Mathematical formulation
This paper proposes a calculation method that requires less calculation
time and provides the resin distribution in both the screw axial and
circumferential directions. Our method combines the Hele–Shaw flow and
2.5D FEM methods to calculate the pressure distribution from which the
velocity, strain rate, temperature, and degree of fill distribution are
calculated. The original FAN method is based on a flattened screw and
barrel for modeling the flow behavior, and requires a certain
approximation when the cylindrical shape of the screw and barrel is
transformed to that of a flat plane. As such, it is limited to the
degree of fill in the axial direction of the screw. Our approach
addresses the screw and barrel as is, in a cylindrical coordinate
system. The Hele–Shaw flow model simplifies the equation of continuity
and motion. This approximation reduces the number of variables to be
determined from four (vr ,v θ, v z, p ) to one
(p ) for each finite element. As the number of variables becomes
1/4, the calculation time can be reduced.
The mathematical formulation consists of three parts: (1) The equations
of continuity and motion of flow in the annulus are formulated in the
cylindrical coordinate system based on the Hele–Shaw flow model. (2)
The equations are then converted to equations of the FEM. The difference
between the flight and disc screws can be considered by changing the
height of the elements. The FEM solves the equations and determines the
pressure of each element. (3) The pressure distribution is updated by
our newly developed scheme: down-wind pressure update scheme. If the
pressure of one element is negative or zero, the element is considered
to be empty. If the pressure is positive, the element is filled with
resin. The pressure distribution is adjusted to correspond with the
theoretical value of the degree of fill as derived from the balance of
drag and pressure-driven flow. Finally, it is determined that each
finite element is filled or empty, and the resin distribution is
obtained.
Meijer et al. described that the fluid moves in a figure ‘8’ motion
around the periphery of the screw.26 Therefore, it is
reasonable to neglect the flow in the self-wiping section and apply the
Hele–Shaw flow approximation to a cylindrical coordinate system.
Hele-Shaw flow in cylindrical
coordinates