ENTROPIC MEASURES OF MIXING IN APPLICATION TO POLYMER PROCESSING

Transcription

1 ENTROPIC MEASURES OF MIXING IN APPLICATION TO POLYMER PROCESSING by KIRILL ALEMASKIN Submitter in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Advisor: Dr. Ica Manas-Zloczower Department of Macromolecular Science and Engineering CASE WESTERN RESERVE UNIVERSITY January, 2005

2 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of candidate for the Ph.D. degree *. (signed) (chair of the committee) (date) *We also certify that written approval has been obtained for any proprietary material contained therein.

3 DEDICATION This work is dedicated to my beloved parents for their love and support.

4 1 TABLE OF CONTENT Table Of Content 1 List Of Tables. 3 List Of Figures 4 Acknowledgements. 7 Abstract... 8 CHAPTER ONE: Introduction Mixing in Polymer Processing Scope of the Work Works Cited: CHAPTER TWO: Entropic Mixing Characterization Shannon Entropy as a Dynamic Measure of Mixing Entropy of Different Species Present in the System Works Cited: 28 CHAPTER THREE: Numerical Procedure Introduction Finite Element Solution of the Flow in Single Screw Extruders Particle Tracking Works Cited: CHAPTER FOUR: Index For Simultaneous Dispersive And Distributive Mixing Characterization In Processing Equipment Dispersive Mixing - Kinetic Model d Flow Simulation of a Single Screw Extruder Results and Discussion Conclusions Works Cited:... 50

5 2 CHAPTER FIVE: Entropic Analysis Of Color Homogeneity Color Homogeneity as a Measure of Distributive Mixing Experimental Procedure Quantitative Analysis of Color Homogeneity Results and Discussion Conclusions Works Cited: 75 CHAPTER SIX: Color Mixing In The Metering Zone Of A Single Screw Extruder: Numerical Simulations And Experimental Validation Introduction Simulation of Color Mixing Numerical Results Particle Distributions Entropic Analysis Conclusions Works Sited:. 92 CHAPTER SEVEN: Summary And Future Work Summary Future Work Works Cited: 109 APPENDIX A APPENDIX B BIBLIOGRAPHY 145

6 3 LIST OF TABLES CHAPTER SIX Table 6.1 Extruder design parameters Table 6.2 Fluid rheological parameters.. 93 Table 6.3 Boundary conditions Table 6.4 Experiment vs. simulation: extrusion rate comparison.. 93

7 4 LIST OF FIGURES CHAPTER FOUR Figure 4.1 Spherical system of coordinates for an eroding agglomerate Figure 4.2 The finite element mesh for s single screw extruder.. 53 Figure 4.3 Particle spatial distributions at different cross sections along the extruder length Figure 4.4 Evolution of relative Shannon entropy along the extruder length.. 55 Figure 4.5 Erosion kinetics for Newtonian fluid. 56 Figure 4.6 Comparison of the erosion kinetics for Newtonian and power law fluids 57 Figure 4.7 Evolution of Shannon entropy along the extruder length Figure 4.8 Evolution of entropy and population distribution along the extruder length 59 Figure 4.9 Weight factors for different size fractions (based on 20 size fractions between the smallest and the largest sizes). 60 Figure 4.10 Evolution of the mixing index in the case of Newtonian fluid along the extruder length when operating the extruder at zero throttle ratio 61 Figure 4.11 Evolution of mixing index for the case of the power law fluid.. 62 Figure 4.12 Comparison between the mixing efficiency of Newtonian and the power law fluids at zero throttle ratio Figure 4.13 Evolution of the mixing index along the extruder length when operating the extruder at negative throttle ratio. 64 Figure 4.14 Comparison between the mixing efficiency of Newtonian and the power law fluids at negative throttle ratio 65

8 5 CHAPTER FIVE Figure Saran conventional screw: 6 pitches in the melting zone, 8 pitches in the tapering zone, and 7 pitches in the metering zone 76 Figure 5.2 Pressure profile inside the extruder 76 Figure 5.3 Screw crash extrusion sample from the metering section.. 77 Figure 5.4 Extrudate slices used for the analysis. 77 Figure 5.5 Grayscale representation of the color images. 78 Figure 5.6 Red channel intensity map. 79 Figure 5.7 Green channel intensity map.. 80 Figure 5.8 Blue channel intensity map 81 Figure 5.9 Evolution of color homogeneity index based on the grayscale image analysis 82 Figure 5.10 Evolution of color homogeneity index based on the Red channel map analysis Figure 5.11 Evolution of color homogeneity index based on the Green channel map analysis.. 84 Figure 5.12 Evolution of color homogeneity index based on the Blue channel map analysis CHAPTER SIX Figure 6.1 Mesh design (a overall, b detailed).. 94 Figure 6.2 Middle section of the mesh chosen for particle tracking 95 Figure 6.3 Experimental sample chosen to describe initial particle distribution.. 96

9 6 Figure 6.4 Initial particle position in the mesh 96 Figure 6.5 Slice generation on ZX and ZY planes. 97 Figure 6.6 Particle distributions at XZ and YZ cross sections of the extruder mesh within the four consecutive pitches Figure 6.7 Comparison between experimental sample cuts and corresponding numerical slices.. 99 Figure 6.8 Index of color homogeneity Figure 6.9 Distribution of modified conditional probabilities p * c/j over 10 bins for slices Figure 6.10 Distribution of modified conditional probabilities p * c/j over 1000 bins for slices

10 7 ACKNOWLEDGEMENTS I would like to sincerely thank my advisor, Dr. Ica Manas-Zloczower, for her wonderful guidance throughout my research work, endless patience, and permanent support during the years of my education at Case Western Reserve University. I would like to thank Dr. Miron Kaufman (Physics Department, Cleveland State University) for his valuable ideas on entropies and entropic analysis. The accomplishment of this work would not have been possible without his help. I would like to thank the National Science Foundation for providing financial support for this research under the grant DMI I would like to thank the Dow Chemical Company for providing technical assistance in the color mixing experiments. I would like to thank Ohio Supercomputer Center for the use of their computational facilities. I would like to thank all my colleagues and group members, especially Alberto Scurati, Prasad Gopalkrishnan, and Marco Camesasca, for their support and fruitful suggestions.

11 8 Entropic Measures of Mixing in Application to Polymer Processing ABSTRACT by KIRILL ALEMASKIN Mixing is an important component in most processing operations including but not limited to polymer processing. Generically, mixing refers to a process that reduces composition nonuniformity. Since the entropy is the rigorous measure of disorder or system homogeneity, in this work we will explore various ways to employ the entropy to characterize the state of mixing in a multi-component system. The various species can be initially present in the system or they can evolve as a result of a dispersive mixing operation involving a cohesive minor component. Computer simulation of agglomerate dispersion and sequential distribution of all particles obtained in the system allows us to evaluate the overall mixing efficiency of processing equipment. Evaluation is based on a specific mixing index, calculated using the Shannon entropies for different size fractions. The index can be tailored to give preference to different particle size distributions, thus relating the quality of mixing to specific properties of the final product. We propose a Shannon entropy based index of color homogeneity to assess color homogeneity as well as deviations from a standard/ideal color. We illustrate the concept by analyzing ABS polymeric samples obtained in a single screw extruder by mixing blue and yellow polymer pellets. Alternatively the proposed technique can be

12 9 employed to assess the efficiency and degree of distributive mixing attained in polymer processing equipment. We present numerical simulations for an ABS resin extrusion in an industrial conventional single screw extruder. Based upon the flow field patterns obtained in the simulations, a particle tracking procedure was employed to obtain information about the spatial distribution of particle tracers of two colors. Results of the simulation were compared with experimental data obtained under similar extrusion conditions. To evaluate the degree of color mixing and color homogeneity for the system, we also employ the entropy based index of color homogeneity for two species populations.

13 10 CHAPTER ONE Introduction 1.1 MIXING IN POLYMER PROCESSING In polymer processing operations the quality of the final product is highly dependent on mixing. A significant number of theoretical and experimental efforts were undertaken to predict the dependence of mixing quality on material parameters, design, and processing conditions [1]. Yet, a big challenge remains finding out criteria which would allow for mixing optimization in situ or serve to predict mixing quality in numerical simulations. It is important to distinguish two major mixing mechanisms. When mixing two immiscible fluids or solid agglomerates within a fluid, a first step is the reduction in size of the minor component (fluid droplets or solid clusters) in the matrix/major fluid. This mechanism is referred to as dispersive mixing and the criterion to be used in assessing mixing quality should be related to the minor component size distribution. Dispersive mixing performance is generally assessed in terms of specific characteristics of the flow field. Manas-Zloczower and Tadmor [2] looked at the distribution of the number of passes over the flights in a single screw extruder (regions of high shear stress), as a way to assess dispersive mixing performance. Cheng and Manas-Zloczower [3] used the relative strength of elongational flow components and shear stress distributions in the equipment to characterize dispersive

14 11 mixing performance. Several researchers [4-8] have adopted the same criteria of flow strength (sometimes labeled as the flow number) and/or shear stress distributions [9] to evaluate dispersive mixing. The local and total dispersive mixing performance were analyzed numerically using also two indices, namely the relative strength of elongational flow components and the number of passage distribution [10]. While shear stress and flow strength volumetric distributions are important factors in overall dispersive mixing characterization, more insight can be gained by following the change in these parameters along particle/minor component trajectories. Wang and Manas-Zloczower [11] used the concept of temporal distributions of shear rate/stress and flow strength to characterize dispersive mixing. A second step in obtaining a homogeneous mixture is to spread the minor component throughout the matrix. This step is referred to as distributive mixing and the criterion to be employed in judging mixing quality should be related to the spatial distribution of the minor component. In order to assess the spatial distribution of the minor component, gross uniformity and the intensity of segregation [1] are traditional tools for distributive mixing characterization. Galaktionov et al. [12] employed a mapping approach to study distributive mixing in Kenics static mixers and made use of a flux-weighed, sliceaveraged, discrete intensity of segregation to characterize mixing in continuous equipment in contrast to the previously used area or volume weighed measures for the intensity of segregation used in 2D or 3D closed prototype flows [13, 14]. Shearer and Tzoganakis [15, 16] measured experimentally distributive mixing in twin screw extruders by employing a mixing limited interfacial reaction between polymer tracers.

15 12 A mixing measure employed to illustrate the mixture structure and point out to unmixed regions is the scale of segregation, originally suggested by Danckwerts [17]. Researchers have used various statistical tools to characterize distributive mixing. Among them, pairwise correlation functions [18], the coefficient of variation (a normalized measure of the standard deviation) [19], the standard deviation sigma and the maximum error amongst average concentrations of finite-sized samples [20]. Recently Renyi entropies have been used as rigorous measures of various aspects of distributive mixing [21, 22]. One of the important aspects of distributive mixing much related to visual quality and consumer satisfaction is the color homogeneity of the polymer product. Examples can be found in automotive, textiles, food, packaging to mention just the most prominent in the plastics and rubber industry. One commonly used method to assess color homogeneity is comparison to a standard for qualitative visual inspection. Color homogeneity can also be assessed using spectroscopic methods. When colored samples are obtained from an experiment, they can be analyzed with particular type spectrometers to give color coordinates and/or other color characteristics of the sample. Color homogeneity can also be assessed using computer image analysis and employing standard direct Red-Blue-Green (RGB) correlations in the studied sample. However, quantitative assessment of color homogeneity as an aspect of mixing along the extruder length still remains a challenge.

16 13 Over the years, many researchers have used various experimental techniques designed to facilitate mixing analysis in processing equipment. Maddock [23] in the late 50 s proposed a technique of visual screw mixing analysis, known as screw freezing or screw crash which has been extensively used since its inception. Benkreira at. al [24] showed that mixing primarily occurs during the melting phase of extrusion and little improves thereafter. In a conventional Archimedean screw, during the melting stage, breakage of the solid bed in the cross channel direction can occur. This creates solid fragments which flow downstream not melted thus reducing mixing quality [25]. To overcome this problem several screw designs have been proposed, including the barrier melting screws [26, 27] and Maddock-style mixers [28, 29]. Other extruder designs implement the idea of forcing the material to flow through regions with repeatedly changing clearances, such as the energy transfer (ET) screws [30] or the Stratablend mixing screw [31]. These designs are mostly useful in enhancing melting efficiency and distributive mixing of the screw. Many fundamental analyses of mixing in extrusion have focused on the melt conveying zone [32-37]. While conventional screws do not provide high mixing efficiency, the application of mixers designed to reorient the flow streamlines and improve mixing in the metering zone, such as pin-type mixers [38-41], or the introduction of spatially periodic barriers in the screw to generate chaotic flow features [42], enhance mixing. There are a large number of distributive mixing devices for single screw extruders developed over the years designed primarily to impart substantial shear strain and also provide material reorientation [43-49].

17 14 Mixing continues past the metering zone of the extruder as the polymer flows through the die. Special dies have been designed to promote mixing, one example being the spiral mandrel die for annular products [50, 51]. Numerical simulations provide another opportunity to study mixing in complex geometries such as extruders without performing the actual experiments. 2-D and 3-D numerical studies of mixing efficiency in the pin mixing section of an extruder have been carried out by Yao et al. [40, 41]. Hwang et al. studied a similar pin region with a quasi 3-D analysis technique [52]. Similar analyses were applied to study mixing efficiency of chaotic screws [42, 53]. Quantification of color homogeneity in experimental samples can be done via image analysis. On the other hand, in numerical simulations, the coordinates of colored particle tracers can be directly computed and therefore direct quantitative measures of color homogeneity can be applied. 1.2 SCOPE OF THE WORK This thesis consists of seven chapters. The entropic analysis of mixing, projected as a tool for process control and optimization, is the key point of this work. Chapter 2 gives an introduction to the entropic characterization of mixing, and shows ways of judging mixing via multiple probability events. Chapter 3 is dedicated to the description of numerical procedures used in this thesis. Introduction to mesh development, numerical solution for the flow field via

18 15 discretization and finite element method, as well as the particle tracking procedure is outlined. The details for each procedure will be discussed in the following chapters. Conventionally processing equipment has been assessed separately for its dispersive and distributive mixing efficiency. Similarly, the final product has been judged separately for its dispersive and distributive mixing quality. In Chapter 4 of this thesis we introduce a new approach to simultaneously characterize dispersive and distributive mixing by monitoring particle erosion along their trajectories as well as their spatial distribution in the system. We express the extruder performance in terms of an overall index of mixing efficiency and apply this index to compare the performance of extruders with different designs under different processing conditions using both Newtonian and shear-thinning fluids. Color homogeneity is a second focus of the thesis. We attempt to quantify color homogeneity in two ways. First, as described in Chapter 5, we propose an entropic measure to assess color homogeneity and deviations from an ideal color based on the image analysis of experimentally obtained extrudate samples. We show that the proposed measure can be applied to polymer processing as a tool for in situ process control. Second, we perform numerical simulations of color mixing in extruder by tacking particles of two colored populations (Chapter 6). Based on the particle spatial distributions, we compute an index of color homogeneity using Shannon entropy. We propose this index as a tool for process optimization in polymer color mixing. Chapter 7 contains conclusions and recommendations for future research.

19 WORKS CITED: 1. Z. Tadmor, and C. G. Gogos, Principles of Polymer Processing, John Wiley & Sons, Chichester, Brisbane, Toronto, Singapore (1979). 2. I. Manas-Zloczower and Z. Tadmor, The Distribution of Number of Passes Over the Flights in a Single Screw Melt Extruders, Adv. Polym. Tech., 3, 213 (1983). 3. J. Cheng and I. Manas-Zloczower, Hydrodynamic Analysis of a Banbury Mixer, Polym. Eng. Sci., 29, 701 (1989). 4. P. J. Gramann and T. A. Osswald, Simulating Polymer Mixing Processes Using the Boundary Element Method, Int. Polym. Process., 7, 303 (1992). 5. P. J. Gramann, M.d.P. Noriega, A. C. Rios and T. A. Osswald, Understanding a Rhomboidal Distributive Mixing Head Using Computer Modeling and Flow Visualization Techniques, SPE Antec Tech Papers, 43, 3713 (1997). 6. C. H. Yao and I. Manas-Zloczower, Influence of Design on Mixing Efficiency in a Variable Intermeshing Clearance Mixer, Int. Polym. Process., 12, 92 (1997). 7. A. C. Rios, P. J. Gramann, T. A. Osswald, M.d.P. Noriega and O. A. Estrada, Experimental and Numerical Study of Rhomboidal Mixing Sections, Int. Polym. Process., 15, 12 (2000). 8. C. Rauwendaal, T. A. Osswald, P. Gramann and B. Davis, Design of Dispersive Mixing Devices, Int. Polym. Process., 14, 28 (1999). 9. T. Ishikawa, T. Amano, S. Kihara and K. Funatsu, Flow Patterns and Mixing Mechanisms in the Screw Mixing Element of a Co-Rotating Twin-Screw Extruder, Polym. Eng. Sci., 42, 925 (2002).

20 W. Yao, M. Mishima, K. Takahashi, Numerical Investigation on Dispersive Mixing Characteristics of MAXBLEND and Double Helical Ribbons, Chem. Eng. J., 84, 565 (2001). 11. W. Wang, I. Manas-Zloczower, Temporal Distributions: The Basis for the Development of Mixing Indexes for Scale-up of Polymer Processing Equipment, Polym. Eng. Sci., 41, 1068 (2001). 12. O. S. Galaktionov, P. D. Anderson, G. W. M. Peters and H. E. H. Meijer, Analysis and Optimization of Kenics Static Mixers, Int. Polym. Process., 18, 138 (2003). 13. O. S. Galaktionov, P. D. Anderson, P. G. M. Kruijt, G. W. M. Peters and H. E. H. Meijer, A mapping approach for three-dimensional distributive mixing analysis, Comput. Fluids, 30, 271 (2001). 14. P. G. M. Kruijt, O. S. Galaktionov, P. D. Anderson, G. W. M. Peters and H. E. H. Meijer, Analyzing Mixing in Periodic Flows by Distribution Matrices: Mapping Method, AIChE J. 47, 1005 (2001). 15. G. Shearer and C. Tzoganakis, The Effect of Kneading Block Design and Operating Conditions on Distributive Mixing in Twin Screw Extruders, Polym. Eng. Sci., 40, 1095 (2000). 16. G. Shearer and C. Tzoganakis, Relationship Between Local Residence Time and Distributive Mixing in Sections of a Twin-Screw Extruder, Polym. Eng. Sci., 41, 2206 (2001). 17. P. V. Danckwerts, The Definition and Measurements of Some Characteristics of Mixtures, Appl. Sci. Res. A, 3, 279 (1953).

21 H. H. Yang, T. Wong, and I. Manas-Zloczower, Flow Field Analysis of a Banbury Mixer in: Mixing and Compounding of Polymers, I. Manas- Zloczower and Z. Tadmor (Eds.), Hansen Publishers, p. 189 (1994). 19. B. C. Hutchinson, A. C. Rios and T. A. Osswald, Modeling the Distributive Mixing in an Internal Batch Mixer, Int. Polym. Process., 14, 315 (1999). 20. C. L. Tucker and G. W. M. Peters, Global Measures of Distributive Mixing and Their Behavior in Chaotic Flows, Korea-Australia Rheol. J., 15, 197 (2004). 21. W. Wang, I. Manas-Zloczower and M. Kaufman, Characterization of Distributive Mixing in Polymer Processing Equipment using Renyi Entropies, Int. Polym. Process., 16, 315 (2001). 22. W. Wang, I. Manas-Zloczower and M. Kaufman, Entropic Characterization of Distributive Mixing in Polymer Processing Equipment, AICHE J., 49, 1637 (2003). 23. B.H. Maddock, A Visual Analysis of Flow and Mixing in Extruder Screws, SPE J., 15 (5), 383 (1959). 24. H. Benkreira, R. W. Shales, M. F. Edwards, Mixing on Melting in Single Screw Extrusion, Int. Polym. Process., 7, 126 (1992). 25. M. A. Spalding, K. S. Hyun, Troubleshooting Mixing Problems in Single-Screw Extruders, SPE-ANTEC Tech. Pap., 229 (2003). 26. C. Maillefer, British Patent, 964,428 (1964). 27. C. Maillefer, Screw for Extrusion Apparatus, U. S. Patent, 3,358,327 (1967). 28. B. H. Maddock, An Improved Mixing Screw Design, SPE J., 23 (7), 23 (1967). 29. G. LeRoy, Apparatus for Extrusion of Thermoplastics, U. S. Patent, 3,486,192 (1969).

22 C. I. Chung, R. A. Barr, Energy efficient extruder screw, U. S. Patent, 4,405,239 (1983). 31. J. D. Frankland, Extrusion screw U. S. Patent, 4,639,143 (1987). 32. Z. Tadmor, I. Klein, Engineering Principles of Plasticating Extrusion, Krieger Publishing Company, Malabar, Florida (1978). 33. G. Lidor, Z. Tadmor, Theoretical Analysis of Residence Time Distribution Functions and Strain Distribution Functions in Plasticating Screw Extruders, Polym. Eng. Sci., 16, 450 (1976). 34. R. T. Fenner, Development in the Analysis of Steady Screw Extrusion of Polymers, Polymer, 18, 617 (1977). 35. H. Lappe, H. Potente, Throughput Behavior of Single Screw Extruders, SPE- ANTEC Tech. Pap., 174 (1983). 36. C. Rouwendaal, Throughput-Pressure Relationship for Power Law Fluids in Single Screw Extruders, SPE-ANTEC Tech. Pap., 30 (1985). 37. M. Gupta, T. H. Kwon, 3-D Flow Analysis of Non-Newtonian Viscous fluids Using Enriched Finite Elements, Polym. Eng. Sci., 30, 1420 (1990). 38. R. A. Barr, C. A. Chung, Auger-Type Extruder, U. S. Patent, 3,487,503 (1970). 39. R. V. DeBoo, C. B. Heard, Methods of and Apparatus for Advancing and Working Thermoplastic Materials, U. S. Patent, 3,762,693 (1973). 40. W. G. Yao, K. Takahashi, K. Koyama, G. C. Dai, Design of a New Type of Pin Mixing Section for a Screw Extruder Based on Analysis of Flow and Distributive Mixing Performance, Chem. Eng. Sci., 52, 13 (1997). 41. W. G. Yao, S. Tanifuji, K. Takahashi, K. Koyama, Mixing Efficiency in a Pin Mixing Section for a Single-Screw Extruder, Polym. Eng. Sci., 41, 908 (2001).

23 T. H. Lee, T. H. Kwon, A New Representative Measure of Chaotic Mixing in a Chaos Single-Screw Extruder, Adv. Polym. Tech., 18, 53 (1999). 43. G. M. Gale, Extruder mixer, U. S. Patent, 4,419,014 (1983). 44. G. M. Gale, An Evaluation of Mixing Devices for Extruders Fed with Masterbatched Additives and Polymer Blends, SPE ANTEC Tech. Pap., 109 (1983). 45. R. Brzoskovski, J. L. White, W. Szydlowski, N. Nakajima, K. Min, Modeling Flow in Pin-Barrel Screw Extruders, Int. Polym. Proc., 3, 134 (1988). 46. Y. Jabushida, R. Brzoskovski, J. L. White, N. Nakajima, Flow of Rubber Compound in a Pin Barrel Screw Extruder, Int. Polym. Proc., 4, 219 (1989). 47. F. E. Dulmage, Plastics Mixing and Extrusion Machines, U. S. Patent, 2,753,595 (1956). 48. R. L. Saxton, Extruder Mixing Screw, U. S. Patent, 3,006,029 (1961). 49. W. L. Kruger, Dispersing Screw Yields Homogenous Melts at High Output Rates, Plast. Eng., 37 (10), 35 (1981). 50. C. Rauwendal, Polymer Extrusion, Hanser Publishers, Munich, Vienna, New York (1986). 51. W. Michaeli, Extrusion Dies. Design and Engineering Computations, Hanser Publishers, Munich, Vienna, New York (1984). 52. W. R. Hwang, K. W. Kang, T. H. Kwon, Dynamical systems in pin mixers of single-screw extruders, AIChE J., 50, 1372 (2004). 53. W. R. Hwang, T. H. Kwon, Dynamic Modeling of a Chaos Single-Screw Extruder and Its Three-Dimensional Numerical Analysis, Polym. Eng. Sci., 40, 702 (2000).

24 21 CHAPTER TWO Entropic Mixing Characterization 2.1 SHANNON ENTROPY AS A DYNAMIC MEASURE OF MIXING Innovative applications of entropic measures are applied in many fields of science, such as statistical mechanics [1, 2], biology and medicine [3-5], cognitive science [6, 7], economics [8, 9], geology [10, 11], to list just a few examples. In pattern recognition, Cheng at al. [12, 13] introduced the entropy based fuzzy homogeneity approach and the method of homograms applied to image threshold and segmentation. Entropic image analysis has also been used to assess the level of image compression as was shown by Tavakoli [14]. Recently a new application of entropic analysis of mixing quality was introduced by Wang et al. [15, 16]. He utilized the approach of calculating Renyi entropies, including the Shannon entropy, as a means to assess the quality of distributive mixing. Shannon entropy [17] is defined in terms of the probabilities p j of M possible outcomes: M S = p ln p (2.1) j= 1 j j Equation 2.1 is the unique representation of the entropy, up to a multiplicative positive constant, as S is to satisfy the following requirements: the lowest entropy (S = 0) corresponds to one of the p's being 1 and the rest being zero (considering

25 22 lim pjln p j = 0 ); the largest value for the entropy is achieved when all p's are p j 0 equal; and S is additive over partitions of the outcomes. If we concentrate only on distributive mixing, we divide the space of interest in M equal volume bins and then estimate each probability p j by the particle concentration in bin #j. The worst distributive mixing in the system (when all particles are in one bin) is characterized by S = 0 and the best mixing (all bins contain equal number of particles) is characterized by the maximum entropy S = ln(m) (see Appendix A). The number of bins is an important parameter as it determines at what scale of observation we are evaluating the quality of mixing. In previous work done in our group [15, 16] it has been shown that the number of bins chosen to describe the quality of distributive mixing is an important parameter as it defines the scale of observation (the magnifying glass ) at which one looks at the system. The smaller the number of bins is, the larger the scale of observation at which the observer evaluates the quality of particle distribution. At a very small scale (M > N, where N is the overall number of particles) some bins will be depleted of particles and the particle distribution will thus differ from the uniform distribution. Thus, the number of particles in the system sets an upper limit for the number of bins to be used for system characterization. 2.2 ENTROPY OF DIFFERENT SPECIES PRESENT IN THE SYSTEM If the mixing process involves particles of different species (characterized for example by color or size) the outcomes are determined not only by their overall

26 23 spatial distribution (i.e. position of the particles in space) but also by the particle properties (such as particle size). We then rewrite Eq. 2.1 as: S = C M c= 1 j= 1 p c, ln p (2.2) j c, j Here the index c labels the particle species, the index j labels the spatial bin and p c,j is the joint probability to find a particle of species c in bin j. C is the total number of species present in the system and M is the number of spatial bins. According to Bayes theorem, the joint probability for a particle to be located in bin j and to be of species c is given by p c, j = p j / c pc (2.3) or p = p p (2.4) c, j c/ j j In Eq. 2.3 p j/c is the probability of finding a particle in bin j conditional on being of species c, and p c is the probability for species c (irrespective of location). In Eq. 2.4 p c/j is the probability of finding particle of species c conditional on bin j, and p j is probability for bin j (irrespective of species). By substituting Eq. 2.3 into Eq. 2.2 we obtain: S = C M [ ( p j c pc ) ( p j / c pc )] c= 1 j= 1 / ln (2.5) and following S = C p M C M [ p j / c p j / c ] [ pc ln pc ] ln p (2.6) c c= 1 j= 1 c= 1 j= 1 j / c

27 24 Since M p j j= 1 / c = 1, the Eq. 2.6 reduces to: S = C M C pc [ p j / c p j / c ] [ pc ln pc ] c= 1 j= 1 c= 1 ln (2.7) Equation 2.7 can be compacted as follows: S = S ( location) + S( species) (2.8) species where: C S ( location) p S ( location) species c c c= 1 = (2.9) c M [ p j / c ln p j / c ] S ( location) (2.10) j= 1 C [ p c ln p c ] S( species) (2.11) c= 1 The first term in the right hand side of Eq. 2.8, S species (location), is the entropy associated with particle locations conditional on species and averaged over all species. This entropy, in fact, shows how well different species are distributed throughout the system by taking into account their relative concentration. Since S c (location) ln(m), to get a relative entropy which takes values between 0 and 1 we divide this entropy by ln(m): S rel species ( location) C p S ( location) c c c= 1 (2.12) = ln ( M )

28 25 The second part of the Eq. 2.8, S(species) gives an idea how equally the species are represented in the system and maximizes only when all p c = 1/C. From a practical standpoint in certain cases, such as for instance in systems where agglomerate dispersion occurs, we are mostly concerned with the distribution of species in space, and prefer not to have equal concentration of all species. In the following discussion (Chapter 3) we therefore will use only the left hand size of the Eq. 2.8, S species (location), to characterize the mixing of eroding particles, and measure the quality of distributive mixing of each species (based on the particle size). We then will develop an index that characterizes simultaneously the spatial mixing of each species and a material property, as reflected in a preference for a certain size particles. This index is based on replacing in Eq. 2.9 the p c s with Gaussian distribution weights centered on the preferred size. By using the second part of Bayes s theorem, Eq. 2.4 and following the same steps as above we get: M S = p js j( species) + S( locations) j= 1 (2.13) where C = c/ j c/ j c= 1 S ( species) - p ln p j (2.14) S( locations) = - p j ln p j M (2.15) j= 1

29 26 S j (species) is the entropy of species intermixing at the location of bin j and S(locations) is the entropy associated with the overall spatial distribution of particles irrespective of species. If the species are distinguished by color, Eq can also be written as: S = S ( colors) + S( locations) (2.16) locations where S ( colors) = p S ( colors) (2.17) locations j j j= 1 M S locations (colors) is a spatial average of the entropies associated with color species intermixing conditional on location. It is maximized for the particular homogeneous state characterized by: p c/j = 1/c in each bin j ranging from 1 to M. This entropy is important if one is interested in judging the quality of intermixing between species, in this case particles of different colors. The maximum value of S locations (colors) is ln (C); thus to obtain an index of color homogeneity based on this entropy with values between 0 (no color intermixing) and 1 (ideal intermixing), we normalize this spatial average entropy by ln (C): S rel locations ( colors) = M j= 1 ps j j( colors) ln ( C) (2.18) The second part of Eq. 2.13, S(locations), characterizes overall distribution of particles in the system. This term can be important for a discrete system, where one would expect particles to occupy just some of the bins. However, in completely filled

30 27 systems, where all the bins are occupied by particles, this term is pretty much constant and can be omitted from the calculations. In Chapter 4 we illustrate the use of S rel locations (color) as an index of color homogeneity to analyze extrudate samples obtained by mixing yellow and blue polymeric concentrates in a single screw extruder. In Chapter 5 we employ this index as a tool to measure the quality of color mixing in a numerical simulation of mixing in a single screw extruder.

31 WORKS CITED: 1. D. F. Styer, Insight into entropy, Am. J. Phys., 68, 1090 (2000). 2. A. Brandenberger, V. Mukhanov, T. Prokopec, Entropy of the Gravitational Field, Phys. Rev. D, 48, 2443 (1993). 3. A. O. Schmitt, H. Herzel, Estimating the Entropy of DNA Sequences, J. Theor. Biol., 188, 369 (1997). 4. Y. Zimmer, R. Tepper, S. Akselrod, A Two-Dimensional Extension of Minimum Cross Entropy Thresholding for the Segmentation of Ultrasound Images, Ultrasound in Med. & Biol., 22, 1183 (1996). 5. R. Smolikova, M. P. Wachowiak, J. M. Zurada, An Information-Theoretic Approach to Estimating Ultrasound Backscatter Characteristics, Computers in Biol. and Med., 34, 355 (2004). 6. T. C. Devezas, J. T. Corredine, The Nonlinear Dynamics of Technoeconomic Systems: An Informational Interpretation, Tech. Forecast. Soc. Change, 69, 317 (2002). 7. J. Hale, The Information Conveyed by Words in Sentences, J. Psycholinguistic Research, 32, 101 (2003). 8. A. Golan, G. Judge, L. Karp, A Maximum Entropy Approach to Estimation and Inference in Dynamic Models or Counting Fish in the Sea Using Maximum Entropy, J. Economic Dynamics and Control, 20, 559 (1996). 9. R. C. Campbell, R. C. Hill, Predicting Multinomial Choices Using Maximum Entropy, Economics Lett., 64, 263 (1999). 10. J. D. Phillips, P. A. Gares, M. C. Slattery, Agricultural Soil Redistribution and Landscape Complexity, Landscape Ecology, 14, 197 (1999).

32 P. O. Bodun, S. Shibusawa, A. Sasao, K. Sakai, H. Honaka, Dredged Sludge Moisture Prediction by Textural Analysis of the Surface Image, J. Terramechanics, 37, 3 (2000). 12. H. D. Cheng, C. H. Chen, H. H. Chiu, H. Xu, Fuzzy Homogeneity Approach to Multilevel Thresholding, IEEE Trans. Image Process., 7, 1084 (1998). 13. H. D. Cheng, X. H. Jiang, J. Wang, Color Image Segmentation Based on Homogram Thresholding and Region Merging, Pattern Recognition, 35, 373 (2002). 14. N. Tavakoli, Entropy and Image Compression, J. Visual Communication and Image Representation, 4, 271 (1993). 15. W. Wang, I. Manas-Zloczower and M. Kaufman, Characterization of Distributive Mixing in Polymer Processing Equipment using Renyi Entropies, Int. Polym. Process., 16, 315 (2001). 16. W. Wang, I. Manas-Zloczower and M. Kaufman, Entropic Characterization of Distributive Mixing in Polymer Processing Equipment, AICHE J., 49, 1637 (2003). 17. C. E. Shannon, A Mathematical Theory of Communication, The Bell System Technical J., 27, 379 (1948).

33 30 CHAPTER THREE Numerical Procedure 3.1 INTRODUCTION Due to constant modification of computer hardware towards faster machines with much higher memory capabilities, and constant development of computational packages designed for computations in a broad range of applications, numerical simulations become more and more appealing for engineers as compared with experimental studies, which require a lot of energy and material expenditures to produce necessary data. In polymer processing an array of factors, such as the variety of designs, freedom in choosing processing conditions, diversity of materials to be processed become a big challenge for experimental research. New methods of simulation, such as finite element modeling on the other hand, allow one to eliminate unnecessary wastage of materials, energy, and time required to build up experimental pilot equipment. The large number of publications on numerical simulation in polymer processing witnesses the importance of numerical methods for research and the interest of the industrial sector towards these innovative procedures. Selected examples covering just the last four years [1-15] are illustrating applications of numerical simulations in extrusion and mixing, injection, compression and blow molding, just to mention a few sectors of polymer processing.

34 31 In previous work done in our group [16-23], numerical simulations have also been extensively used to study complex three-dimensional systems, such as batch and continuous polymer mixing equipment. 3.2 FINITE ELEMENT SOLUTION OF THE FLOW IN SINGLE SCREW EXTRUDERS The finite element method is traditionally used among engineers in problems requiring discretization of a continuous field described by tensorial equations. A full description of the method is given by many authors [24-27]. In computational fluid dynamics, three major equations are to be discretized: continuity, motion, and energy [28]. As applied to our particular task, which is the isothermal viscous flow, the energy equation can be omitted from the calculation and, therefore the final solution of the flow is combined from the solution of the continuity equation for incompressible fluids: v = 0 (3.1) and the equation of motion relaxed for the body forces: Dv ρ Dt = P τ (3.2) In these equations v is the velocity vector, ρ is the fluid density, t is time, P is pressure, and τ is the stress tensor. To solve these two equations in application to a three-dimensional, fully-developed, steady-state flow in a single screw extruder, we used a fluid dynamics package

35 32 FIDAP. No slip boundary conditions on the solid surfaces were applied. To facilitate the particle tracking procedure, rotation of the barrel in the opposite direction relative to the screw rotation was applied. 3.3 PARTICLE TRACKING Based on the solution for the flow field, obtained using FIDAP, a particle tracking algorithm can be employed to observe the motion of particle tracers within a flow field. The particles (or tracers) are considered to be massless points, so that their presence in the system does not affect the flow field or the motion of other particles. We neglect inertia and gravity forces. We also consider that particles are continuously injected in the flow field at their initial positions. The position of a particle at any specific time t can be found by integrating the velocity vector in the flow filed: 1 0 t 1 X ( t ) = X( t ) + V( t) dt (3.3) t 0 Here X(t 0 ) is the position of the particle at time t 0, X(t 1 ) is the position of the particle at time t 1 = t 0 + dt, V(t) is particle velocity vector, and dt is the integration time step. The integration time step was chosen to be twice as small as the time necessary to cross the smallest element in the mesh in any direction (X, Y, or Z) with the highest velocity in that direction (V x, V y, or V z ).

36 33 We employed a 4 th order Runge-Kutta integration procedure [29] to obtain particle positions according to Eq Particular designs of the extruder mesh, processing conditions, and material parameters, as well as the initial particle distributions will be described in details in the following chapters. The developed software algorithm used for the particle tracking procedure is shown in Appendix B. The algorithm tracks particles within the velocity flow field obtained by FIDAP finite element simulation. It also incorporates the dispersion mechanism, which will be explained in Chapter 4.

37 WORKS CITED: 1. S.-W. Kim, L.-S. Turng, Developments of Three-Dimensional Computer-Aided Engineering Simulation for Injection Moulding, Modelling Simul. Mater. Sci. Eng., 12, s151 (2004). 2. A. Cheung, Y. Yu, K. Pochiraju, Three-Dimensional Finite Element Simulation of Curing of Polymer Composites, Fin. Elem. Analysis Des., 40, 895 (2004). 3. N. Clemeur, R.P.G. Rutgers, B. Debbaut, Numerical Simulation of Abrupt Contraction Flows Using the Double Convected Pom Pom Model, J. Non- Newtonian Fluid Mech., 117, 193 (2004). 4. Y. Huang, C. R. Gentle, J. B. Hull, A Comprehensive 3-D Analysis of Polymer Melt Flow in Slit Extrusion Dies, Adv. Polym. Tech., 23, 111 (2004). 5. V. L. Bravo, A. N. Hrymak, J. D. Wright, Study of Particle Trajectories, Residence Times and Flow Behavior in Kneading Discs of Intermeshing Co- Rotating Twin-Screw Extruders, Polym. Eng. Sci., 44, 779 (2004). 6. J.-C. Yu, X.-X. Chen, T.-R. Hung, F. Thibault, Optimization of extrusion blow molding processes using soft computing and Taguchi s method, J. Intell. Manuf., 15, 625 (2004). 7. X.-T. Pham, F. Thibault, L-T. Lim, Modeling and Simulation of Stretch Blow Molding of Polyethylene Terephthalate, Polymer Engineering And Science, 44, 1460 (2004). 8. H Tang, L. C. Wrobel, Z. Fan, Fluid Flow Aspects of Twin-Screw Extruder Process: Numerical Simulations of TSE Rheomixing, Modelling Simul. Mater. Sci. Eng., 11, 771 (2003).

38 35 9. R. K. Connelly, J. L. Kokini, 2-D Numerical Simulation of Differential Viscoelastic Fluids in a Single-Screw Continuous Mixer: Application of Viscoelastic Finite Element Methods, Adv. Polym. Tech., 22, 22 (2003). 10. O. S. Galaktionov, P. D. Anderson, G. W. M. Peters, H. E. H. Meijer, Mapping Approach for 3D Laminar Mixing Simulations: Application to Industrial Flows, Int. J. Numer. Meth. Fluids, 40, 345, (2002). 11. O. S. Galaktionov, P. D. Anderson, P. G. M. Kruijt, G. W. M. Peters, H. E. H. Meijer, A Mapping Approach for Three-Dimensional Distributive Mixing Analysis, Computers And Fluids, 30, 271 (2001). 12. R. E. Khayat, C. Plaskos, D. Genouvrier, An Adaptive Boundary-Element Approach for 3D Transient Free Surface Cavity Flow, as Applied to Polymer Processing, Int. J. Numer. Meth. Eng., 50, 1347 (2001). 13. K. M. Pillai, C. L. Tucker, F. R. Phelan, Numerical Simulation of Injection/Compression Liquid Composite Molding. Part 2: Preform Compression, Composites: Part A, 32, 207 (2001). 14. R.-Y. Chang, W.-H. Yang, Numerical Simulation of Mold Filling in Injection Molding Using a Three-Dimensional Finite Volume Approach, Int. J. Numer. Meth. Fluids., 37, 125 (2001). 15. K. M. Pillai, C. L. Tucker, F. R. Phelan, Numerical simulation of injection/compression liquid composite molding. Part 1. Mesh generation, Composites: Part A, 31, 87 (2000). 16. H.-H. Yang, I. Manas-Zloczower, Flow Field Analysis of the Kneading Disc Region in a Co-Rotating Twin Screw Extruder, Polym. Eng. Sci., 32, 1141 (1992).

39 H.-H. Yang, I. Manas-Zloczower, Analysis of Mixing Performance in a VIC Mixer, Int. Polym. Proc., 9, 291 (1994). 18. C. Wang, I. Manas-Zloczower, 3-D Flow Simulations of a Cavity Transfer Mixer, Int. Polym. Proc., 9, 46 (1994). 19. C. Wang, I. Manas-Zloczower, Modeling and Experimental Study of the Flow in a Simplified Cavity Transfer Mixer, Int. Polym. Proc., 11, 115 (1996). 20. H. F. Cheng, I. Manas-Zloczower, Study of Mixing Efficiency in Kneading Discs of Co-Rotating Twin Screw Extruders, Polym. Eng. Sci., 37, 1082 (1997). 21. C. H. Yao, I. Manas-Zloczower, Influence of Design on Mixing Efficiency in a Variable Intermeshing Clearance Mixer, Int. Polym. Proc., 12, 92 (1997). 22. H. F. Cheng, I. Manas-Zloczower, Influence of Design on Mixing Efficiency in the Kneading Disc Region of Co-Rotating Twin Screw Extruders, J. Reinf. Plast. Compos., 17, 1076 (1998). 23. W. Wang, I. Manas-Zloczower, Temporal Distributions: The Basis for the Development of Mixing Indexes for Scale-up of Polymer Processing Equipment, Polym. Eng. Sci., 41, 1068 (2001). 24. O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, McGraw-Hill, London, New York (1989). 25. K.H. Huebner, E.A. Thornton, The Finite Element Method for Engineers, John Wiley & Sons, New York (1982). 26. K-J. Bathe, E.L. Wilson, Numerical methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs NJ (1976). 27. J. E. Akin, Application and Implementation of Finite Element Methods, Academic press, London, New-York (1982).

40 R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport phenomena, John Wiley & Sons, New York (2002). 29. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, (2 edition), Cambridge University Press (1992). nd

41 38 CHAPTER FOUR Index for simultaneous dispersive and distributive mixing characterization in processing equipment 4.1 DISPERSIVE MIXING - KINETIC MODEL The dispersion of solid particles under hydrodynamic stresses has been studied quite extensively in the past. Manas-Zloczower and Feke [1] have analyzed agglomerate separation in linear flow fields and have shown that pure elongational fields are the most efficient in particle separation. They have also defined a dimensionless group, which scales the magnitude of the viscous forces acting on the agglomerate relative to its tensile strength. Rwei et al [2-4] have observed two distinct modes of dispersion of carbon black agglomerates in a simples shear flow field, namely erosion and rupture, depending on the shear stress applied. Simple kinetics laws for the erosion mode have been established. Bohin et al. [5] have postulated that the rate of erosion is proportional to the excess of hydrodynamic force acting on the agglomerate relative to its cohesive strength: dr = K ( β F F h c) (4.1) dt Here dr is negative as the radius of the agglomerate decreases with time, K is a proportionality factor that is related to the internal structure of the agglomerate, and β is a proportionality factor that reflects the fraction of the overall hydrodynamic force

42 39 that bears on the fragment. As established before by Bagster and Tomi [6], in a simple shear flow filed at low Reynolds numbers, the hydrodynamic force experienced by an agglomerate at the mid-plane is defined as: Fh = µγπ R (4.2) where µ is the fluid viscosity, γ is the shear rate and R is the radius of the agglomerate. The cohesive force, on the other hand, can be expressed as: F c = HN (4.3) b where H is a mean interparticle force, and N b is the number of the particle bonds to be broken [7]. Levresse at al. [8] have analyzed the forces acting on spherical caps of the agglomerate, rather than just the normal force at the mid-plane. The hydrodynamic force was expressed in terms of the surface and orientation of the cap. The modes of dispersion, which can be either: a) no dispersion, b) erosion, c) erosion and rupture, and d) rupture, can be expressed via the fragmentation number F a, defined by Ottino et al. [9] as: F a F F h = (4.4) c While no dispersion takes place at F a 1, erosion occurs at 1 F a F a crit, rupture becomes possible at F a > F a crit, when hydrodynamic force remarkably exceeds the strength of the agglomerate.

43 40 Based on Bohin s kinetic model of erosion, and Levresse s expression for the hydrodynamic force acting on the agglomerate cap, Scurati et al [10] generalized the erosion kinetics model in linear simple shear flow fields: f d 1 { ( ψ ) } K sin 21 cos dr π ψ γσ 5µγ = dt R d 2 2 f 0 0 c R 2 d 1 sin cos sin f d θ θ ϕ R 2σ c 0 1sinψ0 f ( ) (4.5) In this equation ψ 0, θ, and ϕ define the size and orientation of the spherical cap (Fig. 4.1) upon which the hydrodynamic force is acting (the highest hydrodynamic force acts upon the agglomerate cap at θ = 45 ), d f is the mass fractal dimension, corresponding to a certain internal agglomerate structure (agglomerates of uniform structure have a mass fractal dimension d f = 3), K is a fitting parameter obtained through an erosion experiment, R 0 and R are the initial agglomerate radius, and the agglomerate radius at a time t, and σ is the agglomerate cohesivity. Wang and Manas-Zloczower [11] have shown one way of monitoring the history of stresses acting on a particle along its trajectory by plotting temporal stress distributions. They have also modified the expression for the hydrodynamic force (Eq. 4.2) considering deviations from a simple shear via the flow strength parameter λ: λ = γ γ + ω (4.6) where γ and ω are the rate of deformation and vorticity tensors correspondingly. With that Equation 4.2 is modified into: Fh 2 = 5λµγπ R (4.7)

44 41 This modification of the hydrodynamic force results in the final expression for the erosion kinetics: f d 1 { ( ψ ) } dr Kπsin ψ γσ 5λµ γ 21 cos = dt R d 2 2 f 0 0 c R 2 d 1 sinθcosθsin ϕ R f d σ c 0 1sinψ0 f ( ) (4.8) In the following discussion we will use the modified Scurati erosion kinetic model (Eq. 4.8), and Wang s approach of tracing the shear stresses/strains acting on the particle at each time step, accounting also for changes in fluid viscosity and flow strength parameter along the particle trajectory D FLOW SIMULATION OF A SINGLE SCREW EXTRUDER A fluid dynamics package FIDAP was used to simulate the three-dimensional, isothermal, steady flow in a four pitch single screw extruder. No slip boundary conditions on the solid surfaces were applied. The operating condition of the screw rotation corresponds to an RPM of 180. The finite element mesh of the extruder was composed of quadrilateral elements, giving a total number of nodes of Fig. 4.2 shows the finite element mesh for the extruder. We used the following physical parameters of the extruder: barrel diameter D b = 5 cm, diameter of the screw root D s = 2.8 cm, clearance distance δf = 0.02 cm, flight thickness e = 0.2 cm. The design dimensions here are typical for a single screw extruder [12]. We studied the influence of the throttle ratio (pressure flow to drag flow) Q p /Q d on the extruder performance. Two cases were considered: pure drag flow (Q p /Q d = 0) and

45 42 flow with back pressure such that Q p /Q d = -1/2. Also to analyze the influence of the extruder design on mixing performance, we performed the simulation for three different extruders with helix angles θ b = 10, 17, and 24 degree. The pitch distance of the extruder L s was adjusted accordingly to be 2.8 cm, 4.8 cm, and 7.0 cm. Simulations were performed for a Newtonian fluid of viscosity µ 0 = 1160 Pa s and a power-law model fluid with a Newtonian plateau: τ µ γ if γ γ 0 0 = n 1 m γ γ if γ > γ0 The parameters of the power law model were chosen to match the rheological behavior of low density polyethylene (LDPE) melt with a power law index n = 0.59, a consistency index m = 4680, and the shear rate for the onset of shear thinning γ = 30 0 s -1. Particle tracking, as described in Chapter 3, allows us to observe the motion of particle tracers within the flow field. The particle tracking procedure allows us to define the particle position at each time step and estimate the rate of deformation, viscosity, and flow strength at that position. 4.3 RESULTS AND DISCUSSION We first employ entropy to characterize distributive mixing of 9300 particles along the extruder length. An example of particle spatial distributions at various cross

46 43 sections along the extruder length is shown in Figure 4.3. We assume continuous feeding of particles at the extruder entrance and thus steady-state conditions for all the results presented in this work. We employed relative Shannon entropy (normalized by ln M) to characterize the quality of distributive mixing at these cross sections, and therefore the dynamics of distributive mixing: S rel = M j= 1 p ln p ln j ( M ) j (4.9) Figures 4.4a and 4.4b show relative Shannon entropies calculated based on the particle spatial distributions along the extruder length. We calculated the entropy based on the number of bins M = 4700, which is approximately half of the total number of particles in the system. From Fig. 4.4 it can be concluded that a change in helix angle shows little effect on distributive mixing for both Q p /Q d = 0 and Q p /Q d = To study dispersive mixing in the system we now introduce 100 agglomerates at the entrance plane to the extruder. These agglomerates are exposed to hydrodynamic forces as they travel along the extruder length and consequently will erode according to Equation 4.8. During erosion we assumed that parent agglomerates and eroded fragments maintain a spherical shape. The initial agglomerate radius was set at 0.5 mm, while the radius of the smallest eroding fragment was set at 0.11 mm. A simple mass balance shows a maximum of 93 particles to be generated from a single agglomerate thus rendering a maximum of 9300 possible fragments obtained during erosion. We assume agglomerates of fractal dimension d f = 2.7.

47 44 Figure 4.5 illustrates the erosion kinetics in extruders of different designs at two processing conditions when using a Newtonian fluid. Reducing the helix angle and operating the extruder at negative throttle ratios seems beneficial. This result can be explained primarily in light of an increasing residence time in the extruder when decreasing the helix angle and operating at negative throttle ratios. A negative throttle ratio will also positively affect the shear rate/stress distribution in the system and consequently enhance dispersion kinetics. Also, by changing the fluid in the system to one exhibiting a shear thinning behavior while maintaining all the other parameters unchanged (thus allowing the use of the same dispersion kinetic model), erosion is impeded (Figures 4.6a and 4.6b). Due to the shear thinning behavior of the power law fluid, it s viscosity drops down when subjected to the regions of shear rates higher than 30 s -1. Therefore, compared to the Newtonian fluid, the power law fluid generates lower hydrodynamic forces in these regions, and the agglomerate erosion slows down. Referring back to Equation 2.10, we can now assess not only the spatial distribution of all particles present in the system, but we also can look at the quality of distribution for a specific particle population. These populations are created based on particle physical dimension (radius) as a result of the erosion process. Figure 4.7 shows the evolution of the Shannon entropies associated with the different size species, S c (location) c = , for a particular extruder design and processing conditions. Fraction 20 is defined as the initial agglomerate with the largest radius R = 0.5 mm. Fractions 2-19 represent partially eroded agglomerates diminishing in size as the fraction number c goes down. Fraction 1 corresponds to small particles with R = 0.11 mm that erode from the agglomerates.

48 45 The plot starts with a low value of entropy for the largest size fraction (fraction 20) present in the system (initial agglomerates). As these particles advance along the extruder length they get distributed and thus their entropic value increases. However as another fraction (fraction 19) appears in the system, fraction 20 is subjected to attrition and consequently its entropic value decreases. Similar behavior is observed for all intermediate species generated in the system as a result of erosion. The ultimate particle size (fraction 1) is constantly accumulated in the system and as particles of these species continue to distribute, entropic value for this size fraction shows a constant growth. At a given scale of segregation, the entropic value, which characterizes the distribution of the specific population, will be highly dependent on the concentration of particular species in the system. If the number of bins chosen to characterize our system is high enough to account for the maximum number of particles to be generated in the system (e.g particles in our study), at low concentration of a certain fraction, its entropy is low regardless how well particles of this fraction are distributed. To illustrate this point, in Figure 4.8 we superimpose the plot for the entropies and the plot reflecting the change in particle population balance. We can distinguish a good correlation between the loci of maxima in both entropy and concentration, except for the initial size agglomerates. The entropy of species describes the distribution for particles of that particular size only. However, for practical purposes it is important to evaluate the quality of distributive mixing for different size particles. Furthermore, optical or mechanical properties of the final product dictate which size particles are more important. Consequently, based on S species (location) (Eq. 2.9), we define an index that is a

49 46 weighted average of the entropies associated with different species, with weights f c which reflect the relative importance of different sizes for the physical specifications of the final product: C 1 I = fcsc( locations) (4.10) ln( ) M c = 1 The weights are positive numbers that must add up to unity: C fc = 1 (4.11) c= 1 0 f 1 c In the limiting case when only one particular size is important and the others are not, the weight of the favored size is equal to unity and the rest are equal to zero. In our work we follow the evolution of mixing quality in the single screw extruder with an index defined in Equation 4.10 based on Gaussian weights: f Aexp ( r r ) 2 c opt c = 2 = C A exp 2σ ( ) r c r opt 2σ 2 = 1 c 2 1 (4.12) The favored size is r opt and the width of the distribution function is σ. The largest weight f is equal to the parameter A which is chosen such as to satisfy the condition expressed in Equation Note that by increasing the preferred distribution width σ, we attribute less significance to the optimum size (smaller value for A). In this study we consider two case scenarios: one in which the optimum particle size corresponds to the smallest particle size generated during erosion (particles of radius cm) and one in which the optimum particle size is chosen at an intermediate

50 47 size of cm. The value of σ was chosen to be cm. Figure 4.9 shows the weight functions f c for these two scenarios based on 20 size fractions. Figure 4.10a shows the evolution of the mixing index calculated for three extruder designs operated at a zero throttle ratio in the Newtonian fluid, when the preferred size in the system is the smallest particle size. The fast increase of the index from zero to a value of about 0.35 accounts for the appearance of the smallest size fraction (0.011 cm) in the earliest stages of mixing. The three extruder designs show similar performance in spite of the delayed erosion at higher values of the helix angle. Apparently the dominant factor in this case seems to be the overall distribution process. The small fragments produced from the initial agglomerates tend to move in the fluid close enough to their parents for a long time, thus creating local spots of high particle concentration and impeding on overall mixing quality. On the other hand, when the preferred distribution is shifted to the intermediate particle size, the mixing index is primarily dominated by the concentration of the preferred specie and correlates well with the dispersion kinetics curves. This is illustrated in Figure 4.10b. Due to faster erosion at smaller helix angles, the fastest achievement of the maximum of the index is observed for the extruder with 10 degree helix angle. The extruder with 24 degree helix angle shows the slowest achievement of the maximum of the index out of three extruder designs. Similar tendencies are observed when agglomerates dispersion and distribution is taking place within a power law fluid (Fig. 4.11a and 4.11b). In this case however, as the level of dispersion is lower than in the case of a Newtonian fluid, the index indicates less overall mixing efficiency. This is illustrated in Figure 4.12 where mixing indices for Newtonian and power law fluids are compared based upon the case

51 48 of the extruder with 10 helix angle. Again, it is consistent with the shear thinning behavior of the power law fluid, due to which the viscosity of the fluid and, therefore, the hydrodynamic force acting on the agglomerates, is lower in the regions of high shear rates, compared to the hydrodynamic force generated in the Newtonian fluid. Other researchers have also reported that shear-thinning behavior of the fluid can result in a decrease in mixing efficiency [13-15]. Similar observations can be made when operating the extruder at a negative throttle ratio as shown in Figures 4.13a and 4.13b. In general, operating at a negative throttle ratio enhances mixing, both dispersive and distributive. When the preferred size is the smallest one present in the system, a comparison of Figures 4.10 and 4.13 reveals a slight increase in mixing index when operating at negative throttle ratio. Exception to the rule shows the extruder with a helix angle of 10, for which the mixing index does not change with throttle ratio. For an intermediate preferred size, the maximum in mixing index is shifted early on in the extruder length when operating at a negative throttle ratio. This and the fact that extruders with smaller helix angles show consistently better performance point out to a dominant dispersive mixing effect on the general outcome. Fluid rheological behavior affects the overall mixing performance with the Newtonian fluid showing better results as illustrated in Figure CONCLUSIONS We adapted the Shannon entropy to quantify mixing processes, which take place in a single screw extruder by simultaneously accounting for dispersive and distributive

52 49 mixing. A mixing index based on a weighted average of the entropies associated with different species present in the system (as defined by particle size) was developed. This index can be tailored to reflect various physical properties of the system as affected by a preferred particle size. The new mixing index has the potential to be used for equipment design and process optimization as demonstrated here in the study of the influence of helix angle (design) and throttle ratio (processing conditions) on the evolution of the index along the extruder length.

53 WORKS CITED: 1. I. Manas-Zloczower, D. L. Feke, Analysis of Agglomerate Separation in Linear Flow Fields, Intern. Polym. Proc., 2, 185 (1988). 2. S. P. Rwei, I. Manas-Zloczower, D. L. Feke, Observation of Carbon Black Agglomerate Dispersion in Simple Shear Flows, Polym. Eng. Sci., 30, 701 (1990). 3. S. P. Rwei, I. Manas-Zloczower, D. L. Feke, Characterizatiion of Agglomerate Dispersion by Erosion in Simple Shear Flows, Polym. Eng. Sci., 31, 558 (1991). 4. S. P. Rwei, I. Manas-Zloczower, D. L. Feke, Analysis of Dispersion of Carbon Black in Polymeric Melts and Its Effect on Compound Properties, Polym. Eng. Sci., 32, 130 (1992). 5. F. Bohin, I. Manas-Zloczower, D. L. Feke, Kinetics of Dispersion for Sparse Agglomerates in Simple Shear Flows: Application to silica Agglomerates in Silicone Polymers, Chem. Eng. Sci., 51, 5193 (1996). 6. D. F. Bagster, D. Tomi, The Stresses within a Sphere in Simple Flow Fields, Chem. Eng. Sci., 29, 1773 (1974). 7. S. W. Horwatt, D. L. Feke, I. Manas-Zloczower, The Influence of Structural Heterogeneities on the Cohesivity and Breakup of Agglomerates in Simple Shear Flows, Powder Technol., 72, 113 (1992). 8. P. Levresse, I. Manas-Zloczower, D. L. Feke, Hydrodynamic Analysis of Porous Spheres with Infiltrated Peripheral Shells in Linear Flow Fields, Chem. Eng. Sci., 56, 3211 (2001). 9. J. M. Ottino, P. DeRoussel, S. Hansen, D. V. Khakhar, Mixing and Dispersion of Viscous Liquids and Powdered Solids, Adv. Chem. Eng., 25, 105 (1999).

54 A. Scurati, I. Manas-Zloczower, D. Feke, Model and Analysis for Kinetics of Agglomerate Erosion in Simple Shear Flows, ACS, Rubber Div. Meeting, paper 52, GA (2002) 11. W. Wang, I. Manas-Zloczower, Temporal Distributions: The Basis for the Development of Mixing Indexes for Scale-up of Polymer Processing Equipment, Polym. Eng. Sci., 41, 1068 (2001). 12. Z. Tadmor, C. G. Gogos, Principles of Polymer Processing, John Wiley & Sons, Chichester, Brisbane, Toronto, Singapore (1979). 13. T. C. Niederkorn, J. M. Ottino, Chaotic Mixing of Shear-Thinning Fluids, AIChE J., 40, 1782 (1994). 14. F. H. Ling, X. Zhang, Mixing of a Generalized Newtonian Fluid in a Cavity, J. Fluids Eng., 117, 75 (1995). 15. P. D. Anderson, O. S. Galaktionov, G. W. M. Peters, F. N. van de Vosse, H. E. H. Meijer, Mixing of Non-Newtonian Fluids in Time-Periodic Cavity Flows, J. Non-Newtonian Fluid Mech., 93, 265 (2000).

55 52 Figure 4.1. Spherical system of coordinates for an eroding agglomerate

56 53 Figure 4.2. The finite element mesh for s single screw extruder

57 54 Figure 4.3. Particle spatial distributions at different cross sections along the extruder length.

58 55 (a) (b) Figure 4.4. Evolution of relative Shannon entropy along the extruder length

59 56 Figure 4.5. Erosion kinetics for Newtonian fluid Model parameters used in Eq. 4.8: ψ 0 = 10, θ = 45, ϕ = 90, K = Pa -1 m -1, σ c = 1000 Pa.

60 57 (a) (b) Figure 4.6. Comparison of the erosion kinetics for Newtonian and power law fluids

61 58 Figure 4.7. Evolution of Shannon entropy along the extruder length

62 59 Figure 4.8. Evolution of entropy and population distribution along the extruder length

63 60 Figure 4.9. Weight factors for different size fractions (based on 20 size fractions between the smallest and the largest sizes)

64 61 (a) (b) Figure Evolution of the mixing index in the case of Newtonian fluid along the extruder length when operating the extruder at zero throttle ratio

65 62 (a) (b) Figure Evolution of mixing index for the case of the power law fluid

66 63 Figure Comparison between the mixing efficiency of Newtonian and the power law fluids at zero throttle ratio

67 64 (a) (b) Figure Evolution of the mixing index along the extruder length when operating the extruder at negative throttle ratio

68 65 Figure Comparison between the mixing efficiency of Newtonian and the power law fluids at negative throttle ratio

69 66 CHAPTER FIVE Entropic Analysis of Color Homogeneity 5.1 COLOR HOMOGENEITY AS MEASURE OF DISTRIBUTIVE MIXING Color homogeneity or, in more general terms, property homogeneity is achieved through mixing. Thus color homogeneity assessment may become a tool in evaluating mixing efficiency in processing and equipment design optimization. When dealing with continuous plastic processing such as extrusion, in situ quantitative assessment of the color homogeneity would allow for better and faster product quality control. This quantitative assessment, however, is not an easy task and still poses many challenges. There are several major quantitative characteristics, which are used to quantify the degree of distributive mixing in a system. These are gross uniformity and intensity of segregation [1], scale of segregation [2], and striation thickness [3]. Prediction of minor component spatial distribution inside the extruder by using a particle tracking technique in numerical simulations, allows mixing characterization through pairwise correlation functions [4, 5]. All these measures do not completely answer the question how homogeneous the system is, as they cannot distinguish among multiple species of the minor component, such as color particles. On the other hand, due its the additive properties, the

70 67 informational Shannon entropy is a convenient measure for intermixing of multiple species, and therefore becomes a valuable tool in assessing color homogeneity. In this chapter we propose an entropic measure, based on an image analysis, to assess color homogeneity and deviations from an ideal color and employ this measure as a tool to assess distributive mixing efficiency in a single screw extruder. The idea of applying entropy to image analysis is not new. As an example, in the area of pattern recognition, an entropy based fuzzy homogeneity approach and the method of homograms applied to image threshold and segmentation was introduced by Cheng at al. [6, 7]. The concept of homogram in this works is used to express information on homogeneous properties among pixels in an image. In the following we will show how a similar approach of entropic image analysis can be used to obtain information about color homogeneity. Unlike Cheng s homogram approach, we will calculate the entropy directly from the grayscale and RGB maps of sample images. We use this approach to implicitly evaluate distributive mixing in an extrusion process. 5.2 EXPERIMENTAL PROCEDURE Mixing experiments using pellets of an acrylonitrile butadiene styrene (ABS) resin of two colors (yellow and blue) were performed at The Dow Chemical Company.

71 68 The pellets were premixed in proportion 20/1 for yellow/blue by weight. The prepared mixture was extruded in a single screw extruder with the Saran screw at Dow Chemical Co. Figure 5.1 shows the major features of the design of the screw. Extrusion was performed at the rotational speed of 60 RPM with a fully open extrusion valve. Pressure transducers, located along the extruder axis allowed for measuring the pressure in the barrel at different axial positions. Figure 5.2 shows the pressure profile in the extruder at these conditions (vertical red line indicates the start of the metering section). A steady-state polymer melt flow condition was ascertained by no apparent change of the extrudate color. Once the steady-state flow inside the extruder was achieved, the screw crash experiment was performed according to Maddock [8]. The extruder was stopped and the melt was allowed to cool down for 24 hours until solid. Then the screw was pulled out of the barrel and the solidified melt was cut along the screw for the analysis. Figure 5.3 shows the extrudate from the screw crash experiment. The images of 7 slices obtained in the metering section of the single screw extruder (each corresponding to one turn of the screw) are shown in Fig QUANTITATIVE ANALYSIS OF COLOR HOMOGENEITY While qualitative inspection of the cuts shows the clear progression towards better color intermixing, we will analyze the color homogeneity of the 7 consecutive cuts of the extrudate sample in terms of an index of color homogeneity based on a previously

72 69 defined relative spatial average of the entropies associated with species intermixing conditional on location (Eq. 2.18): Index S ( colors) = color rel locations M j= 1 p js j( colors) ln ( C) (5.1) Here C is the total number of color species present in the system. In our case, as we deal with only blue and yellow colors, C = 2. Colored pictures can be converted into grayscale images, which in turn can be examined for their homogeneity. In this case the intensity of grey color in each pixel/bin can be considered as a combination of a certain number of white and black particles, the relative concentration of which can be used as estimators of the probabilities. Alternatively, one can analyze the colored pictures by employing the standard redgreen-blue (RGB) primary colors at each pixel of the image. Thus by using a digital camera and standard image processing software (e.g. Adobe Photoshop ), a matrix of primary colors intensities at each pixel of the image can be generated. At a 24 bit image color depth, the intensity for each primary color/channel ranges from 0 to 255. The channels can be analyzed separately (monochromatic images) with the color intensities interpreted as mixtures of white and black particles summing up to 255 at each pixel. Thus, the intensity of 0 corresponds to 0 white and 255 black particles, while intensity of 255 corresponds to 255 white and 0 black particles. The relative concentrations of white and black particles serve to calculate the respective probabilities.

73 70 The overall color homogeneity will depend on how evenly black and white particles are distributed throughout the whole image. Equal number of black and white particles per each pixel means uniform intensity (level of gray) throughout the whole image, which brings the index of color homogeneity to 1. The maximum number of bins for the calculation will correspond to the number of pixels on the image. Thus at higher image resolutions we can estimate the index of color homogeneity at smaller scales of observation. Furthermore, we can analyze the images by comparison with an ideal color. In the case of grey scale images this entails choosing a particular combination of black and white particles to obtain the ideal grey, which is not necessarily obtained at equal concentration of black and white particles. In the case of color images, we need to choose an ideal color translated into different ideal intensities for the three RGB channels. In turn, each of these ideal intensities is viewed as a combination of black and white particles summing up to 255. To account for the different black/white ratios in the ideal case scenario, we take the p c/j in Eq (with c standing for black or white) to be: p p * black / j * white / j = = n black / j ideal black black / j ideal black white/ j ideal white white / j / n n / n + n / n n and / n ideal white n / n + n / n black / j ideal black white / j ideal white (5.2) where n black/j and n white/j are the number of black and white particles in bin j and n ideal black and n ideal white are the number of black and white particles corresponding to the ideal combination. One can think of n ideal black groups of black particles and of n ideal white groups of white particles, and then Eq. 5.2 provides the probabilities that

74 71 such a group in bin #j is black or white, respectively. Note that the index color is one, i.e. maximum, when in each bin the ratio of black to white particles is the same and is equal to the preferred gray shade: n black/j /n white/j = n ideal black / n ideal white (5.3) Thus the color index defined in Eq. 5.1 constitutes a simultaneous measure of color homogeneity and of closeness to the ideal gray shade. Finally, in analyzing the images for color homogeneity and deviation from an ideal color, one can exclude surface defects or image corners from the analysis by attributing such regions exclusion intensity. 5.4 RESULTS AND DISCUSSION Figures show four sets of pictures, representing the converted grayscale and the three split RGB channels (RGB monochromatic intensity maps) of the original color images. In order to calculate an index of color homogeneity, one can define an ideal color or one can choose as the ideal the average value for the region of interest. Moreover, the definition of ideal can be with respect to the grayscale image or with respect to the color image. In the latter case, the definition of ideal entails identifying the relative intensities of the split channels which make up for the ideal color. In our analysis we have chosen n ideal white /n ideal black based on our visual perception of the best green color that can be extracted from our images. Once this best color was identified, we were able to obtain the combinations n ideal white /n ideal black for the

75 72 grayscale images and for RGB channels. In case of grayscale images, the combination of n ideal white /n ideal black was chosen to be 63/192. In the RGB analysis we choose the relative intensities translated as n ideal white /n ideal black to be 15/240 for the red split channel, 95/160 for the green split channel and 25/230 for the blue split channel. It is important to mention that as these combinations were not chosen based on the average values for the entire region of study, the index of color homogeneity does not necessarily reach 1.0 at the largest scale of observation (M = 1). Figure 5.9 shows the results obtained using the grayscale image. The index of color homogeneity shows a general trend of improvement from the first to the last slice reflecting color intermixing progression along the extruder line. At small scales of observation ( bins), we observe relative declines in the index of color homogeneity at the 5 th and 7 th slices. Examination of the grayscale image (Fig. 5.5) reveals a slightly brighter large region on the right hand side of the slice #5, as well as visible non-uniformities for the slice #7. These features of the images do not play a significant role when the analysis is carried out at larger scales of observation ( bins). At the large scales, the overall distribution of bright/dark regions appears to be quite uniform for slices 4-7. Another interesting observation is the slight decline of the index at the 2 nd slice depicted at large scales of observation. This in fact reflects that the large bright portions are more evenly distributed throughout the first image than they are for the second one. In general, although the evolution of the index of color homogeneity along the extruder line reflects the correct trend of color upgrading, the analysis shows an overall low sensitivity in depicting slight changes in the quality of the images, as the overall deviations of intensity at the grayscale level are not so pronounced.

76 73 The color homogeneity measurements based on the intensities of the split RGB channels provide more opportunity in the selection of a higher sensitivity analysis. In the example taken here, with a color scheme based on mostly yellow and green, the red channel provides better sensitivity (higher contrast). The intensity deviations of the red channel are more pronounced when moving between yellow (high intensity of red) and green (low intensity of red) colors. Indeed, the intensity maps for the red channel depicted in Fig. 5.6 support this notion. As the contrast between bright and dark regions is more pronounced on the red channel, the sensitivity of the color homogeneity index increases as shown in Fig The overall evolution of the index along the extruder length shows similar trends with the analysis based on the grayscale images. To complete the analysis, we computed the indices of color homogeneity for the green and blue channels and the results are shown in Fig and In the experiment presented here, the green channel map is apparently not much different from the grayscale representation of the pictures. Overall, the green channel is slightly brighter than the grayscale one, and shows some less marked intensity deviations, which translates into a slightly less sensitivity for the index of color homogeneity with an analogous evolution trend. In contrast to all the previous observations, the evolution for the blue channel based index appears to be quite flat from the 1 st to the last cut. The blue channel does not contribute significantly to the color images in the green/yellow palette. In fact, all 7 cuts show quite low, evenly distributed intensity of blue (Fig. 5.8). The plunge of the index at the 5 th cut can be explained by a slightly higher overall intensity of this slice

77 74 compared to the other ones. This may very well be an artifact in the analysis. At this point it is important to stress that in order to minimize artifact, one should consider using a photo cell with standardized light conditions and a polarizing filter when taking the pictures for analysis. It is also possible to exclude the undesirable dependence on external light conditions in the analysis by using HIS (hue-intensitysaturation) maps rather than RGB maps. While the compression of the image will affect the color information at different pixels, and as the result, the index of color homogeneity, it is also important to use uncompressed images for the analysis. 5.5 CONCLUSIONS In this work we proposed a method of quantitative analysis for distributive mixing in polymer extrusion based on an entropic measure of color homogeneity for the extrudate. We illustrate the method by assessing the dynamics of distributive mixing in an industrial single screw extruder used to mix yellow and blue concentrates of ABS resin. While the technique requires some tuning (such as better ways to chose the ideal combinations of intensities in the grayscale or split RGB channels analysis, as well as a standard shooting procedure for taking the color images), the method offers a quantitative, inexpensive way for industrial mixing quality control and optimization.

78 WORKS CITED: 1. Z. Tadmor, C. G. Gogos, Principles of Polymer Processing, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore (1979). 2. P. V. Danckwerts, The Definition and Measurements of Some Characteristics of Mixtures, Appl. Sci. Res. A, 3, 279 (1953). 3. J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sydney (1989). 4. H. H. Yang, T. Wong, I. Manas-Zloczower, Flow Field Analysis of a Banbury Mixer in Mixing and Compounding of Polymers, I. Manas-Zloczower, Z. Tadmor, (Eds.), Hansen Publishers, 189 (1994). 5. T. Avalosse, Numerical Simulation of Distributive Mixing in 3-D Flows, Macromol. Symp., 112, 91 (1996). 6. H. D. Cheng, C. H. Chen, H. H. Chiu, H. Xu, Fuzzy Homogeneity Approach to Multilevel Thresholding, IEEE Trans. Image Process., 7, 1084 (1998). 7. H. D. Cheng, X. H. Jiang, J. Wang, Color Image Segmentation Based on Homogram Thresholding and Region Merging, Pattern Recognition, 35, 373 (2002) 8. B.H. Maddock, A Visual Analysis of Flow and Mixing in Extruder Screws, SPE J., 15 (5), 383 (1959).

79 76 Figure Saran conventional screw: 6 pitches in the melting zone, 8 pitches in the tapering zone, and 7 pitches in the metering zone Figure 5.2. Pressure profile inside the extruder

80 Figure 5.3. Screw crash extrusion sample from the metering section Figure 5.4. Extrudate slices used for the analysis

81 Figure 5.5. Grayscale representation of the color images

82 Figure 5.6. Red channel intensity map

83 Figure 5.7. Green channel intensity map

84 Figure 5.8. Blue channel intensity map

85 82 Figure 5.9. Evolution of color homogeneity index based on the grayscale image analysis

86 83 Figure Evolution of color homogeneity index based on the Red channel map analysis

87 84 Figure Evolution of color homogeneity index based on the Green channel map analysis

88 85 Figure Evolution of color homogeneity index based on the Blue channel map analysis

89 86 CHAPTER SIX Color Mixing in the Metering Zone of a Single Screw Extruder: Numerical Simulations and Experimental Validation 6.1 INTRODUCTION In Chapter 5 the procedure for color mixing analysis based on the experimental samples was described. The samples were obtained by the screw crash experiment, pioneered by Maddock [1], and were analyzed by employing image analysis. Finally, based on the entropic image analysis, indices of color mixing were calculated. The analysis, introduced in Chapter 5 is important as an experimental technique to evaluate color mixing efficiency in polymer processing equipment. It can also be adjusted for direct, in situ measurements. In this chapter we present a similar method of entropy calculation applied to color mixing in a numerical analysis. As it was shown in Chapter 4, numerical simulations can be successfully used to analyze equipment mixing performance. There we employed the particle tracking technique to estimate the degree of dispersive and distributive mixing via entropic measures. Now we want to apply the numerical procedure to evaluate the degree of color mixing of two particle populations: blue and yellow. We also want to compare the results of the simulation with the experimental data.

90 SIMULATION OF COLOR MIXING An eight pitch extruder mesh matching the metering section of a Saran screw extruder was designed. The mesh consisted of elements. Table 6.1 shows some of the important design parameters for the mesh. The overall design of the mesh and a detailed view are shown in Figures 6.1a and b. The parameters characterizing the power law rheological behavior of the fluid matched those for ABS resin at 260 C. The Newtonian viscosity, power law index and the critical shear rate for onset of shear thinning behavior for this resin are summarized in Table 6.2. To validate the simulation accuracy, we compared extrusion rates obtained at three different RPMs with those observed experimentally. Table 6.3 shows the values for RPM and pressure difference as boundary conditions for the simulation for each set. The comparison shows good correlation between the experimental data and the simulation results with error lying within the 8% range (Table 6.4). The last column in the table shows the values for the throttle ratio, i.e. the ration between pressure and drag flow rates. Within the selected range of pressure differences, the pressure flow significantly affects the overall flow rate in the case of 30 RPM (almost one third of the drag flow). Its influence, however, becomes less pronounced in the case of 60 RPM, and almost negligible at 90 RPM. To analyze the dynamics of particle motion, distribution, and intermixing, a particle tracking procedure was employed. We employed this procedure for the simulation

91 88 results obtained at 60 RPM to match the experimental conditions for the screw crash experiment as described in Chapter 5. Four pitches in the middle section of the mesh (Fig. 6.2) have been used for the particle tracking to avoid errors related to the less precise flow field solution at the inlet and the outlet sections of the extruder. Two different particle populations (blue and yellow species) have been tracked in order to examine the degree of color intermixing within the extruder. Figure 6.3 shows the color pattern of the ABS extrudate cut in the metering section of the extruder obtained during the screw crash experiment, which was chosen as the initial color particle distribution for the particle tracking procedure. While it is not possible to distinguish between yellow and blue particles in the green region of the cut, we assume that the green color is best represented by a random distribution of yellow and green particles in combination with each other. We neglected a big dark blue spot on the picture, which was attributed to an unmelted pellet of blue ABS resin. We also neglected the defect in the left portion of this slice, as it was attributed to an air bubble formed in the melt. These defects have just the local influence, therefore we do not expect them to significantly affect the overall mode of mixing. Thus the initial distribution of colored ABS resin was assumed to be only yellow in the left half of the slice and only green in the right half. To mimic this color distribution we randomly placed yellow particles across the whole channel of the extruder, and only 1000 blue particles filling out just the right half of the channel (at the pushing flight). A ratio of 20:1 yellow to blue particles was kept to match the ratio between yellow and blue ABS resin concentrates in the

92 89 experiment. Figure 6.4 shows the initial position of blue and yellow particles within the extruder mesh. 6.3 NUMERICAL RESULTS PARTICLE DISTRIBUTIONS As in the previous chapter, we wanted to analyze particle spatial distributions and the color intermixing at the planes of interest, which correspond to the analyzed experimental slices. We generated these slices by taking the snapshots of intersections of particle trajectories with the ZX and ZY planes as shown on Fig Figure 6.6 shows the snapshots of particle distribution at these slices along 4 consecutive pitches of the extruder. Circulatory motion of the particles in the screw channel, as described by Ottino et al. [2], generates striations between the yellow and blue tracers, which decrease in thickness along the channel. The mode of mixing becomes more complex in the flight regions due to the complex geometry of the flights. Figure 6.7 shows comparison between the colored particle tracers distributions obtained by particle tracking with the correspondent polymer cuts from the screw crash experiment. While experimentally obtained samples show more complex mixing features compared to the numerical results, one can clearly identify a similarity between the two by observing the striation formation in the experimental samples. These similarities are particularly pronounced at the corners, where tracers reorient when approaching the flights.

93 ENTROPIC ANALYSIS We analyzed the quality of color intermixing between the yellow and blue particles with a measure defined using informational Shannon entropy, as discussed before in Chapter 5 (Eq. 5.1) The results of the analysis for the seventeen slices shown in Fig. 6.6 are displayed in Fig We used four different numbers of bins M for the analysis, starting with 10 bins and up to 1000 bins, to describe mixing at different scales of observation. The initially high values of the index are explained by a very good intermixing between yellow and blue particles in the right half of the channel of the initial slice. As the mixing progresses, the index shows a trend of increasing color species intermixing at all scales of observation. At the largest scale (M = 10) the index shows a very fast increase as blue particles start occupying all ten bins within just 3 slices. Figure 6.9 shows the distribution of p * c/j (as defined by Eq. 5.2) for the yellow and blue particle tracers over 10 bins at the 4 first slices. The distribution on the right hand * * side shows the ideal distribution at p yellow/j = p blue/j = 1/2. On the first slice, blue particles are mostly distributed over the half of the channel within 5 bins. Slices 2-4 show how blue particles rapidly redistribute over the other 5 bins, rendering the modified conditional probabilities for yellow and blue species closer to the ideal ones. At smaller scales (M = 1000), however, the index has room to increase further, as there are still many bins occupied solely by the yellow particle tracers as shown in Fig The difference between these results in fact emphasizes the importance of

94 91 the scale of the observation. While at a large scale (such as the visual scale) mixing efficiency might appear satisfactory already in the earlier stages of the process, on a more microscopic level, within the first four pitches mixing is shown to be still inappropriate. 6.5 CONCLUSIONS We conducted numerical simulations of mixing for two colored particle populations within four pitches of the metering section of a conventional single screw extruder. The results of the simulation were interpreted in terms of particle spatial distributions over ZX and ZY slices along the extruder helix. As shown in the past by other researchers, mixing in the extruder occurs primarily due to the circulatory motion inside the channel. This motion results in striation formation between the colored populations with a progressive decrease in the striation thickness along the extruder channel. Mixing patterns obtained from experiments exhibit a more complex behavior due to experimental artifacts. However, qualitative comparison with the particle tracking results shows good agreement in terms of the mode of mixing emphasizing a decrease in the striation thickness. Also we observe similarities between mixing patterns at the corners where particles reorient approaching the flights. To assess the dynamics of mixing between two colored particle populations we employed an entropic measure, namely the index of color homogeneity. As the index is susceptible to a chosen scale of observation, it can give insight into mixing quality both at macro and microscopic levels.

95 WORKS CITED: 1. B.H. Maddock, A Visual Analysis of Flow and Mixing in Extruder Screws, SPE J., 15 (5), 383 (1959) 2. J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sydney (1989)

96 93 Table 6.1 Extruder design parameters Lead Distance Barrel Diameter Flight Thickness Flight Clearance Ls (cm) D b (cm) e (cm) δ f (cm) Table 6.2 Fluid rheological parameters Newtonian Viscosity µ 0 (Pa s) Power Law Index n Critical Shear Rate γ 0 (s -1 ) Table 6.3 Boundary conditions Set Number RPM Pressure Difference P (MPa) Table 6.4 Experiment vs. simulation: extrusion rate comparison Set Number Extrusion Rate Q (g/s) Experiment Simulation Error (%) Throttle Ratio Q p /Q d

97 94 (a) (b) Figure 6.1. Mesh design (a overall, b detailed)

98 95 Figure 6.2. Middle section of the mesh chosen for particle tracking

99 96 Figure 6.3. Experimental sample chosen to describe initial particle distribution Figure 6.4. Initial particle position in the mesh

100 97 Figure 6.5. Slice generation on ZX and ZY planes Total number of slices generated: 17

101 98 Pitch Number: Slice Number: Figure 6.6. Particle distributions at XZ and YZ cross sections of the extruder mesh within the four consecutive pitches

102 99 Slice # Figure 6.7. Comparison between experimental sample cuts and corresponding numerical slices

103 100 Figure 6.8. Index of color homogeneity

104 101 Figure 6.9. Distribution of modified conditional probabilities p * c/j over 10 bins for slices 1-4

105 102 Figure Distribution of modified conditional probabilities p * c/j over 1000 bins for slices 1-4