Thixoforming : Semi-solid Metal Processing

With the solution of the Equations 10.1, the volume power density can be calculated:
_F ðr; tÞ ¼ 1 H
k
2
0
d 2
pffipffiffiffiffiffiffi ! 2
j 2r
J1 d
pffipffiffiffiffiffiffi ¼ gðtÞf ðrÞ
j 2RB
J0 d
ð10:2Þ
with
and
sffiffiffiffiffiffiffiffiffi
2
d ¼
mkw
gðtÞ ¼
N 2
c
kd 2 L 2
c
^I 2
c ðtÞ; f ðrÞ ¼
J 1
J 0
pffipffiffiffiffiffiffi !
j 2r
d
pffipffiffiffiffiffiffi j 2RB
d
where J0 and J1 are modified Bessel functions of first kind, zero- and first-order
respectively, d is the penetration depth, which depends on the frequency and
the magnetic material parameters; the penetration depth specifies how deep the
magnetic field penetrates the billet; Nc is the number of turns, Lc is the coil length and
^I c is the coil current peak value and is the control variable of the converter.
The magnetic material parameters (magnetic permeability and electrical conductivity)
are temperature dependent. Therefore, the magnetic behaviour changes during
heating. A more detailed description of the induction process can be found in [26–28].
The solution of the Maxwell equations is not correct for the heating of steel because
ferromagnetic material has a nonlinear relationship between flux density and
magnetic field intensity. To consider this circumstance, the volume power density
in Equation 10.2 has to be multiplied with 1.47 [28].
Figure 10.6 shows the dependence of the volume power density on the ratio of
the penetration depth to the billet radius. If the penetration depth is small compared
with the billet radius, the induced power is concentrated at a small surface area only.
This behaviour is called the skin effect [27]. Thus, a higher frequency leads to a higher
energy entry at the surface areas. In order to compensate forthe radiation losses at higher
temperature,itisnecessarytochoosetheconverterfrequencytobeashighaspossible[26].
10.3.1.1 Heat Transfer Equations
In the following, we discuss the partial differential equation that describes the
heating process of the billet. For more details on the heat transfer, see [29]. We assume
that the heat transfer occurs mainly in the radial direction because the coil geometry
ensures a homogeneous temperature distribution along the height. The heat
conduction through the billet can be described by the following partial differential
equation in cylindrical coordinates:
qWðr; tÞ
rcp
qt
1 q
¼
r qr
lr qWðr; tÞ
qr
10.3 Heating and Forming Operationsj375
2
þ _Fðr; tÞ ð10:3Þ