Heating of Ferromagnetic Materials up to Curie

Temperature by Induction Method

Ing. Dušan MEDVEĎ, PhD.

Pernink, 26. May 2009

FEI

TECHNICAL UNIVERSITY OF KOŠICEFACULTY OF ELECTRIC ENGINEERING AND

INFORMATICSDepartment of Electric Power Engineering

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

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

Mathematical description of induction heating

zrot1

rot JA

A

t

11 20r0

t

AJ e

2ee

1J

q

egraddiv qt

c

(1)

(2)

(3)

(4)

(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

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.

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

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.

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) [  ]

0r

mr 1 (7)

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,):

11 20rr(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

[  ]

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):

S

S

S

SS

S

20r

2arctg

2arctg

B

H

B

HH

H

B

(9)

where: BS saturation magnetic flux density; constant for given

material; [T]HS saturation magnetic intensity; constant for given

material; [A.m-1];H input magnetic intensity, i.e. independent quantity;

[A.m-1]

[  ]

Approximation of correction function ()

Correction function () can be simplified for example by hyperbola:

1)( C

c

c

(10)

where: temperature, at which is determined r(); [°C]

c constant, 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

[  ]

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

1

2

2

2

2

ba

b adjacent half-axis of ellipse; [  ]a  main half-axis of ellipse; [  ]() correction function; [  ]C temperature of magnetic change, Curie temp.; [°C]

2

C

1

Correction function ():

(12)[ ]

Correction function () can be simplified also by exponential:

(13)

were: c constant 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.

)( C1 ce [ ]

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):

2

CS

S

S

SS

S

r 11

2arctg

2arctg

1

B

HB

HH

HB

-approximation of r() by hyperbola by (10) a r20(H) by (9):

1)(

.1

2arctg

2arctg

1C

S

S

S

SS

S

r c

cB

HB

HH

HB

-approximation of r() by exponential by (13) a r20(H) by (9):

)(

S

S

SS

S

rC11

2arctg

2arctg

1

cS eB

HB

HH

HB

(14)

(15)

(16)

[  ]

[  ]

[  ]

Mathematical model of induction heating

Induction heating is the complex of electromagnetic, thermal and metallurgical processes.

Mathematical model of electromagnetic field

t

D

JH

t

B

E

0 B

E

(Ampere law) (17)

(Faraday law) (18)

(Gauss law) (19)

ED 0r HB 0r EJ (21)

(20)

Modification of Maxwell equations

HH 0r21

j EE 0r

2

r

1

j

AJA

jz2

0r

1

AJAA

jyx z2

2

2

2

0r

1

AJAAAA

jrzrrr z22

2

2

2

0r

11

(21)

(22)

(23)

(24)

Modification of expression (22) for 2D rectangular axis system:

Modification of expression (22) for polar axis system:

Mathematical model of thermal field

(25) eqt

c

0

n

e

1q

rr

rrzztc

eqzzyyxxt

c

Modification of expression (25) for 2D rectangular axis system:

Modification of expression (25) for polar axis system:

(26)

(27)

(28)

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

Finite difference method (FDM)

x

xxxx

000

2

0000

2

x

xxxxxx

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)

Solution of 1D thermal field

c

q

x

tx

ct

tx

e

2

2 ,,

2e

e

Jq [W.m-3]

(31)

1-dimensional thermal field inside of planar board:

where the magnitude of current density Je changes in dependence on penetration depth of electromagnetic wave from the source to charge according to expression

(32)

where qe means the internal source – induced specific power per volume unit, which is determined from the current density dissipation in electromagnetic field.

(33)

0,2

2

2

0r

e

,

1

2

Hf

x

reH

Hf

J

Solution of 1D thermal field

(34)

1-dimensional thermal field in cylindrical charge:

c

q

r

tr

rr

tra

t

tr

e2

2 ,1,,

(36) jxJ

jxJH

aJ

20

12e

2

where a is coefficient of thermal diffusivity

ca

[m2.s-1] (35)

The value of internal source qe can be determined from the current density dissipation in electromagnetic field and electric conductivity

2e

e

Jq

where:

[W.m-3]

Solution of 1D thermal fieldBoundary and marginal conditions in cylindrical charge:

r

rjnjn

o,1,

0,0,1

rjj

n

• temperature on the cylinder surface by radius r2, i.e. in the node n:

• temperature in center of cylindrical charge (r = 0), i.e. in the node 1:

• stability condition of solution :

2

122

cr

t

r

at

(37)

(38)

(39)

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

Solution of thermal field by FEM

c

q

yxa

te

2

2

2

2

(42)

(41)

(40)

d,fd

1d

1d yxw

nw

ayy

w

xx

w

atwI

n

Solution of equation (40) by finite weight residues by integration and respecting boundary conditions:

where:

tttFKM

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:

tt

tttt

(43)

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)

AJAA

j

yx z2

2

2

2

0r

1

SAJAjy

A

x

AF

S r

d2

1z

222

0

AjJ e

AjJJ z

xBy

A

yBx

A

22

yx BBB r0

B

H

AjE

(48)

(49)

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:

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 d2 = 10 cm, (d = d2/2 = 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 I1 = 2050 A,number of inductor turns per 1 m of length N11 = 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 sheating time tk = 540 s

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

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

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

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

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

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

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

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

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

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

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

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

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 I1 = 2050 A,number of inductor turns per 1 m of length N11 = 49,current frequency in inductor f = 50 Hz,

parameters of charge: charge radius r2 = 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 sheating time tk = 640 s

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

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

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

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

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

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

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

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

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

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

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

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

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 I1 = 2050 A,number of inductor turns per 1 m of length N11 = 49,current frequency in inductor f = 1000 Hz,

parameters of charge: charge radius r2 = 5 cm,relative permeability r = 1,electric conductivity  = 1,38106 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 sheating time tk = 600 s

Fig. 33 Nodes network of charge division

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.)

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

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

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.

Thank you for you attention