Flat plate model

If the flat plate model for the catalyst pellet as shown in Figure 3.2, Volume 3, is assumed, then a material balance gives ... 

However, with heat transfer between the interior and exterior of the pellet (made possible by temperature gradients resulting from an exothermic diffusion limited reaction) the selectivity may be substantially altered. For the flat plate model the material and heat balance equations to solve are ... 

Figure 40 summarizes the different geometries employed in sonovoltammetry including the side-on approach (Eklund et al., 1996). In this latter case a flow over a flat plate model gave good agreement with experiment assuming solution velocities of ca. 100 cm s" were obtained in solution. These were attributed to acoustic streaming (Marken et al., 1996a), as shown in Fig. 41. These observations prompt a further consideration of mass transport effects.

Flat Plate Model. Modelled as a smooth flat plate for, say, Rejj = 10, the average skin friction coefficent would be 0.003, and... 

By using a flat plate model of drag and pressure flow in blocks of kneading discs, Loomans et al. (29) used Cad-Cam analysis to define flow patterns along the downstream flow channels of the unwrapped moving barrel surface. Figures... 

In the flat plate model, the barrel moves over the unwrapped channels of the stationary shafts. By using simple vector analysis, the magnitude and direction is determined for forward Ut and reverse flow vectors that are axial components of the peripheral velocity U (nDN) due to rotation speed N of the intermeshing twin screws. 

With the equations developed so far, the solids conveying performance can be analyzed as a function of screw geometry and polymer properties. The effect of channel depth on solids conveying rate is shown in Fig. 7.12 the results are from the flat plate model. 

It should be noted, that the effect of the channel depth on the solids conveying rate as determined from a flat plate analysis is different when the curvature of the channel is taken into account. In reality, when the channel depth increases, the area of the flight flanks will increase but the area of the root of the screw will decrease this decrease is not taken into account in the flat plate model. The contact area between the differential element of the solid bed and the barrel is ... 

In the flat plate model, the contact area between the solid bed and the screw is ... 

Both the helix angle cp and the down-channel incremental distance AZj are different in the curved plate model. The flat plate model considerably overestimates the contact area between the solid bed and the screw. Therefore, when the curvature is taken into account, the solids conveying rate will increase in a monotonic fashion with the channel depth as long as the pressure rise is sufficiently small. This demonstrates that the assumptions underlying a model have to be critically evaluated each time the model is used to analyze the influence of a certain parameter. 

Fig. 8.4(a) of Section 8.2.3. In this case, the screw contact area in the flat plate model becomes ... 

The channel curvature is negligible (flat plate model)... 

It is clear that Eq. 7.419 is quite close to Eq. 7.420 the difference is in the tangential velocity term. Equation 7.393 uses v, while Eq. 7.419 uses v. The difference between these two velocities is determined by the screw diameter D and the channel depth H. In most plastic extruders, the channel depth is about 0.05 D this will result in a difference between v, and v, of about 10%. Thus, there is a difference in the tangential velocities and shear rates when we take the screw moving rather than the barrel moving in the flat plate model. However, when the channel depth is small relative to the screw diameter, the difference is not substantial. 

The conveying process in single screw extruders is generally analyzed by using the flat plate model see Chapter 7. A similar analysis can be used in CSCO extruders when both screws are rolled onto a flat plate as shown in Fig. 10.21. 

The first theoretical study of the conveying process in non-intermeshing twin screw extruders was made by Kaplan and Tadmor [17]. They simplified the actual geometry (see Fig. 10.42) to a flat plate model. 

The flat plate model involves three plates the two outside plates representing the screw surface and the middle plate representing the barrel see Fig. 10.43. 

This is the screw characteristic for the Newtonian case, and as shown in the simple one-dimensional flat plate model described at the start of this section, it consists of drag and pressure flow terms. The ratio of pressure to drag flow rates (this is sometimes called the throttle ratio) is... 

FIGURE 8.23 Flat plate model for the closely intermeshing selfwiping twin-screw extruder. 

The starting point for the analysis is the flat plate model of the extruder shown in Figure 8.18. It is assumed that steady-state conditions exist and that the material properties such as viscosity, thermal conductivity, and heat capacity do not depend significantly on temperature. It is postulated that... 

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