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Figure 1. Schematic of a screw channel perpendicular to the flight edge showing the width of the channel at the barrel and the depth of the channel.

Many commercially available computer codes use the pseudo-Newtonian [1] model for metering channel calculations. Other models have been developed using different techniques including three dimensional finite element analysis (FEA) [4], two dimensional FEA with a rotating barrel [5], a channel model with advanced specifications and a rotating barrel [6], and a model with a rotating screw [7]. For the most part, these models were verified using extrusion data from relatively small diameter machines with screw channels that have small aspect ratios. Moreover, these FEA codes are typically difficult to use and take a long time to run on a fast computer.

Previous research has shown that the standard pseudo-Newtonian model must be used with caution when predicting the pressure profiles in extrusion screws. As previously stated, many commercially available codes use this method. For most commercial single-flighted, plasticating screws, the pseudo-Newtonian method is acceptable for calculating the pressure gradients, causing maximum errors of about 15% for power law fluids with power law indices between 1 and 0.75. These commercial single-flighted screws have h/Ws ranging from 0.04 to about 0.10. For this h/W range, the pressure gradient errors increase to a range of 40% to 120% for power law fluids with indices between 0.7 and 0.4. Higher errors will occur for h/Ws that are higher than the range above and for power law fluids with indices less than 0.4. The main component of the error is caused by a drag flow rate that is too high, causing the pressure gradient to be improperly compensated such that the overall flow rate is satisfied; i.e., according to Equation 1.

A 500 mm diameter, melt-fed extrusion process was previously simulated [2]. The h/W for this double-flighted screw was about 0.12 in the metering channel. The process was simulated using the pseudo-Newtonian method and a three dimension finite difference method. The simulation results were then compared to experimental data and are shown in Figure 2. For a LDPE resin, the pressure from the pseudo-Newtonian method was about 1.7 times the actual value measured. The error [3] was caused by the non-Newtonian nature of the LDPE resin and an h/W of 0.12. The relatively high h/W was the result of a double-flighted screw with deep channels, designs that are typical for large-diameter, melt-fed extruders.

The pseudo-Newtonian model over predicts the drag flow rates and pressure gradients for the channel for most conditions [3]. This over prediction is caused in part by the utilization of drag flow shape factors (F

Figure 2. Simulated axial pressure profile for a 500 mm diameter extruder running 11,800 kg/h at 46 rpm for a 0.8 MI LDPE resin [2]. The experimentally determined pressure at 5.6 diameters was 6.4 MPa.

The goal of this paper is to determine the correction function for the rotational flow (historically known as drag flow) term such that the pseudo-Newtonian method provides an acceptable rotational flow rate and pressure gradient.

where Q is the rate of the extruder, Q

where ρ

The z component (downstream direction parallel to the flight) of the screw velocity at the barrel wall V

where N is the screw speed in rpm, D

Figure 3. Shear viscosity for the power law fluids used for this study.

The rotational flow rates were calculated by setting the pressure gradient (∂P/∂z) to zero. The ratio of the rotational flow rate calculated using the numerical method to that of the pseudo-Newtonian method is defined as F

Since the pseudo-Newtonian method was based on a constant viscosity for the entire channel, any factor that would cause the viscosity to vary in the channel may cause the Fc to be different from 1. The factors known to cause the viscosity to vary include the h/W ratio, n, and the average shear rate in the channel. To be consistent with the physical process, the average shear rate in the channel was calculated based on screw rotation as follows:

where D

The F

Figure 4. F

The F

An additional factor needed to be included to make the correlation between the channel geometry and the correction factor. This geometric factor was required since a correlation with different lead lengths and the number of flight starts could not be predicted by h/W alone. The parameter added here was the ratio of the channel width perpendicular to the flight at the barrel wall to the diameter of the barrel or W/D. A series of numerical experiments were performed with constant aspect ratios h/W of 0.12 and 0.16. The width ratio W/D was changed by using different combinations of lead lengths and the number of flight starts. For these experiments, the lead length was varied between 70 and 130 mm using single-flighted, double-flighted, and triple-flighted geometries. The combination allowed W/D to vary from 0.2 to 1.1. The correction factors for these geometries are shown in Figures 5 and 6 for h/Ws of 0.12 and 0.16, respectively. The channel depth was varied to maintain the h/W. Screws with width ratios greater than 0.6 were constructed using single-flighted geometry and varying the lead length from 70 to 130 mm. Double-flighted screws were used to produce W/D ratios of 0.4 and 0.5 while triple-flighted screws were used to produce W/D ratios of 0.2 and 0.3.

Figure 5. F

Figure 6. F

As shown by Figures 5 and 6, the F

The quality of the F

Figure 7. Parity plot showing the quality of the F

The standard deviation for the 102 numerical data verses the predicted fit for F

Once F

This corrected rotational flow rate should be very close to the actual rotational flow for the channel geometry. The calculation of the pressure gradient using Equation 1 is now much more accurate since the pressure flow rate (flow due to a pressure gradient) is now more accurate; i.e., Q

The extruder was relatively long, but only the first 5.6 diameters needed to be simulated to demonstrate the accuracy of the methods. For this extruder, the feed pipe was 500 mm in diameter and thus forced the first diameter of the extruder to be at a temperature and pressure of 225

A pressure transducer was positioned in the barrel at 5.6 diameters downstream of the start of the screw. For a screw speed of 46 rpm and a rate of 11,800 kg/h, the pressure at this location was measured at 6.4 MPa. The barrel temperature for this data set was at 190

The process was simulated using the pseudoNewtonian method with and without the F

Figure 8. Simulated axial pressure profile for a 500 mm diameter extruder running 11,800 kg/h at 46 rpm for the 0.8 MI LDPE resin [2] using the pseudo-Newtonian method with and without the F

The F

Although the analysis here was performed using a power law viscosity model, other models could be used. For other viscosity models, the power law value n would be calculated using two reference shear rates, one higher and one lower than the shear rate calculated using Equation 8. These high and low shear rates and viscosity data would be used to determine a local n value.

The flow in screw channels is complicated and nonlinear, and thus the components of rotational flow and pressure flow cannot be separated. Potente [12] states that flows and pressure gradients should only be calculated using a 3D method because of the limitations of the Newtonian model. The engineering approach presented here allows the quick calculation of the rotational flow rate for non-Newtonian resins, and provides an improved estimate of the pressure gradient in the channel.

2. M.A. Spalding, G.A. Campbell, F. Carlson, and K. Nazrisdoust, SPE-ANTEC Tech. Papers, 52, 792 (2006).

3. M.A. Spalding and G.A. Campbell, SPE-ANTEC Tech. Papers, 54, 262 (2008).

4. M.A. Spalding, J. Dooley, K.S. Hyun, and S.R. Strand, SPE-ANTEC Tech. Papers, 39, 1533 (1993).

5. E.E. Agur and J. Vlachopoulos, Polym. Eng. Sci., 22, 1084 (1982).

6. H. Potente, “Single-Screw Extruder Analysis and Design,” chapter 5 of “Screw Extrusion,” ed. J.L. White and H. Potente, 2003.

7. G.A. Campbell, P.A. Sweeney, N. Donatula, and Ch. Wang, Int. Polym Process., XI, 199 (1996).

8. W.D.B. Robinson, Doctoral Dissertation, Univ. of Delaware, (1960).

9. V.L. Metzner, “Handbook of Fluid Dynamics,” V.L Streeter, ed., Section 7, 28, McGraw-Hill, (1961).

10. J.A. Wheeler and E.H. Whissler, Am Inst. Chem. Engrs. J., 11, 207 (1965).

11. S. Middleman, Trans.Soc. Rheol., 9, 83 (1965).

12. H. Potente, SPE-ANTEC Tech. Papers, 44, 3604 (1998).

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