Numerical simulation of the flow field in the mixing section of a screw extruder by the lattice Boltzmann model
Introduction
A single-screw extruder is commonly used in polymer processing. The mixing performance of the extruder considerably influences the quality and morphology of the final product and so the flow field in the mixing section has been studied by a number of authors to gain a better understanding of the process. Yao et al., 1996, Yao et al., 1997 performed simulations using the finite difference method (FDM) which were found to compare favourably to a visualization experiment involving the flow of high-viscosity corn syrup in a rectangular cavity with a moving top boundary. Further work was performed by Horiguchi et al. (2003) using the lattice gas method (LGM). Qualitatively the FDM and LGM determined the same oval-shaped flow pattern as was observed in the visualisation experiment. The simulations differed in that small circulation was observed in the bottom left and right corners of the LGM simulations; whereas no circulation is evident at the bottom left-hand corner in the FDM results. Horiguchi et al. (2003) also considered a quantitative comparison with theory. This indicated that the LGM produced a more accurate representation of the flow field compared to the FDM, however there was still a discrepancy between the LGM simulation and the theoretical expression for the velocity.
The success of the LGM in simulating flow in a single-screw extruder has motivated the present study using the lattice Boltzmann model (LBM) which has been developed from the LGM in order to overcome a number of difficulties. The main aim of this study is to demonstrate the ability of the LBM to accurately simulate flow in the mixing section of a screw extruder. Using a coordinate system attached to the rotating screw, a screw extractor can be modelled as a cavity flow problem with one moving wall and three stationary walls. The velocity field can then be divided into two components: the two-dimensional cross-section component and the stream-wise component. One of the main applications of computer simulations is to investigate different geometries in order to maximise mixing. In this study a mixing section without pins was selected and the two-dimensional cross-section component considered to enable the result to be compared with alternative numerical, experimental and analytic results. (Yao et al., 1997, Horiguchi et al., 2003, McKelvey, 1962).
Section snippets
Numerical model
The LBM (Chen and Doolen, 1998) has evolved from the LGM. Despite the success of the LGM (Biggs and Humby, 1998; Rivet and Boon, 2001), there are a number of difficulties in applying the LGM and this has lead to the development of the LBM. Below a number of these problems and their solution in the LBM are briefly considered, with particular attention to the simulations presented here and the work of Horiguchi et al. (2003) (HTY).
In the LGM individual ‘particles’ are considered to move on the
The lattice Boltzmann model
The LBM is well described in the literature, see for example (Chen and Doolen, 1998, Wolf-Gladrow, 2000, Succi, 2001). Here we give a brief description of the technique. The simulations are performed on a fixed underlying lattice, in the simulations presented here a square lattice with diagonals is used. Vectors , are defined along the eight link directions of the lattice for as well as a null vector, , see Fig. 1. At each site 9 distribution function are defined, one in each of
Results and discussion
Simulations of a single-screw extractor were performed using the LBM. Although the flow field in a single-screw extruder consists of two components, only the flow field perpendicular to the spiral direction is considered. This is to enable a direct comparison with previous work and to enable comparison with theory. The computational grid had L grid point in the x-direction and H grid points in the y direction and two difference sets of values for L and H were selected. One had and ,
Conclusion
The flow field in the mixing section of a single-screw extractor has been simulated using the LBM. The x-component of the velocity over a cross-section was compared with theory and with LGM and FDM simulations. The results show that the LBM provides a more accurate realisation of the flow field compared to previous simulations using the LGM and the FDM. The results also show no evidence of a small circulation in the bottom corners of the mixing section. The LBM has the same ability as the LGM
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