199 mathematical modeling of mold-filling and solidification - Acta ...

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 206
where T l and T s are the liquidus and the solidus temperatures (K), respectively. Another widely
used expression is the Scheil equation, which assumes uniform solute concentration in the liquid
but no diffusion in the solid [23]:
−1/1−k ⎛ T
o
m − T ⎞
f s = 1 − ⎜ ⎟
Tm
T
⎝ − l ⎠
(12)
where T m is the melting point of the solvent metal (K),and k o is the equilibrium partition
coefficient of the alloy.
Eq. (10) defines the heat flux [3] which is released during liquid cooling and
solidification. In classical models, the heat generated after solidification is assumed to be zero,
i.e., for T< T s , Q & = 0. However, experimental investigations showed that lattice defects and
vacancies are condensed in the already solidified part of the crystal and the enthalpy of the solid
increases and thus the latent heat will decrease. Based on this fact, another way to represent the
change of the solid fraction during solidification can be written as [3]:
2
⎡ π ( T − T ) ⎤
( T − T ) + ( Ts
− T )(1 − cos
l
l
l ⎢ ⎥
π
⎣ 2( Ts
− Tl
)
f =
⎦
s
( Tl
− Ts
)(( 1−
2 / π )
(13)
Combining Eqs. (7), (9) and (10), the new unsteady state heat conduction equation
can be expressed in the form [4; 23]:
∂ ( ρ c´
T )
= ∇.( k∇
T)
∂ t
(14)
At the range of temperatures where solidification occurs for binary metallic alloys, the
physical properties will be evaluated taking into account the amount of liquid and solid that
coexists in equilibrium at each temperature, as shown by Eqs. (9), (10) and (14).
3. Conclusions
A comprehensive theoretical basis regarding equations of fluid mechanics and heat
flow has been shown. Like other CFD problems, the momentum equation, the continuity
equation, and the energy equation are the partial differential equations that are essential in the
numerical analysis of the casting process. The moving boundary liquid metal/air is modeled
using the VOF method to trace the free surface in mold filling processes. The latent heat release
during the unsteady solidification phenomenon is taken into account as a source term in the
energy equation. The implementation of such equations is aimed to furnish important
information such as filling dynamics and the thermal field along the casting process.
Acknowledgements
The authors acknowledge financial support provided by CNPq (The Brazilian
Research Council), FAPESP (The Scientific Research Foundation of the State of São Paulo,
Brazil) and CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior.