Pressure drag flow

Simple pressure/drag flow. Here we treat an idealization of the down-channel flow in a melt extruder, in which an incompressible viscous fluid constrained between two boundaries of infinite lateral extent (2). A positive pressure gradient is applied in the X-direction, and the upper boundary surface at y - H is displaced to the right at a velocity of u(H) - U this velocity is that of the barrel relative to the screw. This simple problem was solved by a 10x3 mesh of 4-node quadrilateral elements, as shown in Figure 1. 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

As discussed in the previous section, it is convenient to consider the output from the extruder as consisting of three components - drag flow, pressure flow and leakage. The derivation of the equation for output assumes that in the metering zone the melt has a constant viscosity and its flow is isothermal in a wide shallow channel. These conditions are most likely to be approached in the metering zone. 

The total output is the combination of drag flow, back pressure flow and leakage. So from (4.3), (4.7) and (4.8)... 

If the clearance between the rolls is small in relation to their radius then at any section x the problem may be analysed as the flow between parallel plates at a distance h apart. The velocity profile at any section is thus made up of a drag flow component and a pressure flow component. 

The streamlines of this flow are shown by Peters and Smith (12). In this case, the effective thickness of this layer appears to be about equal to the gap with the wall, indicating a pressure flow about equal to the drag flow. It can be calculated that this would increase the maximum shear rate on the fluid passing under the agitator blade by a factor of seven. 

FIGURE 8.5 Drag flow between parallel plates with the upper plate in motion and no axial pressure drop. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

Which is better for isothermal chemical reactions, pressure driven flow or drag flow between flat plates Assume laminar flow with first-order chemical reaction and compare systems with the same values for the slit width (2Y=H), length, mean velocity, and reaction rate constant. 

Over the twentieth century, the mbber industry has developed special rheometers, essentially factory floor instmments either for checking process regularity or for quality control purposes, for instance, the well-known Mooney rheometer (1931), the oscillating disk rheometer (1962), and the rotorless rheometer (1976). All those instmments basically perform simple drag flow measurements but they share a common feature During the test, the sample is maintained in a closed cavity, under pressure, a practice intuitively considered essential for avoiding any wall slip effects. Indeed it has... 

Figure 35.34 shows a slight dependency of the pressure buildup on the calender hne speed, which equals the circumferential roll speed. The general shape of the pressure curve can be understood as follows. A converging drag flow yields a pressure buildup until a barrier has been passed. The material left (=upstream) from the pressure maximum will take part in the roUing bank flow. The material between the pressure maximum and the clearance of the calender flows by means of the drag flow and pressure flow. Each material volume element wfll pass the clearance. At the position where the pressure vanishes the sheet will be taken apart from one of the rolls. 

Figure 1. Simulation of drag/pressure plane flow.

Knowledge of the geometry and mathematical description of a screw Is required to understand the analysis of the functional sections of the screw and the troubleshooting of case studies. In Chapter 1 the geometry and mathematical descriptions are presented. Also In this chapter, the calculation of the rotational flow (also known as drag flow) and pressure flow rates for a metering channel Is Introduced. Simple calculation problems are presented and solved so that the reader can understand the value of the calculations. 

Two driving forces for flow exist in the metering section of the screw. The first flow is due just to the rotation of the screw and is referred to as the rotational flow component. The second component of flow is due to the pressure gradient that exist in the z direction, and it is referred to as pressure flow. The sum of the two flows must be equal to the overall flow rate. The overall flow rate, Q, the rotational flow, 0 and the pressure flow, Qp, for a constant depth metering channel are related as shown in Eq. 1.12. The subscript d is maintained in the nomenclature for historical consistency even though the term is for screw rotational flow rather than the historical drag flow concept. 

The degradation ribbon at the merger of the flows occurs because of the crosschannel flow of material from the region between the solid bed and the screw root to the melt pool. As shown by Fig. 6.35, this flow is relatively large. As previously stated, the flow occurs because of pressure-induced flow and the dragging of fresh material under the solid bed by the backwards motion of the screw root. This process is consistent with the physics presented for screw rotation. The flow fields developed for a barrel rotation system would not create the low-flow region such as shown in Fig. 6.37. 

The generalized Newtonian model over-predicted the rotational flow rates and pressure gradients for the channel for most conditions. This over-prediction was caused in part by the utilization of drag flow shape factors (FJ that were too large. Then in order for the sum of the rotational and pressure flows to match the actual flow In the channel, the pressure gradient was forced to be higher than actually required by the process. It has been known for a long time [9] that the power law... 

The velocity component in the z direction for drag flow and pressure flow are provided in Eqs. 7.86 and 7.87. This equation uses Eqs. 7.23 and 7.27 for N/Ws that are less than 0.1. If the channel aspect ratio H/Wli, greater than 0.1, then Eqs. 7.23 and 7.27 should be used. The generalized solution using Eqs. 7.23 and 7.27 is provided in Appendix A7. 

Spalding, M. A. and Campbell, G.A., The Accuracy of Standard Drag Flow and Pressure Gradient Calculations for Singie-Screw Extruders, SPE ANTEC Tech. Papers, 54, 262 (2008)... 

The z-direction velocity is in the downstream direction and parallel to the flight edges and described by Eq. A7.7. The z-direction flow is considered to be the combination of pure pressure flow and pure drag flow. The boundary conditions for Eq. A7.7 are as follows for barrel rotation ... 

Eq. A7.7 is provided by Eqs. A7.18 and A7.19 [pages 190 to 199 in reference 1]. For barrel-rotation-driven flow in the z direction for the transformed reference frame, the drag flow and pressure flow components are as follows ... 

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