge (in nodes: 56, 58, 60, 64, 66) (  r = 1,  = const., c = const., = const.)

58. Fig. 35 Dependence of temperature arrangement in particular places of cylindrical charge in time t = 600 s (  r = 1,  = const., c = const., = const.)

59. Fig. 36 Dependence of current density in particular places of cylindrical charge (  r = 1,  = const., c = const., = const.)

60. Conclusion This contribution dealt with the induction heating of ferromagnetic materials up to Curie temperature, but the accent was given to respecting of relative permeability change by the temperature. The designed algorithm of induction heating solution allows to solve the given problem as strong coupled problem with respecting of non-linear thermal-dependent material quantities. The resulting thermal and electromagnetic field was analyzed according to shape of the thermal characteristics and material properties of charge. It is possible by the suggested method of modeling of that type coupled problem to get the better calculation accuracy of particular fields and so to get the reduction of operational costs for supply energy by the modification of technological procedure of induction treatment.

61. Thank you for you attention