(508g) Frame Invariance in the Simulation of Single Screw Extruder

Introduction

With
a production volume of 280 million tons per year the polymer industry is one of
the major fields in chemical industry. Polymer extrusion is probably the most
important operation in polymer-processing being used for example for reactions
and forming processes. Understanding and predicting the flow in an extruder is
therefore an important task in view of defining and improving the final product
quality.

Often the kinematics are created by keeping the screw stationary
and rotating the barrel instead of rotating the screw, which is done mainly
because this results in a more simplified modeling approach. The question of
whether this simplification can be made and the obtained results are identical
is still under examination, see for example Campbell et al. [1] and Rauwendaal
et al. [2]. The conclusions are commonly drawn for simplifying geometries and
conditions such as unwrapped channels, in which the curvature is neglected [1].
In this work we aim at clarifying this issue by doing a rigorous CFD-modeling and
showing whether the results can be transferred into each other both in two and
three dimensions. We therefore developed a technique, which rotates the whole
mesh and subsequently corrects all variables according to the given rotation.

Transport
of a passive scalar

The kneading and mixing behavior of the extruder is of major
interest. The behavior can be determined by examining the transport of a
passive scalar according to:

where c is the concentration, U is the velocity and t is time. In
Fig. 1 the mesh used for the two-dimensional analysis is given. Furthermore,
the concentration fields when rotating the extruder or the barrel are shown,
wherein the bottom half of the extruder was assigned with the passive scalar.
It can be seen, that the concentration fields match each other perfectly and
there is frame invariance. Furthermore, the results are in accordance with the
DEM analysis of Conelly and Kokini [3].

Fig.
1: Two-dimensional mixing analysis. From left to right: Mesh used for analysis,
results when rotating the extruder, results when rotating the barrel, results
of Conelly and Kokini [3], right: This work.

Kinematics

Fig. 2 shows streamline plots for both cases. When rotating the
extruder, the primary velocity field is obtained, wherefore if the barrel is
rotated, the secondary flow field is revealed. Fig. 2 clearly proves that the
velocity fields can be transferred into each other by v_re = v_rb
? ω x r (cf. the two figures on the right hand side).

Fig.
2: Two-dimensional kinematics. From left to right: Streamlines for rotating the
extruder, streamlines for rotating the barrel, velocity magnitude and vectors
when rotating the extruder, transformed velocity magnitude and vectors when
rotating the barrel by v_re = v_rb ? ω x r.

We extended the simulations towards three dimensional simulations
using a mesh of over 3,000,000 cells as being shown in Fig. 3a. The results
shown in Fig. 3b prove that the results can be interchanged into each other
even in three dimensions. The red line shows the velocity profile in the
centerline when rotating the barrel. When doing the transformation v_re
= v_rb ? ω x r the resulting velocity profile (red marks) almost
perfectly matches the profile when rotating the extruder (back line) and almost
no deviation can be perceived.

a)                                                                                                    b)

Fig.
3: a) Mesh used for three-dimensional computations. b) Velocity profiles in the
centerline: black: rotating screw, red line: rotating barrel, red marks: rotating
barrel with transformed velocity by v_re = v_rb ? ω x r.

Conclusions
and future work

In this work we
proved that the results for the velocity and a passive scalar are identical
when rotating the screw and rotating the rod. This was done using a rigorous
CFD modeling approach. One major advantage when rotating the barrel is the
computational efficiency: The results are twice as fast as the simulations when
rotating the screw. This is because each of the nodes needs to be updated at
every time step according to the current position of the screw when rotating
the screw, which results in a high computational cost. Currently we are
evaluating whether frame invariance also applies to non-isothermal flows and
here in particular to the heating term due to viscous dissipation.

References

[1]      
G. A. Campbell, M. A. Spalding and F. Carlson, Prediction of melt
temperature rise in single-screw pump extruders, ANTEC 267 (2008).

[2]      
C. Rauwendall, T. A. Osswald, G. Tellez, P. J. Gramann, Flow
Analysis in screw extruders ? effect of kinematic conditions, Int. Polym. Proc.
4 (1998) 327-333.

[3]      
R. K. Conelly, J. L. Kokini, Examination of the mixing ability of
single and twin screw mixers using 2D Finite Element Method simulation with
particle tracking, J. Food Engr. 79(3) (2007) 956-969.