Modeling of polymer flows in melt spinning

In a melt spinning process, the two different types of polymer flow start to develop as soon as the polymer pellets or granules come into contact with the screw. Initially, tangential flow is developed due to the rotation of the screw but axial flow may also occur due to the pressure difference across the duct, or when the polymer flow comes into contact with the screw heUx (Tadmor and Gogos, 1979). 

The new analytical models yielded encouraging results however, the models were valid only for Newtonian fluids, and as such cannot be used for real polymeric fluids (which are primarily non-Newtonian). 

Booy (1963) has considered the effects of curvature and helicity of the screw and derived a model known as the curved channel model. This model was further generalized by Yu and Hu (1998) to calculate the velocity profile and flow rates. The results were compared with the parallel plate models and found to be very similar in drag and pressure flows. However, these models were more accurate in the case of deeper screw channels such as twin-screw extruders or some special single-screw extruders. The details of modeling of polymer flows in various stages of the screw are given by Tadmor and Gogos (1979). 

After passing through the screw, the molten polymer flows into the capillary holes or slots of the spinneret, where it is pushed into a small duct to achieve the desired shape and size as determined by the spinneret. In general, the hole or slot shapes may be classified in two categories circular and non-circular. In the case of circular shapes (Bird, 1977) the equations of motion involving non-Newtonian flows can be solved by assuming polymeric flows as power law fluids. The flow rates of such fluids are shown in Equation [4.7]. 

For non-circular shapes, the equations of motion may result in nonlinear partial differential equations, which are difficult to solve analytically. Therefore, approximate methods such as the variational method (Kantorovich and Krylov, 1958) are generally used for solving non-Newtonian flow problems. Schechter (1961) used the application of the variational method to solve the non-linear partial differential equations of pressure drop and flow rate of the polymer for non-circular shapes such as a rectangle or square. Moreover, Mitsuishi and Aoyagi (1969 1973) used similar methods for other non-circular shapes such as an isosceles triangle. The results were based on the Sutterby model (1966), which incorporates a viscosity function based on the rheological constants. Flow curves with pressure drop and flow rate for both circular and non-circular shapes were generated and the results were compared with the power law model. 

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