Screw rotation theory

Eqs. 7.22 and 7.24 represent the velocities due to screw rotation for the observer in Fig. 7.9, which corresponds to the laboratory observation. Eq. 7.25 is equivalent to Eq. 7.24 for a solution that does not incorporate the effect of channel width on the z-direction velocity. For a wide channel it is the z velocity expected at the center of the channel where x = FK/2 and is generally considered to hold across the whole channel. The laboratory and transformed velocities will predict very different shear rates in the channel, as will be shown in the section below relating to energy dissipation and temperature estimation. Finally, it is emphasized that as a consequence of this simplified screw rotation theory, the rotation-induced flow in the channel is reduced to two components x-direction flow, which pushes the fluid toward the outlet, and z-direction flow, which tends to carry the fluid back to the inlet. Equations 7.26 and 7.27 are the velocities for pressure-driven flow and are only a function of the screw geometry, viscosity, and pressure gradient. 

As mentioned in the introduction to this chapter this is a necessary condition when approximating the cylindrical screw in the Cartesian coordinate system. The screw rotation theory, New Theory line, predicts that the rate should constantly increase as the channel gets deeper. When a fixed positive pressure occurs for the screw rotation model, the New Theory with Pressure line, the predictions fits the data very well for all H/Ws. Thus for modern screw designs with deeper channels, reduced energy dissipation, and lower discharge temperatures, the screw rotation model would be expected to provide a good first estimation of the performance of the extruder regardless of the channel depth for Newtonian polymers. 

The double integral represents the nonzero terms of the dissipation rate tensor as adapted by Middleman [61] and Bernhardt and McKelvey for adiabatic extrusion [62]. The nontensorial approach was adopted by Tadmor and Klein in their classical text on extrusion [9]. In essence these are the nonzero terms of the dissipation rate tensor when it is applied to the boundary of the fluid at the solid-fluid interface. In the following development this historic analysis was adopted for energy dissipation for a rotating screw. In this case the velocities Ui are evaluated at the screw surface s and calculated in relation to screw rotation theory. The work in the flight clearance was previously described in the literature [9]. The shear... 

Theory Development for Melting Using Screw/ Rotation Physics... 

Because it is more complicated to solve the moving boundary problem for the rotation of the screw, the barrel rotation models described above have been extensively adopted and investigated. In practice the screw is rotated and not the barrel. The barrel rotation theory has several limitations when describing the real extrusion process, so correct interpretation of the calculated results based on barrel rotation becomes necessary. Most screw design practitioners, with substantial previous design experience, make major adjustments in design specifications to obtain effective correiations. 

Figure 7.13 Comparison of literature drag and screw rotation for deep channels [45], The experimental data for screw rotation and barrel rotation and the theory lines were for screws with a 7° helix angle...

Experimental and simulation results presented below will demonstrate that barrel rotation, the physics used in most texts and the classical extrusion literature, is not equivalent to screw rotation, the physics involved in actual extruders and used as the basis for modeling and simulation in this book. By changing the physics of the problem the dissipation and thus adiabatic temperature increase can be 50% in error for Newtonian fluids. For example, the temperature increase for screw and barrel rotation experiments for a polypropylene glycol fluid is shown in Fig. 7.30. As shown in this figure, the barrel rotation experiments caused the temperature to increase to a higher level as compared to the screw rotation experiments. The analysis presented here focuses on screw rotation analysis, in contrast to the historical analysis using barrel rotation [15-17]. It was pointed out recently by Campbell et al. [59] that the theory for barrel and screw rotation predicts different adiabatic melt temperature increases. 

C. Vermilyeo, J. Chem. Phys. 28 1254 (1956). Screw dislocation theory kinetics of rotation. 

Crystal symmetries that entail centering translations and/or those symmetry operations that have translational components (screw rotations and glides) cause certain sets of X-ray reflections to be absent from the diffraction pattern. Such absences are called systematic absences. A general explanation of why this happens would take more space and require use of more diffraction theory than is possible here. Thus, after giving only one heuristic demonstration of how a systematic absence can arise, we shall go directly to a discussion of how such absences enable us to take a giant step toward specifying the space group. 

The theory developed up to this point is based on a model where the screw is stationary and the barrel rotates around the screw. It is assumed that the flow that results is the same as when the barrel is stationary and the screw rotates in the opposite direction. This assumption was considered valid for over fifty years until several workers challenged this assumption, first in the early 1990s [272-276], and then more recently [323]. Because of the importance of this issue we will critically analyze this assumption to determine to what extent the assumption is correct. Flow will be analyzed using the parallel plate assumption with either the barrel or the screw considered moving. Flow will also be analyzed without the parallel plate assumption, using a cyhndrical coordinate system, again considering both cases. This analysis is based on a study by Osswald et al. [281]. 

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