Two-dimensional power law analysis

Figure 8.12(a) compares the results of simultaneous optimization from the modified Newtonian analysis and the two-dimensional power law analysis, obtained by numerical computations [2]. 

Optimum helix angle versus reduced axial pressure gradient, resulting from two-dimensional power law analysis (numerical)... 

It can be seen that the values of the optimum helix angle from the modified Newtonian analysis are about 10 to 20% above those from the two-dimensional power law analysis. This indicates thatthe results from the modifiedNewtoniananalysis, Eq. 8.44, may be more appropriate than the results from the one-dimensional power law analysis. The use of one-dimensional power law analysis leads to errors when the helix angle is substantially above zero, as discussed in Section 7.4.2. Therefore, one would like to use a two-dimensional power law analysis. However, there are no analytical solutions for this case. From numerical computations, it is possible to develop a plot of optimum helix angle as a function of a dimensionless axial pressure gradient g , where g° is ... 

It can be seen that the optimum helix angle is strongly dependent on the power law index. Comparison with results from the two-dimensional power law analysis, shown in Fig. 8.13, indicates a reasonable agreement when the power law index is larger than one-half (n > 0.5), but relatively large differences when the power law index is smaller than one-half (n < 0.5). 

Figure 8.14(b) can be compared directly to Fig. 8.12(b) showing results from a two-dimensional power law analysis. The agreement between the two sets of results is quite reasonable. 

Summarizing, it can be concluded that for simultaneous optimization of the channel depth and helix angle, the modified Newtonian analysis yields reasonably accurate results when compared to the two-dimensional power law analysis. The important equations are Eq. 8.39 for the optimum helix angle and Eq. 8.40 for the optimum channel depth. The results of simultaneous optimization from a one-dimensional power law analysis are less accurate than the modified Newtonian analysis. 

From Figs. 7.62 through 7.65, it becomes clear that the Newtonian output-pressure gradient relationship is unacceptably inaccurate when the power law index of the polymer melt is less than 0.8. The one-dimensional power law (1-DPL) output-pressure gradient relationship is accurate only for small helix angles. Thus, for accurate results, a two-dimensional power law (2-DPL) analysis should be used. However, the 2-DPL analysis does not yield analytical solutions numerical techniques have to be... 

The values of the laminar boimdary layer thickness and of the frictional drag are not very sensitive to the form of approximations used for the velocity distribution, as illustrated by Skelland [1967] for various choices of velocity proflles. The resulting values of C(n) are compared in Table 7.1 with the more reflned values obtained by Acrivos et al. [1960] who solved the differential momentum and mass balance equations numerically the two values agree within 10% of each other. Schowalter [1978] has discussed the extension of the laminar boimdary layer analysis for power-law fluids to the more complex geometries of two- and three-dimensional flows. 

The next section wiii start with an analysis of melt conveying of isothermal fluids. This wiii be foiiowed by a non-isothermal analysis of melt conveying of cases that allow exact analytical solutions. More general analyses of the effect of temperature on flow will be discussed in more detail in Chapter 12 on modeling and computer simulation. In the next section, melt conveying of Newtonian fluids and non-Newtonian fluids will be analyzed. The non-Newtonian fluids will be described with the power law equation (Eq. 6.23). The effect of the flight flank will be discussed and the difference between one- and two-dimensional analysis will be demonstrated with particular emphasis on the implications for actual extruder performance. 

The difference between the one-dimensional and the two-dimensional analysis will increase with increasing helix angle and reducing power law index. From a practical point of view, the use of a two-dimensional analysis becomes important when large helix angles and strongly non-Newtonian fluids are analyzed. The equation of motion in the down-channel direction is the same as used before see Eq. 7.194. A similar expression has to be used for the cross-channel direction. The shear stress profiles can be written as ... 

As mentioned before, these results should be the same as the results from the onedimensional analysis as shown in Fig. 7.62. By comparing the two figures, it is clear that the results are virtually identical. This confirms the accuracy of the analytical solution for the one-dimensional flow of a power law fluid. The difference is generally less than 1%, except at small values of the dimensionless pressure gradient g° < 0.1. The loss of accuracy at low pressure gradients was predicted earlier and should not pose a serious problem in the analysis of real extrusion problems. 

It should be noted that fundamentally it is not entirely correct to take an expression derived for a Newtonian fluid and insert a power law viscosity form into it. However, if this simplification is not made, the analysis becomes much more complex and analytical solutions much more difficult to obtain, if not impossible. Results of the analytical solutions have been compared to results of numerical computations for a two-dimensional flow of a power law fluid. In most cases, the results are within 10 to 20% [2]. It should be noted that the results are exact when the power law index is unity, i. e., for Newtonian fluids. However, if the optimum depth and helix angle for a pseudo-plastic fluid are calculated using expressions valid for Newtonian fluids only, very large errors can result, particularly when the power law index is about one-half or less. It is therefore very important to take the pseudo-plastic behavior into account, because the large majority of polymers are strongly non-Newtonian. 

It should be noted that the optimum helix angle in Fig. 8.12(b) is not the result of simultaneous optimization of channel depth and helix angle, but optimization of the helix angle only. Figure 8.13 shows the optimum helix angle versus dimensionless down-channel pressure gradient as again determined from a two-dimensional analysis of a power law fluid. 

Part wall thickness can now be predicted quite accurately with finite element analysis. The physical sheet is replaced with a two-dimensional mesh of triangular elements and nodes, which is then mathematically deformed imder increasing load. When the nodes touch the electronic surface of the mold, they are affixed. Force continues to increase imtil all or most of the elements are rendered immobile. Currently the Ogden power-law model is used as the pol5mier elastic constitutive equation or response to applied load (28). 

Taking into account some ambiguity in the choice of the hydration shell width D, it is reasonable to estimate its effect on the temperature of the percolation transition. Such analysis was performed in Ref. [566] for various choices of D from 3.8 to 5.4 A. Such variations of D were also useful for an accurate location of the percolation threshold at every temperature studied. Depending on the chosen value of D, the number of water molecules in the hydration shell varied up to about a factor of two. Due to the increasing number of water molecules in the hydration shell, a percolation transition occurs at some value of D, particular for each temperature studied. Example of a percolation transition at constant temperature is shown in Fig. 131. With increase in the thickness of hydration shell, larger deviations from a strict 2D to 3D percolation transition may be expected. The respective power laws for ns at the 2D and 3D percolation thresholds are shown in Fig. 131. Obviously, the behavior of ns allows the location of the percolation threshold between = 147 and JVw = 153 without any assumption about the dimensionality of the transition. This means, in particular, that atT = 300 K, water network around a peptide is spanning, if all water molecules within hydration shell of 4.75 A width are considered as hydration water. 

It is illuminating to consider two limiting cases. When [X"]o = 0, eqn [24] gives the now familiar power law (eqn [25]), which was first derived by Fischer from a dimensional analysis approach and subsequently modified to the present fe,-including form by Ohno et al. using this approach. 

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