Introduction

Twin-screw extruder are widely used in polymer processing due to their good mixing capabilities, high flexibility and high productivity. Unfortunately, numerical modeling of the flow in twin-screw extruder is extremely difficult due to the complex screw movement. This is further complicated by the presence of viscoelastic material behavior, two-phase flow in case of partially filled twin-screw extruder and non-isothermal effects caused by viscous dissipation or external heating and cooling. Our research is directed towards developing a numerical method to model this problem within the open-source software OpenFOAM®.

Preliminary research

Our preliminary research focused on developing a viscoelastic VoF-model [1] and a conditionally volume-averaged viscoelastic two-phase model [2] with extension to a conservative level-set method [3] in order to be able to capture the transient movement of the free-surface in partially filled twin-screw extruder. Due to the presence of viscoelasticity special care has to be taken to developing a highly stable and accurate solution method in order to overcome the well-known High-Weissenberg-Number-Problem (HWNP). This was done with a semi-implicit formulation for the viscoelastic constitutive equation [4] and a logarithmic reformulation [5]. A module for handling non-isothermal effects including viscous dissipation is available [6] and research on the temperature rise in single-screw extruder has also already been done [7].

Moving Immersed Boundary Method

(a) (b)

Fig. 1: Background mesh (a) and screws as immersed objects in triangulated surface file format (stl-files) (b).

In this work we focus on capturing the screw rotation with a Moving Immersed Boundary Method (IBM). A background mesh consisting solely of regular hexahedra is used (cf. Fig. 1a). The immersed objects, which are the barrel and the two screws (cf. Fig. 1b), are embodied by triangulated surfaces in stl-file format. Therefore, the task of meshing, which is often the most time consuming task in a CFD study, is completely superfluous. In order to keep the cell count small and still achieve a sufficient resolution of wall-near regions, which is particularly important to resolve the flight clearance, an adaptive mesh refinement is used, see Fig. 2a. Source terms are introduced into the immersed boundary cells (shown in red in Fig. 2b) of the discretized equations in order to directly enforce either the Dirichlet or Neumann boundary condition of the dependent variables at the triangulated surfaces of the immersed objects. Rotation of the screws is achieved by updating the position of the stl-files at the beginning of each time-step.

(a)

(b) (c)

Fig. 2: Two-dimensional cross-section of the domain with adaptive mesh refinement of the wall-near regions (immersed boundary cells indicated red) (a), predicted velocity magnitude and velocity vectors (b) and pressure field (c).

Preliminary results of a two-dimensional transient simulation are presented in Fig. 2b and 2c for the flow of a 8-mode Giesekus fluid, which was fitted to a HDPE at 150°C. The screws have a diameter of 8 cm and are co-rotating at ω = 10 rad/s. The flight clearance was assumed to be 1/50 of the screw diameter. In the snapshot of the velocity magnitude one can recognize the high velocity between the screws, which is due to the fluid being pushed through the screws because of continuity. Furthermore, large velocities and velocity gradients are present in the flight clearance, where the polymer is being sheared extensively. In the remaining fluid regions the polymer solely undergoes a solid-body rotation with hardly being sheared or elongated.

References

[1] Habla, F., Marschall, H., Hinrichsen, O., Dietsche, L., Jasak, H., Favero, J. L., *Numerical
simulation of viscoelastic two-phase flows using OpenFOAM®*,
Chem. Eng. Sci. 66 (2011) 5487-5496.

[2] Habla, F., Dietsche, L., Hinrichsen, O., *Modeling and simulation of conditionally
volume averaged viscoelastic two-phase flows*, AIChE
J. 59 (2013) 3914-3927.

[3] F. Habla, C. Waas, L. Dietsche, O. Hinrichsen, *An
Improved Conditionally Volume Averaged Viscoelastic Two-Phase Model for Simulation
of Transient Droplet Deformations under Simple Shear*, Chem. Eng. Sci. 126
(2014) 32-41.

[4] Habla, F., Obermeier, A., Hinrichsen, O., *Semi-implicit stress formulation for
viscoelastic models: Application to three-dimensional contraction flows*, J.
Non-Newtonian Fluid Mech. 199 (2013) 70-79.

[5] Habla, F., Tan,
M. W., Haßlberger, J., Hinrichsen,
O., *Numerical simulation of the
viscoelastic flow in a three-dimensional lid-driven cavity using the
log-conformation reformulation in OpenFOAM®*, J.
Non-Newt. Fluid Mech. 212 (2014) 47-62.

[6] Habla, F., Woitalka, A., Neuner, S., Hinrichsen, O., *Development
of a methodology for numerical simulation of non-isothermal viscoelastic fluid
flows with application to axisymmetric 4:1 contractions flows*, Chem. Eng.
J. 203 (2012) 772-778.

[7] Habla, F., Obermeier, S., Dietsche,
L,. Kintzel, O., Hinrichsen,
O., *CFD Analysis of the frame invariance
of the melt temperature rise in a single-screw extruder*, Int. Polym.
Proc. (2013) 463-469.

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