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• the dependence of melt viscosity on shear and temperature,

• resin thermal properties,

• screw geometry, and

• process conditions.

Melt temperature vs. extruder length is provided in a succinct format that vividly illustrates the melt temperature development process.

A few of the factors that can be investigated about the process are:

• Is there excessive residence time?

• Is the temperature uniformity near optimum?

• Are the barrel zone temperatures set properly?

It will be shown how the UMTD, a single diagram of temperature vs. length, universally illustrates these conditions of importance to good product quality.

• The change in energy is assumed to be proportional to the product of the resin melt specific heat and melt temperature change.

• The viscosity of the resin melt is assumed to follow the Carreau-Yasuda model for shear strain rate. See Appendix 2.

• The viscosity (Appendix 2) is assumed to follow an exponential function of temperature. The reference point is the viscosity modulus at the barrel wall temperature which ties the exponential function closely to the viscosity vs. temperature data curve.

• Heat transfer with the barrel is assumed to be a function of the heat transfer coefficient between the melt and the barrel, which is known to be a function of flight clearance and screw speed [2-6].

• A known fixed screw-geometry consists of a helical channel formed by a flight with a small clearance.

• Screw speed, fixed barrel wall temperature, and flow rate are known.

• Thermal properties of the resin melt are known constants.

• Thermally steady-state equilibrium and an adiabatic screw are assumed.

The melt section length of the extruder is divided into “melt zones”. Melt zones are defined as any length of the melt section that has constant barrel zone temperature and fixed screw geometry. A change in either one of these constitutes the beginning of a new melt zone.

1. θ, Temperature

2. χ, Axial Position

3. N

4. N

All four of the dimensionless groups are mathematically defined in Appendix 1 along with the governing equation. Each group is defined as uniquely having a variable of the process as a factor and so named. For example, the first group is the temperature number, because it is the only group of the set that has temperature explicitly as a factor (equation 5). The temperature number is on the ordinate of the UMTD to solely represent average melt temperature. Similarly, the abscissa is represented by the uniquely defined axial position number, equation 6. Therefore, the UMTD axes are proportional to temperature vs. extruder axial position so as to appear in form and shape to data for temperature vs. axial position. Therefore, the UMTD is easily related to actual machine operation and interpretation.

The UMTD will be shown to have two domains. The first domain represents conditions for heating the melt, and the second for cooling the melt. Conditions for heating the melt are most common for plastic extrusion. They will always occur when the melt temperature entering a melt zone is less than the fully developed melt temperature for that melt zone.

Figure 1. The basic UMTD for Melt Heating. Units are all dimensionless. An adiabatic barrel solution, θA, is shown as a limiting condition. An example for a single melt zone is shown by the dashed lines.

The beginning of a melt zone, χ

The end of the melt zone then occurs when the barrel zone temperature or the screw geometry is changed. In Figure 1 a change in axial position, ∆χ = 0.75, is shown that is calculated based on the actual length, L, of the melt zone. This change in axial position, ∆χ, is then added to the initial position, χ

Figure 2 gives the axial position of the beginning of fully developed melt temperature as a function of heat transfer and shear strain numbers. The practical importance of this is that undue residence time will be added to the melt at greater axial length. Longer residence time is often associated with poor product quality, such as discoloring.

Figure 2. Axial Position Number at which Fully Developed Melt Temperature Begins and Fully Developed Temperature. They are a funtion of only heat transfer and shear strain numbers.

The fully developed temperature shown on the right axis of Figure 2 is a key value for each melt zone. The fully developed temperature will not change once this axial position given by Figure 2 is reached without changing process parameters or entering a new melt zone.

The hyperbolic shape of the curves of Figure 2 illustrates that heat transfer coefficient and shear strain have a point of diminishing returns as the heat transfer is increased and/or shear strain is decreased. That is, at high values of N

The fully developed temperature does occur in large metering extruders that are designed to only pump melt at contant temperature with the purpose of stabilizeing flow disturbances. The screw is normally of fixed channel dimensions and helix angle. Heat must be removed over the entire length of the extruder to maintain the constant melt temperature, and that energy is provided by the motor.

Figure 3. Upper and Lower Limits of Melt Temperature Heating

In Figure 1, the temperature function for an adiabatic barrel is shown as θ

Also, note in Figure 3 that there is a “narrows” between the upper and lower limit at χ = 1. The significance is that at this point the variation in melt temperature is minimal. The consequence is that the melt temperature is least sensitive to variations of any of the other variables for χ at or near a value of 1. This would be a good condition to occur at the end of the extruder to minimize variations in product temperature uniformity. Figure 3 also shows that above a value of χ= 1 the melt temperature will always be hotter than the barrel wall temperature (θ = 0).

Figure 4. Melt Cooling The temperature of the melt entering the melt zone is greater value than the prevailing fully developed melt temperature.

Cooling of the melt is not typically done in plastic extrusion since this would be needed as a result of overheating the melt. Overheating is avoided as it would lead to degradation of the resin. Cooling the melt stream is also very inefficient because viscous heating requires substantially colder barrel zone temperature, and this would lead to excessive temperature gradients and possible resin freeze-out on the barrel walls. Melt zones can be established so as to avoid needless melt cooling with the aid of the UMTD.

Figure 5. The Complete UMTD Melt heating and melt cooling shown for comparison.

1. bulk temperature,

2. barrel wall temperature,

3. heat flux at the barrel wall, and

4. the assumption of an adiabatic screw

provide four independent parameters used to calculate the temperature distribution of the melt with a cubic function. See Appendix 3 for details.

Data for the exit melt temperature distribution for a 31.75 mm, 16/1 L/D extruder were measured [7] with a thermocouple bridge spanning a 25.4-mm diameter exit pipe. The resin was LDPE, and Figure 6 shows the data [7]. Also shown is the exit melt temperature cubic profile calculated from the UMTD analysis as described above, and good agreement is noted. The prediction compares well with an iterative approach used later for the same test setup [8].

Figure 6. Temperature Distribution for LDPE Model based on UMTD analysis of bulk temperature distribution.

The cubic function used here to model the temperature profile will demonstrate a known variety of melt temperature profiles. The UMTD predicts the average temperature, but for a given average temperature dramatic differences can occur in the melt temperature profile predicted by the cubic function.

For the above example a hypothetical doubling of the viscosity modulus, η

Figure 7. An “M” Shaped Profile The average product temperature for the model is the same as for the data.

The practical significance is that an average product temperature is subject to a large variety of exit temperature profiles. Using the UMTD to model the average temperature leads to calculation of the temperature profile with the cubic function. A large thermal profile difference in two processes can be identified even though they have the same average product temperature.

It is also notable that the melt temperature distribution so calculated is a constant (uniform) for values of axial dimension, χ, where θ=0. Therefore, thermal uniformity is optimized for these values of the axial position.

2. Finite temperature limits are established for the melt with the UMTD.

3. The UMTD shows the melt can be either heated or cooled with an extruder.

4. Cooling of the melt with an extruder is demonstrated to be very inefficient by the UMTD.

5. Conditions for optimum product temperature uniformity are clearly defined by the UMTD.

6. Data show melt temperature distribution can be accurately calculated with the results of the UMTD.

7. Conditions for fully developed melt temperature are identified.

2. S. J. Derezinski, “Heat Transfer Coefficients in Extruder Melt Sections”, Conference Proceedings, ANTEC ’96, Society of Plastic Engineers, 1996, pp. 417-421

3. Jepson, C.H., “Future Extrusion Studies”, Industrial and Engineering Chemistry, Vol. 45, No. 5, May 1953, pp. 992-993.

4. McCabe, Warren, and J. C. Smith, Unit Operations of Chemical Engineering, Third Edition, McGrawHill, Inc., 1976, pp.378-280.

5. Tadmor, Zehev, and I. Klein, Engineering Principles of Plastic Extrusion, Van Nostrand Reinhold Company, 1970, p. 259.

6. Rauwendaal, C., “The Effect of Flight Clearance on Extruder Performance”, Society of Plastic Engineers Technical Papers Volume XXV, ANTEC 89, 1989, pp. 108-110.

7. McCullough, T. W. and R. T. Hilton, “Predicting Melt Temperatures Using an Adjustable, Exposed Tip Thermocouple”, Conference Proceedings, ANTEC ’92, Society of Plastic Engineers, 1992, pp. 927-930.

8. McCullough, T. W. and M. A. Spalding, “Predicting Actual Temperature Distributions in a Polymer Stream Using an Adjustable, Exposed-Tip Thermocouple Assembly”, Conference Proceedings, ANTEC ’96, Society of Plastic Engineers, 1996, pp. 412-416.

9. Stevens, M. J. and J. A. Covas, Extruder Principles and Operation, Chapman and Hall, Second Edition, 1995, pp. 358-359.

10. Chung, Chan I., Extrusion of Polymers, Theory and Practice, Carl Hanser Verlag, Munich 2000, p. 32.

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

Adiabatic Barrel Condition, set N

The viscosity is assumed to follow the Carreau-Yasuda Equation for shear combined with an exponential function for temperature as follows:

where the value of the viscosity modulus ηw, at the known barrel wall temperature, T

The four factors that result from the temperature analysis are used to estimate the melt temperature distribution as follows. The distribution is assumed to follow a cubic function as

Bulk melt temperature is known, so

Barrel wall temperature (x=0) is known, so

Heat flux at the barrel wall (x=0) is known, so

An adiabatic screw is assumed (x=H), so

Equations 14-17 are solved for the 4 coefficients of the cubic equation 13 (a, b, c, and d) to give the temperature profile.

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