Thixoforming : Semi-solid Metal Processing

10.3.1.2 Modelling of Inductive Heating with the Finite Difference Method (FDM)
For simulating the heat transfer, we have to discretize the heat transfer (Equation
10.3) in time and over the radius. The billet is split into different scales via the
radius. This yields the following equation: for more details, see [29]:
ðrcpÞ k
i
k þ 1
Wi þ l k W
i
k
Dt
W k
i
¼ lk
i þ 1 lk
i 1
2Dr
i þ 1 2Wki
þ Wki
1
Dr2 þ P k i
þ lk
!
k
i 1 W
ri
i þ 1 Wk
i 1
2Dr
ð10:7Þ
where P k
i is the induced power in the scale i at the time-step k. The induced power is
the integral of the volume power density over the scale i:
P k
i ¼
ri þðDr=2 ri Dr=2
_F ðr; tiÞdr¼
N 2
Sp
kd 2 L 2
Sp
^I ðtiÞ Sp
ri þðDr=2 ri Dr=2
10.3 Heating and Forming Operationsj377
J 1
J 0
pffiffiffiffiffi 2jr
d
pffiffiffiffiffi RB
2j
d
2
dr ð10:8Þ
A similar approach to model the heat flow can be found in [30].
For the inductive heating of aluminium, the temperature dependences of the
material properties can be neglected and the calculation of the heat transfer is
considerably easier.
10.3.1.3 Control of Inductive Heating
The reproducible heating of the billet into the semi-solid state is a very important part
of the production of thixoforging parts. The results of the subsequent forming step
depend heavily on the quality of the heating. The billet should have a uniform
temperature distribution in order to obtain good forming results. At the same time,
the billet should be heated to the target temperature as fast as possible. However, it
must be guaranteed that the outer area of the billet does not begin to melt prematurely.
With the existing control system for the heating process [20], it is possible to follow
a predefined power over time trajectory. This trajectory has to be found in timeconsuming
and expensive test series. This method, like all open-loop control
strategies, is not robust against any changes in the conditions of the production.
Hence reproducibility cannot be guaranteed. Another problem is that in an industrial
production environment it is not possible to equip each billet with thermocouples to
measure the temperature. Therefore, closed-loop control is not feasible. For this
reason, two approaches were developed to avoid the need for thermocouples. The first
approach is to measure the temperature at the surface of the billet with a pyrometer
and to detect the entry into the melting phase with the characteristic slope change of
the temperature. The second approach is to calculate the current trajectory needed for
the induction unit from a predefined temperature trajectory at the middle of the billet.
This method is the so-called flatness-based control. In the following, these two
approaches are described.