Thixoforming : Semi-solid Metal Processing

376j 10 Thixoforging and Rheoforging of Steel and Aluminium Alloys
Figure 10.6 Volume power density dependence on radius and
ratio of penetration depth to billet radius [26].
where _Fðr; tÞ is the power volume density as shown above, c p is the specific heat
capacity, l is the heat conduction coefficient and r is the density of the alloy. The
specific heat capacity and the heat conduction coefficient are temperature dependent;
see Section 10.4.
At the beginning of a heating cycle, the billet temperature is supposed to be
uniformly equal to the constant environment temperature We, which leads to the
following initial condition:
Wðr; 0Þ ¼We 0 r RB ð10:4Þ
Because of the rotational symmetry with respect to the point r ¼ 0, no heat transfer
through the axis can occur. This leads to the first boundary condition:
qWðr ¼ 0; tÞ
qr
¼ 0 ð10:5Þ
The second boundary condition is determined by the heat transfer from the jacket to
the environment. This heat transfer is a combination of radiation and convection.
The second boundary condition is described by
l qW RB; ð tÞ
qr
¼ a½WðRB; tÞ
WeŠ
es W 4 ðRB; tÞ
W 4
e
ð10:6Þ
where a is the heat transfer coefficient, e is the emissivity factor, We is the temperature
of the environment and s is the Stefan–Boltzmann constant. In the case of the
heating of steel, the losses caused by radiation are much larger than through
convection. The emissivity factor and the heat transfer coefficient are temperature
dependent; see Section 10.4. Therefore, the heat transfer is a highly nonlinear
problem.