263628 Frame Invariance in the Simulation of Single Screw Extruder
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.  and Rauwendaal et al. . The conclusions are commonly drawn for simplifying geometries and conditions such as unwrapped channels, in which the curvature is neglected . 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 .
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 , right: This work.
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 vre = vrb – 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 vre = vrb – 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 vre = vrb – 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.
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 vre = vrb – 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.
 G. A. Campbell, M. A. Spalding and F. Carlson, Prediction of melt temperature rise in single-screw pump extruders, ANTEC 267 (2008).
 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.
 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.