Drag flow shape factors

The fraction of the channel that is partially full,/, can be approximated using Fig. 6, which is based on results developed by Squires (1958). The drag flow shape factor Fu is given by... 

Fig. 6. Drag flow shape factors for partially filled channels in a single-screw extruder.

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... 

Fd Drag flow shape factor in a screw extruder (6.3-20)... 

Fj Drag-flow shape factor for co-rotating disk processor (Example 6.12)... 

Note that the shape factors plotted in Fig. 6.13 are a function of only the II/W ratio. The effect of the flight on the pressure flow is stronger than that on drag flow. When the ratio Il/W diminishes, both approach unity. In this case, Eq. 6.3-19 reduces to the simplest possible model for pumping in screw extruders, that is, isothermal flow of a Newtonian fluid between two parallel plates. 

Fig. 6.13 Shape factors for drag and pressure flows from Eqs. 6.3-20 and 6.3-21.

Now it is clear that the drag- and pressure-flow components need to be multiplied by appropriate shape factors.18 The total pressure rise, APt, depends on its length Lt ... 

We now take the drag and pressure flow terms in Eq. 9.2-5 and substitute the relevant numerical values. We assume a square pitched screw, neglecting the difference between mean and barrel surface helix angle, and neglecting shape factors and flight clearance. We further assume that flight width is 10% of the barrel diameter. We can make these simplifying assumptions because, at this point, we only wish to select the barrel diameter and the screw speed. The channel width can be expressed in terms of the screw diameter as follows ... 

The aerodynamic diameter dj, is the diameter of spheres of unit density po, which reach the same velocity as nonspherical particles of density p in the air stream Cd Re) is calculated for calibration particles of diameter dp, and Cd(i e, cp) is calculated for particles with diameter dv and sphericity 9. Sphericity is defined as the ratio of the surface area of a sphere with equivalent volume to the actual surface area of the particle determined, for example, by means of specific surface area measurements (24). The aerodynamic shape factor X is defined as the ratio of the drag force on a particle to the drag force on the particle volume-equivalent sphere at the same velocity. For the Stokesian flow regime and spherical particles (9 = 1, X drag... 

To account for the shape effects during the flow of nonspherical particles, Fuchs (1964) defined the shape factor % as the ratio of the actual drag force on the particle FD to the drag force F g on a sphere with diameter equal to the volume equivalent diameter of the particle ... 

Drag coefficient Dimensionless coefficient factor representing the effect of shape and flow-field characteristics on the drag exerted on a body immersed in a flow. 

However, the curve of the sphere drag coefficient has some marked differences from the friction factor plot. It does not continue smoothly to higher and higher Reynolds numbers, as does the / curve instead, it takes a sharp drop at an of about 300,000. Also it does not show the upward jump that characterizes the laminar-turbulent transition in pipe flow. Both differences are due to the different shapes of the two systems. In a pipe all the fluid is in a confined area, and the change from laminar to turbulent flow affects all the fluid (except for a very thin film at the wall). Around a sphere the fluid extends in all directions to infinity (actually the fluid is not infinite, but if the distance to the nearest obstruction is 100 sphere diameters, we may consider it so), and no matter how fast the sphere is moving relative to the fluid, the entire fluid cannot be set in turbulent flow by the sphere. Thus, there cannot be the sudden laminar-turbulent transition for the entire flow, which causes the jump in Fig. 6.10. The flow very near the sphere, however, can make the sudden switch, and the switch is the cause of the sudden drop in Q at =300,(300. This sudden drop in drag coefficient is discussed in Sec. 11.6. Leaving until Chaps. 10 and 11 the reasons why the curves in Fig. 6.22 have the shapes they do, for now we simply accept the curves as correct representations of experimental facts and show how to use them to solve various problems. 

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