Optimum helix angle

Similarly, we can solve for the optimum helix angle, , by setting the variation of eqn. (6.19) with respect to the helix angle to zero, dQ/d = 0. The helix angle is embeded in the channel length, L, term... 

It is interesting to point out that for a die with a pressure flow through a slit, or sets of slits, the optimum channel depth is directly proportional to the die gap. Decreasing the die gap by a certain percentage will result in an optimum channel depth that is reduced by the same percentage. To determine the optimum helix angle we can re-write eqn. (6.23) for this specific application,... 

The velocity at any as indicated by Eq. 6.3-25 is a function of the helix angle 6 and attains a maximum value at 0 rc/4. Hence this is also the optimum helix angle for... 

Equation 7.85 allows the calculation of the optimum barrel helix angle if the screw geometry is known and if the various coefficients of friction, internal and external, are known as well. The solution for the optimum helix angle is not completely analytical because Aj contains a term that is dependent on the barrel helix angle. This can be solved by initially guessing a value of Wb, then calculating cpj according to Eq. 7.85. The Wb can be calculated with ... 

This results in the following optimum helix angle cp tan2(p = -... 

At the optimum channel depth H, the optimum helix angle becomes ... 

Chapter 8) shows that 30° happens to he the optimum helix angle for Newtonian fluids with respect to output. This means that at a helix angle of 30°, the shortest residence time is achieved, provided the depth of the channel is optimum as well. 

The optimum depth depends on the diameter, screw speed, power law index, consistency index, pressure gradient, and helix angle. The optimum helix angle for output can be determined from ... 

By using the same equations for the melt conveying rate, the optimum helix angle for output rate cp( has to be determined from the following equations ... 

The optimum helix angle now has to be determined from 2cos>t-2g (cos9 )"" =1... 

Figure 8.9 shows the optimum helix angle as a function of the dimensionless down-channel pressure gradient for n = 1 and n = 0. 

Optimum helix angle versus dimensionless do A/n-channel pressure gradient... 

It may be more interesting to optimize the channel depth and helix angle simultaneously. This can be done by inserting Eq. 8.31 into Eq. 8.33. After some calculations, the optimum helix angle can be determined to be ... 

The details of the derivation can be found in an article on screw design of two-stage extruder screws [2], The optimum helix angle is only a function of the power law index and the reduced flight width. Figure 8.10 shows the optimum helix angle as a function of the power law index at various values of the reduced flight width. 

The simplicity of Eq. 8.39 makes it a useful and convenient expression for optimizing the geometry of the melt conveying zone of an extruder. For polymer melts with a power law index in the range of 0.3 to 0.4 and typical values of the flight width, the optimum helix angle is about 22 to 24°. 

Unfortunately, the optimum channel depth is dependent on many more variables than the optimum helix angle. The latter depends only on the power law index and reduced flight width. In addition to these variables, the optimum channel depth also depends on the screw speed, screw diameter, consistency index, and pressure gradient. This means that it is not possible to design a universally optimum screw geometry. Thus, one has to determine the most likely operating parameters that the screw is likely to encounter and design for those parameters. 

When the depth and helix angle are optimized simultaneously, Eq. 8.42 is inserted into Eq. 8.43. The resulting optimum helix angle for simultaneous optimization of the pressure-generating capacity is ... 

Comparing this result to the optimum helix angle for output cp Eq. 8.39, it is clear that the two expressions are exactly the same, thus ... 

A complete solution of Eq. 8.49(a) may be rather involved however, it can be seen quite easily that A, = 0 is a solution of Eq. 8.49(a). Thus, the channel depth is optimized when A. = 0, i.e., when the velocity gradient becomes zero at the screw surface. The optimum helix angle is obtained by taking the first derivative of output V and setting the result equal to zero. This results in the following expression ... 

Inserting Eq. 8.49(d) into Eq. 8.49(c) yields the solution of the optimum helix angle ... 

To compare this result with the result from the modified Newtonian analysis, Eq. 8.44, the optimum helix angle can be written as ... 

It is clear that the one-dimensional power law analysis yields higher values of the optimum helix angle than the modified Newtonian analysis, except when the power law index is unity. 

Optimum helix angle versus power law index, resulting from modified Newtonian analysis and one-dimensional power law analysis... 

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 ... 

The plot of the optimum helix angle versus dimensionless axial pressure gradient is shown in Fig. 8.12(b). 

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. 

The optimum helix angle can be determined by taking the first derivative of output V with respect to the helix angle and setting the result equal to zero. This results in the following expression for the optimum helix angle (p ... 

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). 

A very simple expression for the optimum helix angle was obtained by Rauwendaal [82]. The optimum helix angle in degrees can be expressed as ... 

The optimum helix angle as a function of reduced axial pressure gradient according to Eq. 8.54(b) is shown in Fig. 8.14(b). 

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. 

The helix angle in the feed section will also have an optimum value for which the solids conveying rate reaches a maximum. This is obvious if one realizes that a zero-degree helix angle results in zero rate and a 90° helix angle also results in zero rate. Thus, somewhere between zero and 90°, the solids conveying rate will reach a maximum. The optimum helix angle can be determined from ... 

The optimum helix angle cp for power consumption can be determined by setting ... 

This equation does not have an obvious simple, analytical solution. One can find the solution either graphically or hy using a numerical technique, such as the Newton-Raphson method. Figure 8.21 shows the optimum helix angle as a function of channel depth for a 50-mm extruder, running at 100 rpm. 

Optimum helix angle versus channel depth... 

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