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The solubility limit for the solute elements is given by:
ܥ
כ
ൌ
ܣ
݁
ି
்
( 2 )
where
T
is the temperature in Kelvin.
The formation of the intermetallic compounds is generally defined by a combination of
the previous information and the particle composition. For example for Al
3
FeSi
2
phase,
precipitation occurs when:
ܥ
ி
ܥ
ி
כ
ή
ቆ
ܥ
ܥ
כ
ቇ
ଷ
ͳ
( 3 )
The value on the right hand side is adjusted to better fit the literature data. In some
cases, the simple combination of concentration over solubility limit is not sufficient and
another formula is used as for Al
2
CaSi
2
:
ܥ
ή
ܥ
ଶ
ܣ
ଶௌଶ
݁
ି
ಲమೌೄమ
்
( 4 )
For the quaternary phase Al
6
CaFe
4
Si
8
, the lack of data has led to ignore the temperature
dependence. Nevertheless, it was checked that the transformation occurs in the proper
temperature range as described by Anglezio [2].
In addition, one has to compute the composition of the particles. The composition of
the particles can be:
•
fixed (usually stoichiometric, e.g. Al
2
CaSi
2
)
•
temperature dependent (e.g. SiB
3
vs. SiB
6
)
•
liquid composition dependent (e.g. FeSi2 that is actually (Al, Ca)-FeSi
2.4
)
The solidification of Si and liquid enrichment is, in the current version, simply
computed assuming no diffusion in the solid phase and infinite diffusion in the liquid
phase. The implementation of a more realistic model accounting for finite diffusion is
planned for later releases.
At the solidification of cell
n
, for all elements
i
, the mass balance is according to:
݂
ିଵ
ܥ
ିଵ
ൌ ݂
ܥ
߮
߮
ǡ
( 5 )
where
݂
is the liquid fraction remaining in the whole domain when step
n
is solidified,
ܥ
is the "averaged" concentration of specie
i
in the remaining liquid at step
n
,
߮
is
the mass fraction of phase j in the cell n, and
߮
ǡ
the composition of the phase
j
.
Algorithm
The solidifying grain is discretized into 200 cells (with geometric progression).The
calculation starts by initializing the concentration and the solid fraction. The liquidus
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