Cross-channel flow

Elbirli, B., Lindt, J.T., Gottgetreu, S. R., and Baba, S.M., Mathematical Modeling of Melting of Polymers in a Single-Screw Extruder, Polym. Eng. ScL, 24, 988 (1984) Lindt, J.T. and Elbirli, B., Effect of the Cross-Channel Flow on the Melting Performance of a Single-Screw Extruder, Polym. Eng. ScL, 25, 412 (1985)... 

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. 

Figure 8.14 Two-dimensional flows in a screw channel with a 6/14 = 1 and operating in extrusion mode. The region shown is the shaded area in Fig. 8.13. Cross-channel flow lines and the corner circulation of the Moffat eddy are visible in the schematic...

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 ... 

Mixing in single screw extruders. Mixing caused by the cross-channel flow component can be further enhanced by introducing pins in the flow channel. These pins can either sit on the screw as shown in Fig. 3.29 [9] or on the barrel as shown in Fig. 3.30 [15]. 

Cross Channel Flow in a Single Screw Extruder... 

The cross channel flow is derived in a similar fashion as the down channel flow. This flow is driven by the x-component of the velocity, which creates a shear flow in that direction. However, since the shear flow pumps the material against the trailing flight of the screw channel, it results in a pressure increase that creates a counteracting pressure flow which leads to a net flow of zero1. The flow rate per unit depth at any arbitrary position along the 2-axis can be defined by... 

If we combine the flow generated by the down channel and cross channel flows, a net flow is generated in axial or machine direction (w ) of the extruder, schematically depicted in Fig. 6.5. As can be seen, at open discharge, the maximum axial flow is generated, whereas at closed discharge, the axial flow is zero. From the velocity profiles presented in Fig. 6.5 we can easily deduce, which path a particle flowing with the polymer melt will take. 

A full analytical solution of the cross channel flow vx(x,y) and vy x, y), for an incompressible, isothermal Newtonian fluid, was presented recently by Kaufman (18), in his study of Renyi entropies (Section 7.4) for characterizing advection and mixing in screw channels. The velocity profiles are expressed in terms of infinite series similar in form to Eq. 6.3-17 below. The resulting vector field for a channel with an aspect ratio of 5 is shown... 

Fig. 6.11 Vector field of the cross-channel flow of an incompressible isothermal Newtonian fluid in a channel with an aspect ratio of 5. [Reprinted by the permission from M. Kaufman, Advection and Mixing in Single Screw Extruder—An Analytic Model, The AIChE Annu. Tech. Conf. Meeting Proc., San Francisco (2003).]...

Distributed Parameter Models Both non-Newtonian and shear-thinning properties of polymeric melts in particular, as well as the nonisothermal nature of the flow, significantly affect the melt extmsion process. Moreover, the non-Newtonian and nonisothermal effects interact and reinforce each other. We analyzed the non-Newtonian effect in the simple case of unidirectional parallel plate flow in Example 3.6 where Fig.E 3.6c plots flow rate versus the pressure gradient, illustrating the effect of the shear-dependent viscosity on flow rate using a Power Law model fluid. These curves are equivalent to screw characteristic curves with the cross-channel flow neglected. The Newtonian straight lines are replaced with S-shaped curves. 

Let us next consider the simple isothermal drag flow (dP/dz = 0) of a shear-thinning fluid in the screw channel. The cross-channel flow, induced by the cross-channel component of the barrel surface velocity, affects the down-channel velocity profile and vice versa. In other words, the two velocity profiles become coupled. This is evident by looking at the components of the equation of motion. Making the common simplifying assumptions, the equation of motion in this case reduces to... 

P.E. Arratia, C.C. Thomas, J. Diorio, and J.P. Gollub. Elastic instabilities of polymer solutions in cross-channel flow. Phys. Rev. Lett., 96 144502, 2006. 

These interesting devices consist of a tube or duct within which static elements are installed to promote cross-channel flow. See Figure 8.5 and Section 8.7.2. Static mixers are quite effective in promoting radial mixing in laminar flow, but their geometry is too complex to allow solution of the convective diffusion equation on a routine basis. A review article by Thakur et al. (2003) provides some empirical correlations. The lack of published data prevents a priori designs that utilize static mixers, but the axial dispersion model is a reasonable way to correlate pilot plant data. Chapter 15 shows how Pe can be measured using inert tracers. 

The net flow rate in the cross-channel direction should be zero if it is assumed the leakage over the flight is negligible. The cross-channel flow rate is ... 

Figure 7.127 Cross-channel flow and resulting shear deformation...

Other mixing screws have been developed in the past to disrupt the solid bed and mix unmelted with melted material. The double wave screw shown in Fig. 8.80 breaks up the solid bed and mixes the material by forcing a cross-channel flow by the cyclic variation in channel depth. The principle of the double wave screw was used by Barr in his energy transfer (ET) screw [90]. The ET section is basically a double wave section with occasional undercuts in both flights to force a cross-channel mixing between the two channels. Modeling of the ET mixer is discussed in Section 12.4.3.2 see also Figs. 12.23 to 12.25. 

The drag-induced pressure gradient in the clearance g<,2 can be related to the drag-induced pressure gradient in the channel gn by stating that the net cross-channel flow equals the flow through the clearance ... 

Whilst the layers nearest the screw and barrel are spiralling around the top and bottom of the channel to give good lamina shear mixing, the area centred on the stationary (with respect to cross channel flow) two-thirds depth layer travels down channel at a faster rate. Furthermore a spiralling particle spends less of its time above the 2/3 h level than below it. 

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