Hu Dong-dong，Chen Jin-nan
Screw extruders are widely used for compounding and blending in polymer processing. In recent years，intermeshing co-rotating three-screw extruders are developed based on the traditional intermeshing co-rotating twin-screw extruders，in order to bring about more efficient mixing effect over traditional counterparts. Com-pared to the twin-screw extruders，the three-screw extruders enjoy more flexible arrangement of screw positions，more intermeshing regions and higher output/energy ratio，which probably promises a large scope for future development. So far，little attention is devoted to flow simulations in three-screw extruders，because the flow field is more complicated and the time-dependent flow boundaries pose a great difficulty for mesh generation. In this work，with finite element package POLY-FLOW and its mesh superposition technique，the mesh generation problem is easily solved and the simulation of the full 3D flow in such complicated-configured extruders is achieved.
As for the three-screw extruders，the screw layout could be configured as three forms，i. e. the linear/parallel type，isosceles triangle and equilateral triangle. A numerical investigation of the flow in the kneading block region of the following three types of extruders is presented in this work:
① the intermeshing co-rotating three-screw extruders with screw layout in a equilateral-triangle shape (triangular for short)；
② the intermeshing co-rotating three-screw extruders with screw layout in a linear-type shape (parallel for short)；
③ the traditional intermeshing co-rotating twinscrew extruders (twin for short).
Based on the consecutive quasi-steady velocity fields，the flow patterns in these machines are visuall. ized，and the pumping and mixing characteristics，are analyzed and compared.
1 Geometry Specifications and Finite ElementMeshing
To ensure three screws not interfering with each other in the triangular-arranged three-screw extruder，three-tip screw element is used in our study.The same geometric parameters for the three machines are used as follows；screw root diameter 42.0mm，screw tip diameter 53.0mm，barrel diameter 53.2mm and centerline distance 48.0mm.For the kneading block，the stagger angle of the discs is 30°，the disc width 6mm and the length of the kneading block 30mm.The mesh superposition technique (MST) is used to simplify the mesh generation of the flow domain (Fig.1).This method is particularly robust and convenient since it avoids any remeshing requirement.
Fig.1 Screw configurations and finite element meshing with MST method
2 Mathematical Models
The same mathematical model is used to simulate the flow fields in the three extruders. The polymer melts are considered as incompressible and pure viscous non-Newtonian fluid；the flow is quasi-steady，laminar and isothermal；the flow channel is fully filled；inertia and gravitational force neglighible.
Form these assumptions，the continuity and momentum equations along with the generalized constitutive equation are reduced to
where u is the velocity vector； p the pressure； τ the stress tensor；
the shear rate and D the deformation rate tensor，respectively.
The rheology of the fluid is described in terms of a Carreau model:
whereη0is the zero shear rate viscosity；λcandn are Carreau model’s parameters.
The model fluid used is low density polyethylene(LDPE)，with its Carreau-Model parameters fitted as follows；η0＝19500Pa·s；λc＝5.5，n＝0.52.
No-slip boundary conditions are used for the screw surfaces and chamber walls. Fluid elements are stationary on the chamber wall and move with and angular velocity equivalent to the specified revolution per minute on the screw surface. For all the cases studied in this work，the rotational speed of screws is maintained at 100r/min. We also specified the axial pressure difference between the entrance and exit planes of the flow domain.
After the velocity field is calculated，we also track the trajectories of fluid particles at a time step of 1/30s (corresponding to the time of 20° of screw rotation in this simulation).Then POLYSTAT is used to visualize these trajectories and to do statistical treatment on them.All the simulations are carried out on the SGI-O2 (R10000) workstation with POLYFLOW3.7.It has to be pointed out that because of computer limitations the time step here is too large for accurate tracking of the fluid particles.Also，the mesh for the flow domain is not fine enough.But as the first and preliminary investigation of the flow in three-screw extruders，out work would still render some meaningful results for practical reference.In the near future，we will try to shorten the time step for particle tracking simulations and recalculate the flow fields with refined meshes.
3 Results and Discussions
3.1 Flow Patterns Visualization with ParticleTracking Technique
In the theoretical research of various screw extrud-ers，the accurate knowledge of flow patterns is of significant importance in understanding the conveying and mixing mechanism of screw elements. The numerical visualization of flow patterns in these extruders would have a great advantage over the traditional flow-visualization experiments. In our previous work，the trajectories of fluid particles in twin screw extruders and triangulartype three-screw extruders had been presented. Suppose a total of 1000 particles are placed at the entrance of the kneading block region for each type of extruder；tracking of these particles in 18s yields the trajectories displayed in Fig.2 and Fig.3. Similar to the flow patterns in twinscrew extruders，the predominant flow pattern in the intermeshing co-rotating three-screw extruders is also an alternating rotational axial motion (Fig.2a and Fig.3). For the triangular-type machines，some particles have a transient stroll in the central region (Fig.2b)，and still some incidentally revolve around one screw during their travels (Fig.2c). As three screws are just partly rather than fully intermeshed，the open channels exist between three intermeshing screws. This causes a small amount of particles to stray from the mainstream.
Fig.2 Particle trajectories in the kneading block region of triangularly-arranged intermeshingco-rotating three-screw extruders (represented by bold lines)
Fig.3 Particle trajectories in the kneading block region of parallelly-arranged intermeshingco-rotating three-screw extruders (represented by bold lines)
3.2 Pumping Characteristics of the Three Types ofExtruders(www.xing528.com)
The volumetric flowrate as a function of the axial pressure difference is plotted in Fig.4. The change in output with pressure gradient is more pronounced for the three-Screw extruders by comparison with the twin-screw extruders，with the parallel slightly below the triangular，which means the three-screw extruders have larger pumping capacity over twin-screw extruders. On the other hand，the twin-screw extruder may enjoy more stable flowrates if the pressure gradient fluctuates slightly，as the slope of its screw characteristic curve is less than that of its three-screw counterparts.
Fig.4 Volumetric flowrate vs. axial pressure difference
We also calculated the volumetric flowrates at different screw rotational angles when zero axial pressure difference is imposed. The result is plotted in Fig.5. The more stable pumping operation is found in the order of triangular>twin>parallel in terms of flowrate fluctua-tion with screw rotation.
Fig.5 Volumetric flowrate vs. rotational degreeof screws with Δp＝0
3.3 Instantaneous Mixing Characteristics of theThree Types of Extruders
Fig.6 shows the profiles of shear rate distribution at a cross-section of the flow field domain. All three machines show very similar characteristics in that the shear rates in the intermeshing region and screw tip region are prominently higher than in other areas. Yet we could not evaluate the mixing performance of three machines from the shear rate contours. In this work，three mixing indices，i. e. the average shear rate，average characteristic shear stress and average elongational flow index，are used to characterize the mixing capacity of the kneading discs. All of these indices are based on the instantaneous flow field and their definitions are given as follows:
Average shear rate is defined as
Average characteristic shear stress is given as
Average elongational flow index is given as
Fig.6 Profiles of shear rate distribution on cross-sections of three machines
The profiles of these mixing indices vs. axial pressure difference are plotted in Fig.7，Fig.8 and Fig.9 respectively. In Fig.7，the average shear rate over the whole flow field domain is in the order of parallel>triangular≈twin；while Fig.8 shows that the average characteristic shear stress is in the sequence of parallel>twin>triangular. All curves in Fig.7 and Fig.8 reach aminimum value at zero axial Pressure difference，and a negative pressure gradient corresponds to a higher average characteristic shear stress than an equivalent positive pressure gradient. Fig.9 indicates that the average elongational flow index takes the sequence of twin>parallel>triangular，yet the differences are not very distinct between three machines. Judging from these results it seems that in all three cases，the triangular extruder shows inferior mixing performance. However，it is worthwhile to note that these mixing indices are instantaneous values which do not provide information about cumulative effects on the polymer melt. What is more，the higher average mixing index doesn′t ensure a homo-geneous mixing quality over the whole flow field.
Fig.7 Profile of average shear rate vs.axial pressure difference
Fig.8 Profile of average characteristic shear stressvs. axial pressure difference
Fig.9 Profile of average elongational flow indexvs. axial pressure difference
Fig.10 Cumulative residence time distributionin three types of extruders
3.4 Cumulative Residence Time Distribution
Based on the 1000 trajectories，POLYSTAT is used to calculate the cumulative residence time distribution (RTD) for the three types of extruders respectively.The result is shown in Fig.10.The three curves almost overlap for residence time exceeding 4s.In order to make the difference more disincentively，we clipped the residence time range from (0~ 18s) to (0~ 4s).From Fig.10，a very striking feature for the three triangularly-arranged extruder is that the curve has a small shoulder at the time range between 0.25s and 0.40s，which means almost 20% of the 1000 particles has a residence time less than 0.40s，and this part of polymer melt flows more quickly throughout the flow domain and gets poor axial mixing.
In fact，different from its parallelly-arranged coun-terpart，there is a so-called central region between the three screws in the triangularly-arranged extruder. The flow in the central region has been analyzed using particle tracking technique in our previous work，which reveals that the particles released in the central region of this machine travels much more quickly than those initially distributed in other parts of the flow channel. Thus we could come to such a conclusion that the plateau at the start section of the cumulative RTD curve rightly corresponds to the faster flow in the central region of the triangular type extruder.
4 Conclusions
In this study，3-D isothermal flow simulations of a non-Newtonian fluid in two types of three-screw extruders and a twin-screw extruder have been carried out by using Polyflow.
The numerical results indicate that the flow patterns in three-screw extruders are similar to those in twinscrew extruders. The triangular-arranged three-screw extruder has the largest pumping capacity and also the highest extrusion stability in terms of flowrate fluctuation with screw rotation.
Several instantaneous mixing indices have been proposed to describe the mixing capacities of the kneading blocks in the three machines. As a tentative conclusion，the parallel-type machine offers the highest average shear rate and characteristic shear stress over the whole flow field domain. It has to be pointed out that the cumulative mixing characteristic is another important aspect that measures the mixing capacity of screw machines，however，to calculate such indices，much more memory and CPU resources are required. Further elaborate investigations are to be carried out in out later presentations.
Cumulative residence time distributions for the three machines indicate that a small amount of particles in the triangular-arranged three-screw extruder has a shorter residence time，which rightly corresponds to the faster flow in the central region of this type of extruder.
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