同向旋转双螺杆挤出机聚合物熔体输送的数值模拟优化方案

Peng Jiong，Chen Jin-nan

Twin screw extruders are widely used in polymer processing. It is important to understand the flow of materials around the rotating twin screws. Many defects of the final products are due to a low mixing quality. One type of twin screw extruder，the intermeshing co-rotating twin screw extruder，is a high-speed machine used primarily in compounding applications.

In the last 20 years，the simulation of the flow in co-rotating twin screw extruders got a lot of attention. Due to the complex geometry in the vicinity of the interscrew region and the large deformations induced by the two rotating screws，simulating the flow around the twin screw has always been a difficulty and challenge. Many researchers limited their investigation to a 2-D flow field analysis for Newtonian or non-Newtonian fluids. However，these reports based on 2-D flow analysis could not take into account the real flow pattern because of disregard of the helix angle and various kinds of leakage flows.

Nguyen and Lindt expanded the flow analysis into the 3-D case in a non-intermeshing twin screw extruder，they simplified the flow channel using zero-helix angle screws. 3-D simulations of the flow field in co-rotating twin screw extruders were undertaken by Sastrohartono et al. In their simulations，the screw is supposed fixed，while the barrel moves in the direction opposite to screw motion，but it is physically impossible to have a barrel moving in a “figure eight” shape. Li Peng et al. developed a 3-D flow modeling of kneading block region in co-rotating twin screw extruders.

The aim of the present work was to study the distributions of the velocity，temperature and pressure in the real flow domain of the twin screw extruder and the influences of the screw speed on the flow rate of the extrusion.

1 Numerical Modeling

Molten polymers are non-Newtonian fluids，in the process of extrusion，the shear rates in the polymer melt vary greatly，commonly in the range of 10~10³s^－1. In the simulation of the non-isothermal flow，since the vis cosityηis dependent upon the shear rateand the tem peratureT，the following constitutive equation is used:

whereη₀is the zero shear rate viscosity，is the refer ence shear rate，n is the power law index，b is the temperature coefficient of viscosity，andT₀is the reference temperature.

The complex flow is simplified by the assumption that the screw channel is fully filled with a steady flow of an incompressible fluid. The Reynolds number of the flow is very small.

Considering the assumptions，the Continuity equation is satisfied，

And the components of the equation of momentum are reduced to

where p is the pressure， κis the thermal conductivity of the fluid， ρ the density and c_pthe specific heat， Φ is the heat source，which arises due to viscous heating，given by

The shear rateis given by

The shear rate greatly influences the temperatures in the extruder because of viscous dissipation.

The governing equations are solved in the flow domain to determine the velocity components in all the three directions. The energy equation is coupled with the equations of motion through viscosity. Because the governing equations are highly non-linear，the analytical solutions cannot be obtained. By means of finite element method，the numerical results of the velocity，temperature，pressure，and shear rate in the flow domain were obtained.

2 Boundary Conditions and Finite ElementMesh

No-slip boundary conditions on the screw surfaces and barrel walls were used. In order to calculate the natural flow produced by the rotation of the screws，in our simulations，the barrel is fixed，and the screws rotate counter-clockwise in the barrel at a rotational speed of 10，50，100，150，200r· min^－1. The temperature of the polymer melt at the inlet is 453K. In order to study the viscous heating in the twin screw，the barrel walls and the screws are assumed to be adiabatic. At the entry and exit，we impose vanishing forces.

Geometry specifications for a ZSK-30 extruder are as follows: barrel diameter is 30.85mm，screw tip diameter is 30.70mm，screw root diameter is 21.30mm，screw lead is 28.00mm，and center distance of the screws is 26.20mm.Two screws lie adjacent to each other in a barrel casing whose cross section will be in a figure eight pattern.The materials move towards the die with the help of screw flights and the relative movement between the barrel and the screw.In order to save memory and CPU time，the flow domain is limited to one flight.The cross section of the twin screw extruder is shown in Fig.1.The mesh superposition technique is used to simplify the mesh generation of the flow domain (see Fig.2).The elements used were hexahedron elements with 8 nodal points in each element.The total number of the Nodal points for the geometry is 14136，and the total number of elements is 10176.Material properties used in the simulation are given in Tab.1.

Fig.1 The dimensions of the cross section (in mm)(www.xing528.com)

Fig.2 The finite element mesh of the flow domain

Tab.1 Material properties used in the simulation (T＝453K)

3 Numerical Results and Discussion

A numerical study of the 3-D polymer melt flow and heat transfer in a co-rotating twin screw extruder has been carried out. The numerical scheme incorporated velocity components in all the three directions and calculated the resulting pressure，temperature，viscosity and viscous heating.

The pressure contours at the cross section ofz＝14mm are presented in Fig.3. The pressure gradient in the intermeshing region is higher than that in the other areas. And the maximum pressure and the minimum pressure occur in the intermeshing region.

The velocity vectors at the cross section ofz＝14mm，are presented in Fig.4. In the intermeshing region，the velocity vectors are not all in the same direction，and this is one of the reasons why the twin screw is much more efficient in polymer mixing than the single screw.

The viscosity profiles in the flow channel are presented in Fig.5. We find that the viscosity in the center of the flow channel is much larger than that in the other parts of the flow channel，for the shear rate in these regions is much smaller. In the intermeshing region，the viscosity is very small for the high shear rate in these regions.

Fig.3 Pressure contours at the crosssection ofz＝14mm(inMPa)

The viscous heating in the flow channel is presented in Fig.6，the viscous heating is large in the intermeshing region and the region near the barrel wall because of the large shear rate in these regions.

The temperature profiles on the plane ofy＝－13mm are shown in Fig.7. The temperatures in the intermesh-ing region and near the barrel wall are found to be higher than that in the other parts of the extruder for the viscous heating in those regions is much larger. Therefore，in polymer extrusion，the cooling of the barrel in those regions is necessary to prevent the degradation of the polymer.

Fig.4 Velocity vectors at the cross section ofz＝14mm

Fig.5 Viscosity profiles in the flow channel (in Pa·s)

Fig.6 Viscous heating in the flow channel (in MW·m^－3)

Fig.7 Temperature profiles on the planey＝－13mm(mK)

The flow rates at different screw speeds are shown in Fig.8. For incompressible polymer melt，the flow rate is almost proportional to the screw speed.

Fig.8 The flow rate at different screw speeds

4 Conclusion

This simulation provides understanding of the nonisothermal non-Newtonian viscous polymer flow in a ZSK-30 twin screw extruder. These results are important for the optimization of the melting and mixing processes in the extruder and the design of corotating twin screw extruders. The simulation of the polymer extrusion in the kneading elements will be a subject of a forthcoming article.