Download 19.- Modeling Electromagnetic Fields in Induction Heating

Case 19
Modeling the Electromagnetic Fields in Induction
Heating
Induction heating treatments are used on steel components to produce surface hardening
by self-quenching of the heated surface. Coaxial coils are used to treat components with
cylindrical symmetry. High frequency alternating current is made to flow through the inductor coils. Through the action of the associated fluctuating magnetic field, oscillating
eddy currents are induced in the treated component without the need of electrical contact.
The induced currents together with the electrical resistance of the material result in localized heating by Joule effect. The resulting temperature field is then directly related to the
electro-magnetic parameters of the system.
The objective of modeling is to produce a mathematical representation of the induction
heating process by first determining the induced current distribution in the component.
Ultimately, one would like to produce a predictive capability capable of assisting in process
optimization and new process design.
The formulation of the problem requires statement of the electromagnetic field (Maxwell’s)
equations in the time-harmonic form neglecting displacement fields
∇ × E = −jωB
∇×H=J
∇·B=0
∇·D=0
where E, H, B and J are, respectively, the electric field, the magnetic √
field, the magnetic
flux density and the current density vectors, ω is the frequency and j = −1.
Further, since the magnetic field density can be represented in terms of a magnetic
potential A by B = ∇ × A one has
∇2 A = −µJ
1
where µ is the permeability. From the above the vector potential inside a volume v can be
expressed as
A=
µ
4π
Z
v
J
dv
r
Integrating the first Maxwell equation over the area a, using Stokes theorem and the
above yields
I
L
I
J
µ Z J
dl + jω
dvdl = Uapp
σ
r
L 4π
where Uapp is the externally applied scalar electromagnetic potential, σ = J/E is the electrical
conductivity and L is the length of a current carrying path. This equation is valid for each
closed current carrying path Lk .
Introducing the mutual inductance Mj,k defined as
Mi,k =
µ I dlj dlk
4π
rik
where rik is the straight line distance between points i and k, the above equation becomes
Z
J k Lk
+ jω Ji Mi,k da = Uk
σk
a
where the subscripts i and k refer to current carrying loops i and k, respectively.
The above is an integral equation that can be solved numerically by first subdiving
the domain of interest into a finite number of current carrying loops and then adding all
individual contributions. Specifically, consider an axisymmetric system consisting of a part
to be induction treated and the coaxial inducting coils. Next, subdivide the workpiece and
the coil into a set of n current carrying loops in the r− direction and a set of m loops in
the z− direction. For simplicity, let all current carrying loops have the same square cross
sectional area h2
Let the current density passing through the loop labelled i, k be Ji,k = J(ri , zk ). From the
above, this current plus the result of the collective influence of the currents passing through
all other loops in the system must equal the scalar potential at the loop, i.e.
Ji,k Ri,k + jω
XX
n
m
Mi,k,m,n Jm,n =
Ui,j
h2
where Ri,k = Lk /σh2 .
Since, there is no applied potential in the workpiece, the equations there are
Ji,k Ri,k + jω
XX
n
m
Mi,k,m,n Jm,n = 0
2
The mutual inductance in this case is given in closed analytical form by
q
Mi,k,m,n = µ0 h (n − k)2 + (i + m − 1)2 [(1 − p2 /2)Kp − Ep ]
where
p2 =
4(i − 1/2)(m − 1/2)
(n − k)2 (i + m − 1)2
and Kp and E(p) are the elliptic integrals of first and second order, respectively. The selfinductance is given by
Li,k,i,k = µ0 h(2i − 3/2)[(1 − p21 /2)K(p1 ) − E(p1 )]
with
p21 =
(2i − 3/2)2 − 1/4
(2i − 3/4)2
The above is a system of algebraic equations that can be solved by standard numerical
linear algebra methods.
A sample of results is shown in the attached figures.
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