Heating of Ferromagnetic Materials up to Curie Temperature by ...

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Heating of Ferromagnetic
Materials up to Curie
Temperature by Induction
Method

Ing.
Dušan MEDVEĎ
, PhD.

Pernink
,
26
.
May

200
9

T
E
C
H
N
I
C
K
Á

U
N
I
V
E
R
Z
I
T
A
K
O
Š
I
C
E
F
E
I
TECHNICAL UNIVERSITY OF
KOŠIC
E

FACULTY OF ELECTRIC ENGINEERING AND
INFORMATICS

Department 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

z
rot
1
rot
J
A
A














t


















1
1
20
r
0
t





A
J


e
2
e
e
1
J



q


e
grad
div
q
t
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
) [



]

0
r




m
r
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
,

)
:
















1
1
20
r
r
(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


(

)

景rs瑥敬

㐰4

[



]

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
S
S
S
20
r
2
arctg
2
arctg
B
H
B
H
H
H
B
























(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
]
;


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


b
a



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
.



)
(
C
1








c
e
[


]

The final expression of relative permeability dependence
on



and

H

approximated by various curves

-
approximation of


r
(

) by

quarter
-
ellipse

by

(
11
) a

r
20
(
H
)

by

(9):




2
C
S
S
S
S
S
S
r
1
1
2
arctg
2
arctg
1





















































B
H
B
H
H
H
B
-
approximation of


r
(

) by
hyperbola

by

(
10
) a

r
20
(
H
)

by

(9):




















































1
)
(
.
1
2
arctg
2
arctg
1
C
S
S
S
S
S
S
r
c
c
B
H
B
H
H
H
B






-
approximation of


r
(

) by
exponential

by

(
13
) a

r
20
(
H
)

by

(9):






)
(
S
S
S
S
S
r
C
1
1
2
arctg
2
arctg
1















































c
S
e
B
H
B
H
H
H
B
(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
J
H
t






B
E
0



B




E
(
Ampere law
)



(17)

(
Faraday law
)



(18)

(Gauss
law
)



(19)

E
D



0
r


H
B



0
r


E
J



(21)

(20)

Modification of Maxwell equations

H
H







0
r
2
1




j
E
E







0
r
2
r
1




j
A
J
A













j
z
2
0
r
1
A
J
A
A

























j
y
x
z
2
2
2
2
0
r
1
A
J
A
A
A
A






























j
r
z
r
r
r
z
2
2
2
2
2
0
r
1
1
(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)



e
q
t
c















0



n

e
1
q
r
r
r
r
z
z
t
c







































e
q
z
z
y
y
x
x
t
c

























































Modification of expression

(
2
5)
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
x
x
x
x






0
0
0













2
0
0
0
0
2
x
x
x
x
x
x
x
















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
t
x
c
t
t
x














e
2
2
,
,

2
e
e
J
q

[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)









0
,
2
2
2
0
r
e
,
1
2

























H
f
x
r
e
H
H
f
J
Solution of 1D thermal field

(34)

1
-
dimensional thermal field in

cylindrical charge
:







c
q
r
t
r
r
r
t
r
a
t
t
r

























e
2
2
,
1
,
,
(36)





j
x
J
j
x
J
H
a
J







2
0
1
2
e
2
where

a


is coefficient of thermal diffusivity

c
a





[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


2
e
e
J
q

where
:

[W.m
-
3
]

Solution of 1D thermal field

Boundary and marginal conditions in
cylindrical charge
:

r
r
j
n
j
n

















o
,
1
,
0
,
0
,
1






r
j
j



n


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


:





2
1
2
2










c
r
t
r
a
t




(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
y
x
a
t
e
2
2
2
2
(42)

(41)

(40)



















































d
,
f
d
1
d
1
d
y
x
w
n
w
a
y
y
w
x
x
w
a
t
w
I
n






Solution of equation

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

where
:











t
t
t
F
K
M







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
:







t
t
t
t
t
t













(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)

A
J
A
A

























j
y
x
z
2
2
2
2
0
r
1
S
A
J
A
j
y
A
x
A
F
S
r
d
2
1
z
2
2
2
0




































A
j
J







e
A
j
J
J







z
x
B
y
A




y
B
x
A




2
2
y
x
B
B
B


r
0




B
H
A
j
E





(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 charge
vsádzke
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(

⤬)


=



⤬)
c

=

const
.,


=

捯湳


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

Fig
. 10
Dependence of temperature arrangement on heating time in particular locations of charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 11
Dependence of current density dissipation on heating time in particular places in the charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 12
Dependence of current density dissipation in particular locations in charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 13
Dependence of temperature in particular charge locations

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 14
Dependence of internal source dissipation in particular charge locations

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 15
Dependence of relative permeability on heating time in particular charge locations

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 16
Dependence of relative permeability in particular locations of charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 17
Dependence of electric conductivity on heating time in particular locations of charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 18
Dependence of electric conductivity in particular location of charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 18
Dependence of current density dissipation on temperature at particular selected times

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 19
Dependence of relative permeability on charge temperature in particular heating times

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 20
Dependence of electric conductivity on charge temperature in particular heating times

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Calculation of

1
-
dimensional coupled problem in
cylindrical charge with respecting material parameters
change

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳
琮t

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

Fig
. 21
Dependence of temperature on heating time in particular locations of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 22
Dependence of current density dissipation on heating time in particular locations of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 23
Dependence of current density dissipation in particular locations of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 24
Dependence of temperature arrangement in particular places of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 25
Dependence of internal source dissipation on heating time in particular places of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 26
Dependence of relative permeability on heating time in particular places of cylindrical charge
(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 27
Dependence of internal source dissipation in particular places of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 28
Dependence of relative permeability in particular places of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 29
Dependence of electric conductivity on heating time in particular places of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 30
Dependence of electric conductivity in particular places of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 31
Dependence of internal source dissipation in particular places of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Fig
. 32
Dependence of current density on relative permeability in particular locations of cylindrical charge

(

r

=

f(

⤬)


=



⤬)
c

=

const
.,


=

捯湳t


Calculation of

2
-
dimensional coupled problem in
cylindrical charge without respecting material properties
change

(

r

=

1,


=

捯湳t
⸬.
c

=

const
.,


=

捯湳t


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

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,


=

捯湳t
⸬.
c

=

const
.,


=

捯湳t


Fig
. 35
Dependence of temperature arrangement in particular places of cylindrical charge in time

t

=

600

s
(

r

=

1,


=

捯湳t
⸬.
c

=

const
.,


=

捯湳t


Fig
. 36
Dependence of current density in particular places of cylindrical charge

(

r

=

1,


=

捯湳t
⸬.
c

=

const
.,


=

捯湳t


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