Thixoforming : Semi-solid Metal Processing

with the liquid density rL, the isotropic pressure p the extra stress tensor for the liquid
phase SL and the Newtonian viscosity of the liquid phase hL. The specific interface
friction force q is proportional to the slip velocity difference:
q _ ¼ CLS f S u _ S u _ L ¼ CLS f S
f S
u _ u _ L ð6:21Þ
For the high solid fractions occurring during thixoforming, the momentum conservation
equation of the liquid phase reduces to Darcy s law:
u _ S u _ L ¼ f L
CLS f S
r _ p ð6:22Þ
The non-isothermal phenomena are described by the energy equation, coupled with a
temperature–enthalpy and solid fraction relation, which is used in the following
form:
Dh
Dt ¼ r_ ðlrTÞþhB _g 2 ðT ðT ;hðTÞ¼f S cSrSdT þ 1 f S cLrLdT þ 1 f L
T ref
T ref
ð6:23Þ
with enthalpy h, thermal conductivity l and temperature T; cS and cL are the specific
heat capacities of the solid and the liquid phase, respectively. The time evolution of
the solid fraction, fS, is described by the mass conservation equation for the solid
phase:
qf S
qt þr_ f u S _ S ¼ d~ f S DT
dT Dt
6.2 Numerical Modelling of Flow Behaviourj199
ð6:24Þ
where the right-hand side describes the rate of phase change based on the Scheil
equation. Influencing the solid fraction, the thermal changes are then reflected in the
rheological Equation 6.1, which affects the flow of both phases.
6.2.2.2 Discretization Methods (Numerical Solution Techniques)
Discretization methods are numerical techniques, approximations for solving partial
differential equations. that is, methods of approximating the differential equations by
a system of algebraic equations for the variables at some set of discrete locations in
space and time. There are three distinct streams of numerical solution techniques:
finite difference (FD), finite volume (FV) and finite element (FE) methods.
Finite difference methods describe unknowns f of the flow problem by means of
point samples at the node points of a grid coordinate lines. The truncated Taylor
series expansions are often used to generate finite difference approximations of
derivatives of f in terms of point samples of f at each grid point and its immediate
neighbours. Those derivatives appearing in the governing equation are replaced by
finite differences. yielding an algebraic equation for the values of f at each grid point.
The finite volume method is based on the control volume formulation of analytical
fluid dynamics, where the domain is divided into a number of control volumes, while
the variable of interest is located at the centroid of the control volume. The governing