ON THE 3-D EDDY CURRENT FIELD COUPLED TO THE HEAT TRANSFER OF INDUCTION HEATING OF A SLAB L. Gong, R. Hagel, K. Zhang, and R. Unbehauen Lehrstuhl fiir Allgemeine und Theoretische Elektrotechnik Universitat Erlangen-Niirnberg Cauerstrasse 7, D-91058 Erlangen, Germany
Absb.act - In this paper a semi-analytical method is proposed for solving the 3-D eddy current field problem coupled to the steady-state heat transfer in an inductor-slab system that is part of an induction heating process. The frequency of the inductor current is 1OOkHz. The heat transfer problem of the moving slab is solved by Galerkin’s FEM. Experiments performed in a German factory have confirmed the results obtained by the method proposed. The results also indicate the possibility of using induction heating to compensate for the non-uniform heating effect of the strip casting process so that a uniform temperature distributionwill be obtained.
where T is the temperature, K; is the thermal conductivity, cp is the thermal capacity, p is the mass density, U is the velocity of the slab and q represents the heat source which results from the eddy current power loss. To solve the steady-state heat transfer equation, the calculation is performed inside the slab by using Galerkin’s 3-D E M . Since the calculation is accomplished in 3-D space, the method can be considered a further development of the 2-D approach in [l], however for the linear case.
I. INTRODUClTON
A conducting and permeable slab uniformly moving along its longitudinal direction ( z -axis in Cartesian coordinates) is heated by an inductor idealized as a line loop current (Fig. 1). The cross-section of the slab is a thin rectangle in the x - y plane. The influence of the velocity on the constitutive relations is not taken into account. The electromagnetic field due to the inductor current is considered as the incident field. The total field in the air is represented as the superposition of the incident field and the scattered field. The scattered field in the air as well as the total field in the slab are governed by homogeneous Helmholtz equations. The homogeneous Helmholtz equations for the phasors of the magnetic flux density and the electric field strength are
11. ALGORITHM
Eddy current field
Due to the symmetry of the slab inductor system only a quadrant of the space is to be considered which is divided into four regions as shown in Fig. 2 where d << w .Assuming an infinitely long slab, we apply the Fourier transform with respect to the z-variable to the components B,, E , (v = x ,y ,z ). The transformed components are m
E,
= JB,e
-jkz”dz,
-m
m
E,
=
r E , e -Jkz”dz -m
v2B+k2B= 0,
(1) The Helmholtz equations for the components are
v 2 E+ k 2 E = 0 ,
(2)
respectively, where k2 = - jc+(a+ jut,,), CJ is the electric conductivity, ,U is the permeability, E, is the air permittivity and w is the angular frequency. To avoid difficulties of discretization in the numerical methods owing to the very small penetration depth of the skin effect and the open boundary, separation of variables is applied. The governing equation of the steady-state heat transfer is
0-7803-3008-0195 $4.000 1995 IEEE
(4)
where k, is the variable of the FoEier_tresform.-As is known, the transverse components B, ,B y , E, and Ey can be eFressend in terms of the longitudinal components B, and E, :
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B" and E". Since have
E, must be continuous at x = d , we -.
Bf"
_The left-hand side of (13) is
-cos(k,,d)
where = R, = -k;+jk?, k: = - w p ( a + j w c , ) in the slab and R = R, = - k ; + k t , k,2 = - o/&((a,+ j u E 0 ) in the air. Equations (4) and ( 5 ) are to be considered in every region mentioned above. Taking @to account the symmetry of the situation the solutions B, and ,!?, in regions I and I1 are
=
D, sin (k,,x) sin (ky,y ) ,
(7)
-e
c2 A
-kx2d
1
cos(kyly). ( 1 4 4
To write the right-hand side of (13) in the same mathematical form, with U t j v = kyl we obtain from (A-4a) the coefficient a - while due to symmetry b = 0 - and then from (A-3) (c.o. Appendix) -.
Bf" [ z ] x = d
=
a cos(ky,y).
From (14a,b), we obtain Cl -cos(kx,d) P
E:
-
-
c,
-e
-kx2d
A
Since also Gy is continuous at x
= d , we
=
a .
(15)
get
As above the right-hand side of (16) can be written as b sin (k, ,y ) to obtain a further-hear equation for C, and D, analogously to (15). Since Ey and are continuous at x = d , the boundary conditions where the eigenvalues kxl, kyl and k,, are complex numbers. Substituting (6) into (4), then (8) into (4) and considering that the ratio of the lengths of the sides w and d of the cross-sections equals k,, /&,, the three eigenvalues are obtained:
k,,
=
dk:t
k$- k,Z .
Boundary conditions due to the continuity of they - and z components of the magnetic and the electric field strength are established to determine the unknown constants where the incident field has been calculated by using Biot-Savart's law after applying the discrete F y r i e r tr_asform (DFT) to this field. In region 11, the fields B and E are decomposed into the incident parts ginand E'" and the scattered parts
can be established where it has been considered that no z component of the incident electric field exists. The righthand side of (17) can be written as Z i cos ( ky,y ). In this way (17) and (18) provide two further linear equations for D,, C, , D, . Equations (15) - (18) are finally used to evaluate the four unknowns C,, C, , D, and D, . For region IV, the algorithm is similar to that of region 11. In the case of d << w ,the electromagnetic fields in regions 111 and IV decay rapidly, therefore the calculations for regions I11 and IV are neglected. Next, after applying the inverse DFT, the heat source is calculated as the eddy current power density in the shell. Steady-state heat transfer problem
The length of the slab is limited. Equation (3) is multiplied by a shape function .k and integrated by parts accord-
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incident fields is easy; third, analytical formulas are available for the field variables. The computation of the heat transfer problem is based on Galerkin's FEM. 2. The 3-D calculation not only shows that at the corners of the cross-section of the slab the intensity of the eddy current field is maximum, but it also means that the induction heating can compensate for the heating effect of the strip casting process so that a uniform distribution of the temperature can be obtained because the strip casting process yields a higher temperature at the center.
ing to Galerkin's FEM [2], i.e.
REFERENCES
where R is the field domain within the slab, 8 0 is the boundary of 0, n is the unit vector outward normal to a 0 and U = U e , . The integration in (19) is performed by Gauss quadrature. The FE algebraic equation corresponding to (19) in matrix form is
[l] L. Gong, R. Hagel, and R. Unbehauen, "On the nonlinear eddy current field coupled to the nonlinear heat transfer,"IEEE Transactions on Magnetics, vol. 29, pp. 1546-1549, March 1993. [2] P.P. Silvester and R.L.Ferrari,Finite elements for electrical engineering. Cambridge: Cambridge University Press, 1992.
APPENDIX
where [TI is the vector of the node temperatures, [S J is the stiffness matrix and [Q] is a vector whose entries are the power densities of the prisms used as finite elements. Because the slab moves along the z -axis, the U -dependence makes [SI non-symmetric. However, due to the low velocity of the slab, (20) can be solved by Gauss elimination without numerical oscillation.
If a function F (y ) is given and if it is known that F (y ) can be put into the form
F ( y ) = a c o s ( y ( u + j u ) ) + bsin(y(u + j u ) )
the problem arises how to calculate the coefficients a and b where U , U ,y E W are specified. Multiplying (A 1) by cos (y ( U + j U)) and integrating from any real c, to co + p withp = 2x/u one obtains the equation
EXAMPLE
The illustrative example is based on I = 100A, f = 100 kHz,U =lm/s, 2d = 3mm, 2w = lm, pr = 29, g =4.1 X lo4 S/m, go= 3 X 10 -I4 S/m, K = 54.5W/(Km), cp = 464 J/(kgK), p = 7800kg/m3, the ambient temperature To= 20째C and the calculated penetration depth dp = 0.47". The distributions of the magnitudes of B and T are shown in Figs. 3 and 4. The temperature of the slab is 20째C before entering the inductor and rises to 114째C after induction heating. It is to be noticed that in Fig. 4 the origin of the coordinate system has been shifted in the z -direction. The main part of B is the component B, which has even symmetry with respect to the z -axis. The distribution of the magnitude of B against the z-axis shows the skin effect clearly. The distribution of T at the surface of the slab is higher than near the center. These results are confirmed by experiments carried out in a Germany factory. CONCLUSIONS
(Al)
a d,,
+ b d,,
=
Fl
(A 2)
where CO
+P
d,, = S c o s 2 ( y ( ~ + j u ) ) d ,y CO
CO
+P
d,, = / - C O S ( Y (+Uj u ) ) s i n ( y ( u
+ ju)) dy
CO
and CO
+P
F, = / - F ( Y ) C O ~ ( Y ( ~ + ~ ~ ) ) ~ Y . CO
Multiplying (A 1) by sin (y ( U + j U)) and integrating from any real co to c, + p, wherep = 2 x / u , one obtains ad,, + bd,, = F, where
1. Based on the longitudinal method, a semi-analytical algorithm for calculating the eddy current field problem of an infinitely long slab is presented. The advantages are: first, the above algorithm avoids difficulties involved with a very small penetration depth; second, the calculation of the
CO
+P
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(A 3)
d,,
=
Solving (A 2) and (A 3 ) gives
+P
CO
s s i n z ( y ( u + j u ) ) dy
a = Fl d22 D - F242 , b = F2dll D - F , d21
(A 4ab)
CO
and CO
F2
=
with the determinant D = d,, d,, - d,, d,,
+P
s F ( y ) s i n ( y ( u + ju)) dy
f 0.
.
CO
I
1
I
I
1.4 1
g
2 x=d-dp
1.2
-
x=d
-
v
1.0
-
2
0.8
-
1 c
0.6
3 .A
2d
/ /
slab
-
Y
t 0.0
inductor
0.00
0.05
0.10
0.15
I
I
0.20
0.25
I
I
0.30 0.35 0.40
z-axis (m)
Fig. 1 Slab and inductor
Fig. 3: Distribution of the magnitude of the magnetic flux density against z
120 100
al
s
60
a, L
a
$
40
â&#x201A;Ź-
20 0
0.3 0.4 0.5 0.6 0.7 0.8 z-axis (m) Fig. 4: Distribution of the temperature against z
0.0
Fig. 2 Field regions
0.1
0.2
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