SCALING LAWS AND FRACTAL SCREW DESIGNS TOWARDS SINGLE PELLET EXTRUSION

Transcription

1 SCALING LAWS AND FRACTAL SCREW DESIGNS TOWARDS SINGLE PELLET EXTRUSION D.O. Kazmer Department of Plastics Engineering, Univ. Mass. Lowell Abstract Analyses and screw design concepts are presented to control the plastication path of materials in single screw extrusion. The methodology uses analytical solutions for transient heat conduction with shear heating. The design concept uses multiple channels in the transition and feed sections to provide improved control of extrusion scale-up. Two screw fractal designs are presented for comparison with general purpose and barrier screw designs. Introduction Extrusion is the workhorse of the plastics industry. Not only is extrusion used in the conversion of pellets to finished goods (tubing, sheet, film, profiles, etc.), but it is also used in most commercial compounding and polymerization processes. There are many extrusion screw designs with significant advances including the early rubber screws, metering screws with varying depth flights, melt barrier screws, and various mixing screws. Screw design guidelines are well-known [-3] but based on rules of thumb and often inaccurate analysis that leads to suboptimal designs. Rauwendaal [4] provides a comparative analysis of well-known rules for estimating shear rate, melt conveying rate, residence time, shear strain, conductive and dissipative melting capacity, solids conveying rate, power consumption, and specific energy consumption. Covas [5] has implemented a multi-objective optimization methodology using these rules with weighting preferences to develop Pareto optimal sets balancing multiple performance criteria. Critically, Covas found the results from experimental validation with the implemented screw designs to be inferior due to the poor accuracy of the underlying models and simulations. Given a reference extruder, scale-up rules seek to define the screw geometry and operating conditions to provide the same flow and heat transfer conditions to yield extrudates with the same characteristics. Scale up rules were first defined by Carley and McKelvey [6] who suggested that melt flow rate and power consumption vary with the square of the extruder diameter. Direct application of the scale-up rules, however, results in several secondary considerations. As shown in Figure for screw diameters equal to and, first order analysis indicates that the volumetric throughout Q at the same screw rotational rate varies by a factor of 8, suggesting that the scaling rule for throughput should vary with the cube of the diameter. Figure : Cross-section of two screws showing geometric scaling relationship between diameter,, area, A, velocity, v, and throughput, Q Table provides first order relations for other extrusion performance measures including the shear rate, residence time, and heat conduction. Given the nonuniform scaling relations between the different performance measures, more advanced scaling corrections have been developed including, for example: channel depth and melt temperature by Maddock [7]; corrected zone lengths by Potente [8]; color/additive mixing by Manas- Zloczower [9]; and reactive extrusion by Ortiz-Rodriguez [0]. However scientific these research efforts may seem, these scale-up methodologies rely on coarse process models and so rarely provide scaled replication of the reference extruded product. Table : Scaling relations for various extrusion performance measures at same screw RPM Performance Measure Underlying Relation Flow rate Q Av Scaling Law 3 Shear rate kv Residence time t L v Heat conduction Fo k t L Reflecting on conventional screw design and operation, one may marvel on the complexity of the process from the polymer s perspective. However, the conveyance and melting of the thermoplastic feedstock remains neither efficient nor consistent. It seems that improved extrusion processes could be more reliably developed with lower equipment and processing costs. A fractal screw design and methodology is next described to provide the same processing for each pellet regardless of extruder scale. SPE ANTEC Anaheim 07 / 55

2 Design The fundamental issue with the scaling rules of Figure and Table is that the underlying relations do not follow the same relations. For example, flow rate will increase with the cube of the extruder scale while heat conduction will decrease with the square of the scale. Other relations, such as with shear rate and residence time should be independent of scale but are in fact, much more complex due to conveyance and recirculation dynamics of the material(s) being processed. Spalding has conducted significant extrusion research related to plastication [- 5] including screw freezing experiments. The results demonstrate the multiphase nature of the plastication process with two main conclusions: () little work and heat transfer occurs in the first 50% of plastication, and () there remains significant variation in the melt dispersity, temperature, and residence time throughout the process. Ideally, the material(s) being processed will undergo the same (and hopefully optimal) processing history. The theoretical melting of a single pellet in an extrusion screw was recently analyzed [6] considering transient heat conduction with a convention boundary condition and three sources of internal generation. The results suggest that the theoretical melting time for a pellet is on the order of seconds and more optimal extrusion screw designs should be possible by controlling the screw channel dimensions relative to the size of the feedstock. Reflecting on the scaling rules of Table, the plastics engineer realizes that one approach to providing the same (and hopefully optimal) processing history regardless of scale is to process the material with the same or equivalent extrusion channel geometry. Thus, the intent is to literally maintain the same or rationally adjusted channel height and width across different extruder sizes. As the extruder diameter and length increase, the size of the channel relative to the size of the screw decreases such that extrusion screws may be designed with multiple channels. There are two concepts that guide the extrusion screw design. First, Mandelbrot [7] recognized that selfrepeating patterns, or fractals, occur often in nature. Figure provides one example of a generator in which a line is replaced by four smaller lines. The generator is then recursively applied to each of the four smaller lines to generate a subsequent design with sixteen lines, to which the generator pattern may again be applied. The resulting geometry can be quite complex, and sometimes replicate objects found in nature. Mandelbrot s work was not directed to generation of images, but rather the modeling of physical systems such as Brownian motion [8] and mechanical fracture [9]. The fractal concept can also be applied to screw design in which a larger channel section, such as in the feed section, can be divided into multiple smaller downstream channels to achieve some goal(s). Figure : Fractal generator and recursive shapes The second concept to guide the extrusion screw design is that of axiomatic design. A basic premise of axiomatic design [0] is that each functional requirement (FR) is directly controlled by one design parameter (DP). Axiomatic designs have the significant benefit that any functional requirement can be directly changed through modification of a design parameter without (potentially negatively) influencing other functional requirements. Figure 3 provides a qualitative design matrix linking screw design parameters with extrusion functional requirements. It is observed that there is significant coupling between the screw design parameters and extrusion functional requirements. Changing most DPs has both beneficial and negative impacts on the FRs. Figure 3: Design matrix indicating lack of control While Figure 3 is arguably qualitative, it is demonstrative that there is no one to one mapping between the DPs and FRs. However, the concept of axiomatic design is helpful in considering the development of extrusion screws with the added degrees of freedom provided by multiple channels so that the material can be more optimally processed. SPE ANTEC Anaheim 07 / 56

3 Analysis Materials processed in extruders typically have compressibility behaviors that are a function of temperature and pressure. For example, Figure 4 was plotted using coefficients from the Autodesk/Moldflow database for a double domain Tait equation fit to specific volume data for Braskem H503 (a grade of polypropylene (PP) with a melt flow index of 3.5 g/0 minutes). As illustrated, the specific volume of a material increases with the temperature of a material and changes phase from a semi-crystalline solid to an amorphous melt around 30 C. A plastication path through the extruder is also shown. about the screw centerline, a mean lead, P, and number of rotations, n, equal here to. The arc length for this channel section can be approximated as: L n R P () Figure 5: Section of screw channel with geometric definition and conversion to prismatic channel Figure 4: Pressure-volume-Temperature (PvT) behavior of PP with plastication path and definition of and For analysis purposes, the variables v, P, and T are used to define the specific volume, melt pressure, and melt temperature. Chi,, is the material s coefficient of volumetric thermal expansion defined as the derivative of specific volume with respect to temperature at constant pressure and equal to /C at the target,. Beta,, is the material s compressibility defined as the negative of the derivative of specific volume with respect to pressure at constant temperature and equal to /MPa at the target. It is interesting to note that the relatively similar values of and might be advantageously used to control the melt temperature and melt pressure of a material as it is being processed. The relationship between pressure and temperature may be directly determined by application of the chain rule from calculus. Since is the derivative of specific volume with respect to temperature and is the derivative of specific volume with respect to pressure, the derivative of temperature with respect to pressure may be calculated as -(dv/dp)/(dv/dt) = -/. For the data of Figure 4, dt/dp =.535 C/MPa. Figure 5 depicts an isometric view of a screw channel section. The geometric definition of the extrusion channel section follows a curved centerline having a radius, R, In Figure 5, the channel section has an outer diameter D equal to 0 mm and a lead equal to 8 mm. The upstream section has a depth H equal to.8 mm and a width W equal to 6.4 mm while the downstream section has a depth H equal to. mm and a width W equal to 3.6 mm. With these dimensions, the material in the channel section has a mean radius R equal to 9 mm and a centerline length of approximately 59 mm. The defined geometry of Figure 5 can be laid flat or unrolled to facilitate modeling of the material being processed in a prismatic channel. For numerical simulation of complex channel geometries, the prismatic channel(s) can be modeled as a series of rectangular cuboids, each having dimensions that are representative of their section of the screw channel(s). Figure 6: Definition of rectangular cuboid with temperature and velocity boundary conditions Figure 6 shows a top and section view of the rectangular cuboid defined by length L, width W, and depth H of a material processed in the channel section. The pressure, P, and flow rate, Q, at upstream and downstream sections may be known by inspection of an existing extrusion process or estimated through solution of the SPE ANTEC Anaheim 07 / 57

4 channel geometry according to known or desired boundary conditions. The top surface corresponds to the outer surface of a material being processed in a channel section of the extrusion screw; this outer surface is bounded by the bore of the extruder barrel and so is ordinarily stationary. The bottom surface corresponds to the inner surface being processed in a channel section of the extrusion screw; this inner surface is bounded by the outer surface of the extruder screw and so is ordinarily rotating. The flow relations for material processed in extrusion are well known. The radial velocity, tangential to the screw location, is equal to d RPM, where d is the diameter of the screw at the bottom of the channel and RPM is the number of screw rotations per minute. A material s velocity, v, in the direction down the channel is approximately equal to: v d RPM cos () where is the helix angle as a function of the lead, P, is: P d tan (3) The downstream velocity of a material in the channel is a function of the depth direction, y, as []: v dp dl H dp dl v y y y H where dp/dl is the derivative of the pressure of a material being processed with respect to the length direction, and is the apparent viscosity of a material being processed. A material being processed will have a shear rate, s, as a function of y that is the derivative of the downstream velocity with respect to the depth direction: (4) dv v dp dl H dp dl s y y (5) dy H Kazmer [6] investigated viscous heating phenomena in polymer processing with closed form solutions. As described therein, the rate viscous heat generation can be estimated as the product of the viscosity, square of the shear rate, and volume. The total energy generated by viscous heating is equal to the rate of viscous heat generation times the time duration, t, for which a material is being viscously heated. The change in temperature is equal to the total heat generated by viscous heating divided by the product of the mass and the specific heat, C P. For a material being processed with mass, m, of a material being processed in the channel with length, width, and depth, of L, W, and H, the volume will be the product of L, W, and H, and the mass will be the product of the volume and the density,. With the time duration, t, estimated as the channel length divided by the downstream velocity of a material in the channel, the change in the bulk temperature of a material, T, can be estimated as: s LWH t sl T (6) LWH C vc This relation provides a useful estimate of the change in a material temperature as a function of shear heating. For example, consider a section of a metering channel for the screw design of Figure 5. For an extrusion screw with an outer diameter of 0 mm and a channel section having a depth of mm, the helix angle can be calculated as 7.7 degrees relative to the channel centerline. The bottom of the channel is located at a different radial location relative to the screw centerline than the curved centerline. Given that the bottom of the channel is located at a distance of 8 mm from the screw centerline, the helix angle is more accurately estimated as 9.7 degrees. Then the downstream velocity of a material near the base of the channel is 9,00 mm/minute or 39 mm/s. For a channel depth of mm, the apparent shear rate is 60 /s. The time for a material to traverse the channel section length is 0.9 s. Given a target melt temperature of C, the apparent viscosity is 476 Pa s. For a target melt pressure of MPa, the specific volume, at the upstream section is.303 corresponding to a density of kg/m 3. If the specific heat is 800 J/kg C, then the temperature change, T, of a material being processed in the channel is estimated as.05 C. The temperature increase would cause thermal expansion of a material in the channel and a concurrent pressure increase. For the analysis described with respect to Figure 4, the pressure increase is estimated as T times dp/dt =.05 C.535 C/MPa =.6 MPa. These effects are quite significant given that the estimated temperature change and concurrent pressure change are for only a small section of the total screw channel; the effects of internal viscous heating would be much larger for longer channels, higher screw speeds, and thinner channels. Temperature changes due to heat transfer between the material, barrel, and screw may also be analyzed. Near the feed throat, the extruder is actively cooled to some temperature T. Suppose that the die of the extruder screw is set to T. Then by Fourier s law of conduction, the rate of energy transfer, de/dt, due to thermal conduction down the length of the extrusion screw is approximately: P de R k T T (7) dt L Consider an extrusion screw with an outer diameter of 0 mm and a length of 50 mm produced of stainless steel 40 with a heat conduction coefficient, k, of 4.9 W/m. If T and T are 4C and C, respectively, then the rate of heat transfer by conduction can be estimated as 53 W. P SPE ANTEC Anaheim 07 / 58

5 Results The above formulae may be analyzed to determine the channel design as a function of the channel length to obtain the desired target temperate and pressure. Clearly, errors in the analysis are prevalent and include: () variances in the true polymer rheology from constitutive models, () complex flows of melts and multiphase materials, (3) inappropriate modeling of velocity and temperature boundary conditions, (4) variation in the material feedstocks as a function of mechanical and thermal degradation, and (5) many others. The point of the above analysis is not to provide a final, quantitative answer but rather provide qualitative insights into the governing phenomena. The author is working on new material models and meshless computer simulations (see, for example, []) to model the flow and melting of pelletized feedstock but these research efforts are not yet validated. Accordingly, the suggested strategy is to rely on common screw design guidelines but then apply the foregoing analysis principles with fractal designs to improve screw performance. Figure 7 provides a design of a 38 mm (.5 inch) diameter screw to be used for film extrusion; the designation xx4 indicates channel in the feed section, channels in the transition section, and 4 channels in the metering section. The channels in each section of the screw are purposefully designed to achieve a specific objective. Feed Section: In the feed section, the screw uses a single channel to efficiently load the channel with pelletized and granulated feedstock from the feed throat. The lead, channel width, and channel depth in the feed section are, respectively, 00%, 90%, and 0% of the screw diameter. This feed section design follows standard screw design guidelines to achieve pumping of the feedstock to downstream portions of the screw. Transition Section: After four turns of the screw, the feed channel is split into two transition channels. With the same flight thickness of 0% and new helix angle of 0, each transition channel is 48% of the screw diameter with a depth that transitions to 8% of the screw diameter after ten turns. The objectives are two-fold: () to physically break up the solidified bed and thus impart more physical work on the feedstock by the flights, and () to provide a uniform and greater amount of shear on the processed material. Here, the channels are purposefully designed with specific lengths, widths, and depths per the foregoing analysis to achieve a target operating temperature. Metering Section: The metering section introduces an additional set of flights at a helix angle of 4, with each having a metering channel having a channel width of 5.% of the screw diameter and a final channel depth of 9% of the screw diameter. The metering section transitions to a slightly deeper channel to avoid excessive temperatures while developing the higher pressures required for film extrusion. Optional Mixing Section: During discussion about the fractal screw design concepts, both Spalding and Malloy separately and independently suggested the use of an intermediate mixing section between the transition and metering sections to provide homogenization of the melt. One simple mixing section design is shown in Figure 8. Such use of a mixing section can help to homogenize melt variations from the slightly different transition channel lengths while also ensuring uniform flow in identical metering sections. Figure 7: Design of a xx4 screw for a 38 mm diameter extruder with application to film extrusion Figure 8: Design of a xx4 screw in Figure 7 with intermediate mixing section SPE ANTEC Anaheim 07 / 59

6 Conclusions The described analysis and design concepts form the basis of a methodology for controlling the plastication path in extrusion. The described concepts are especially relevant to larger extrusion screws that may provide multiple channels without excessive helix angles that become problematic for melt pumping. The described designs are currently being machined and will be compared to general purpose and barrier screw designs using an instrumented 38 mm (.5 inch) extruder. Performance measures include the output volumetric flow rate, melt pressure, melt homogeneity, residence time, and energy efficiency. In the long term, the research should lead to new methods for analyzing and designing complete extrusion systems including extrusion screws, dies, and other control subsystems. Acknowledgements The author gratefully acknowledges the support of Mark Spalding, Robert Malloy, and U.S. Army/Natick Labs. References [] R. T. Fenner, Extrusion Screw Design: Newnes- Butterworth, 970. [] J. L. White and H. Potente, Screw Extrusion: Science and Technology Hanser Gardner Publications, 003. [3] T. P. Womer, Single Screw Design: 0 Years of Society of Plastics Engineers' Practical and Theoretical Technical Papers. DESTech Publications, 006. [4] C. Rauwendaal, "Scale up of single screw extruders," Polymer Engineering & Science, vol. 7, pp , 987. [5] J. Covas and A. Gaspar-Cunha, "Extrusion Scaleup: An Optimization-based Methodology," International Polymer Processing, vol. 4, pp. 67-8, 009. [6] J. Carley and J. McKelvey, "Extruder scale-up theory and experiments," Industrial & Engineering Chemistry, vol. 45, pp , 953. [7] B. H. Maddock, "Extruder scale up by computer," Polymer Engineering & Science, vol. 4, pp , 974. [8] H. Potente, "Existing scale-up rules for singlescrew plasticating extruders," International Polymer Processing, vol. 6, pp , 99. [9] K. Alemaskin, I. Manas Zloczower, and M. Kaufman, "Color mixing in the metering zone of a single screw extruder: numerical simulations and experimental validation," Polymer Engineering & Science, vol. 45, pp. 0-00, 005. [0] E. Ortiz-Rodriguez and C. Tzoganakis, "Scalingup a Reactive Extrusion Operation: A Onedimensional Simulation Analysis," International Polymer Processing, vol. 5, pp. 4-50, 00. [] K. S. Hyun and M. A. Spalding, "Bulk density of solid polymer resins as a function of temperature and pressure," Polymer Engineering & Science, vol. 30, pp , 990. [] K. S. Hyun, M. A. Spalding, and J. R. Powers, "Elimination of a restriction at the entrance of barrier flighted extruder screw sections," Journal of plastic film and sheeting, vol., pp , 995. [3] M. A. Spalding, K. S. Hyun, S. R. Jenkins, and D. E. Kirkpatrick, "Coefficients of dynamic friction and the mechanical melting mechanism for vinylidene chloride copolymers," Polymer Engineering & Science, vol. 35, pp , 995. [4] A. J. Bur, S. C. Roth, M. A. Spalding, D. W. Baugh, K. A. Koppi, and W. C. Buzanowski, "Temperature gradients in the channels of a single screw extruder," Polymer Engineering & Science, vol. 44, pp , 004. [5] A. Altınkaynak, M. Gupta, M. Spalding, and S. Crabtree, "Melting in a single screw extruder: experiments and 3D finite element simulations," International Polymer Processing, vol. 6, pp. 8-96, 0. [6] D. O. Kazmer, "Single Pellet Extrusion," Society of Plastics Engineers Annual Conference, Extrusion Division, 06. [7] B. B. Mandelbrot, The fractal geometry of nature, vol. Macmillan, 983. [8] B. B. Mandelbrot and J. W. Van Ness, "Fractional Brownian motions, fractional noises and applications," SIAM review, vol. 0, pp , 968. [9] B. B. Mandelbrot, D. E. Passoja, and A. J. Paullay, "Fractal character of fracture surfaces of metals," 984. [0] N. P. Suh, "Axiomatic design theory for systems," Research in engineering design, vol. 0, pp , 998. [] C. Rauwendaal, "8.4. The Standard Extrusion Screw," in Polymer Extrusion, ed Munich: Hanser, 986, pp [] D. O. Kazmer. (Published Feb., 06 at Single Pellet Extrusion with a Fractal Screw Design. SPE ANTEC Anaheim 07 / 60