Transformation screw

Figure 7.1 Extruder screw transformation a) schematic of a screw inside a barrel, and b) an unwrapped channel showing the transformation with the helical length Z and the channel width W. The screw is shown moving and the barrel is stationary...

The spatial transformation matrix, is usually written as a single genoal tiansfamation, as shown in Equation 2.5. Howevo-, as demonstrated by Feath-erstone in [9], significant savings can be obtained by using two screw transformations instead, the first on the x-axis, followed by a second on the new z-axis. Thus, the transformation, X, may be written ... 

Multiplication of one screw tiansfcMmation with a general spatial vector requires (10 scalar multiplications, 6 scalar additions). Thus, the complete transformation of a genoal vector requires a total of (20 multiplications, 12 additions) if two screw transformations are used. The product of a genoal transformation between adjacent coordinate systems and a general vector, on the otho- hand, requires (24 multiplications, 18 additions). 

The number of scalar operations required by each of the four methods presented in this chapto have been calculated explicitly. As a specific example. Table 3.6 lists the computations required by the Modified Composite-Rigid-Body Method for the case of an A -link manipulator with simple revolute and prismatic joints. Note that the computations listed for the transformation matrix, % i, correspond to the use of two screw transformations as discussed in the previous section. 

A number of theories have been put forth to explain the mechanism of polytype formation (30—36), such as the generation of steps by screw dislocations on single-crystal surfaces that could account for the large number of polytypes formed (30,35,36). The growth of crystals via the vapor phase is beheved to occur by surface nucleation and ledge movement by face specific reactions (37). The soHd-state transformation from one polytype to another is beheved to occur by a layer-displacement mechanism (38) caused by nucleation and expansion of stacking faults in close-packed double layers of Si and C. 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

Figure 7.2 Transforming the Cartesian reference frame a) cylindrical cross section of the screw and barrel with flow out of the surface of the page, and b) the unwound rectangular channel with a stationary barrel and the Cartesian coordinate frame positioned on the screw, is the velocity of the screw core in the z direction and it is negative...

Figure 7.10 Transformed (Lagrangian) frame for the analysis of extruder fluid flow. Here the reference frame is positioned on the bottom of the screw channel. The observer on the frame would see the barrel move with the component velocities of and V, ...

Substituting Eq. 7.18 into Eq. 7.3 and solving Eqs. 7.1 and 7.3 for V, 14, and Vp, the solution for the transformed boundary condition problem Is obtained, and the equations are shown by Eqs. 7.21, 7.23, and 7.26. These equations physically represent the flow due to rotation and pressure in the transformed frame of reference in Fig. 7.10. Equation 7.21 is the velocity equation for the x-direction recirculatory cross-channel flow for the observer attached to the screw, and Eq. 7.23 is the apparent velocity in the z direction for the observer attached to the moving screw. 

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. 

Down-channei veiocity KizI in the transformed (Lagrangian) frame is provided by Eq. 7.23. This equation provides the veiocity in the z direction due to the rotation of the screw. 

Equation 7.23 is the iiterature expression for the down-channei veiocity in the z direction for barrei rotation physics. This down-channei veiocity Va i in the transformed reference frame must be converted back to the iaboratory frame by adjusting for the motion of the frame using Eq. 7.20. The down-channei veiocity due to the rotation of the screw in the iaboratory frame is as foiiows ... 

A three-dimensional simulation method was used to simulate this extrusion process and others presented in this book. For this method, an FDM technique was used to solve the momentum equations Eqs. 7.43 to 7.45. The channel geometry used for this method was essentially identical to that of the unwound channel. That is, the width of the channel at the screw root was smaller than that at the barrel wall as forced by geometric constraints provided by Fig. 7.1. The Lagrangian reference frame transformation was used for all calculations, and thermal effects were included. The thermal effects were based on screw rotation. This three-dimensional simulation method was previously proven to predict accurately the simulation of pressures, temperatures, and rates for extruders of different diameters, screw designs, and resin types. 

There is not an analytical velocity function for the y-direction velocity at the flights, so the wide channel approximation is used for demonstration purposes with a pressure gradient of zero. Using the equation developed previously for screw rotation for a very wide shallow channel, the transformed Lagrangian form of is the same as the laboratory form for barrel rotation and is as follows ... 

Traditionally the fluid mechanics of the extrusion process are summarized by the simple plate model illustrated in Fig. A7.1 and as described in Section 7.4. The motion of the screw is unchanged, but the reference frame has been moved to transform the problem to a fixed boundary problem for the observer. The flow in the rectangular channel is reduced into the x-direction flow across the channel and the z-direction flow down the channel. 

Equation A7.13 is the cross-channel flow in the transformed (Lagrangian) frame and concludes the derivation of Eq. 7.18. Equation A7.13 also applies to a physical device where the barrel is actually rotated. Transforming Eq. A7.13 to the laboratory (Eule-rian) reference frame as follows for a physical device where the screw is rotated ... 

Equation A7.50 is the generalized velocity in the z direction due to any combination of barrel rotation and screw core rotation in the transformed reference frame. 

The solution for screw rotation is obtained by moving Eq. A7.51 from the transformed frame back to the laboratory (Eulerian) frame ... 

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