MODELING FLOW, MELTING, SOLID CONVEYING AND GLOBAL BEHAVIOR IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS. A Dissertation.

Transcription

1 MODELING FLOW, MELTING, SOLID CONVEYING AND GLOBAL BEHAVIOR IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS A Dissertation Presented to the The Graduate Faculty of the University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Qibo Jiang August, 008

2 MODELING FLOW, MELTING, SOLID CONVEYING AND GLOBAL BEHAVIOR IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS Qibo Jiang Dissertation Approved: Accepted: Rlawhdeorlawhdeorlawhdorlawheo Advisor Dr. James L. White rlawhdeorlawhdeorlawhdorlawheo Department Chair Dr. Sadhan C. Jana Rlawhdeorlawhdeorlawhdorlawheo Committee Member Dr. Avraam I. Isayev rlawhdeorlawhdeorlawhdorlawheo Dean of the College Dr. Stephen Z.D. Cheng Rlawhdeorlawhdeorlawhdorlawheo Committee Member Dr. Sadhan C. Jana rlawhdeorlawhdeorlawhdorlawheo Dean of the Graduate School Dr. George R. Newkome Rlawhdeorlawhdeorlawhdorlawheo Committee Member Dr. Zhenhai Xia rlawhdeorlawhdeorlawhdorlawheo Date Rlawhdeorlawhdeorlawhdorlawheo Committee Member Dr. Xiaosheng Gao ii

3 ABSTRACT Intermeshing counter-rotating twin screw extruders are widely applied in polymer processing industry, especially in compounding and PVC profile processing. However, the design of this type of machines is generally based on experiences and error-and-try. In addition, most of the investigations on intermeshing counter-rotating twin screw extruders were made on the melt conveying region. There is a lack of adequate study on a complete extrusion process to this type of machines. In this study, models were developed to simulate the extrusion processes, including solid conveying, melting and metering, evaluate the performance of intermeshing counter-rotating twin screw extruders, and optimize the design of machines and operating conditions. Experiments were carried out on a laboratory modular intermeshing counterrotating twin screw extruder to observe solid conveying, the melting process and the global behavior of this type of machine. The solid bed is formed in the solid conveying region. The inter-screw region plays a dominant role in the melting process. Based on our observations, models were developed to describe both the solid conveying and the melting process. Based on hydrodynamic lubrication theory, a melt conveying model was developed to characterize the pumping capacity of screw elements in intermeshing counter-rotating twin screw extruders. The effect of screw channel aspect ratio (screw channel depth / width) was incorporated into the melt conveying model to improve the iii

4 prediction of screw pumping capacity. Calculations were made to investigate the effect of geometrical parameter on screw pumping capacity. Models of solid conveying, the melting process and melt conveying were integrated together and a global composite model was developed to characterize the whole intermeshing counter-rotating twin screw extrusion process. The global model is intended for both flood fed and metered starved fed conditions. This is the first composite model designed for this type of machines. Simulations and experiment results were compared and it was found that they match very well. This global model was further successfully developed into user-friendly software, which is used to design, test and optimize intermeshing counter-rotating twin screw extruders. iv

5 ACKNOWLEDGEMENTS My first gratitude goes to my academic advisor, Dr. James Lindsay White for his invaluable guidance and constant encouragement during the course of this dissertation. I also gratefully acknowledge the Committee members, including Dr. A. I. Isayev and Dr. S. C. Jana from the Department of Polymer Engineering, Dr. X. Gao and Dr. Z. Xia from the Department of Mechanical Engineering, for their considerate recommendations to this dissertation. I also would like to thank Dr. G. Wang from the Department of Mechanical Engineering, and my friends in the Department of Polymer Engineering for precious discussions. The help of Prof. K. Wilczynski of the Warsaw University of Technology and Mr. J. Stasiek of the Metalchem Institute of Plastics Processing of Torun, Poland is acknowledged. Unconditional love from my parents and brother and sister has been unbreakable supports. v

6 TABLE OF CONTENTS LIST OF TABLES...xiii LIST OF FIGURES... xiv CHAPTER I. INTRODUCTION... II. BACKGROUND AND LITERATURE SURVEY Classification of Twin Screw Extruders... 4 vi Page. Development of Intermeshing Counter-Rotating Twin Screw Pumps and Thermoplastic Extruders Early Studies of Flow Mechanisms Experimental Studies of Screw Pumping and Mixing Modeling Flow Hydrodynamic Lubrication Theory Lubrication Theory Analyses Calendering Leakage Flow Pressure Flow Leakage Flight Leakage Effects Flow Analysis Network (FAN) Method and Hong-White Model Finite Element Simulations Energy Balance and Heat Transfer Solid Conveying Studies... 34

7 .8 Melting Studies Melting in Single Screw Extruders Melting in Co-rotating Twin Screw Extruders Melting in Counter-rotating Twin Screw Extruders Global Composite Extrusion Models Screw Extrusion Computer Software Introduction Single Screw Extruder Co-rotating Twin-screw Extruders III. EXPERIMENTAL: EQUIPMENT, MATERIALS AND METHODS Introduction Apparatus Materials Differential Scanning Calorimetry (DSC) Rheological Properties Melt Densities and Bulk Densities Experimental Studies Solid Conveying Melting Experiments Global Behavior of Total Screw Screw Configurations IV. EXPERIMENTAL STUDIES OF INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUSION Introduction vii

8 4. Solid Conveying Experiments Thick Flight Screws Thin Flight Screws Melting Experiments Thick Flight Screws Thin Flight Screws Melt Conveying and Global View of Screws Thick Flight Screws Thin Flight Screws Complex Screws Discussion and Interpretation Melting and Fill Status Melting Location, Initiation and Mechanism Effect of Operating Conditions Feed Rates under Flood Fed Conditions Conclusions V. SOLID CONVEYING IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS Introduction Solid Conveying Model for Thick Flight Screws Pumping Capacity Determination of the C-chamber Force Balances and Pressure Profiles in the C-chamber Region Flow Rate in the Inter-screw Region Force Balances and Pressure Profiles in the Inter-screw Region viii

9 5..5 Feed Rate Predictions Solid Conveying Model for Thin Flight Screws Pumping Capacity Determination Fill Factor Determination Energy Balance Bulk Temperature Rise in the C-chamber Bulk Temperature Rise in the Inter-screw Region Initiation of Melting Calculations Starved Fed System Flood Fed System Discussions and Conclusions VI. THE MELTING PROCESS IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS Introduction Model for the Melting Process Model of Melting in the Inter-screw Region Model of Melting between Screw and Barrel (the C-chamber) Calculations and Discussions Effect of Feed Rate on The Melting Process Effect of Screw Rotation Speed Effect of Leakage Effect of Clearance on The Melting Process Effect of Channel Depth... 5 ix

10 6.5.6 Effect of Friction Coefficient on Screw Surface Conclusions... 8 VII. MELT CONVEYING IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS Introduction General Metering Model Model for the Region between Screw and Barrel Model for the Inter-screw Region between One Screw Root and the Other Screw Flight Model for the Inter-screw Region between both Screw Roots Flux Balance Energy Balance for the Metering Region Model for the Region with Positive Displacement Model for the Region without Positive Displacement Considerations fo Screw Design The Effect of Ratio of Flight Width/Channel Width The Effect of Screw Clearance The Effect of Aspect Ratio (Channel Depth / Width) The Effect of Helix Angle Effect of Aspect Ratio on Pumping Basic Idea Determination of Real Flow Rate in Three Dimensional Model Determination of Flow Modification Shape Factors Summary x

11 VIII. SIMULATION OF GLOBAL BEHAVIOR OF INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS Introduction Computation Procedure for Global Model Determination of Feed Rate under Flood Fed Conditions Case Studies for Starved Fed Machines Effect of Screw Configuration Design Effect of Operating Conditions Effect of Materials Effect of Other Important parameters Comparison with Experimental Results Thick Flight Screw Designs with Shear Elements (A) and (D) Thin Screw Designs (B) and (E) Mixed Screw Design (C) Summary... 6 IX. SOFTWARE DEVELOPMENT Introduction Determination of Power Consumption, Torque, SEC and Average Residence Time Power Consumption Torque Determination The Specific Energy Consumption The Average Residence Time Outline and Structure of Software Sample Calculations... 4 xi

12 9.5 PVC Profile Extruders of Varying Screw Diameter Scale-up and Other Insustrial Applications Summary X. SUMMARY AND RECOMMENDATIONS Summary Solid Conveying and Melting Experiments Solid Conveying Model Melting Model Melt Conveying Model Global Model User-friendly Software Development Recommendations REFERENCES APPENDICES APPENDIX A. SOLID CONVEYING IN C-CHAMBER APPENDIX B. FORCE BALANCE IN INTER-SCREW REGION xii

13 LIST OF TABLES Table Page 3. Leistritz LS30.34 GG twin screw extruder specifications Leistritz LS30.34 twin screw extruder screw elements Materials used in this study Materials properties Material properties, operating conditions and geometry used in calculations Parameters used in studying screw pumping capacity Parameters used in calculations Extruder configurations Material properties and operating conditions Comparison of simulations for different operating conditions Screw geometrical details and code assigned in calculations Screw geometrical details used in calculations Screw geometrical details and codes assigned in scale-up calculations Screw configurations for scale-up calculations...53 xiii

14 LIST OF FIGURES Figure Page. Classes of twin screw extruders [30]...5. Wiegand's 874 machine for dough sheet [4] Olier intermeshing twin screw machine [34] Intermeshing twin screw extruder kneading pump of Kiesskalt et al. [36, 37] Burghauser-Leistritz kneading pump [40] Pasquetti design of intermeshing counter-rotating twin screw extruder [5]....7 A Kestermann intermeshing counter-rotating twin screw extruder [56-58]....8 Montelius patent drawing for explanation of operating mechanism [5] Screw geometry of two intermeshing counter-rotating screws [78] Shearing lubrication flow...0. Leakage flow terms in intermeshing counter-rotating TSE [8].... The calendering leakage General intermeshing C-chamber view in coordinate system at U n [0] Screw pumping characteristics of partially intermeshing counter-rotating twinscrew elements according to White and Adewale [0] Modular barrel configuration of Leistritz LS30.34 GG twin screw extruder Shear viscosity of HDPE (MP / GA655-66) as a function of shear rates Shear viscosity of PP (Equistar 8800-GK) as a function of shear rates Shear viscosity of PVC as a function of shear rates Schematic of the machine and the screw used for solid conveying study...55 xiv

15 3.6 Schematics and photographs of screw configurations used in solid conveying experiments, (A) Thick flight elements, and (B) Thin flight elements Screw configurations used in experiments Schematic of feeding system in counter-rotating twin screw extruders Solid conveying in thick flight elements in flood fed conditions, (A)Top view and (B) Down View Solid conveying in screw configurations shown in Figure 3.6 (B) (thin flight elements) under flood fed conditions, (A)Top view and (B) Down View Photographs of the thick flight screws pulled out from the machine after extrusion of PVC at the feed rate 6 kg/hr and the screw speed 30 rpm for thick flight screw configuration with PVC, (A) Top view and (B) Down View Photographs of the thick flight screws pulled out from the machine after extrusion of PVC at the feed rate 6kg/hr and the screw speed 60 rpm for thick flight screw configuration with PVC, (A) Top view and (B) Down View Photographs of the thick flight screws pulled out from the machine after extrusion of PVC at the feed rate 3 kg/hr and screw speed 30 rpm for thick flight screw configuration with PVC, (A) Top view and (B) Bottom View Photographs of the thick flight screws pulled out from the machine after extrusion of powder HDPE at screw speed 30 rpm Photographs of the thick flight screws pulled out from the machine after flood fed extrusion of powder HDPE at the screw speed 30 rpm Photographs of the thick flight screws pulled out from the machine after flood fed extrusion of powder HDPE at the screw speed 0 rpm Photographs of the thick flight screw configuration pulled out from the machine after extrusion of HDPE pellets at screw speed 30 rpm under flood fed condition, (A)Top view, (B) Down View Photographs of the thin flight screws pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/h and the screw speed 60 rpm, (A) Top view and (B) Down View Photographs of the thin flight screws pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/h and the screw speed 0 rpm, (A) Top view and (B) Down View...7 xv

16 4.3 Photographs of the thin flight screws pulled out from the machine after extrusion of HDPE pellets at the feed rate 4 kg/h and the screw speed 60 rpm, (A) Top view and (B) Down View Photographs of the thick flight screw configuration without shear elements pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/hr and the screw speed 60 rpm, (A) Top view, (B) Down View Photographs of the thick flight screw configuration with two shear elements pulled out from the machine after extrusion of HDPE pellets at the screw 8 kg/hr and the screw speed 60 rpm, (A) Top view, (B) Down View Photographs of the thick flight screw configuration with two shear elements pulled out from the machine after extrusion of HDPE pellets at the feed rate 4 kg/hr and the screw speed 60 rpm, (A) Top view, (B) Down View Photographs of the thick flight screw configuration with two shear elements pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/hr and the screw speed 0 rpm, (A) Top view, (B) Down View Melting location, melting length and fill status of screw configuration (A) under different processing conditions and with different materials Melting location, melting length and fill status of screw configuration (D) under different processing conditions Melting location, melting length and fill status in screw configuration (B) at the feed rate of PP 0 kg/hr and the screw speed 00 rpm Melting location, melting length and fill status in screw configuration (E) under different processing conditions with pellet HDPE Melting location, melting length and fill status in screw configuration (C) at the feed rate of PP 0 kg/hr and the screw speed of 00 rpm Measured feed rates under flood fed conditions for thick flight screws Solid conveying model in an intermeshing counter-rotating TSE Schematic of a closed C-chamber in solid conveying Schematic of geometry calculation in the calendering region Forces loading and schematic of the inter-screw region Predictions of feed rates under flood fed conditions for thick flight screws...95 xvi

17 5.6 Schematic of the open C-chamber in solid conveying Mass balances and schematic of the inter-screw region where Q and Q are the rate of solid conveying from left and right the C-chambers Velocity diagram for the second section of thin flight screws Melting initiation and propagation in the C-chamber Melting mechanisms: in the inter-screw region; between screw and barrel.. 6. The melting process between screw and barrel Effect of feed rates on the melting process at screw speed of 60 rpm Effect of screw speed on the melting process at feed rate 0 Kg/hr, where filled symbols represent both mechanisms and unfilled symbols are for the melting process considering the calendering effect only Effect of leakage on the melting process at G = 0 Kg/hr and N = 60 rpm Effect of leakage on the melting process at G = 0 Kg/hr and N = 60 rpm Effect of clearance on the melting process at G = 0 Kg/hr and N = 60 rpm Effect of channel depth on the melting process at G = 0 Kg/hr and N = 60 rpm Effect of friction coefficient at screw surface on the melting process at G = 0 Kg/hr and N = 60 rpm Modular Leistritz elements, (A) thick flight and (B) thin flight Calculated domains for screw elements Model mean flux balance Flow Chart for non-newtonian calculations Schematic of one screw element of counter-rotating twin screw extruders The effect of the ratio of flight width/channel width on pumping at n= The effect of the ratio of flight width/channel width on pumping at n= The effect of the ratio of flight width/channel width on pumping at n= The effect of the ratio of flight width/channel width on pumping at n= xvii

18 7.0 The effect of clearance on screw pumping capacity at n= The effect of clearance on screw pumping capacity at n= The effect of clearance on screw pumping capacity at n= The effect of clearance on screw pumping capacity at n= The effect of aspect ratio on pumping capacity at n= The effect of aspect ratio on pumping capacity at n= The effect of aspect ratio on pumping capacity at n= The effect of helix angle on screw pumping capacity at n= The effect of helix angle on screw pumping capacity at n= The schematic of channel flow when considering flights effect Modification factors vs. H/W by considering the aspect ratio effect (n=0.4) Modification Factor vs. power-law index with the H/W effect (H/W=0.4) Screw characteristic curves for screw FF--30 with /out the aspect ratio effect (H/W = 4/4) Screw characteristic curves for FD--30 without flight effect (H/W = 4/5) Screw characteristic curves for FD--0 without flight effect (H/W = 4/0) Schematic of the closed C-chamber in solid conveying Procedure for the calculation of Global model Predicted feed rates under flood fed conditions for screw design (A) Simulations for screw design (A) under flood feeding of powder HDPE and screw speed 60 rpm Simulations for screw design (A) under flood feeding of powder PVC and screw speed 60 rpm Simulations for screw design (A) under flood feeding of powder PVC and screw speed 30 rpm...80 xviii

19 8.7 Simulations for screw design (A) under flood feeding of powder PVC and screw speed 45 rpm Screw configurations used in case studies Simulations for screw design (A) at metered starved feed rate 0 kg/hr of pellet PP and screw speed 00 rpm Simulations for screw design (B) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm Simulations for screw design (C) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm Simulations for screw design (D) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm Simulations for screw design (E) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm Simulations for screw design (A) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm Simulations for screw design (A) at feed rate 5 kg/hr of pellet PP and screw speed 00 rpm Simulations for screw design (A) at feed rate 0 kg/hr of powder PVC and screw speed 00 rpm Simulations for screw design (A) at feed rate 0 kg/hr of pellet HDPE and screw speed 00 rpm Simulations for screw design (A) at feed rate 0 kg/hr of PP and screw speed 00 rpm (marker represents solid heat transfer coefficient at barrel surface 300 W/m K and screw surface 00 W/m K, while solid line represents the original case) Simulations for screw design (A) at feed rate 0 kg/hr of PP and screw speed 00 rpm (marker represents melt heat transfer coefficient at barrel 000 W/m K and screw 500 W/m K, while solid line represents the original case) Simulations for screw design (A) at feed rate 0 kg/hr of PP and screw speed 00 rpm (solid line represents fs=0.0; represents fs=0.6; and represents fs=0.) Comparison of screw design (A) between experiments and simulations xix

20 8. Comparison of screw design (D) between experiments and simulations Comparison of screw design (B) between experiments and simulations at feed rate 0 kg/hr and screw speed 00 rpm with PP pellets Comparison of screw design (E) between experiments and simulations Comparison of screw design (C) between experiments and simulations at feed rate 0 kg/hr and screw speed 00 rpm with PP pellets The front page of software, Akro-Counter-Twin Screw Extruder Select or edit screw machine category View detailed machine design or results Machine design for sample calculations Simulations for fill factor, pressure profiles, temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 50 rpm Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 00 rpm Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 50 rpm Simulations for fill factor, pressure profiles, temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 00 rpm Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 5 kg/hr of PP and screw speed 00 rpm Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 5 kg/hr of PP and screw speed 00 rpm Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 00 rpm A typical screw design for production of profile with PVC Machine design for sample calculations Simulations for flood fed fill factor, pressure and temperature profiles, melting status and power consumption at N = 30 rpm. Estimated feed rate = 4 kg/hr...43 xx

21 9.5 Simulations for flood fed fill factor, pressure and temperature profiles, melting status and power consumption at N = 45 rpm. Estimated feed rate = 5 kg/hr Simulations for flood fed fill factor, pressure and temperature profiles, melting status and power consumption at N = 60 rpm. Estimated feed rate = 85 kg/hr Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at screw speed 30 rpm and feed rate of 00 kg/hr PVC Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at screw speed 30 rpm and feed rate 50 kg/hr PVC Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 30 rpm with a diameter at 34 mm machine Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 40 rpm with a diameter at 34 mm machine Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 50 rpm with a diameter at 34 mm machine Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 30 rpm with a diameter at 68 mm machine Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 40 rpm with a diameter at 68 mm machine Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 50 rpm with a diameter at 68 mm machine Predicted feed rates under flood fed condition for PVC and at various screw speeds...60 xxi

22 CHAPTER I INTRODUCTION Twin-screw extruders (TSE) play a major role in polymer processing technology and are widely used in blending, compounding, devolatilization, reactive extrusion and profile extrusion. Of the twin screw extruder family, fully intermeshing counter-rotating twin screw extruders have the best pumping ability because of their positive displacement character. The intermeshing counter-rotating twin screw extruder for thermoplastics is an outgrowth of twin screw pump developed for oils in 90s [, ]. These machines were applied to profile extrusion of polyvinyl chloride with Maschinenfabrik Paul Leistritz, now Leistritz AG of Nuremberg Germany, the first manufacturer in the 930s [3, 4] when polyvinyl chloride pipes and profiles were needed for infrastructure. These efforts were greatly expanded by many new industrial firms in the post World War II period [5-9]. Later in the 960s, modular intermeshing counter-rotating twin screw extruders were devised by Maschinenfabrik Paul Leistritz [0] to compete with the metered starved fed modular co-rotating machines being produced by Werner and Pfleiderer for compounding []. Different types of screw and mixing elements were assembled in these machines for better blending and compounding. Japan Steel Works subsequently introduced a modular intermeshing counter-rotating twin screw extruder with modular elements of different

23 designs in the 980s. In more recent years American Leistritz has developed a third design [, 3]. Along with the historical development in the design, manufacture and applications of intermeshing counter-rotating twin screw extruders, the operating mechanisms were investigated and theoretical models were developed to describe the extrusion process. The forward pumping of filled C-chambers seemed to be well understood since 870s [4] for intermeshing counter-rotating twin screw extruders. And it was until 90s that the mechanism of positive displacement was clearly explained mathematically [5]. Later, studies were carried out to improve the predictions on pumping capacity by specifying the leakages [6-]. However, the above models were based on geometrical analysis and experiences. Numerical methods [3-6] were subsequently applied to characterize the pumping behavior by conducting flow analysis to the flow fields inside screw channels. The above studies were focused on melt conveying region. Intermeshing counterrotating twin screw extrusion, however, consists in another two regions, solid conveying region and melting region. There is little literature [8, 7-9] available on the melting process in intermeshing counter-rotating twin screw extruders. In most recently, a Newtonian model was developed to predict the initiation and length of melting region by Wilczynski and White [7]. Until now, no attention has been paid to the solid conveying region. The purpose of this dissertation was to investigate and characterize solid conveying, the melting process and melt pumping behavior of intermeshing counter-rotating twin

24 screw extruders, as well as to develop a global integrated model that predicts the overall performance of a twin screw extruder for extruder design. This involves the development of user-friendly software. Solid conveying and the melting processes are individually studied and modeled to determine the position for melting. We developed a method of design for intermeshing counter-rotating twin screw extruders which includes solid conveying, melting and metering regions. This was done for both metered starved fed modular machines used for compounding and flood fed machined screw machines intended for profile and pipe extrusion. There have been similar efforts for flood fed single screw extrusion and starved fed intermeshing corotating twin screw extrusion. This dissertation was the first effort for intermeshing counter-rotating twin screw extruders. 3

25 CHAPTER II BACKGROUND AND LITERATURE SURVEY. Classification of Twin Screw Extruders Usually there are two methods to classify twin-screw extruders. Based on the type of relative screw rotation direction, they can be classified as co-rotating (both screws rotate in the same direction) or counter-rotating (both screws rotate in the opposite directions). According to the interaction of the two screws, there are two types of twinscrew extruders: intermeshing (partially, fully) and non-intermeshing (separated, tangential). For intermeshing twin-screw extruders, the separation between the screw axes is less than the outer screw diameter. These classifications are shown in Figure.. Conical as well as parallel screw types of twin screw extruders have been developed. In the parallel twin screw extruder, the external screw diameter and inter-axial distance remain constant over the entire screw length, whereas, in the conical case, they decrease continuously in the direction to the die. 4

26 . Development of Intermeshing Counter-Rotating Twin Screw Pumps and Thermoplastic Extruders Intermeshing counter-rotating twin-screw extrusion machines for thermoplastics evolved out of twin-screw pumps in the 0 th century. Intermeshing twin-screw pumps were first used for lubricating oils. An 874 U.S. patent by LIoyd Wiegand [4] gave the first explicit description of a fully intermeshing counter-rotating twin screw pump, shown in Figure.. It was used for extrusion of baking dough. Figure. Classes of twin screw extruders [30] 5

27 The early description on the technology of counter-rotating twin screw extruders is mainly found in patent applications. It was with Kiesskalt [3] in 97 and Montelius [3] in 933 that the first published technical papers appeared. Summaries of the origins and technology development have been given by Janssen [8] and White [33]. Figure. Wiegand's 874 machine for dough sheet [4] The most important advantage of intermeshing counter-rotating twin-screw extruders is their positive displacement mechanism. The positive displacement refers to the axial displacement of the closed constant volume C-shaped chamber filled with fluid that exists in these machines. So the throughput can be determined by the geometry and processing parameters, such as screw speeds. The intermeshing counter-rotating twinscrew pump or extruder has been an important tool in pumping viscous oils for a hundred years. It was also applied in rubber compound extrusion for its advantage in the superior 6

28 output uniformity, compared to single screw extruders. One of the earliest applications was described in an anonymous 9 patent [34] to the French Olier Company on the extrusion of rubber profiles with a largely intermeshing counter-rotating twin screw machine. The so-called die-molding machine is shown in Figure.3. Figure.3 Olier intermeshing twin screw machine [34] The Swedish engineer, Montelius [35] gave one of the first clear discussions of the C-chamber in intermeshing counter-rotating twin screw extruders in a 99 patent application. Montelius [5] also discussed the influence of screw designs on pumping efficiency in a 95 patent application. He argued that the most effective twin-screw pump has two screws with different numbers of thread starts. Thus, a single threaded 7

29 screw matched with a double threaded screw is better than two single threaded or two double threaded screws. Since the 930s these machines have been developed for extrusion of polyvinyl chloride pipes and profiles. Intermeshing counter-rotating twin screw extruders were commercially developed by Maschinenfabrik Paul Leistritz for the extrusion of PVC profiles from the 930s. The 930s also witnessed the development of new type counter rotating twin-screw extruders by introducing continuous kneading into the screw design to enhance mixing and compounding. The applications were coal-oil masses and ceramic masses, in addition to rubber compounds. One basic design was given by Kiesskalt et al of the IG Farbenindustrie [36, 37], where screw flight thickness increases along the screw axis, shown in Figure.4. Figure.4 Intermeshing twin screw extruder kneading pump of Kiesskalt et al. [36, 37] 8

30 Subsequently, modifications in design were made to improve the kneading and homogenization [38-40]. Figure.5 shows a modified kneading pump by Franz Burghauser and Paul Leistritz of Maschinenfabrik Paul Leistritz [40]. It is seen that this fully closed and intermeshing screw machine was designed to lower the C-chamber volume along the pumping direction by decreasing steadily the inter-flight distance. Hence material is forced to move over the flights, resulting high shearing and better mixing. Hartner [4] and Handle and Maschinenfabrik [4] described screws which had transitions in them including a section with decreasing inter-flight distance and flight thickness toward the outlet. Similar tight fitting screw designs were found from the patents by Helstrup [43] and Marshall [44]. One interesting application of this design was patented by Schmitz [45] of Dupont in 944. It was used to remove carbon disulfide from coagulated cellulose xanthate. The barrel of the twin screw machine is perforated and the decreasing C-chamber volume forces the fluid out through the perforations. Kiesskalt and Borgwardt [46] described a continuous intermeshing counterrotating machine for homogenization of mixtures to produce emulsions in a 938 patent application. The first commercially produced intermeshing counter-rotating continuous mixer/masticater, called the Knetwolf, was developed in 940 and patented in 94 by W. Ellermann of Friederich Krupp [47]. The screws had diameters of 400 mm. They were used to masticate polyisobutylene. 9

31 Figure.5 Burghauser-Leistritz kneading pump [40] There were significant efforts to improve these machines in the period from 950. For instance, the evolution of L/D ratios in twin-screw extruders increased from 8 to over 30 in the past half century [48]. More publications can be found in the 950s, which described technology developments in Germany in twin screw extruders from 930 to 950 [49-5]. Many new companies began to introduce new machines in the 950s. Among these machines were Schloemann-Siemag s Pasquetti Bitruder, the Mapre extruder, Trudex machine, Kestermann extruder, and the Anger conical extruders. The Pasquetti Bitruder is based on 950 patent of Carlo Pasquetti [5] of Vanese-Masnao, Italy, which describes a machine 0

32 divided into segments, each of which have the number and pitch of screw threads decreasing in the flow direction as shown in Figure.6. The Mapre extruder from Nouvelle Mapre SA of Diekirch [8, 9, 53-55], Luxemberg, involved a screw with a rectangular profile and a progressively increasing flight thickness, along the screw axis. In the Trudex machine the section furthest from the die was closely intermeshed without compression while near the die there was a section of helically gashed discs [9]. Various designs of the Kestermann extruder were developed by Selbach. He designed new complex screws in which a new flight grows along the length, which penetrates into the thick flights existing at the feed end of the extruder (Figure.7). The details of this machine are given in patents by Selbach [56-58] of Kestermann in the 960s. The Anger conical screw extruder was developed by Anger Plastic Verarbeitungsmaschinen GmbH (APM). This machine included the screws where the pitch and flight thickness varied along the length of the screw [59]. The primary use of this machine was as a screw pump. Slices in screw flights were also introduced to develop mixing. Figure.6 Pasquetti design of intermeshing counter-rotating twin screw extruder [5]

33 Figure.7 A Kestermann intermeshing counter-rotating twin screw extruder [56-58] The first published experimental study of extrusion of rigid polyvinyl chloride on an intermeshing twin screw extruder was given by Tanner [60] on a Leistritz machine in the 950s. This was followed by studies of Doboczky [6, 7] and Tenner [6]. To compete with the new Werner & Pfleiderer modular co-rotating extruder, which was being used for compounding, Maschinenfabrik Paul Leistritz in the late 960s developed modular intermeshing counter-rotating twin-screw machine for compounding. This machine was invented by Helmut Tenner [6] and was intended for compounding. A detailed description of GG modular counter-rotating twin-screw extruders is given in papers by Thiele [] of American Leistritz, Tenner [6] and by Thiele et al.[63]. In the 970s, more companies became involved with intermeshing counter-rotating twin screw machines. Cincinnati Milacron acquired both of the Austrian Anger and AGM firms and continued the Menufacture of their conical twin screw machines in Vienna[64].

34 Reifenhauser [65] obtained Schloemann AG s Pasquetti Bitruder and continued to produce their machine with a modular design. Reifenhauser introduced special mixing elements. Thyssen Plastic Maschinen GmbH [66] absorbed the Kestermann machinery company which in turn was taken over by Battenfeld. Krauss-Maffei entered the Menufacture of conical twin screw machines [6, 67]. Japan Steel Works introduced modular intermeshing counter-rotating twin-screw extruders of different designs from the 980s. A new design was developed by Thiele [3] of America Leistritz in the 990s, but the detailed design was not fully disclosed. The modular machines were later applied in devolatilization [68] and reactive extrusion [69-7]. Dey and Biesenberger [69] used a Leistritz fully intermeshing counter-rotating twin screw extruder for the polymerization of methyl methacrylate. Gadzenveld and Janssen [73-75] and Gadzenveld et al. [70] took the fully intermeshing counter-rotating twin screw extruder as a reactor and used it for a number of polymerizing systems, then modeled both free radial and condensation reaction processes. In Gadzenveld and Janssen numerical model, all the C-chambers were taken as a series of perfectly mixed continuous stirred tank reactors moving on a conveyor belt. B.H. Lee and White [7] used a modular intermeshing counter-rotating twin screw extruder to form a polyetheramide triblock copolymer in a one-step, solvent-free method. Comparison between reactive extrusion with different types of extruders and a batch mixer in the polymerization of caprolactam was carried out by B.H. Lee and White [7]. It was found by B.H. Lee and White [7] that the conversion of twin screw extruders with proper mixing segments is much higher than the internal mixer until very high mixing 3

35 times. This was seen as higher conversion in twin screw extruders than an internal mixer for the same mean residence time. The modular counter-rotating twin screw extruder for compounding seems most widely used in North America. The machined shaft machines for polyvinyl chloride pipe and profiles are widely used in Europe and also Asia..3 Early Studies of Flow Mechanisms The basic understanding of the mechanisms of counter-rotating twin screw extruders first appeared in patents, instead of the scientific literature. In an 874 patent application [4] of LIoyd Wiegand and a 95 patent application of Holdaway [76], and the later patent application of Montelius [5, 35], the forward pumping of filled C- chambers seemed to be well understood. It was until the 95 patent application by Montelius [5] that, a clear mathematical discussion was presented. In this patent, the mechanism of positive displacement was clearly explained, shown in Figure.8. The volumetric displacement by a single revolution of screw B is, ( A ) V = S S A A where A s is the S-shaped bore volume of the screw casing, A is the cross-sectional area of the single thread screw A and A is the cross-sectional area of the double thread screw B. All of these cross-sections are taken as right angles to the axis, and S represents the pitch of screw. 4

36 Figure.8 Montelius patent drawing for explanation of operating mechanism [5] Kiesskalt [3] as early as 97 gave a modified expression for forward pumping capacity, Q = mv c Q leak where m is the number of thread starts, V c shows the total C-chamber volume and Q leak is a backward leakage flow. However, an explicit development of this and associated quantification would first seem to appear in a monograph by Gerhardt Schenkel [77] of 959. A full view of the C- chamber is shown in Figure.9. 5

37 Figure.9 Screw geometry of two intermeshing counter-rotating screws [78]. Figure.9 shows the geometry of two counter-rotating intermeshing screws. The intermeshing of two screws with left-handed and right-handed flights produces a closed C-shaped chamber which is advanced by the rotation of the screws towards the die. The extrusion rate through an intermeshing counter-rotating twin screw extruder is simply the product of the total volume V c of the C-shaped chambers by the screw rotation rate, i.e., Q = mnv c

38 where Q is the volumetric throughput, m is the number of the screw thread starts, N is the screw speed and V c is the associated the C-chamber volume per screw. Schenkel [79] first gave this engineering analysis of the intermeshing counter-rotating multiple screw extruders for thermoplastics in his book in Doboczky [6, 7] of Paul Leistritz in 965 wrote the chamber volume Vc, which is given by V c Vb Vsr mv f mvcal = m where V b is the volume of a barrel half over one pitch. V sr is the volume of the screw root, V f is the-volume of the screw flight, V cal is the volume of the intermeshing region, V b = πr S ( R H ) S V sr = m V f = R R H b () r πrdr where R is screw radius, H is channel depth, S is the pitch, b(r) is the width of the screw flight at position r. If the screw elements are not fully intermeshing, the situation is much more complex..4 Experimental Studies of Screw Pumping and Mixing The first experimental study of the intermeshing counter-rotating twin screw extruder may trace back to a 956 paper by Tanner [60]. Axial temperature profiles, polymer distribution and the softening behavior were investigated by the screw removal 7

39 method on a specially constructed Leistritz twin-screw machine for PVC profile extrusion. A later experimental study on pumping characteristics was reported by Doboczky [6, 7] on a Leistritz PVC profile extruder. He concluded that an intermeshing counterrotating twin-screw extruder has about three times the output capacity of a single screw extruder of the same size and screw speed. Compared to Schenkel s [79] theoretical prediction using C-chamber dimensions, the experimental throughput was in the range of 50 to 90% of the predictions, with an average of 70%. Subsequently, in the same year Markhenkel [80] of Schloemann AG published experimental studies of pressure profiles along the length intermeshing counter-rotating twin screw machines. Again the material was polyvinyl chloride. Menges and Klenk [8] and Klenk [0, ] found that the range of output is 34-4%, based on experiments with a Schloemann AG Pasquetti Bitruder profile extruder. They argued that the throughout is only dependent upon screw speed. Janssen and co-workers [9] found that throughput increased linearly with screw speed and the specific output (Q/N) seemed to be independent of pressure developed over a wider range of melt temperatures and screw speeds. They used a Pasquetti type machine and polypropylene. They hypothesized that the pressure development was associated more with the level of starvation than the operating parameters of the machine. The accumulated pressure rise was lower at higher screw speeds and higher temperatures, due to the lower viscosity of polypropylene obtained with higher melt temperatures. Higher screw speeds cause higher melt temperatures. 8

40 Counter-rotating twin screw extruders have been found to have narrow residence time distribution characteristics. Janssen and Smith [8] presented experimental study on residence time with Newtonian fluid. Subsequently residence time distributions and devolatilization were studied by Sakai [68, 83] with a modular intermeshing Japan Steel Works Tex65 counter-rotating twin-screw extruder. The feed was starved for this study. Residence time distributions were subsequently investigated, either flood fed or starved fed, by Janssen, et al. [84], Rauwendaal [85], Wolf et al. [86], Potente and Schultheis [87] and Shon et al. [88]. They all found that the residence time distributions are narrower than that of single screw extruders and other types of twin screw extruders. Lim and White [89] characterized the mixing of polyamide-6 and polyethylene and phase morphology development along the axis of the screw for different modular screw configurations on a Leistritz twin-screw extruder. They determined the effect of different modular elements on both the positions of melting and mixing. Cho and White [90] carried out a comparison between co-rotating and counter-rotating twin-screw extruders on blending processes of polyethylene and polystyrene using a modular JSW 30mm twinscrew extruder. They found melting happened more rapidly in intermeshing counterrotating than in co-rotating twin screw extruder..5 Modeling Flow Besides the experiment studies introduced above, theoretical models were developed to describe and predict the melt flow in intermeshing counter-rotating twin screw extruder as well. 9

41 .5. Hydrodynamic Lubrication Theory The lubrication approximation is widely used in analyzing the flow of highly viscous fluid in narrow moving gaps. Osborne Reynolds [9] first applied this method to study the flow of lubricating oils in a journal bearing. Usually the depth of the screw channel is shallow, much smaller than the length and width. Polymer melts flowing in extruder screws have high viscosities. The lubrication approximation has been applied in modeling most extruders including intermeshing counter-rotating twin-screw extruders. x x H(x,x 3 ) x 3 Figure.0 Shearing lubrication flow Take a flow in the -3 plane of Cartesian coordinate system as example, where H shearing is in the -direction (Figure.0). Due to << x H and <<, Cauchy x 3 laws of motion can be simplified and written as, 0

42 p 0 = x σ + x p 0 = x 3 σ + x where shear stress σ ij is given as v i σ ij =η x j For a Newtonian fluid, viscosity η is a constant. The top surface moves relative to the bottom with velocity components U and U 3. The narrow gap varies slowly in -3 plane. Substituting Equation -0 into Equations -8 and -9, and integrating, then the velocity field is obtained. The flow fluxes per unit width are given by integration, q U H p x H H = H v = vdx = η q 3 U H p x H 3 H = H v3 = v3dx = η Inserting these equations into the continuity equation, given as x H 3 p + x x 3 H 3 p = 6η ( U H ) + ( U 3H ) x3 x x3 This is the famous Reynolds equation. The flow field, using Eqs. - and -, can be determined by the pressure field.

43 .5. Lubrication Theory Analyses The C-chamber theory was developed in early studies to model flow in counterrotating twin screw extruders [4, 5, 3, 35, 76]. However, if one looks into closely at the geometry in Figure., it is seen that the leakage should be considered. This was noted quite early by Kiesskalt [3, 46]. Doboczky[6, 7] was the first to model leakage flows. Klenk [0] and Janssen [8] subsequently carried out investigations on this. The volumetric throughput was given as, Q = Q Q Q Q C cl pl ft where Q C is the C-chamber flow, Q cl is the calendering leakage flow between the screws, Q pl is the pressure(tetrahedron) leakage flow and Q ft is the flight leakage flow. Figure. Leakage flow terms in intermeshing counter-rotating TSE [8]

44 .5.. Calendering Leakage Flow The calendering leakage flow refers to the flow between the gaps of two screws. It is widely discussed by investigators in the 960s and 970s. Doboczky [6, 7], Klenk [0, ] and Janssen [8, 9, 6, 9, 93] discussed the effect of the calendering leakage. The calendering region is shown in Figure.. The force balance is given as, p 0 = x σ + x with boundary conditions, v ( h) = U A and v (+ h) = U B where h (h<<h) is the gap between the screws which is much smaller than the channel depth. And U A and U B are the linear velocities of the two screws, defined as, and = π ( D H )N. U B U A = πdn Figure. The calendering leakage 3

45 The velocity for Newtonian fluid is given as, v ( x ) U = + U B U A U B x h p x + H η x H A Integrating from h to +h, the total flow is obtained as Q CL h 3 h = W v( x ) dx = ( U A + U B ) Wh h 3η p x If we approximate x h = ho R where h o is the minimum nip and R is the screw radius, we may integrate Equation -8. Janssen et al gave a form, 3 Rho 3Qc S = πη H tanθ ( U + U ) h 3 o h o 4 p where θ is the flight helix angle..5.. Pressure Flow Leakage Dobockzky [6, 7] and Janssen [8] have considered the pressure flow backward through the tetrahedron gap between the flanks of the screws. Janssen gave the expression, Q PL.8 H 3 = ψ D p η D where ψ is the slope in radians of the screw flight, D is the screw diameter and p is the pressure difference between opposite the C-chamber. 4

46 .5..3 Flight Leakage Effects Flight leakage occurs in a single screw extruder as well. It is due to the combined effects of drag and pressure flow over the tips of screw flights. Dobockzky[7] and Janssen[8] studied this leakage. The flight leakage flow Q FL over a screw flight tip of length e is given as, where Q FL δ F = δ l C f 0 0 v dx 3 dx 3 L cδ p F f = ULcδ F sinθ η e is the clearance between flight and barrel surface, e is flight thickness in the direction perpendicular to screw channel direction, p f is the pressure drop over a single flight caused by the pressure distribution across the screw channel and the pressure build up along the screw axis to the die..5.3 Flow Analysis Network (FAN) Method and Hong-White Model The Flow Analysis Network Method was first introduced by Tadmor et al. [94] to simulate flow in dies and the filling flow of rectangular molds with inserts. White and his group [95-98] have generalized this method to flows with moving boundaries, such as the flow in twin-screw extruders. FAN is a simple lattice-type Finite Difference Method (FDM), which is formulated from the Reynolds equation. This method is powerful numerical scheme for solving two dimensional flow problems, especially where intricate but closely spaced geometrical configurations impose serious difficulties such as flows in injection molds and extruders. The basic idea is to divide the region into cells and carry out flux balances on these cells. 5

47 When there is no source flow inside one cell, from the continuity equation, the flux balance equals to zero, [ q( i, j + ) q( i, j ) ] + [ q( i +, j) q( i, j) ] = 0 j i where j and i are the node length in i and j directions, respectively. FAN was first applied to co-rotating twin screw extruder by Szydlowski and White [95-98]. David and Tadmor [99] later applied a cylindrical coordinate based FAN formulation to model the batch and continuous twin rotor mixers and extruders. Hydrodynamic lubrication theory and cylindrical coordinate FAN were later applied in modeling flow in tangential counter-rotating twin screw extruder by Bang and White [00, 0]. White and Adewale [0] developed a general model to describe Newtonian fluid flow in different levels of intermeshing between the screws for a counter-rotating twinscrew extruder. They flattened one of the screws and associated barrel, and then fixed a coordinate system on the screw, shown in Figure.3. After discretizating to the whole calculation domain and applying flux balance to each mesh, by combining the corresponding boundary conditions, an equation between pressure gradient and throughput is generated. FAN method was used to resolve this equation and give screw characteristic curves, such as those shown in Figure.4. However, the positive displacement mechanism of intermeshing counter-rotating twinscrew elements was neglected, which caused significantly under-predicting pumping capacity, especially for thick flight screws. 6

48 πdncosθ-u n πdncosθ-u n πηncosθ-u n h -U n -U n Figure.3 General intermeshing C-chamber view in coordinate system at U n [0] Dimensionless Throughput, [Q/WHU N ]. h/h = 0.0 h/h = h/h = h/h = Tangential Machine Single Screw Extruder Dimensionless Pressure Gradient, [H cosφp/(πηdn(l I +L II ))] Figure.4 Screw pumping characteristics of partially intermeshing counter-rotating twin-screw elements according to White and Adewale [0] 7

49 8 Later, Hong and White[4] introduced positive displacement effects into the FAN method to simulate different screw elements. Both Newtonian and non-newtonian fluid flows[3] were considered in isothermal and non-isothermal cases. By applying hydrodynamic lubrication approximation, the flow analysis model was developed. In this formulation, screw characteristic curves were calculated. A coordinate system fixed on the screw surface moving with the same velocity as the screw was established for the region between screw and barrel. In the inter-screw region, a coordinate system fixed in space was used so that there were no local geometry changes as the screw rotates. Flow rates in the angular and screw axis directions were given as a function of processing parameters. Pressure gradients in the angular and screw axis directions were determined. In cylindrical coordinate, when taking (,, 3) as (z, r,θ), the velocity field has the form, () () θ θ e r v e e r v v r z z + + = The force balances are, after neglecting inertial and tensile stretching terms, ( ) + = + = r v r r r r p r r r r p r r θ θ η θ σ θ ( ) + = + = r v r r r z p r r r z p z zr η σ The components of the stress tensor are taken as = r v r r r θ θ η σ r v z zr = η σ

50 9 For the Newtonian case, viscosity η is a constant. The top surface moves relative to the bottom with velocity components z U and θ U. The position on the top and bottom surface in radial direction is given as R and R, respectively. The narrow gap varies slowly in z-θ plane. Integrating Equations -5 and -6, then the velocity field is given + = r R R r R R R R R R R R R r R r P R R R R R r R R r U v ln ln θ η θ θ ( ) ( ) + = ln ln 4 ln ln R R R R R r R r z P R R R r U v z z η When 0 = z U, it would mean in the geometry of that the screw is fixed in axial direction, the Equation -30 becomes, ( ) ( ) = ln ln 4 R R R R R r R r z P v z η The flow fluxes per unit width are given by integration, ( ) ) ( 8 r r R R k G p R k H R U dr v q θ η θ θ θ = = = = r r r o R R z z k k k z p R R dr rv R q ln 6 4 * 4 * η

51 where H( k ) r k = r r ( ) + ln k kr kr ln ( ) k = r kr G k kr kr R k r = and R R * is a characteristic radius. After fluxes in each directions were determined, they were inserted into Equation -3 replacing i and j with z and θ for cylindrical coordinates, the FAN balance equation becomes θ + [ ] [ q( z, θ + θ ) q( z, θ θ )] + Uθ ( Ro Ri ) ( ) U ( ), θ R R θ θ z o i ( θ + θ, z ) [ q( z +, θ ) q( z, θ )] = 0 z In this formulation, the positive displacement effect was clearly given as ( R o R i ) ( θ θ, z) Uθ ( R o R i ) ( θ + θ z) U θ,, where U θ is barrel velocity, R o and R i are barrel and screw root radii, respectively, ( θ θ,z) and ( θ + θ,z) mean the different elements. When there is no clearance at ( θ + θ,z), U ( ) ( ) 0 R o R i θ + θ, z = twin screw extruder without any leakage flow through the calendering gap. θ, it is an ideal intermeshing When taking an average pressure between two ends of an element, combining the throughput, screw characteristic curves can be generated for different screw elements. A more detailed discussion of Hong and White s model is given in Chapter 7. 30

52 .5.4 Finite Element Simulations Other numerical simulations were conducted to investigate pumping capacity and mixing of screw elements. The finite element method has been applied in the numerical study on the flow of intermeshing counter-rotating twin-screw extruders. The earliest paper involved numerically simulating of the flow field in the calendering gap of a counter-rotating machine with FEA method by Speur et al. [6] in the late 980s. Later in 990s, Li and Manas-Zloczower [5] characterized the flow field in 3-dimensional isothermal flow patterns for an element of the Leistritz machine with the CFM commercial FIDAP FEM software package. However, the C-chamber positive conveying flow mechanism was not properly modeled. Geometrical symmetry was assumed in their study, so only the channel segment of one screw was considered. Mixing performances were studied numerically. Li and Manas-Zloczowe [5] used an FEM software package to investigate the dynamics of distributive mixing by tracking the evolution of particle positions in 3-Dimensional isothermal flow patterns, evaluation of spatial distribution. Another 3-dimensional FEA numerical study on opened the C-chamber was conducted by Kajiwara et al. [03] to compare and evaluate the mixing performance of co-rotating and counter-rotating twin-screw extruders. It was found that the co-rotating twin-screw machine was superior in distribution mixing, while both machines were similar in dispersive mixing. However, there are no experimental results to confirm this argument. 3

53 .6 Energy Balance and Heat Transfer Non-isothermal conditions exist in twin screw extrusion. Generally material is fed in the solid state at room temperature, heated up and melted. When the melting process is completed, molten material is conveyed to the die for further processing at melt state. During this process, material experiences temperature variations. In addition, with increases of extruder in size, the volume of the machine per unit length increases with the square of the scale factor if keeping the same ratio of geometry, and the surface area of the barrel and screw which define cooling capability increases linearly. Internal viscous heat generation becomes increasingly important, as extrusion machines have larger size. So it is meaningful to consider energy balance and analyze heat transfer. Non-isothermal flows were investigated by Chen and White [04, 05] in corotating twin screw extruders, by Bang and White [0] in tangential counter-rotating twin screw machines, and by Hong and White [4] for intermeshing counter-rotating twin screw extruder. Energy balances were also considered in other studies on the numerical simulation of co-rotating twin screw extruders [06-]. In modeling melt flow of extrusion process, energy balances are constructed to present the variations of temperature along screw axis. Enthalpy flow balance for unit mass is given as, Q = Q + H + W out in in in where Q out and Q in are enthalpy flow output and input, respectively, H in is heat flow input, and W in is power input by shear dissipation. 3

54 If the effect of pressure on enthalpy is neglected, enthalpy per unit mass can be expressed only by temperature with, T Q ( T ) = c dt To p Heat flow H in is due to heat transfer from barrel which is related to the local heat transfer coefficients, given as, Ti ( T T ) H = h da m To i o From Equation -36, by coupling Equations -37 and -38, and heat shearing dissipation where shear history is derived from momentum equations, temperature profiles along screw direction can be determined. In 3-D rectangular coordinate system, when lubrication theory is applied, the energy equation is expressed as T T T v 3 ρ m c m v + v3 k 3 x x = + σ + σ x x x v where c m is the heat capacity, ρ m is the material density and k is thermal conductivity, v and v 3 are the velocities in direction and 3, respectively, T is temperature, and σ and σ 3 are the associated shear stresses. 33

55 .7 Solid Conveying Studies Solid conveying zone refers to the region between the hopper and the start of melting. Solid polymer is fed, transported and compressed in this region. Hence this region determines the throughout under flood fed conditions. Generally under starved fed the screw is incompletely filled. When looking into the operating of counter-rotating twin-screw extruders, it is found that the material is in the form of loosely packed state in part region of the C-chamber even for fully filled. Usually a fill factor is used to indicate the filling degree. The understanding of the solid conveying zone is much less developed than the metering zone and even the melting zone. The reason is attributed to a lack of good knowledge of the coefficients of friction which have a complex dependence on temperature, pressure, screw speed, time, and other variables. In single screw extruders, plug-flow is widely used to study the solid conveying. With this method, the solid is taken as a solid plug in cross-section. Maillefer [] was the first to model solid conveying in a single screw extruder. Darnell and Mol [3] improved this analysis subsequently. These early investigations considered isothermal conditions and combined the kinematics of the problem with the force and torque balances on the plug. Solid conveying rate and pressure buildup were determined by these models. Broyer and Tadmor [4] and later Kacir and Tadmor [5] sought to develop a non-isothermal theory of solid conveying. In this work variable material density was considered. 34

56 It became widely accepted that higher coefficients of friction at the barrel surface and lower coefficients of friction at the screw surface give higher outputs due to higher pressure buildup in the solid conveying zone. In the 970s grooved barrel single screw extruders was introduced to make use of this idea [6]. Zhu and Chen [7] studied the plug flow of solid conveying in a transparent barrel in the solid conveying zone and observed that interparticle motions exist and significant compaction occurs only at the end of the solid conveying zone. Later Fang, Chen and Zhu [8] developed a non-plug flow theory for solid conveying. Co-rotating twin screw extruders are always operated under starved fed conditions. A solid conveying model by Carrot et al. [9] considered two conveying mechanisms: one in the upper intermeshing region and the other in the partially filled side channel prior to the lower intermeshing region. An empirical co-relation was assumed for solid conveying in the upper intermeshing region whereas in the side channels the model was closely similar to the model applied in single screw extruders. Bawiskar and White [0, ] also developed a model solid conveying mechanisms and carried out numerical calculations. Around the same time, Potente, Mellisch and Palluch [] developed a physico-mathematical model for solid conveying in co-rotating twin screw extruders. No adequate description exists as yet for solid conveying zone of counter-rotating twin-screw extruders. 35

57 .8 Melting Studies Before polymer materials are fed into the metering region, they must be melted from solid phase to melt state. This region is referred as the melting process. The region of the melting process is one of the main concerns to design the screw machines and determine operating conditions..8. Melting in Single Screw Extruders Maddock [3] was the first to study melting mechanisms in single screw extruders experimentally. He introduced pigments into polyethylene pellets and after the extruder was operated at a steady state, it was suddenly stopped and cooled to room temperature. The screw then was pulled out and the peeled polymer carcasses were sectioned. A melt layer along the barrel and a melt pool along the leading flight were observed. Similar experiments were conducted by Street [4], Tadmor, Duvdevani and Klein [5], Menges and Klenk [6] and Dekker [7]. Tadmor [8] presented the first quantitative model to calculate the solid bed profiles and lengths of melting, based on observations notably of Maddock [3]. Subsequent research [6, 7] has often found different melt/pellet bed configurations. Tadmor s model considered the heat transfer and viscous dissipation, in terms of Maddock s bed configuration. Subsequent modeling took in addition force balances to determine the position of the solid bed in the screw channel[9-3]. 36

58 .8. Melting in Co-rotating Twin Screw Extruders The first study of melting in co-rotating twin screw extruder was in the 980s. A more detailed investigation was reported by Todd [3, 33]. Todd removed and investigated carcasses from the melting section of the machine. He argued that the melting mechanism is attributed primarily to the heat dissipation imparted in particle to particle friction. He also hypothesized that a compression/expansion effect in the intermeshing region between the kneading disc blocks played a key role in the melting. Potente and co-workers [34] subsequently modeled melting in kneading disc blocks [35], representing it as on solid pellets dispersed in a melt pellets phase. They presented calculations to determine melting lengths and profiles in co-rotating twin screw extrusion [36]. Bawiskar and White [0, 37] found that the development of a melt layer along the barrel and a pellet bed at the surface of the screws, so they argued that this hot barrel surface plays a key role in melting and developed a model to predict the melting position and lengths. Essenghir, Yu, Gogos, Todd and Tadmor [38] built on the work of Todd [3, 33] and explained their experimental results on the melting process in terms of dissipative phenomena such as interparticle frictional heat generation, irreversible deformation and breakage of the particulate solid and viscous energy dissipation of the resulted melt or unmolten sold mixtures. Gogos and his co-worders [39-4] argued that melting in modular co-rotating twin screw extruders is induced by heat generated in the bulk by compacted sold pellets movement. Vergnes et al. [43] concluded that a thin film of molten polymer occurs in the beginning of melting and then solid pellets are wetted by the thin film. In the fully developed melting step, the deformed pellets are surrounded by 37

59 molten polymer. So they concluded that melting mechanism from Bawiskar and White [0] is an initiation step and that from Gogos et al. [38] is the propagation step. Jung and White [44, 45] made a most comprehensive experimental study on the melting mechanism of co-rotating twin screw extruder. Three initiation melting mechanisms and four propagation melting mechanisms were observed [3] under different conditions and later modeled mathematically [33]..8.3 Melting in Counter-rotating Twin Screw Extruders Compared with other extruders, including single screw extruders and co-rotating twin screw extruders, the investigations of the melting process in counter-rotating twin extruders receive much less attention. For counter-rotating twin extruders, there is little literature [8, 7, 8] available on the melting process in intermeshing counter-rotating twin screw extruders. Under flood fed condition, Janssen [8] reported the melting process to start at the fifth chamber and finish between three and five chambers, which depending on die pressure. However, due to low screw speeds (less than 0. rpm), conductive heat transfer from the barrel is the main resource of melting. With the increase of screw speed, heating dissipation, such as friction heating and shear heating, would dominate the whole the melting process. Lim and White [89] found that melting occurred more rapidly in intermeshing counter-rotating twin screw extruders than in co-rotating machines. The rate of melting varied with detailed screw design. Cho and White [90] compared melting and mixing between self-wiping co-rotating and intermeshing counter-rotating twin screw 38

60 extruder susing blending polyethylene and polystyrene and confirmed Lim and White s observations that it melts more rapidly in counter-rotating twin screw extruder. Wilczynski and White [7] specifically investigated experimentally the mechanism of melting in a modular intermeshing counter rotating twin-screw extruder with starved fed, and also found that it has a more rapid melting, compared with single screw or corotating twin screw processing. All the experimental results suggest that melting is initiated by frictional work on the pellets by the calendering stresses between the screws. Wilczynski and White [7] also argued that, at higher screw speeds, due to faster transport, the initiation of melting is delayed, and due to higher viscous dissipation, faster melting occurs. Less filled C-chambers result and shorter lengths of pellet beds and the distance along the axis of screws for complete melting is shorter. The length of screws for complete melting increases with feed rate. Wilczynski and White [8] modeled the melting process and made an approximate comparison with experimental results. Two different melting mechanisms were proposed for two regions, the C-chamber and the calendering gap. An adiabatic condition was assumed for the melting process in the calendering gap which means that all mechanical work is transferred into the heat increase of pellets in the calendering gap. The viscous dissipation in the C-chamber was handled by using closely similar approach in single screw extruders and co-rotating twin screw extruders. The final melting rate is the sum of two regions. The model presented was based on Newtonian fluids. The calculation was found to have a 5% discrepancy to experimental results. Even though, this result is much better than single screw models. 39

61 A variant melting model was presented by Wang and Min [9] on the melting process in an intermeshing counter-rotating twin-screw extruder. In this model, the interscrew region was taken as the calendering gap and the viscous dissipation from this region was used to determine the melting process. The melting processes in the C- chamber were not included..9 Global Composite Extrusion Models To characterize the performance of screw extrusion process, the whole process, that is, from screw under hopper to die, should be studied as one united zone, not any single part of them, such as melt conveying or melting. So it is meaningful to combine solid conveying, the melting process and melt conveying together, and to develop a composite model for the whole process. The prediction of the composite model should help designers and process engineers with screw design, process control and optimization, even scale-up consideration. The earliest global composite development for single screw extrusion was given by Tadmor et al. [46] of Western Electric in 967. Since then, many efforts have [47-50] been taken to improve the predictions of these composite models. Forward force and energy balances are calculated in developing a composite model for single screw extrusion. White and Szydlowski [5] first presented a complete simulation of flow in a composite modular twin screw extruder for metered fed conditions for Newtonian fluids in screws and kneading disc block elements in 987. Later in 989, Wang et al. [98] 40

62 introduced a first isothermal composite model for non-newtonian fluids in a total corotating machine in 989. In 994 Chen and White [04] expanded this global composite model to non-isothermal behavior. In 997 Potente and co-workers [5] subsequently developed a computer simulation program for a composite modular co-rotating twin screw extruder to calculate pressure and temperature profiles, local degree of filling, melting behavior, residence times, and characteristics of mixed materials and energy consumption of the machine. In 997, Bawiskar and White [0,, 37, 53] extended Chen and White s composite model [04] and developed a computer simulation program for a composite modular co-rotating twin screw extruder by combining solid conveying, the melting processes and metering process. Similar to theory proposed by White and his group [0,, 37, 53], Vergnes et al. [54] from Ecole de Mines developed a global computer software for polymer flows in corotating twin screw extruders in 998. In 997 Bang et al. [00, 0, 55] developed a composite model for the melt metering region of tangential modular counter-rotating twin screw machines. Fill factor, pressure and temperature profiles along the axis of the twin-screw machines were predicted for various modular screw configurations. Experiments were carried out on the modular machine to verify the predictions. Generally, good agreement with the flow analysis was found. In Hong and White [4, 30] presented simulations of a composite modular intermeshing counter twin screw extruder only based on the melt metering region to 4

63 predict the fill factor, pressure and temperature profiles along the axis of the twin-screw machines with different modular screw configurations. Based on approximate analytical flow and heat transfer models, Canedo from Polytech [] also developed a computer simulation program for modular co-rotating twin screw extruders with similar functions independently in the late 990s..0 Screw Extrusion Computer Software Global composite extrusion models are not convenient to be applied in machine designs and process optimization. Screw extrusion computer software, which was based on those models, was developed and commercialized to promote the applications of global composite extrusion models for single screw extrusion or twin screw extrusion machines..0. Introduction Computer software of single screw extrusion and co-rotating twin-screw extrusion, based on composite model, were developed to improve and optimize the design of the extrusion process for designers and engineers. There are many advantages for extrusion software. Properly modeled extrusion simulation software allows the scale-up from one extruder to another, predict processing problems and optimize screw design. It also helps to analyze how a new screw would perform in one device before acquiring it, or demonstrate to customers that the new screw 4

64 designed will perform much better than the old one. Furthermore, the operating conditions, such as screw speed, and the subsequent effects, such as pressure built or temperature increase due to viscous heating can be analyzed and the optimal conditions can be identified. With the help of software, the number of experiments needed for setting up a new process or machine, or for optimizing an existing process/machine could be dramatically reduced. In addition, the torques and power consumption can be evaluated. Furthermore, both the cost and the mixing quality depend on the geometry of the barrel and the moving screws, the flowing material and the operating conditions. All these parameters can be modified on the computer in order to select the best operating window..0. Single Screw Extruder Klein and Marshall [56, 57] from Western Electric Inc and later Klein and Tadmor [47] of Scientific Process & Research Inc. developed the first and second software package for simulation the plasticating single screw extrusion process in the 960s. Theoretical basis of the software is solid plug transportation, Tadmor and Klein melting model, and classic metering theory. Later in 970s, the Institute for Kunststoff Verarbeitary (IKV) at RWTH Aachen University developed software, BILAN [58], which is based on an analytical two-plate model, for single screw extrusion and predicted the performance, such as temperature and pressure, melting, and the position and shape of the solid bed in the melting section. 43

65 The Institut fur Kunststofftechnik (KTP) at the University of Paderborn [59] developed software tools: REX (simulation of single-screw extruders) in 990s, based on a composite model developed in 970s [60]. Screw channel was unfolded, and Tadmor and Klein s melting model was applied in this software. This tool is not dependent on the type of machine or material. It can be used to simulate any machine and any material, as long as the geometrical data of the machine and the characteristics of the polymer are available. The Polymer Processing Institute s WinSSD Software Package (formerly known as PASS ) [6] is another one developed for the design, performance analysis, trouble shooting, scale-up, and optimization of single screw extrusion processes in 990s. Over 30 industrial companies throughout the world use the WinSSD. Most recently, general commercial software, such as POLYFLOW [6] developed by Fluent Company, is reported to simulate extrusion process of single screw extruder. Flow pattern, temperature field through the flowing material, pressure increase along the screw, residence time, local shear rate, and local stress are predicted by POLYFLOW for designing or acquiring a screw. However, solid conveying and melting were not properly considered. There are other similar software available for single screw extrusion [49, 50, 63, 64]. 44

66 .0.3 Co-rotating Twin-screw Extruders The first developed software to simulate melt flow in co-rotating twin-screw extruders may trace back to 980s, which was constructed at the University of Akron s Institute of Polymer Engineering [04]. Later, this group released the first Dos commercial, PC-based version in 990 and a windows platform version of the personal computer-based AKRON-CO-TWIN SCREW [53] extrusion simulation modeling program for intermeshing modular co-rotating twin screw extruders in 00. AKRON- CO-TWIN SCREW is a powerful, quick and effective scientific tool to gain optimum performance of co-rotating twin screw extrusion. It has a unique capability in scale-up processes from one extruder system to another. Lubrication theory and Flow Analysis Network (FAN) were used to develop this software. Non-Newtonian fluid (power-law model) is assumed for polymer melt. Morex was originally developed for reactive extrusion analysis in intermeshing corotating twin screw extruders at the Institute of Plastics Processing (IKV) at Aachen University of Technology in 99 [65]. Morex is based on a simplified, extended onedimensional analytical flow model that includes components for the intermeshing section, the radial gap and the main flow channel between screw and barrel. Degree of fill, temperature and pressure can be calculated from this software. However, it seems that solid conveying and the melting process are not included in this software. Sigma from the Institute of Plastics Engineering (KTP) of the University of Paderborn [66], is the work of a joint research project between the University of Paderborn and ten industrial companies. According to the project researchers, 45

67 calculations are based on a two-dimensional channel model that works on the assumption of a two-dimensional screw barrel and two-dimensional screw channels. The intermeshing zone of the two screws is integrated in the channel model as an additional channel section of reduced channel width. Melting model is included in this software. The design theory for this software is the same as Akron-Co-Twin Screw [53]. The software is available only to companies participating in the project. The Twin-Screw Extruder Simulator (TXS TM ) software is developed by Canedo of PolyTech [], utilizing research by the Polymer Processing Institute in Newark, New Jersey. This software is based on a -dimensional analytical model for the whole extrusion region. Frication was included to describe the melting process in kneading blocks. The program contains databases of about 50 commercial extruders and several generic resins and fillers. Ludovic twin screw extrusion simulation software [67, 68] developed by Cemef (Centre for Material Forming) and INRA (The French Institute for Agronomy Research) is marketed by Sciences and Computer Consultants (SCC), based in France. Ludovic was updated to a three dimensional FEM module called XimeX by SCC in 005, which can simulate the macro thermo-mechanical behavior in co-rotating twin screw and sing screw extruders, and batch processes. However, the results for co-rotating twin screw extrusion are still shown in one-dimensional. The basic assumptions and procedures of this software is the same to Akron-Co-Twin Screw [53]. 46

68 CHAPTER III EXPERIMENTAL: EQUIPMENT, MATERIALS AND METHODS 3. Introduction The objectives of this chapter are to describe experiments and characterize the behavior of polymer particles in an intermeshing counter-rotating twin screw extruder with the purpose of developing and testing solid conveying models, melting models melt metering and global behavior of this type of continuous extrusion machines. The equipment, materials and experimental setup are presented in this chapter. The apparatus used in our experiments is outlined first, which is followed by materials and their properties. Following these sections, the experimental procedures to be used are outlined. 3. Apparatus A Leistritz modular intermeshing counter-rotating twin screw extruder (LSM30.34) was used in this investigation. This machine has an inside screw diameter of 34 mm and inter screw distance of 30 mm. The ratio of length to diameter is about 8.3. The general specifications of the machine are given in Table 3.. Both the barrel and the screw consist of modular sections. A schematic of the modular barrel is shown in Figure 3.. There is one downstream vent / feed port subsequent to the hopper. The port is at 435 mm from the die. The barrel is air fan cooled. 47

69 Table 3. Leistritz LS30.34 GG twin screw extruder specifications Technical Data Model Screw Diameter, D Inter-Screw Distance Specifications LSM 30.34GG 34 mm 30 mm Screw Length -35D Screw Speed Drive Power Max. Torque Heating Power Per Zone Output Rate Depending on speed Weight Barrel Diameter Screw External Diameter Total Extruder Screw Length Inter-Screw Distance Clearance between Screw Flight and Barrel Channel Depth Helix angle Channel Width Flight Width rpm 7.7kW at 300 rpm 3 Nm W 3-5 kg/hr Kg 35 mm 34 mm 980 mm 30 mm 0.5 mm 4.mm Variable Variable Variable 48

70 The Leistritz machine has screw shafts with modular screw elements. These elements: closely-meshing FD screws with thick flights, free-meshing FF screws with thin flights and additional mixing elements such as slit stowing elements and shearing elements. The detailed geometry of available modular screw elements are shown in Table 3.. Among the accessories for the Leistritz LS30.34 is a mechanical screw pulling assembly, which facilitates screw removal from the barrel. First Hopper Second Hopper 35mm 60mm 45mm 50mm.5mm 0mm 0mm 0mm 0mm 0mm 0mm 0mm 0mm Figure 3. Modular Barrel Configuration of Leistritz LS30.34 GG twin screw extruder 3.3 Materials Both amorphous and crystalline polymer materials were used in our experiments. Powder and pellets of high density polyethylene (Equistar Chemical HDPE, MP and Petrothene GA ) were used. Isotactic polyproplene pellets (Equistar Chemical PP 8800-GK) and Polyvinyl Chloride (PVC, Polyone, grade Oxy. 85 F) powder were used. The PVC resin was 49

71 stabilized by organotin based stabilizer (Thermolite 3S) obtained from Atofina Chemical Inc. Table 3. Leistritz LS30.34 twin screw extruder screw elements 50

72 Table 3.3 Materials used in this study Chemical Name Polyvinyl Chloride Chemical Structure CH CHCl n Abb. Trade Name Supplier PVC Powder Oxy. 85F Polyone Isotactic Polypropylene CH CH n PP PP 8800-GK Equistar High Density Polyethylene CH 3 HDPE CH CH n Powder Microthene MP Equistar High Density Polyethylene HDPE CH CH n Pellets Petrothene GA Equistar 3.3. Differential Scanning Calorimetry (DSC) A TA Instruments differential scanning calorimeter was used to measure the specific heat, the heat of fusion and the melting point. The molten density was measured by capillary rheometer. A vessel was used to determine the bulk density. The detailed results are shown in Table 3.4. Table 3.4 Materials properties Chemical Name Polyvinyl Chloride T g or T m ( o C) Heat of fusion (J/g) Specific Heat (J/(g K)) Bulk Density (g/cm 3 ) Melt Density (g/cm 3 ) 8 (T g ) Isotactic Polypropylene 60 (T m ) HDPE Powder 36 (T m ) HDPE Pellets 36 (T m )

73 3.3. Rheological Properties The shear viscosity-shear rate relationships at different temperatures were measured. The shear viscosities of the four polymer melts investigated are shown in Figure 3. to Figure 3.4. The viscosities at low shear rates, from 0.0 s - to 0 s -, were obtained from steady rate sweep using the ARES (Advanced Rheometric Expansion System) of Rheometric Scientific Inc., with a cone-plate apparatus of dimensions 5 mm diameter and cone angle 0. radian. The viscosities at high shear rates over 0 s - to 0 3 s - were obtained using an Instron capillary rheometer with three dies of L/D equal to 9.3, 9.3, and 8.5 mm, where barrel and die diameters are 9.5 and.6 mm, respectively. We applied standard viscometric calculation procedures including the Bagley plot to eliminate the end pressure loss and the Weissenberg correction for the die wall shear rate. After the data for shear viscosities were obtained at varied shear rates and different temperatures, a power law model was fitted to express the effect of shear rates and temperatures on viscosities (at high shear rates) with the following form, ( T T ) n b o η = K oe γ 3- where T o is a reference temperature, n is the power-law index, γ is the shear rate, b is the temperature sensitivity of viscosity and K o is the consistency index at T o. When the shear rate is low enough, the measured viscosity approaches a constant value, which is zero shear viscosity, taken as the limit value at corresponding temperature. From measurement of viscosities vs. shear rates and temperatures, the power-law index, temperature sensitivity, consistency index and reference temperature are fitted for HDPE, PP and PVC, respectively. 5

74 Viscosity, [Pa-s] o C 80 o C 00 o C Shear Rate, [/s] Figure 3. Shear Viscosity of HDPE (MP / GA655-66) as a function of shear rates 0 5 Shear Viscosity, [Pa-s] o C 00 o C 0 o C Shear Rate, [/s] Figure 3.3 Shear viscosity of PP (Equistar 8800-GK) as a function of shear rates 53

75 0 5 Viscosity, [Pa-s] o C 80 o C Shear Rate, [/s] Figure 3.4 Shear viscosity of PVC as a function of shear rates Melt Densities and Bulk Densities An Instron capillary rheometer was used to measure the melt density by extruding the molten material at a specific volumetric flow rate, Q, and measuring the total weight ( m w ) at a certain time interval (t). The melt density is given as, m w ρ = m tq The bulk density of materials was measured by filling solid material in a cylindrical cup with known volume and weighting. Results we computed are given in Table Experimental Studies Experimental studies were carried out to investigate solid conveying, the melting process and global behavior for intermeshing counter-rotating twin screw extruders. 54

76 3.4. Solid Conveying Solid conveying experiments under metered starved fed conditions, have been carried out in our group by constructing transparent polymethyl methacrylate (PMMA) barrels to conduct flow visualization experiments []. The conveying of pellets was observed at room temperature and recorded by video camera. Under flood fed system, experiments were investigated with different screw configurations by pre-setting barrel temperature at room temperature. Due to the limitations of the machine in power supply (this machine was designed for compounding applications), only half of total screw length was used to investigate solid conveying. The schematic is shown in Figure 3.5. Figure 3.5 Schematic of the machine and the screw used for solid conveying study 55

77 (A) (B) Figure 3.6 Schematics and photographs of screw configurations used in solid conveying experiments, (A) Thick flight elements, and (B) Thin flight elements. In most of our experiments with powdered materials, they were fed from the second hopper, and the die was not attached with the machine, as shown in Figure 3.6, where screw configuration (A) only has thick flight elements and screw configuration (B) has all thin flight elements. It is expected to see the formation of the solid bed when the machine is stopped, and the screw pulled out of the barrel. The results are described in Section 4. with a detailed discussion of material distribution Melting Experiments One of the main focuses of experimental study was to check if starvation happens under flood fed conditions. Under what conditions does it occur? Other concerns are to locate the position and the total length of the melting process. 56

78 In the operating of the extruder, it is heated to a preset temperature. It needs to be recognized that the temperature in the feeding section is lower than the melting temperature to prevent premature melting. In the steady state operation, the machine is shut down and a screw pulling-out technique is used to investigate polymer behavior along the screw axis. To do this, we slightly raise the barrel temperature again to a little higher than the softening temperature and pull out the screw to study polymer distribution and melting status Global Behavior of Total Screw In running counter-rotating twin screw extruders, starvation may occur at different stages, which depend on screw configurations and feed systems. In addition to the solid conveying region and the melting region, it is meaningful to check the global behavior of total screw, such as fill status, as well as temperature and pressure fields Screw Configurations To consider the practical case of industrial productions, different screw configurations are used in experiments. These include thick flight elements, thin flight elements and Leistritz shearing elements. These configurations are presented in Figure 3.7. In screw configurations, design (A) only has thick flight elements. As a comparison, design (B) only has thin flight elements. Design (C) has both thin flight and thick flighed 57

79 elements. Shearing elements are introduced in design (D) and (E), corresponding designs (A) and (B). (A) (B) (C) (D) (E) Figure 3.7 Screw configurations used in experiments 58

80 CHAPTER IV EXPERIMENTAL STUDIES OF INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUSION 4. Introduction The purpose of intermeshing counter-rotating twin-screw extrusion is to transform polymer powder and pellets into profiles or compounds. The initial pellets / powder experience solid conveying, melting and melt conveying. So a complete study on intermeshing counter-rotating twin screw extrusion should include all three regions. The objectives of this chapter are to experimentally investigate solid conveying, melting, and melt metering processes in intermeshing counter-rotating twin screw extrusion. We consider both metered starved fed and flood fed behavior in separate sections. We showed above different screw configurations, as presented in Figure 3.6 and Figure 3.7. They have thick flight and thin flight screw elements, as well as special mixing screw elements. 4. Solid Conveying Experiments Both thick and thin flight elements were used in experiments to investigate the solid conveying status. The feeding process is schematically shown in Figure 4.. When 59

81 we performed the experiment studies on solid conveying, the barrel heater was turned off. So the barrel temperature was at room temperature. This is different from experiments on the melting and metering processes, where the barrel temperature was pre-set at a point higher than the melting temperature of materials. Figure 4. Schematic of feeding system in counter-rotating twin screw extruders 4.. Thick Flight Screws Powder HDPE material was first used in a screw configuration with thick flight elements (as shown in Figure 3.6 (A)) in experiments. Our observations are shown in Figure 4., where the machine was run under flood fed conditions and a screw speed of 30 rpm. The screw channel was observed to be fully filled with a solid bed. Along the screw direction, the solid bed became more compacted, and the movement between 60

82 material particles became impossible. The material remained inside the screw channels when the screws were pulled out of the barrel. Most of materials in the inter-screw region seemed to be softened and melted. Part of the material on the surface of the solid bed was also softened and melted. (A) (B) Figure 4. Solid conveying in thick flight elements in flood fed conditions, (A)Top view and (B) Down View Due to the power limitations of our twin screw extruder, we were not able to perform this experiment with HDPE pellets under flood fed condition. However, some trends were obtained from melting studies to be described in Section 4.3. The solid conveying process under metered starved fed conditions was considered with thick flight screws. The solid material, however, dropped out of screw channels. So 6

83 we could not observe solid conveying separately. This situation will be discussed in Section 4.3 where we investigate solid conveying together with the melting process. 4.. Thin Flight Screws Starved metered feeding was investigated to study solid conveying with thin flight screws experimentally at room temperature. However, the material dropped out of the screw channel when screws were pulled out of the barrel. So we could not observe the distribution of solid materials for the thin flight screw design. We made experiments for thin flight screws under flood fed conditions with HDPE powder, and observed the distribution of materials inside the screw channel, as shown in Figure 4.3. Powder HDPE material was found to fully fill screw channels with a loose compact, and most of the material at the bottom screw channel dropped out of the screws during the pulling of screws out of the barrel. This was different from the behavior of thick flight screws observed above. Along the screw axis direction, materials were not melted or softened. 4.3 Melting Experiments Experiments were carried out to investigate the melting process for intermeshing counter-rotating twin screw extrusion with both thick flight elements and thin flight elements. The barrel temperature was pre-set at a temperature higher than the melting point for crystalline materials or the softening point for the amorphous material. The results with these polymers for both types of elements are presented in this section. 6

84 (A) (B) Figure 4.3 Solid conveying in screw configurations shown in Figure 3.6 (B) (thin flight elements) under flood fed conditions, (A)Top view and (B) Down View 4.3. Thick Flight Screws Experimental studies were previously carried out in our laboratory on the melting process in an intermeshing counter-rotating twin screw extruder of isotactic polypropylene by Wilczynski and White [7]. Our experiments results involved high density polyethylene and polyvinyl chloride (PVC). For both materials, the barrel temperature was set at 70 o C, except the feeding region which was at room temperature. Figure 4.4 shows the screw configuration with thick flight elements, under the condition of metered starved feeding. PVC in the powder form was fed at the rate of 6 63

85 kg/h. The machine was run at screw speed 30 rpm. A small part of powder material was conveyed at the top screw channel. Only the down part of the screw channel near the inter-screw region is fully filled. Along the screw channel direction, the screw channel was partially filled. The softening process was found to be started at a position of 75 mm from the hopper and finished in a length of 0 mm. Increasing the screw speed to 60 rpm for PVC at a fixed feed rate of 6 kg/h, it was found that softening started at a position of 05 mm from the hopper, while the total melting region becomes shorter (finish in a length of 80 mm) as well, shown in Figure 4.5. It was observed that the screw channel under metered starved feeding was partly filled in the solid conveying region, and most of solid particles at down screw channels dropped out of the screw when pulling out the screws. It was found that with the increase of feed rates, the screws were more highly filled, especially at lower screw speeds. For flood fed experiments, we found more of the screw channel was filled. This is summarized in Figure 4.4 to Figure 4.6. In addition, the effect of feed rates on the melting process was investigated. Figure 4.6 shows the result of thick flight screw configuration at a lower feed rate (3 kg/hr) and fixed screw speed (30 rpm). Compared with a higher feed rate shown in Figure 4.4, the starting point of the melting process begins later (the starting position at 5 mm from the hopper), and the total length for melting to be completely melted is shorter (in a length of 30 mm). 64

86 (A) (B) Figure 4.4 Photographs of the thick flight screws pulled out from the machine after extrusion of PVC at the feed rate 6 kg/hr and the screw speed 30 rpm for thick flight screw configuration with PVC, (A) Top view and (B) Down View (A) (B) Figure 4.5 Photographs of the thick flight screws pulled out from the machine after extrusion of PVC at the feed rate 6kg/hr and the screw speed 60 rpm for thick flight screw configuration with PVC, (A) Top view and (B) Down View 65

87 (A) (B) Figure 4.6 Photographs of the thick flight screws pulled out from the machine after extrusion of PVC at the feed rate 3 kg/hr and screw speed 30 rpm for thick flight screw configuration with PVC, (A) Top view and (B) Bottom View HDPE was also investigated in our experiments. Screw configurations with thick flight elements were used to investigate the melting process under flood fed conditions with HDPE powder material. We only focused on the melting process here, there was no die attached and it was open discharge at the end of the screw. The screw configuration shown in Figure 3.6 (A) was used, where HDPE material was fed from the second hopper. The results are shown in Figure 4.7. It is seen that the down section of screw channels melted much faster, compared with the top section. The screw channel of the first pitch is fully filled, while the melting process started at a position of 30 mm from the hopper and was finished in a length of 00 mm. With the initiation of melting, the fully filled screw channel became starved, which caused a pressure reduction to atmospheric. This occurs despite there being no die. The reason must be due to material densification. The molten 66

88 material moved into the forward side of screw channels, where the solid powder materials still piled up at the other side of screw channel. Figure 4.7 Photographs of the thick flight screws pulled out from the machine after extrusion of powder HDPE at screw speed 30 rpm Another screw configuration with a die was used to investigate the effect of thick flight screws on the melting process (Figure 3.7 (A)). This involved the second hopper with only thick flight screw elements FD-3-30, shown in Figure 4.8. Figure 4.8 shows the result for this screw configuration at a screw rotation speed 30 rpm under flood fed condition. The same powder HDPE was used. There was about a 50 mm length of fully filled region at the end of screw. In the solid conveying region, there was around a 30 mm length of fully filled region. After melting, the screw channel became starved, but later became fully filled before the die. For comparisons, the same screw configuration shown in Figure 4.8 was used and run at a lower screw speed, 0 rpm and flood fed with pellet HDPE, shown in Figure 4.9. It was found that most of the screw length was fully filled. The reason seems due to lower 67

89 pumping capacity at lower screw speeds. Starvation also occurs when the melting process starts. Figure 4.8 Photographs of the thick flight screws pulled out from the machine after flood fed extrusion of powder HDPE at the screw speed 30 rpm Figure 4.9 Photographs of the thick flight screws pulled out from the machine after flood fed extrusion of powder HDPE at the screw speed 0 rpm To investigate the melting process thoroughly with a thick flight screw configuration under flood fed conditions, HDPE pellets were also used in experiments. The results are shown in Figure 4.0. The screw configuration and operating conditions 68

90 were fixed (screw speed was 30 rpm and the machine was run under flood fed conditions). Similar phenomena were observed. The melting process started at a position of 30 mm from the hopper and was finished in a length of 80 mm. Starvation occurred with the initiation of melting. Materials in the down section melted much faster than the top section. Separation of the pellets and the molten region occurred as melting progresses. A greater amount of melt is produced from the HDPE pellets than the HDPE powder apparently because of the pellet bed s higher density. (A) (B) Figure 4.0 Photographs of the thick flight screw configuration pulled out from the machine after extrusion of HDPE pellets at screw speed 30 rpm under flood fed condition, (A)Top view, (B) Down View 69

91 4.3. Thin Flight Screws In addition to screw configurations with thick flight elements, thin flight elements were also investigated. These are shown in Figure 4. to Figure 4.3. It is seen that more of the screw channel is filled in thin flight screw configuration. This may be due to the lower pumping capacity of this type of screw designs. The starting position of the melting process for HDPE pellets at a feed rate 8 kg/h and a screw speed 60 rpm is at a position of 50 mm from the hopper and finished in 0 mm of screw length. Compared with the case of thick flight screws, the starting position of melting was delayed. The length of the melting process was around 40 mm. (A) (B) Figure 4. Photographs of the thin flight screws pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/h and the screw speed 60 rpm, (A) Top view and (B) Down View 70

92 (A) (B) Figure 4. Photographs of the thin flight screws pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/h and the screw speed 0 rpm, (A) Top view and (B) Down View (A) (B) Figure 4.3 Photographs of the thin flight screws pulled out from the machine after extrusion of HDPE pellets at the feed rate 4 kg/h and the screw speed 60 rpm, (A) Top view and (B) Down View 7

93 The HDPE pellets fell out of the screw channel when the screws were removed. We are thus unable to comment on the behavior of the solid conveying behavior of the thin flight screws. 4.4 Melt Conveying and Global View of Screws Experiments were conducted to investigate the melt conveying and the global behavior of different screw configurations. Thick flight screws, thin flight screws and complex screws with thick flight and thin flight screws as well as other elements were used Thick Flight Screws The global behavior of intermeshing counter-rotating twin screw extrusion was presented from Figure 4.4 to Figure 4.6 for PVC. The results for HDPE under starved metered fed conditions with screw configuration (D) are shown in Figure 4.4 to Figure 4.7. It is found that with an increase of screw speeds, the melting process is delayed and the length of melting along the screw direction becomes shorter. With the increase of feed rates, the screw length for the melting process becomes longer and melting starts earlier. Following melting, the screw channels are starved. When there is a shearing element in the metering region, the regions of the shearing element and ahead of this region are fully filled. The fully filled length depends on operating conditions and screw pumping capacity. 7

94 (A) (B) Figure 4.4 Photographs of the thick flight screw configuration without shear elements pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/hr and the screw speed 60 rpm, (A) Top view, (B) Down View (A) (B) Figure 4.5 Photographs of the thick flight screw configuration with two shear elements pulled out from the machine after extrusion of HDPE pellets at the screw 8 kg/hr and the screw speed 60 rpm, (A) Top view, (B) Down View 73

95 (A) (B) Figure 4.6 Photographs of the thick flight screw configuration with two shear elements pulled out from the machine after extrusion of HDPE pellets at the feed rate 4 kg/hr and the screw speed 60 rpm, (A) Top view, (B) Down View (A) (B) Figure 4.7 Photographs of the thick flight screw configuration with two shear elements pulled out from the machine after extrusion of HDPE pellets at the feed rate 8 kg/hr and the screw speed 0 rpm, (A) Top view, (B) Down View 74

96 Figure 4.8 shows the results for PVC and HDPE at different feed rates and screw rotation speeds with screw configuration (A), where thick flight elements were used. Figure 4.9 shows the results for pellet HDPE at different feed rates and screw rotation speeds with screw configuration (D), where mixed both shear and thick flight elements were used. 75

97 3 kg/hr at 30 rpm for powder PVC 3 kg/hr at 60 rpm for powder PVC 6 kg/hr at 60 rpm for powder PVC 8 kg/hr at 60 rpm for pellet HDPE Flood fed power HDPE at 60 rpm Solid Conveying Melting Region Starved Melt Conveying Fully filled Region Figure 4.8 Melting location, melting length and fill status of screw configuration (A) under different processing conditions and with different materials. 76

98 8 kg/hr at screw speed 60 rpm for pellet HDPE 8 kg/hr at screw speed 0 rpm for pellet HDPE 4 kg/hr at screw speed 60 rpm for pellet HDPE Solid Conveying Melting Region Starved Melt Conveying Fully filled Region Figure 4.9 Melting location, melting length and fill status of screw configuration (D) under different processing conditions. 77

99 4.4. Thin Flight Screws Figure 4.0 shows the results for PP at the feed rate 0 kg/ hr and the screw speed 00 rpm with screw configuration (B) (barrel temperature at 0 o C), where thin flight elements were used. Solid Conveying Melting Region Starved Melt Conveying Fully filled Region Figure 4.0 Melting location, melting length and fill status in screw configuration (B) at the feed rate of PP 0 kg/hr and the screw speed 00 rpm. 78

100 8 kg/hr at screw speed 0 rpm for pellet HDPE 4 kg/hr at screw speed 60 rpm for pellet HDPE 8 kg/hr at screw speed 60 rpm for pellet HDPE Solid Conveying Melting Region Starved Melt Conveying Fully filled Region Figure 4. Melting location, melting length and fill status in screw configuration (E) under different processing conditions with pellet HDPE. 79

101 4.4.3 Complex Screws Figure 4. shows the results for pellet HDPE at different feed rates and screw rotation speeds with screw configuration (E), where mixed both shear and thin flight elements were used. Figure 4. shows the results for PP at the operating conditions of the feed rate 0 kg/ hr and the screw speed 00 rpm with screw configuration (C), where both thin and thick flight elements were used. Solid Conveying Melting Region Starved Melt Conveying Fully filled Region Figure 4. Melting location, melting length and fill status in screw configuration (C) at the feed rate of PP 0 kg/hr and the screw speed of 00 rpm. 4.5 Discussion and Interpretation The effect of different screw configurations and operating conditions on melting, fill status, and the global behavior were observed from experimental results shown in previous section. We seek to interpret this behavior below. 80

102 4.5. Melting and Fill Status Our observations are summarized and shown in Figure 4.8 to Figure 4. for melting and the global behavior of intermeshing counter-rotating twin screw extrusion, based on experiments at different operating conditions, screw configurations and materials. It was found that the length of the solid conveying region was generally short, especially for crystalline materials. The melting process was finished in 3 to 5 pitches. The intermeshing counter-rotating twin screw extruder had a good pumping capacity, due to its positive displacement. Under normal operating conditions, the filled length before the die was less than pitches. Our observations are consistent with those by Wilczynski and White [7]. PVC has a longer length for both regions of the solid conveying and the melting process. The possible reasons are the lower friction coefficients, the smaller particle size of raw materials and higher leakages Melting Location, Initiation and Mechanism The melting mechanism in intermeshing counter-rotating twin screw extruders, as described by Wilczynski and White [7, 8], is different from that observed in corotating twin screw extruders and single screw extruders. Heat transfer from the hot barrel, heat dissipation in the C-chamber and the inter-screw region all help to melt solid materials. However, among these factors, heat dissipation from the inter-screw region plays a dominant role in the melting process for intermeshing counter-rotating twin screw extruders. Our observations support this. Generally melting initiates in the inter-screw region. When the barrel temperature is higher than the melting point, melting also starts 8

103 from the inter-face between the barrel and the solid bed inside the C-chamber. It was observed that the melting process occurs at a very early stage, less than 5 pitches from the hopper Effect of Operating Conditions Higher screw speeds delay the initiation of melting. The melting process starts earlier with an increase of feed rates. At a higher feed rate, more screw channels are filled, and the pressure buildup is also higher, which in turn causes more friction heating and earlier melting. However, a higher feed rate requires more energy input to melt all the materials. Combining the two effects, it takes a longer length to completely melt the solid materials Feed Rates under Flood Fed Conditions The feed rates under flood fed conditions were studied. The screw design shown in Figure 3.6 (A) was used in experiments. Both powder and pellet HDPE materials were used. The temperature at the first barrel zone was set as 50 o C, and after then, the barrel temperature was set at 70 o C. When the machine was run steadily, the feed rate equals to the outputs. We measured the output and took the values as feed rates. The results are shown in Figure 4.3. With an increase of screw speeds, the flood feed rates increases. The relationship between flood feed rates vs. screw speeds approaches linear behavior. However, it is not a strictly linear. The reason is probably due to the variations of bulk densities. At higher screw speeds, the porosity of materials 8

104 increases, which causes a lower bulk density of solid materials. So the actual feed rates fall below the straight line at higher screw speeds. 0 Powder HDPE Pellet HDPE Feed Rate, G, kg/hr Screw Speed, RPM Figure 4.3 Measured feed rates under flood fed conditions for thick flight screws 4.6 Conclusions Experiments were carried out to investigate solid conveying, melting and the global behavior in an intermeshing counter-rotating twin screw extruder. The solid polymer is not well compacted in thin flight screws, even under flood fed conditions. Thick flight screws have a better compacting under flood fed conditions. However, compacting is not well developed under starved metered fed conditions. Generally the pressure buildup for 83

105 both thick and thin flight screws in the C-chamber is small under starved fed conditions. The pressure buildup in the inter-screw region for thick flight screws is in a higher level, which dominates the melting process. Even if the screw channel is starved and the barrel temperature is at room temperature, melting still can be initiated. For all processes including flood feeding, starvation occurs when melting starts. It originates from the densification of materials. Solid part and molten part were observed to be separated under both flood fed and starved feed conditions in screw channels. Generally the solid conveying region is short. The degree of fill depends on operating conditions. The observed global behavior in intermeshing counter-rotating twin screw extrusion is that the solid conveying and the melting region are short. In our experiments, they were generally finished in 8 L/D screw length or less. Starved flow exists from the position of melting to just in front of the die. Only the region before the die, and nonpumping elements if they are present, are fully filled. It was also noticed that thick flight screws have a better pumping capacity, compared with thin flight screws. 84

106 CHAPTER V SOLID CONVEYING IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS 5. Introduction The unique feature of intermeshing counter-rotating twin-screw extruders is the positive displacement mechanism which makes them different in material transport, melting and metering than single screw extruders and intermeshing co-rotating twinscrew extruders. Depending on application areas, there are two feed methods, flood fed and metered starved fed. The former feed system is mainly applied to profile extrusion where the screw rotation speed is at a relatively low level. Metered starved fed is generally used in compounding and reactive extrusion. Polymers and additives are fed in the known quantity from the hopper state in the form of pellets/granules/powders. The region from below the hopper to the initial melting is classified as the solid conveying region. Generally when designing counter-rotating twin-screw extruders, only elements with pumping capability appear in solid conveying region. Therefore in generating a solid conveying model, only those elements with pumping capacity are considered. Interests in this region are to predict pressure fields and the feed rate for both flood fed and starved 85

107 fed machines as well as the fill level, and the bulk temperature rise before it reaches the melting zone, and the location of initial melting. There are primary two regions of solid conveying, as shown in Figure 5., region A and region B. Region A refers to the inter-screw region and region B is so-called the C- chamber. In general, region A is always fully filled, unless the feed rate is at a very low level. Depending on feed systems, region B may be fully filled or starved. For flood fed machines, region A and region B merge together. Figure 5. Solid conveying model in an intermeshing counter-rotating TSE 86

108 The open C-chamber and the closed C-chamber have different transport mechanisms. The former is primarily drag flow, while the latter is positive displacement. Solid conveying in the C-chamber should be modeled with methods corresponding to screw elements which may have an open C-chamber or a closed C-chamber. 5. Solid Conveying Model for Thick Flight Screws Generally a pair of thick flight intermeshing screws forms a closed C-chamber. The schematic is shown in Figure 5., where W c is the width of screw channel, S is the pitch. Material is fed from the hopper and is entrapped into a series of discontinuous closed C-chambers. So material is transported in a discontinuous form. The primary transport mechanism is positive displacement. Clearance and friction may cause some leakage, if the C-chamber is not perfectly closed and the particle sizes are small enough. Under metered starved fed conditions, the feed rate is known and the C-chamber is not fully filled. Under flood fed conditions, the C-chamber should be fully filled, until melting occurs where the formation of melt from granular pellets or powder lowers the volume of materials. The feed rate can be predicted by the geometry of the C-chamber, screw speed and material bulk density. Pressure fields and temperature fields are determined by force and energy balances. 87

109 U θ H D U W c S Figure 5. Schematic of a closed C-chamber in solid conveying The whole domain is divided into two separate parts: a C-chamber region (Region A) and an inter-screw region (Region B), shown in Figure 5.. Generally the inter-screw region (also called the calendering region) is fully filled and the C-chamber is incompletely filled under metered starved fed conditions. When flood feeding, the C- chamber under the hopper is fully filled and subsequent the C-chamber may be fully filled or incompletely filled, which is determined by the pumping capacity, screw configurations and changes in bulk densities. The output is the sum of both regions. Heat dissipation and heat transfer determine the initiation of melting. Since the solid material is entrapped inside each closed C- chamber, it moves with the C-chamber along screw axis. Every turn of the screw rotation, solid material is transported one pitch distance. 88

110 5.. Pumping Capacity Determination of the C-chamber The flow rate in Region B (the C-chamber) is given as, G c = k G c where G c is the flow rate of the pellet bed, G is the feed rate and k c is the fraction of the polymer that is transported in the C-chambers. According to the C-chamber mechanism, discussed by Montelius [3], Kiesskalt [3] and Schenkel [77, 79], G k can be expressed as, G = ρ c NV occ b where ρ b is the bulk density of solid material, N is the screw speed and V occ is the volume of the C-chamber filled with the solid particles. In a rectangular coordinate system, V occ is given as approximately, V occ = L W H c c where L c is the length of thesolid bed and H is the height of the solid bed. From Eqs. 5- to 5-3, the length of the solid bed can be expressed as, L c = kcg Nρ W H b c From the C-chamber geometry, the maximum length of the pellet bed, L max, can be determined. This is L max = H π i ( α ) R + 89

111 where R i is the screw root radius, α is the overlap angle, which is determined by R o, the inner radius of barrel and the distance between two screw axis L x, L cos = R x α. o If the machine is flood fed, the maximum output is given by G = Lmax NρbWc H k c 5.. Force Balances and Pressure Profiles in the C-chamber Region For thick flight screws, materials are pumped by positive displacement. So the flow rate is determined by geometrical parameters and operating parameters. Force balance and torque balance are not the main issue for this situation. Based on our observations, generally the force balances and pressure profiles in the C-chamber for thick flight screws are unimportnat, because of the development of starvation following melting. The pressure developed reduces to atmosphere, when this occurs. A model has been developed for force balances and pressure profiles in the solid conveying as a reference for future study, if necessary. This is attached in Appendix A Flow Rate in the Inter-screw Region The mass flow rate in Region B is determined from the calendering geometry, as shown in Figure

112 Figure 5.3 Schematic of geometry calculation in the calendering region The cross section area of the inter-screw region is, from the geometrical anaysis, ( R R ) α A cal = Ro sinα cosα o, R + i where R, is the opposite screw radius. If neglecting the clearance between one screw o R flight and the barrel surface, and assuming fully intermeshing, the C-chamber channel depth can be represented as, H = R o R i The overlap angle is then given by, R H o cosα = R o and Equation 5-7 becomes ( H ) H α R R H ( R R ) A cal = 4 + o o o, R i

113 The volume in the inter-screw region for both thick and thin flight screws is, V cal = A cal W c = W c ( H ) H α R R H ( R + R ) o o o, R i where W c is the width of the screw channel. Inside the inter-screw region, the pressure profiles are different between the upper part and lower part. Due to the opposite rotation of two screws, either the upper part or the lower part builds up a higher pressure, which results in different densities in different locations. Here the lower part is taken as the pressure built-up region and the density of solid material is presented as ρ L. The density of solid material at upper part is taken as the same to that inside the C-chamber. If we neglect leakage, the flow rate in the interscrew region, G cal, is, G cal = NW = NV c cal ( ρ + ρ ) R ( ρ + ρ ) α ( ) ( ) L L b b o H R H H o R 4 o, R + R i If we assume the densities are the same, ρ = ρ L b G cal = NV cal ( H ) H α R R H ( R + R ) ρ =. b NWcρb o o 4 o R i The total transported rate G consists of two parts. One is from the C-chamber, G c, and the other is from the inter-screw region, 9 G cal. We have the following expression G = G c + G cal The solid bed length in the C-chamber can be expressed as kg G Gcal Lc = = Nρ W H Nρ W H b c b c

114 Generally the feed rate is larger than G cal, and then the fill factor in the inter-screw region is. The overall fill factor k total is determined by, k total = Lc H + A L H + A max Force Balances and Pressure Profiles in the Inter-screw Region The calender geometry is considered for the inter-screw region, shown in Figure 5.4. The clearance between two screws is. In open C-chambers, is taken as the sum of and channel depth, + h. When the solid bed is assumed, the force balance is applied in the inter-screw region. Figure 5.4 Forces loading and schematic of the inter-screw region 93

115 In the z direction (Figure 5.4), the force balance between pressure and friction is given as F = pdz hdp = The solution is obtained by integrating Equation 5-7, and given in Appendix B, where the pressure is expressed as = R z p p exp f s arctg R where p is the pressure at clearance point (line), R is screw radius (taken as R i for approximation), z is the position from clearance point Feed Rate Predictions From theory presented in Section 5.. and 5..3, the feed rates under flood fed conditions were studied. Our considerations are based on the suggestion of Doboczky [6, 7]. A dimensionless feed rate, which is equivalent to the fill factor, is used to represent our observations of feed rate, defined as * G Q = Nρ b ( V + V ) c cal where G is the measured output, ρ b is the bulk density, V c and V cal are the volumes of the C-chamber and the calendering region. The results are dependent on the value of ρ b used, which may well depend on the pressure in the solid conveying region. The results are shown in Figure 5.5. The lowest feed rate, Q*, for the HDPE powder, is 90% of theoretical value. For the pellets, it is 80%, which means our bulk density is too low. 94

116 At higher screw speeds N, Q* decreases reflecting a lowering of bulk density with N. In experiments, there was no pressurized feeding system. So the porosity could increase at higher screw speeds, which may well cause a lower bulk density of solid material, or the reverse. So the actual dimensionless feed rates are lower at higher screw speeds. Doboczky [6, 7] who first considered this problem found lower values quoted by other investigations. Our model using Equation 5-9 approaches the predictions very well..0 Dimensionless Feed Rate, Q * Powder HDPE Pellet HDPE Screw Speed, RPM Figure 5.5 Predictions of feed rates under flood fed conditions for thick flight screws 95

117 In predicting the feed rate under flood fed conditions for thick flight screws, the effect of leakage was neglected. That s probably another important reason for this discrepancy. However, different from melt conveying, the level of leakage depends on both the machine geometry and material particle sizes, in addition to operating conditions. A scale-up study on the effect of machine geometry and material particle sizes is necessary to describe the leakage. This study will be carried out in the future, when there are machines available to us with different sizes and designs. 5.3 Solid Conveying Model for Thin Flight Screws Generally when thin flight screws appear in the solid conveying region, the intermeshing counter-rotating twin screw extruder works only under metered starved fed conditions. The experimental results in Chapter IV suggest that pressure buildup is small for thin flight screws, even under flood fed conditions. So the pressure developed in the C-chamber under starved metered fed conditions is negligible. Hence, the main concerns in modeling solid conveying are the prediction of temperature profiles along the screw axis and fill factors Pumping Capacity Determination Thin flight screws form open C-chambers, where positive displacement may not be the primary mechanism for solid conveying. Barrel frictional drag may play a significant role in transporting solid material. So it is meaningful to construct a force balance to determine the pumping capacity and pressure development. 96

118 For open C-chambers, the total pumping capacity of screws in solid conveying is determined by screw geometry, operating parameters and material properties. Force balances should be constructed to determine pressure and temperature fields. The schematic of an open C-chamber is shown in Figure 5.6. U H U θ D W Wc S Figure 5.6 Schematic of the open C-chamber in solid conveying Theoretically, the maximum pumping rate for open C-chambers has two parts. One is from the inter-screw region, and the other is from the C-chamber. The maximum feed rate is given as, b ( V V ) G = Nρ c cal where V cal is the volume of the inter-screw region (the calendering region) and V c is the volume of the C-chamber. To have this pumping rate, the screw friction coefficient 97

119 should be zero and clearances are negligible. The actual pumping rate is, however, always less than this maximum value. Compared with thick flight screws, thin flight screws are more complex in screw geometry. We divide the total screw channel into two sections. One section is assumed to work with positive displacement, where the channel width for this section equals to the screw flight width. The same solid conveying model to thick flight screws is applied in this section. Figure 5.7 Mass balances and schematic of the inter-screw region where Q and Q are the rate of solid conveying from left and right the C-chambers 98

120 The second section is the remaining part of the total screw channel with a channel ' width, W = W ( S ) c W c (in Figure 5.6). The cross area of the second section is shown in Figure 5.7. The depth of the inter-screw region equals to the screw channel depth approximately. For this section, it is the barrel frictional drag that conveys the solid material. So we unfold the screw channel and construct force balances to determine the pumping capacity. The velocity for the second section of thin flight screws, V i, shown in the velocity diagram( Figure 5.8), is given [3,, 8], ( φ) sin( θ ) ( φ + θ ) πdn sin V i = sin where θ is the helix angle, V i is the solid conveying velocity, either in the C-chamber or in the inter-screw region, and φ is the solid conveying angle. The solid conveying angle, φ, is determined from the analysis of force balances, given as [], f s H f + s H + f + s ' ' φ fb W f = b W sin θ + f s in the C-chamber region, or [ tan θ ] π φ = f s in the inter-screw region. 99

121 Figure 5.8 Velocity diagram for the second section of thin flight screws 5.3. Fill Factor Determination In starved metered feed system, the feed rate is known. The screw channel is partially filled. The pressure gradient can be expected to be negligible, as is the friction heating in the solid conveying region. Predictions from Sections 5.3. can be used to determine the fill factor. We assume the filled length in the C-chamber for the two sections is the same, so the filled cross areas are the same as well. We also assume that the filled region in the C- chamber is uniformly filled, instead of filled in the back section of the screw channel. The conveying rate by positive displacement is given as, ' G = Nρ ( W W )( A + A ) s b c cal c where G s is the solid conveying rate by positive displacement in the first section, A c is the averaged filled area in the C-chamber, and 00 A cal is defined in Section 5..3.

122 The conveying rate in the second section is given as, G = ρ ( A v + A v ) s b cal cal c c where v cal is the solid bed velocity in the inter-screw region, defined by Equations 5- and 5-3, v c is the solid bed velocity in the C-chamber of the second section, defined by Equations 5- and 5-, and A cal is defined from the geometry analysis, given by Equation 5-7. The total known feed rate in the solid conveying region is the sum of both sections, G = G s + G s Combining Equations 5-4, 5-5 and 5-6, we can obtain the value of A c, the filled area in the C-chamber. Then the total occupied volume in both the C-chamber and inter-screw region, occ V occ, can be calculated, ' ' [( W W )( A + A ) + W ( A A )] V = c cal c cal We also know the total volume by conducting geometrical analyses, which is given c as, V total ' ' [ W A + ( W W ) A + W ( )( R R )] = π α cal c cal c o i If we unfold the screw channel, the total volume is represented approximately as, V ' ' [ W A + ( W W ) A + W ( π )( R R ) H ] total = cal c cal c α o + and the filled volume is represented as, V occ ' ' [( W W )( A + LH ) W LH ] = c cal i 0

123 The actual feed rate is then related to Equation 5-39 by considering the packing density of the solid part. So the total fill factor is given as, V occ k total = Vtotal 5.4 Energy Balance Temperature profile is another important concern. The energy analysis can determine at which point the melting process starts, and the bulk temperature rises along the screw axis Bulk Temperature Rise in the C-chamber The bulk temperature rise of the polymer in the solid conveying region is calculated by performing an energy balance on the solid bed being conveyed in the down channel direction with a velocity v. The energy balance on a down channel increment x becomes, GH x x GH x = qnet + W & where the left hand side is the difference between the rate of enthalpy leaving and entering the increment and q& net is the net rate at which heat is conducted into the increment, and W, the work done. The enthalpy difference is expressed as, GH ( T x ) ( ) + x T x + Q px + x p + GH = GC x x x s x where T is the mean cup mixing temperature, G is the mass flow rate per channel, C s is the specific heat of the soild polymers and p is the pressure. The net rate of heat 0

124 conduction to the solid polymers is due to the heat transfer with the screw and barrel surface and also due to the frictional heating generated at the barrel and screw surface, respectively. Then we have, W W T T q & net = k dx3 k dx s s 3 x + x x W x x 0 W x x H = = where W s x W x x = 0 ( Tb T ) T k dx = h W + q& b b b W k T s x W x x = H dx 3 = h W s s ( T Ts ) + q& s are the rate of heat input to the sold bed in a unit increment expressed in terms of h b and h s, the heat transfer coefficients at the barrel and screw surfaces at temperatures T b and T s respectively. q& b and q& are the rate of frictional heat generated at the screw and barrel surface. So the temperature rise of the solid material can be expressed as GC s dt dx ( Tb T ) + hsws ( T Ts ) p = h W + q& + q& Q b b s b x 5.4. Bulk Temperature Rise in the Inter-screw Region We assume the gap is adiabatic, so all the increase in a heat content of the polymer in the gap equals to mechanical working, and we have, θ b 0 (, ) σ zx πn Ri + Ri o WRdθ = Gp H

125 where the solid melting rate due to friction heating inside the inter-screw gap is given as, dl Gp = ρswh, the specific enthalpy of solid polymer material is given as dt H = C s ( T T ) + λ m s, where we assume the melt temperature equals to the melting point before the melting process finishes, and σ zx is the shear stress resulting from frictional forces in the gap which is proportional to the local pressure. We may take σ p zx = f s where f s is the coefficient of friction and p is the pressure. If we check the melting process in the gap, we find that the melt density is higher than the packed solid density. So we cannot simply apply solid conveying model to determine the pressure here. To be simple, we assume the pressure field from the melting conveying model is valid here. So we have θ b [ C ( T T ) + λ] = σ ( πn )( R R ) WRdθ Gcal s m s zx i + o R , Initiation of Melting In addition to calculating the bulk temperature rise, as outlined in the above section, we wish to also know the local temperature at the interface between the barrel and the solid material in contact with it. When the temperature at this interface exceeds the melting point, melting is initiated and the solid conveying regime ends. 04

126 Heat transfer to this layer is due to the heat conduction from the hot barrel and also due to the frictional heat generation in pressurized regions. A heat balance on this region is of the form T T ρ s Csv = ks + q x x and boundary conditions are of the form x = 0 ( Tb T ) T k s = hb x = 0 x T x = T = where q is the frictional heat generated near the barrel surface. In the unpressurized regions q is negligible. In pressurized regions, q is expressed as q& = v W b r, σ Wb + v 0 0 r,3 σ b The initial condition is T x = T = The solution yields a temperature distribution ( ) the barrel surface. The interfacial temperature is defined as T x, x in the boundary layer at T surf = T ( x,0) When T surf exceeds the melting point, melting is initiated. 05

127 3 Barrel Melt v m Screw Solid bed Figure 5.9 Melting initiation and propagation in the C-chamber As for the inter-screw region, we assume the temperature is uniform. The initiation of melting in this region means that the bulk temperature rises to the melting temperature. The above sections outline a general solid conveying model. By properly simplifying, the general model can be applied to specific regions for different elements. 5.5 Calculations The models developed above give a general description on how to predict solid conveying for thick flight and thin flight screws. Further assumptions could be taken to simplify the specific applications Starved Fed System When the C-chamber is closed, the determination of fill factor should combine both regions, the C-chamber and the inter-screw region, so we have 06

128 k overall G = Nρ b ( V + V ) c cal where G is the total feed rate, V c and V cal are the volumes of the C-chamber and the interscrew region, respectively. In a starved feed system, the main heat resources can be expected from the interscrew region. In the C-chamber, friction heating is negligible, compared with heat conduction from the barrel and the screw. When pressure gradient is neglected, so Equation 5-37 can be written as, G C c s dt dx = h W b b ( Tb T ) + hsws ( T Ts ) In the inter-screw region, the bulk temperature increase is given from Equation So the total bulk temperature is taken as average of two regions: T G Tc GinTcal = G c + where T c and T cal are referred to the bulk temperature at the C-chamber and the interscrew region and determined by Equations 5-38 and The surface temperature rise is calculated from Equation 5-4, by neglecting friction heating. This may be given as [] T surf [ erfc k x ] ( T T ) exp( k x ) = T o b o where k hb = represents the ratio of heat conduction to the inner energy changes k ρ c v s s s of materials. 07

129 5.5. Flood Fed System A similar method to that used under starved fed conditions can be taken here. In this region, however, the fill factor is taken as one and the pressure gradient may not be neglected. In the closed C-chamber, material is entrapped into discontinuous forms, so the pressure gradient may build up to a certain level. In the inter-screw region, the pressure always builds up to a relative high level. Due to our being unable to measure pressure fields in the solid conveying region, and melting happening very early, compared with the single screw extrusion, solid conveying usually involves a very short region. In addition, once melting starts, even the fully filled screw channel becomes starved. Polymer material has a higher density at melt state, compared with solid state. The pressure development in the C-chamber is not considered in this investigation, although pressure development can be calculated qualitatively from the derivation of Sections 5... The pressure development in the inter-screw region is considered and determined by Equation 5-8. With above assumptions, the bulk temperature rise and the surface temperature rise can be determined with the same form in the starved fed system. Experiments show that the inter-screw region is responsible for the initiation of the melting process, so our main concern is to determine the place where melting starts, under the inter-screw melting mechanism. 08

130 5.6 Discussions and Conclusions Based on our observations of solid conveying under both flood and starved metered fed conditions, a solid conveying model was developed. For each type of feed systems, two kinds of screw geometry, the closed C-chamber from thick flight elements, and the open C-chamber from thin flight elements, were considered. For thick flight elements, solid material is entrapped inside discontinuous closed C- chambers, and conveyed along the screw direction. Positive displacement is the only mechanism for solid conveying. The fill factor can be determined by the C-chamber theory. Due to the calendering effect, no matter flood fed or starved metered fed, high pressure is built up in the inter-screw region, which causes severe friction heating. For thin flight elements under starved metered fed conditions, solid material is mainly conveyed at the bottom part of screw channels. Positive displacement and the barrel friction drag are the two main mechanisms for solid conveying. Due to the relative high level of gaps between the two screws, leakage is much higher than those cases of thick flight screws. Pressure buildup is not significant in thin flight elements, so the friction heating is negligible. Experimental studies were conducted to investigate the feed rate under flood fed conditions with thick flight screws for both powder and pellets polymer materials. Corresponding predictions from our model were made and compared with these results. It was found they match very well for both cases. Comparisons on the length of solid conveyings between simulations and experiment results are carried out and discussed in Chapter VIII. 09

131 CHAPTER VI THE MELTING PROCESS IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS 6. Introduction The melting process is one of the most important steps for extrusion processing. Generally materials are fed into the hopper in a solid state, either as pellets or powder. To fulfill the purposes of extrusion, such as compounding, shaping (profiles) or reaction processing, solid materials must be melted first. Few studies [8, 7, 8] have been conducted to describe the melting process in intermeshing counter-rotating twin screw extrusion. This process is different from other extrusion machines, such as single screw extrusion and co-rotating twin screw extrusion. It has been found that the inter-screw region plays the dominant role in melting materials for intermeshing counter-rotating twin screw extrusion. Melting initiates and takes place primarily in the inter-screw region and secondly at the barrel surface. Wilczynski and White [8] presented the first melting model for this kind of extrusion process. In their model, a Newtonian fluid model was used to derive the flow rate from the solid bed to the inter-screw region, and leakage was not considered in the inter-screw region. We expand Wilczynski and White s model to the non-newtonian case, and investigate the effect of leakage and discuss other considerations. 0

132 6. Model for the Melting Process Based on our experimental observations and literature [7], polymer materials were observed to initially form a solid bed with rectangular shape cross-section, which moves along the screw axis inside of a rough C-chamber. The bed is dragged into the inter-screw region where melting initiates [7]. Due to the calendering effect, pressure is developed in the inter-screw region, where intense inter-pellet friction and pellet deformation occurs. This in time leads to melting. Melted material inside the inter-screw region flow back to the solid bed and melt the solid bed. For the C-chamber solid material between screw and barrel, a melt layer is developed between the barrel and the solid bed, if the barrel surface temperature is higher than the melting point. Melted material produced by heat conduction from the barrel may also flows toward the neutral flight. So it appears that the length of the solid bed decreases during the melting process, until it becomes to zero and melting is finished. The two melting mechanisms are shown in Figure 6.. In this figure, S = Rθ. Solid materials are dragged into the inter-screw region and melted and then flow out of the inter-screw region by pressure flow. The flow rate of the solid bed is θ G b = k G c where G b is the total flow rate of the solid bed, G is the mass feed rate and k c is the fraction of the polymer that is transported in the C-chamber. The total flow rate of the solid bed is given as G = ρ b NV sc b

133 where ρ b is the bulk density of solid, N is the screw speed and V sc is the volume of the C-chamber filled with solid material. This may be roughly expresses as, V sc = L W H = φv s s s c where L s is the length of the solid bed, W s is the width of the pellet bed, H s is the height of the pellet bed, and φ is the volume fraction of solid. So L s can be given as, L s k G ρ NW H c = b s s Figure 6. Melting mechanisms: in the inter-screw region; between screw and barrel. 6.3 Model of Melting in the Inter-screw Region This analysis is based upon and extends the analysis of Wilczyski and White [8]. Material is dragged into the inter-screw region from the solid bed and pressure is built up.

134 Heat dissipation is generated by friction or viscous heating during this process. If we assume all the contribution of mechanical work used to generate heat dissipation and the movement of the solid bed along with the screw rotation is insignificant, compared with the screw rotation. An enthalpy balance on the moving bed is θ b 0 (, ) σ zx πn Ri + Ri o WRdθ Hloss = Gp H where G p is the mass flow rate of solid, H is the specific enthalpy change, Hloss is heat loss to screws, θ is the contact angle, W is channel width, R i and io, R are the two screws radii at position contacting with materials, N is the screw rotation speed and σ zx is the shear stress resulting from frictional forces in the inter-screw and is represented by, σ zx = p f s where f s is the coefficient of friction, and p is the pressure. Since solid materials contact with both screw surfaces, we doubled the mechanical work by times R + R in the i o integration term. The specific enthalpy of a polymer melt may be written as s ( T T ) + + c ( T T ) H = c λ m s m f m where cs and c m are the specific heats of the solid and melt. T s and T f are the temperatures of the solid and melt. T m is the melting point and λ is the heat of fusion. If we assume the inter-screw region is adiabatic, i.e. heat loss to screws is neglected, the energy balance equation becomes, 3

135 θ b 0 8, we have (, ) σ zx πn Ri + Ri o WRdθ = Gp H Inserting the specific enthalpy of Equation 6-7 into the energy balance Equation 6- θ b 0 ( +, ) = ( ) + + ( ) σzx πn Ri Ri o WRdθ Gp cs Tm Ts λ cm Tf T m The temperature inside the inter-screw region may become higher than the melting point. When material at this higher temperature flows back to the solid bed, it helps to melt the solid material in the solid bed. When investigating this effect, leakage out of the gap should be considered. The melted materials are pushed back into the solid bed at the rate ( f leak ) Gp, where f leak is the fractional leakage. The ideal case is when there is no leakage, f = 0. leak The melting rate of solid material in the solid bed due to molten material flowing back from the calendering region can be determined. The mass rate of additional material melting in the solid bed is G ' G, given by [ c ( T T ) + ] = ( f ) G c ( T T ) ' s m s λ leak p m f m or ' G = ( f ) leak G p c s c m ( T T ) ( T T ) + λ From Equations 6-9 and 6-, we have m f s m

136 θb σ zx πn( Ri + Ri, o ) WRdθ ' 0 G = ( fleak ) G p Gp cs( Tm Ts) + λ Since G p is dragged from the solid bed in the solid state, the total melting rate, or disappearance rate of the solid bed, due to the inter-screw region should consider both G p (melting in the inter-screw region) and ' G (melting in the region between screw and barrel), given by G ' total = G p + G. θb σ πn( R + R, ) WRdθ G f G G zx i i o 0 total = ( ) leak p + p Gp cs( Tm Ts) + λ From the assumptions in the solid bed, we can express the total melting rate in the inter-screw region as, G dlc = WH dt total ρ s Then the melting rate of the solid bed can be shown as θ b N( R R, ) WRd dl G σ π + θ c = + dt ρ WH G c T T zx i i o p 0 ( f ) leak s p s( m s) + λ When there is no backward leakage, we have θ b ( +, ) σ zx πn Ri Ri o WRdθ dlc 0 = dt ρswh cs( Tm Ts) + λ

137 6.4 Model of Melting between Screw and Barrel (the C-chamber) The schematic of the solid bed in the C-chamber is shown in Figure 6., where, and 3 represent directions in the direction of screw channel, channel depth and channel width, respectively. With screw rotating, the solid bed is conveying along direction. There are two main heating resources in the C-chamber, shear heating and heat transfer from the hot barrel. 3 Barrel Melt Barrel U θ Melt -v s v s Screw Solid bed Screw Solid Figure 6. The melting process between screw and barrel The melting model by Tadmor [8] was applied to analyze the melting process between screw and barrel. Heat balances on the solid bed, at the interface between the 6

138 melt and solid layer, and on the melt film were constructed to determine the melting rate contributed by the C-chamber region. The melting rate is given by [] v s h = b ( T T ) b m σ ρ s melt x = 0 U [ λ + C ( T T )] s σ m 3 melt x = 0 s U 3 q m p x where λ is the heat of fusion, ρ s is the bulk density of the solid bed, C s is the specific heats of the solid, T m, T s and T b are the temperatures of melting point, the solid bed and barrel, respectively. Shear stresses σ and σ melt melt x = 0 3 x = 0, as well as flow rate in direction, q m, can be determined by applying the hydrodynamic lubrication theory in the melt film layer [69]. Densification occurs during the melting process, so the pressure becomes atmospheric, even for the cases under flood fed conditions. Equation 6-7 becomes v s = h b ηo + n + n ( T T ) + [ U + 4U ] b ρ m s H n m [ λ + C ( T T )] s m s A mass balance on the solid implies that dm dt s = ρ v WL s s c where m s is the mass of the solid bed in the screw channel, given as we have m s = ρ WHL. So s c dlc Lc = vs dt H 7

139 6.5 Calculations and Discussions Both mechanisms are considered in the calculations. Combining Equations 6-6, which arises from melting in the inter-screw region, and Equation 6-0 which is associated with the barrel induced melting, the disappearance rate of the solid bed is given as dl dt c = v s Lc H G p + ρ swh ( f ) leak θb σ 0 G p zx NR WRdθ + s ( π ) [ C ( T T ) + λ] s m where vs is given in Equation 6-8. Iterations were used to obtain the solid bed length left. From Equation 6-, it is found that material properties, operating conditions and geometrical data all can affect the melting rates. Among these parameters, operating conditions (screw speeds and feed rates), screw geometrical parameters and friction coefficient at screw surface are of critically important, so parameter studies were carried out to investigate their effects on the melting process. The detailed values are given in Table 6., where a thick flighed element (FD--30, shown in Table 3.) was used as the default design to study the effect of geometrical parameters, material properties was assumed based on HDPE pellets, and three different friction coefficient at screw surface were used as well. 8

140 Table 6. Material properties, operating conditions and geometry used in calculations Quantity Value Density of the solid bed, ρ, g/cm b Density of melt, ρ, g/cm Specific heat of the solid bed, C s, J/g o C. Specific heat of melt, C m, J/g o C. Melting temperature, T o m, C 35 Heat of fusion, λ J/g 45 Friction coefficient at screw, f s Solid temperature, T s, o C 40 Barrel temperature, T b, o C 70 Heat transfer coefficient at barrel, h b, W/cm 0.05 Polymer fraction transported as pellet bed, k p 0.9 Location of nip, Z nip =RS, cm 0.6 Diameter of barrel, R o, cm 3.4 Diameter of screw root, R i, cm.6 Pitch of screw, S, cm 3.0 Viscosity constants, K, Pa/s n 4800 Power law index of melt, n 0.5 Width of screw channel, W, cm.5 Depth of screw channel, H, cm Calendering gap,, cm Rotation Speed, rpm 60 0 Feed Rate, Kg/hr Leakage, f leak, % *Material properties were assumed, which is based on HDPE pellets used in experiments (Chapter III). **Geometrical parameters were based on a thick flight element FD--30. *** Operating conditions were assumed. **** Friction coefficient at screw surface was assumed based on references [8, 70] 9

141 6.5. Effect of Feed Rate on The melting process Figure 6.3 shows the effect of feed rates on the melting process, where filled symbols represent cases by considering both melting mechanisms, while unfilled symbols are cases by taking into account only the melting mechanism in the inter-screw melting region. The number of turns is obtained by rotation speed, N, times operating time, t. The non-dimensional length of the solid bed is defined by the actual length divided by the original length. When both melting mechanisms are considered, more turns or number of pitches are required to melt all the solid bed, with the increase of feed rates. Similarly results were obtained for only one melting mechanism inside the inter-screw region considered..0 Nondimensional Length of Solid Bed Kg/hr 5 Kg/hr 0 Kg/hr 0 Kg/hr 5 Kg/hr 5 Kg/hr Number of Turns Figure 6.3 Effect of feed rates on the melting process at screw speed of 60 rpm 0

142 Pressure gradients in the screw channel are negligible in the development of modeling, the effect of melting mechanism in the inter-screw region remains the same quantitatively, unless the inter-screw regions are not filled with materials. Comparing the melting process at varied feed rates, it is seen that the effect of the C-chamber melting become more and more important with the increase of feed rates. That is due to longer filled screw channel at higher feed rate. If the pressure gradient is not negligible in this calculation, we could expect a more obvious trend of this kind. The reason is due to a higher intense shear heating accomplishing with more filled channel area in the case of higher feed rate Effect of Screw Rotation Speed The effect of screw rotation speed was studied. Figure 6.4 shows the melting process at various screw rotation speeds, where filled symbols represent cases by considering both melting mechanisms, while unfilled symbols are calculations involving only the melting mechanism in the inter-screw melting region. Melting becomes faster and is finished earlier with the increase of screw rotation speeds. The total number of pitches involved in melting becomes smaller at higher screw speeds. The reason is due to more friction and shear heating generated at higher speeds. Another important effect at higher screw speeds is the fill factor. At the same feed rate, the fill factor decreases with rising screw speeds. So the filled screw channel length becomes shorter at higher screw speeds, and it requires less energy for melting.

143 Nondimensional Length of Solid Bed N = 60 rpm N = 60 rpm N = 90 rpm N = 90 rpm N = 0 rpm N = 0 rpm Number of Turns Figure 6.4 Effect of screw speed on the melting process at feed rate 0 Kg/hr, where filled symbols represent both mechanisms and unfilled symbols are for the melting process considering the calendering effect only Effect of Leakage Leakage is another important issue in investigating counter-rotating twin-screw extrusion. Due to the complex geometry, there are mainly four kinds of leakage. Even in fluid mechanics analysis, it is difficult to characterize leakage. To be simplified, only leakages in the calendering region and the inter-flight region were considered and represented as one factor, f leak. Leakage lowers both pumping capacity and melting efficiency. In addition, the solid bed length would be longer under leakage condition. Three different leakage levels were taken into account, 0%, 0% and 0%. Figure 6.5 shows the effect of leakage to the melting process, without considering the effect of

144 leakage on the fill factor. There is no doubt that a longer melting length is needed as leakage becomes worse. Checking back to Equation 6-4, it is found certain modifications are necessary to the solid bed length. According to the C-chamber positive displacement theory, to have the same transportation rate, more material should be filled into the C-chamber, under the condition of leakage, which means the length of the solid bed becomes longer. Inserting the effect of leakage, Equation 6-4 becomes L c kg = ( f ) ρ NWH leak b Nondimensional Length of Solid Bed No Leakage 0% Leakage 0% Leakage Number of Turns Figure 6.5 Effect of leakage on the melting process at G = 0 Kg/hr and N = 60 rpm 3

145 Results with this modification are shown in Figure 6.6. To have the same melting degree, a longer melting region is required, which means the efficiency of the melting process was lowered..0 Nondimensional Length of Solid Bed % leakage without modified L c 0% leakage with modified L c Number of Turns Figure 6.6 Effect of leakage on the melting process at G = 0 Kg/hr and N = 60 rpm Effect of Clearance on the Melting Process Due to the calendaring effect of intermeshing counter-rotating twin-screw extruders, generally high pressure is built up in the inter-screw region, which causes a bowing problem of a pair of screws and rapidly wearing of screw surface and barrel inner surface and then shortens the life of machines. So the clearance between screw flight and barrel surface, or between one screw root and another screw flight, should be designed in reasonable range. 4

146 Figure 6.7 shows the effect of clearance on the melting process, where dimensionless clearances are the calendering gap divided by screw radius, as given in Table 6.. With the increase of clearance, the length of the melting process becomes longer. That is because pressure build-up in the inter-screw region drops quickly and hence the friction heating is reduced. If leakage is considered, much longer melting length would be expected for large clearances. In practical operating, leakage increases much along a larger clearance, which means more screw channel areas are filled with materials and the solid bed length is increased at higher leakages with same feed rate..0 Nondimensional Length of Solid Bed Clearance = 0.0 / 7 Clearance = 0.5 /7 Clearance = 0.0 / Number of Turns Figure 6.7 Effect of clearance on the melting process at G = 0 Kg/hr and N = 60 rpm Effect of Channel Depth Channel depth is one of the most important factors in the design of screws and screw extrusion machines. Figure 6.8 shows the effect of channel depth on the melting 5

147 process at the same screw speed and feed rate, where dimensionless channel depth is defined as the screw channel depth divided by screw radius. To make machines still fully intermeshing, the root diameter of screws should increase, when keeping screw outer diameter and lower channel depth. In the study of the effect of channel depth, without considering leakage on pumping material, the total area of the solid bed length and depth remains the same. Reducing channel depth brings the increase in the solid bed length, which means more shear heating occurs between screw and barrel. Since the feed rate does not change, a shorter screw length is needed to melt solid materials..0 Nondimensional Length of Solid Bed H ' = 4 / 7 H ' = 3 / 7 H ' = / Number of Turns Figure 6.8 Effect of channel depth on the melting process at G = 0 Kg/hr and N = 60 rpm 6

148 6.5.6 Effect of Friction Coefficient on Screw Surface Friction coefficient on screw surface determines the buildup of pressure at the inter-screw region. Our observations showed that the inter-screw region plays a dominant role in the melting process, so it is meaningful to investigate the effect of friction coefficient on screw surface. Figure 6.9 shows the screw lengths for melting at different friction coefficient on screw surface. It was found a longer screw length is required for a smaller friction coefficient on screw surface. With the increase of friction coefficient on screw surface, the length for melting becomes shorter. The reason is due to larger friction heating at higher friction coefficients..0 Nondimensional Length of Solid Bed f s = 0.0 f s = 0.6 f s = Number of Turns Figure 6.9 Effect of friction coefficient at screw surface on the melting process at G = 0 Kg/hr and N = 60 rpm 7

149 6.6 Conclusions The melting process was modeled and a parameter study was conducted. It was found that higher screw speed and lower feed rate both help the melting process to be finished sooner. Leakage delays the melting process and causes longer screw length to finish the melting process. Larger clearance between screw and barrel or between the screws decreases heat dissipation rapidly, and hence delays the melting process. The screw channel depth also affects the melting process. Shallower channel helps the melting process to finish sooner. It lowers, however, the material transport efficiency at the same machine size. So a balanced design of screw channel depth should be considered. In modeling the inter-screw region, the solid bed was assumed to determine the pressure field. Higher intense shear or friction occurs in this region, materials may experience deformation or show elastic-viscous property. So pressures may be in a lower level and not so much friction work is done. The melting process in the inter-screw region is complex. Friction work on solid polymer particles and shearing heating may both occur. The former could dominate at the beginning step of the melting process. With the progress of the melting process, shearing heating could become a main reason for melting, which is due to less solid polymer particles in the inter-screw region. However, this is too complicated and almost impossible to be clarified. Only the friction work was considered here. 8

150 CHAPTER VII MELT CONVEYING IN INTERMESHING COUNTER-ROTATING TWIN SCREW EXTRUDERS 7. Introduction Among the three regions of counter-rotating twin screw extrusion process, metering operating holds a special position. Mixing and pumping mainly occur in this region. To characterize the performance of intermeshing counter-rotating twin-screw extruders, it is most important to develop a model for the metering region. The most comprehensive metering model of this machine has been presented by Hong and White [3, 4]. The pumping capacity of screw elements can be predicted from this metering model. In this chapter, the basic idea of Hong and White s model [3, 4] is applied to investigate the effect of geometry on pumping capacity, such as helix angle, clearance, flight width and channel depth. In addition, the effect of aspect ratio (screw channel depth / channel width), which was neglected in Hong and White s model [3, 4], is investigated in this chapter. The purpose of this study is to investigate the pumping behavior of screws, and supply useful information for optimizing the design of screws. 9

151 30 Nonisothermal non-newtonian fluid flow behavior is considered. The viscosity behavior is expressed with a power law fluid model, and is formulated in a cylindrical coordinate system. In a Cartesian coordinate system, the viscosity is ( ) = n T T b x v x v e K o η (a) and in cylindrical coordinates, ( ) 0 + = n z T T b r v r r r v e K o θ η (b) where K o is a viscosity consistency, b is temperature sensitive index, T o is a reference temperature, n is the power law index. 7. General Metering Model The two major screw elements associated with pumping capacity are shown in Figure 7.. They are thick flight and thin flight elements. Other specially designed elements are also applied in machines, such as the modular Leistritz machine shear elements.

152 Figure 7. Modular Leistritz elements, (A) thick flight and (B) thin flight. When we study the cross section of screw elements in the machine, we find the geometry shown in Figure 7.. This shows the cross-section as divided into two regions: one is between screw and barrel termed Region I, and the other is the inter-screw region which we call Region II. As for Region I, a coordinate system is fixed on the screw surface moving with the same velocity as the screw, and a coordinate system fixed in space is established for Region II. Hydrodynamic lubrication theory is utilized and the Flow Analysis Network (FAN) method is used to obtain results numerically. 3

153 Figure 7. Calculated domains for screw elements 7.. Model for the Region between Screw and Barrel The region between screw and barrel is named as Region I. A cylindrical coordinate fixed on the root of the screw and moving at the same velocity of screw rotation is mounted in each screw, shown in Figure 7., where direction z is along the screw direction, r is normal to the axis in the radial direction and θ is in the circumferential direction. When neglecting the flow in radial direction, the velocity field has the form () r ez v θ () r e θ v = v z + 3

154 33 The continuity equation is 0 = + z v v r z θ θ The force balances for a lubrication flow are [9] ( ) + = + = r v r r r r p r r r r p r r θ θ η θ σ θ a ( ) + = + = r v r r r z p r r r z p z zr η σ b With the coordinate system fixed in the screw-root channel, the boundary conditions on the screw root and the barrel are R i r =, 0 = = z v v θ a R o r =, θ θ π U DN v = = b R o r =, 0 = z v c where D is the diameter of the barrel and N the screw rotation speed. We may integrate Equations 7-4 to give the flow fluxes as ( ) dr r c p r r R dr v q o i o i R R o R R + = = θ θ θ θ η a ( ) dr r c r z p r R R dr rv R q o i o i R R z o R R z z + = = * * η b where the viscosity function of Equation 7-b may be rewritten using Equations 7-4 ( ) n n z T T b r c r z p r c p e K o = θ θ η

155 7.. Model for the Inter-screw Region between One Screw Root and the Other Screw Flight There may be two different zones in the inter-screw region, which is determined by the type of screw elements. For thick flight screw elements, it is only between one screw flight and the other screw root. In addition to this zone, thin flight screw elements have another zone between two screw roots. For the latter case, we discuss in Section For the region between one screw flight and the other screw root, the governing equations are the same forms as above in Region I. A cylindrical coordinate fixed in space is constructed. So the velocities between two coordinate systems have the following relationships: v = v a z I z II v = v + πrn b θ I θ II which are applied in flux balance. The effect of positive displacement will be discussed in details in Section 7..4 with the flux balance. The boundary conditions are given as, r =, v = a R i z r = R i, vθ πri N = U θ R o = b r =, v = c z vθ = πr N = U d r = R o, o, s θ where R o, s refers to the radius of opposite screw at this position. Similarly, the velocities are derived 34

156 35 dr r c p r r R r U v r R i i + + = θ θ θ θ η a dr r c r z p v r R z z i + = η b From the relationship of the velocity components in this inter screw region (region II) to the screw barrel region (region I) shown in Equation 7-8b, the flow fluxes in the inter screw region with a transformed format for flux balances between region I and region II are, ( ) ( ) ( ) ( ) dr r c p r r R R R U R R R U dr U v q o i o i R R o i o i i o R R + + = + = θ θ θ θ θ θ θ η a ( ) dr r c r z p r R R dr rv R q o i o i R R z o R R z z + = = * * η b 7..3 Model for the Inter-screw Region between both Screw Roots For the region between two screw roots, there is little positive displacement conveying of the thick-flight element. If we neglect the positive displacement conveying, we can use a cylindrical coordinate fixed on the screw surface. The governing equations are the same as the other regions. The boundary conditions are R i r =, 0 = = z v V θ a R o r =, 0 = z v b R o r =, ( ) ' θ θ π U N R R v i o = = c

157 36 The velocities z v and θ v become, dr r c p r r v r R i + = θ θ θ η a dr r c r z p v r R z z i + = η b and flow fluxes are ( ) dr r c p r r R dr v q o i o i R R o R R + = = θ θ θ θ η a ( ) dr r c r z p r R R dr rv R q o i o i R R z o R R z z + = = * * η b 7..4 Flux Balance The flow fluxes are balanced over each element of dimensions z and θ which is equivalent to the continuity equation as shown in Figure 7.3. Figure 7.3 Model mean flux balance

158 37 For the region I (between screw and barrel), it has the following form: θ θ θ θ θ θ θ θ θ θ + = * * ), ( ), ( ), ( ), ( R q z q R q z q z z I z z I z z I Z z I a For region II (inter screw), positive displacement should be considered. At the upper zone, fluid entering from the two screws should be accounted for θ θ θ θ θ θ θ θ + + * ( ), ( ), ( ), R q z q z q z z II Z z IL z IR θ θ θ θ θ + = + + * ), ( ), ( R q z q z z II z z II b Here IR q θ and IL q θ are the input fluxes from the right and left screws, respectively. Similarly at the lower zone, fluids exiting to the two screws should be accounted for, = + θ θ θ θ θ * ), ( ), ( R q z q z z II z z II θ θ θ θ θ θ θ θ * ), ( ), ( ), ( R q z q z q z z II Z z IL z IR c In the midst for thick-flight elements with positive displacement, ( ) θ θ θ θ θ θ θ θ + + * ), ( ), ( ' ), ( R q z R R U z q z z II z z i o z II ( ) θ θ θ θ θ θ θ θ + + = * ), ( ), ( ' ), ( R q z R R U z q z z II z z i o z II d where ( ) ( ) [ ] z R R U R R U z i o z i o + ), ( ), ( θ θ θ θ θ θ and ( ) i o II II R R U q q + = θ θ θ ' represent the positive displacement effect in the inter-screw region. The ideal case is that there is no leakage flow through the calendering gap where ( ) 0 ), ( = + z R o R i U θ θ θ or ), ( ), ( z i z o R R θ θ θ θ + + =. If there is no positive displacement, such as region I, ( ) ( ) ), ( ), ( z i o z i o R R U R R U θ θ θ θ θ θ + =.

159 In the midst for thin-flight elements with positive displacement, it has the same formation to Equation 7-5d. There is a region in the midst for thin-flight elements without positive displacement, flow rate Equations 7-4 are inserted into the flux balance, q II II θ ( θ θ, z ) z + qz ( θ, z z ) R * θ II II = qθ ( θ + θ, z) z + qz ( θ, z+ z) R * θ e Recalling the expression of flow fluxes in each direction, the balance equation is a function of pressure with position. When assuming the isothermal case, the Fixed Point Iteration Method [7] was used to solve the pressure field in this non-newtonian case, where Simpson s Composite Integration Method [7] was applied to obtain integration constants in each node. A flow chart is shown in Figure 7.4 for the computation. The results of computations by Hong and White [3, 4] are not be discussed here. 38

160 Newtonian Pressure Distribution P 0 (i,j) Calculate C θ0 C z, using P 0 (i,j) and C θ = C θ0, Cz, = C z0 P(i,j)= P0(i,j) n=, step Step=Step - Calculate new C θ, C z Iterate P(i,j) P(i,j) Convergence? NO NO YES n < n set? YES Record Stop Figure 7.4 Flow Chart for non-newtonian calculations 39

161 7.3 Energy Balance for the Metering Region The twin screw extrusion process is run under non-isothermal conditions; temperature variations play a significant role. In addition, with the increase of extruders in size, the machine volume per unit length increases with the square of the scale factor if keeping the same ratio of geometry, and the surface area of the screw and barrel which define cooling capability increases linearly. Internal viscous heat generation becomes increasingly important as the extrusion machines become larger. To obtain accurate predictions for the velocity profiles, pressure profiles and temperature profiles, force balance equations and energy equations should be coupled and solved together. However, this has a huge requirement for computer system and longer computation time. To be simplified, we determine the velocity profiles and pressure profiles by assuming isothermal case first and then calculate the temperature rise for each modular or pseudo element in a presumed composite intermesh modular counter-rotating twin-screw extruder. The effect of mean temperature rise is accounted for at the next element Model for the Region with Positive Displacement A rectangular coordinate system fixed on the screw surface moving with the same velocity as the screw is established in each screw. We develop the model for the region between screw and barrel first. The flow in radial () is neglected, so the velocity is expressed as, ( x ) e + e v 3 ( x ) 3 v = v 0 + e and it is assumed that there is no inertia and gravitational forces. 40

162 4 The force balance equations for lubrication flow are + = + = 0 x v x x p x x p η σ = + = x v x x p x x p η σ The energy equation has the form [7]: x v x v x T k x T v x T v C + + = + σ σ ρ where C is the heat capacity, ρ is the molten material density and k is thermal conductivity. The mean cup mixing channel temperature T can be determined by integrating Equation 7-9, to be given as [30], ( ) ( ) ( ) ( ) ( ) x p Q dx H U H U W E H W E x T CQ c W s b c = σ σ ρ where ( ) ( ) = W c dx x x T q Q x T 0 3 3, ; 0 q dx x T v x T H j j = ; = W c dx q Q 0 3 ( ) ( ) H x T k H E = ; () () 0 0 x T k E = π cosθ DN U = ; θ π sin 3 DN U = Equation 7-0 can be written as

163 ρcq + W 0 c T x ( σ ( H ) U + σ ( H ) U ) = W h b b ( Tb T ) + Wshs ( Ts T ) 3 3 dx 3 Q c p x where E ( H ) = W b 0 T k x W ( H ) b dx 3 = h b ( Tb T ) E W b T k () 0 dx3 x 0 () 0 = = h T T W s s s and h b and h s are barrel and screw heat transfer coefficients. When integrated with respect to x, the temperature rise in channel direction is obtained. It should be noted that the shear heating is more intensive in the inter-screw region, which accounts for the main heat dissipation. The mean cup mixing temperature in the inter-screw region is given as ρ T ( H ) calwos, cal + E( 0) calws cal CQ cal = E, x W p + cal σ x 0 cal ( σ ( h) U + ( h) U ) dx Q If we assume the temperatures of both screws are identical, the heat transfer terms may be expressed as: E ( H ) cal E( ) cal = hs ( Ts T ) = cal 4

164 It is noted that as the channel widths of both screws are identical, we have ρ T = h ( Ts T )( Wos, cal + Ws cal ) CQ cal s, x W p ( ( h) U ( h) U ) dx Qcal x + σ + σ cal Combining the two regions together, we obtain the mean cup mixing temperature for thick-flight screw elements, CQ ct CQcalT cal = C c + ( Q Q ) T out cal cal ρ + ρ ρ Model for the Region without Positive Displacement For the case of thin-flight elements, there are two different local geometries in the inter-screw region as shown in Figure 7.. First is the closely intermeshed region between the flight of one screw and the root of the other one, which is the same to the case in thick-flight element with positive displacement. The other is the region between the screw roots of both screws, where there is no positive displacement. For the former case, the same method to the region with positive displacement is applied to the analysis of flow field. For the latter case, a cylindrical coordinate moving on the screw is built as the case of screw-barrel region. A cylindrical coordinate system fixed on the screw surface moving with the same velocity as the screw is established in each screw. The flow in the radial direction is neglected, so the velocity is expressed as, () r ez v θ () r e θ v = v z + 43

165 44 The force balances for a lubrication flow are ( ) + = + = r v r r r p r r r r p r r θ θ η θ σ θ a ( ) + = + = r v r r r z p r r r z p z zr η σ b The energy equation has the form: r v r v r r r T r r r k z T v T r v C z rz r z + + = + σ σ θ ρ θ θ θ Integration of equation 7-30 through the r and θ direction, the energy balance becomes, ( ) ( ) [ ] ( ) z p Q d U R R T T R h T T h R f z T CQ z o f r o i i i b b o c z = + θ σ π ρ θ π θ ) ( 0 ) ( 7-3 where c f represents the fraction of the region between screw and barrel, give as α π = c f, and θ θ π π d R q d R z T q z T ch f z ch f z + + = ) ( 0 ) ( 0 ; θ π d R q Q ch f z z + = ) ( 0 ; z R R z q dr z T rv z T o i = ( ) ( ) ( ) T T h R R R r T R k R R E b b ch o o o ch o = = ; ( ) ( ) ( ) T T h R R R r T R k R R E s s ch i i i ch i = = = o i R R dr v q θ θ = o i R R z ch dr rv R q θ DN U π θ =

166 7.4 Considerations fo Screw Design Screw pumping capacity and cup mixing temperature development are predicted by the previously described metering model. The relationship between throughput and pressure gradient can be determined by the metering model, and screw characteristic curves determined as well. These are used to represent screw pumping capacity for different viscous properties of materials. Theory of the metering model is also helpful to optimize the design of screws. The screw schematic is given in Figure 7.5, where D is the screw diameter, H is screw channel depth, δ is the clearance between screw flight and barrel surface, as well as one screw flight and another screw root, φ is helix angle, S is the length of a pitch, B is channel width along screw axis, and W c is the width of screw channel. The flight width along screw axis, W f, can be determined from pitch and channel width, given as ( S ). The above mentioned parameters are all significant in designing screw elements. We can expand the application of the metering model to investigate the effect of these parameters on screw pumping capacity. Wc 45

167 Figure 7.5 Schematic of one screw element of counter-rotating twin screw extruders The effect of clearance, channel width and depth, and helix angle are represented from Figure 7.6 to Figure 7.8. A power law non-newtonian viscosity model was used. Geometrical parameters are summarized in Table 7.. In drawing these figures, dimensionless forms for throughput and pressure gradient were used, and defined as Q * Q = WHL N c n+ * H cosθ p = n K L ( p L) ( π DN) where L c is one screw channel length and L is the screw length. Table 7. Parameters used in studying screw pumping capacity Parameters Dimensionless Values Aspect Ratio of Channel Depth / /5 4/5 6/5 Width ( H/W c ) Helix Angle φ 5.70 o 3.8 o 0.6 o Clearances between screws( δ / D ) 0./34 0.4/34.0/34 Ratio of Flight Width/Channel 5/5 0.5/9.5 6/4 Width (W f /W c )* *where flight width is defined: W = S W f c 46

168 7.4. The Effect of Ratio of Flight Width/Channel Width The ratio of flight width/channel width is one important parameter in determining pumping capacity. The effect of the ratio of flight width/channel width was investigated. The ratios of flight width/channel width were taken as 5:5, 0.5:9.5 and 6:4. When the ratio is 5:5, screw element is thick flight element. The other two cases are thin flight elements. The pumping capacity for three different designs is determined by the analysis in Section 7., and given in Figure 7.6, where the non-newtonian power law index n is 0.5. Thick flight elements were found to have a much better pumping capacity, compared with thin flight elements. However, considering the throughput shown in Figure 7.6 is in dimensionless form, as defined in Equation The actual dimensional throughput for thin flight screws keeps in a relative high level, especially at smaller pressure gradient. For example, when the dimensionless pressure gradient is 0., the dimensional throughput of a screw element with the ratio of flight width/channel width at 6:4 is around half of that with a ratio of 5:5, when transformed into dimensional form. To have the same output, higher screw rotation speed is required for thin flight elements, which causes higher torques and higher power consumptions. Decreasing the pressure gradient, thin flight screws gain better pumping capacity. Figure 7.7, Figure 7.8 and Figure 7.9 show the similar effect of screw channel width on pumping capacity at power law index n=0.3, n=0.7 and n=.0, respectively. Compared with n=0.5 shown in Figure 7.6, pumping capacity decreases with power law index for all three different screw design. The reason is due to shear thinning. At the 47

169 same shear rate, the fluid with lower power indexes has a smaller viscosity, which causes higher leakage and weakens the pumping capacity. Another important parameter in characterizing the screw design is the distribution of residence time. Due to positive displacement, thick flight screws have a narrower distribution of residence time, while the residence time distribution for thin flight screws are broader. Dimensionless Throughput, Q * W f / W c = 5.0 /5.0 W f / W c = 0.5 /9.5 W f / W c = 6.0 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.6 The effect of the ratio of flight width/channel width on pumping at n=0.5 48

170 Dimensionless Throughput, Q * W f / W c = 5.0 /5.0 W f / W c = 0.5 /9.5 W f / W c = 6.0 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.7 The effect of the ratio of flight width/channel width on pumping at n= Dimensionless Throughput, Q * W f / W c = 5.0 /5.0 W f / W c = 0.5 /9.5 W f / W c = 6.0 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.8 The effect of the ratio of flight width/channel width on pumping at n=0.7 49

171 Dimensionless Throughput, Q * W f / W c = 5.0 /5.0 W f / W c = 0.5 /9.5 W f / W c = 6.0 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.9 The effect of the ratio of flight width/channel width on pumping at n= The Effect of Screw Clearance Clearance refers to the gap between screw flight and barrel surface, as well as the gap between one screw flight and the root of another screw. According to the requirement of this being a real machine, the clearance cannot be zero. Large values of the clearance lower the pumping capacity. The reason is due to higher leakage with the increase of clearance. A thick flight element, FD--30, was used to investigate the effect of clearance on pumping capacity. The molten polymer was represented as a non-newtonian powerlaw fluid. Four different power law indexes, 0.3, 0.5, 0.7 and.0, were used to study the clearance effect on screw pumping capacity. Screw characteristic curves for three different levels of clearance were given from Figure 7.0 to Figure 7.3, where the ratios of screw clearance to screw diameter are 0.05/34, 0.0/34 and 0.50/34. 50

172 It is seen that the pumping capacity decreases with an increase of clearance for all three cases. Even at smaller pressure gradient, screw design with the lowest clearance has the highest throughput. Helix angle, channel depth and width are all the same for three cases, it represents that small clearance is necessary to have the best pumping. So a balanced value of clearance should be introduced in designing extrusion machines, by combining the requirement of machine manufacture and operating conditions..0 Dimensionless Throughput, Q * δ /D = 0.05 /34 δ /D = 0.0 /34 δ /D = 0.50 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.0 The effect of clearance on screw pumping capacity at n=0.3 The effect of power law index on pumping capacity may be seen by the comparison of Figures 7.0 to 7.3. With an increase of power law index, the screw element has a better pumping capacity. This is the same as for single screw extruders and co-rotating screw extruders. 5

173 Dimensionless Throughput, Q * δ /D = 0.05 /34 δ /D = 0.0 /34 δ /D = 0.50 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7. The effect of clearance on screw pumping capacity at n=0.5 Dimensionless Throughput, Q * δ /D = 0.05 /34 δ /D = 0.0 /34 δ /D = 0.50 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7. The effect of clearance on screw pumping capacity at n=0.7 5

174 .0 Dimensionless Throughput, Q * δ /D = 0.05 /34 δ /D = 0.0 /34 δ /D = 0.50 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.3 The effect of clearance on screw pumping capacity at n= The Effect of Aspect Ratio (Channel Depth / Width) The effect of aspect ratio on the performance of screw machine is shown in Figure 7.4 to Figure 7.6. If the channel width is fixed, a shallow channel depth means a smaller aspect ratio. Compared with the results shown in Figure 7.4, it is found that a most shallow screw channel with H/D at /34, has a highest dimensionless throughput. When the dimensionless channel depth increases to 6/34, the dimensionless throughput is still at a high value, around 0.75 at n=0.5. With an increase of power law index, the pumping capacity becomes better. Considering the definition of dimensionless throughput in Equation 7-33, a deeper screw channel could pump more materials at the same screw speed. However, the channel depth cannot increase infinitely. There are other 53

175 concerns, such as torque requirements, the potential shearing of screws, energy consumption and mixing. When fixed the channel width and screw diameter, a larger aspect ratio has a deeper screw channel. If the screw speed is also a constant, more materials could be pumped. But a higher pumping rate requires higher torque and more power consumption. Meanwhile, a deeper screw channel is at the expense of screw shaft, and the screw shaft becomes thinner, which means the tolerance of total torque becomes weaker. So a comprehensive consideration should be taken in designing screws and optimizing the aspect ratio. Dimensionless Throughput, Q * H / W c = / 5 H / W c = 4 / 5 H / W c = 6 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.4 The effect of aspect ratio on pumping capacity at n=0.5 54

176 .0 Dimensionless Throughput, Q * H / W c = / 5 H / W c = 4 / 5 H / W c = 6 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.5 The effect of aspect ratio on pumping capacity at n=0.7.0 Dimensionless Throughput, Q * H / W c = / 5 H / W c = 4 / 5 H / W c = 6 / Dimensionless Pressure Gradient, (dp/dx) * Figure 7.6 The effect of aspect ratio on pumping capacity at n=.0 55

177 7.4.4 The Effect of Helix Angle Helix angle can be defined as, φ tan S π D =, where S and D are the pitch and screw diameter, respectively. A best designed helix angle could improve the pumping capacity, as is proved in single screw extruders. However, counter-rotating twin screw extruders are running under positive displacement. So helix angle is not as important as in single screw extruders. Figure 7.7 and Figure 7.8 present the effect of helix angle on pumping capacity at different power law index. Compared with the results in Section 7.4., 7.4. and 7.4.3, it is seen that the power law index has a similar effect on puming capacity. With an increase of power law index, the pumping capacity becomes better. So two power law indexes, 0.5 and 0.7, are used in investigating the effect of helix angle on pumping capacity. The helix angles were taken as 5.70 o, 3.8 o and 0.6 o, respectively. It is found that helix angles have little effect on pumping capacity and all three cases have the similar behavior to our previously calculated screw characteristic curves. The reason is due to thick flight element considered here, which works under positive displacement in conveying molten materials with the rotation of screw. It is interestingly noted that dimensionless throughput seems to be higher at a middle value of helix angle, around 3. o, which corresponds to the ratio of pitch to diameter at 5/34. So there could be an optimum helix angle between 5.70 o and 0.6 o 56

178 which corresponds to the highest pumping capacity. Hence it is meaningful to optimize the helix angle to obtain the highest pumping capacity..0 Dimensionless Throughput, Q * θ = 5.70 o θ = 3.8 o θ = 0.6 o Dimensionless Pressure Gradient, (dp/dx) * Figure 7.7 The effect of helix angle on screw pumping capacity at n=0.5 57

179 .0 Dimensionless Throughput, Q * θ = 5.70 o θ = 3.8 o θ = 0.6 o Dimensionless Pressure Gradient, (dp/dx) * Figure 7.8 The effect of helix angle on screw pumping capacity at n= Effect of Aspect Ratio on Pumping The effect of aspect ratio (screw channel depth to width) was neglected in the above analysis with lubrication approximation, where the aspect ratio was presumed to be smaller enough, such as less than 0., and the fluid flow was considered to be between two infinite plates. However, some designs have deeper screw channel, where the ratio of channel depth to width may be above 0.. So it is meaningful to investigate the effect of aspect ratio on pumping capacity by introducing them into the analysis of lubrication approximation. 58

180 7.5. Basic Idea Compared with the region between screw and barrel, much less fluid is conveyed in the inter-screw region. So we only focus on the region between screw and barrel in investigating the effect of aspect ratio on pumping capacity. When the effect of aspect ratio was neglected, the analysis in Section 7. determines the flow flux in the region between screw and barrel, as shown in Equation 7- and given as q z = Ro Ro rvzdr = * * o R η Ri Ri R p cz ( R r ) r + dr z r Equation 7-35 shows that the flow rate is determined by the geometry of screw channel, material viscous properties, screw speed and pressure field. For a Newtonian fluid, the flow rate in screw axis direction can be simplified as B p Q = AN η z where A and B are the function of geometrical parameters. The first item in Equation 7-36 is the influence of drag flow, and the second is the role of pressure flow. To consider the effect of aspect ratio, we may introduce two factors, F d and F p for drag flow and pressure flow, respectively. Equation 7-36 becomes Q = ANF d B F η p p z The analysis above was constructed under Newtonian fluid flow assumption. However, generally polymer molten flow has non-newtonian behavior. Non-Newtonian should be taken into account. Equation 7-35 becomes approximately 59

181 q z = Ro Ro rvzdr = * * o R η Ri Ri R p cz ( R r ) F r + F dr p z d r where F d and F p are function of geometrical parameter (H/W) and material viscous properties (n, if power-law fluid). It is impossible to develop an analytical expression of F d and F p for power law fluids. So a 3-dimensional numerical model must be constructed to investigate the effect of flights. We address how to determine the two geometrical modification factors in the following discussion Determination of Real Flow Rate in Three Dimensional Model Screw channel is unfolded and a rectangular coordinate system is constructed. Screw is fixed in space and barrel is moving at a velocity of π DN, shown in Figure 7.9. The velocity is expressed as, ( x, y, z) e + v ( x, y, z) e v ( x, y, z) e 3 v = v x y + z The continuity equation is v v x y vz + + = x y z 60

182 6 Figure 7.9 The schematic of channel flow when considering flights effect. When neglecting inertia and gravitational forces, the force balances are z y x x p zx yx xx = τ τ σ 0 z y x y p zy yy xy = τ τ τ z y x z p zz yz xz = σ τ τ 0 where γ η σ & = yz and η is a non-newtonian viscosity function which is a function of both the shear rate and temperature, which may be taken as follows: ( ) = n T T b x v x v e K o η Boundary conditions are given as L V V z H W Y Z X V x α

183 v x = vy = vz = 0 at x = W or x = W a v v = v = 0 at y = 0 ; x = y z v x = πdn sinφ, at v = 0, and v z = πdn cosφ at y = H b y p = P at z = 0 ; p = P at z = c The energy equation has the form: T T T T T T C v v v k ρ + + q& = x y z x y z where q& is shear heating and boundary conditions are T = T o at x = W or x = W a T = at y = 0 and y = H b T o T =, at z = c T We applied the above model in commercial CFD software (Fluent), Fluent, to obtain the flow rate in screw channel direction, Q. Fluent is CFD commercial software designed for fluid flow simulation with finite element method Determination of Flow Modification Shape Factors The real flow rate in 3-dimensional model is obtained in Section Now we discuss on how to determine the corresponding flow rate under lubrication approximation. The same geometry shown in Section 7.5. is used. When making the lubrication approximation and the effect of walls is neglected, the boundary at both walls become, 6

184 dvx dx dv y = dx dvz = dx = 0 at x = W or x = W dt dx = 0 at x = W or x = W The other boundary conditions are the same. We obtain a corresponding flow rate in screw channel direction, Q` with Fluent, the same software used in in Section The ratio of Q`/Q is applied to modify the prediction from lubrication theory. Figure 7.0 and Figure 7. show how to determine the modification shape factors, F d and F p from above calculations. Figure 7.0 represents the effect of geometrical parameter, H/W on the shape factors. It is seen that with the increase of aspect ratio, H/W, both modification factors decrease from.0, which means that the output is overpredicted by lubrication models when the aspect ratio is at a higher value. When H/W is less than 0., both of the modified factors are above 0.9. Further lower the value of H/W, both factors approach to one. So the effect of aspect ratio should be considered in predicting the output, especially when H/W is larger than 0., as shown in Figure 7.0. With the results shown in Figure 7.0 and Figure 7., F d and F p can be determined by fitting an expression, to this data. This was found to be = n 0. 4 F d H W a 0. H = 0.66n b W F p 4 With the above expressions, we can insert F d and F p in our analysis in Section of 7.. Some results are presented from Figure 7. to Figure 7.4. The predictions of 63

185 output of the cases without flights effect are lower than the cases with flights effect. However, the degrees for modification are different at varied screw elements, due to the different aspect ratios of screw channel depth to width..0 F d 0.8 F p Modification Factors, F d, F p H/W Figure 7.0 Modification factors vs. H/W by considering the aspect ratio effect (n=0.4) 64

186 Modification Factors, F d, F p Power Law Index, n F d F p Figure 7. Modification Factor vs. power-law index with the H/W effect (H/W=0.4) 0.4 Dimensionless Throughput, Q * n = 0.3 W ithout Aspect Ratio Effect With Aspect Ratio Effect n = 0.5 n =.0 n = Dimensionless Pressure Gradient, (P/L) * Figure 7. Screw characteristic curves for screw FF--30 with /out the aspect ratio effect (H/W = 4/4) 65

187 Dimensionless Throughput, Q * n = 0.3 n = 0.5 Without Aspect Ratio Effect With Aspect Ratio Effect n = 0.7 n = Dimensionless Pressure Gradient, (P/L) Figure 7.3 Screw characteristic curves for FD--30 with /out the aspect ratio effect (H/W = 4/5) Dimensionless Throughput, Q * n = 0.3 n = 0.5 Without Aspect Ratio Effect With Aspect Ratio Effect n = 0.7 n = Dimensionless Pressure Gradient, (P/L) Figure 7.4 Screw characteristic curves for FD--0 with /out the aspect ratio effect (H/W = 4/0) 66

188 7.6 Summary The main procedures to generate screw characteristic curves are presented in this chapter. Cup-mixing temperature and screw pumping capacity can be determined from these calculations. Our method could be applied in generating screw characteristic curves of general screw designs, which will be used in the development of integrated global model (Chapter VIII and IX). The simulations can help to optimize the design of screw geometry, and study the effect of shear stresses, energy balance and others in the process of scale-up. In the metering model, the effect of screw clearance, flight width, aspect ratio and helix angle were investigated to characterize the behavior of screw pumping capacity. It was found that screw clearance has a significant effect on screw pumping capacity. Smaller clearance brings in higher throughput. Screw flight width, however, has a dominant effect on pumping capacity. Thick flight produces higher pumping capacity. There is an optimum helix angle for the maximum pumping. In lubrication approximation discussed above, the aspect ratio was assumed to smaller enough, and not included into the analysis. For deep screw channels, however, that does not hold the truth. So we investigated the effect of the aspect ratio on pumping capacity. Different screw designs were used in our study. Results were calculated both cases with and without the effect of aspect ratio. It was found that the effect of aspect ratio should not be neglected for some screw designs, such as a channel with larger aspect ratio. So modification should be taken to improve the prediction for screw pumping capacity. 67

189 CHAPTER VIII SIMULATION OF GLOBAL BEHAVIOR OF INTERMESHING COUNTER- ROTATING TWIN SCREW EXTRUDERS 8. Introduction To characterize the performance of an intermeshing counter-rotating twin screw extruder, a complete analysis of the whole process from the hopper till the die, is required, where solid conveying, the melting process and metering are combined into an integrated global model. The important information and parameters, such as pressure profiles, temperature profiles, fill status, melting status, and other concerns along the axis of the twin screw machine, should be predicted by this global model. In this chapter, we present the basic procedure on how to develop an integrated global model to characterize the global performance, by combining the solid conveying model (Chapter V), the melting model (Chapter VI) and the metering model (Chapter VII). With this integrated global model, we made calculations with different screw designs and operating conditions, and compared the predictions with experimental studies presented in Chapter IV and the literature. 68

190 8. Computation Procedure for Global Model In industry, intermeshing counter-rotating twin screw extruders are run under both starved metered feed and flood fed conditions where the feed rate is unknown. The feed system is determined by the application. The compounding machine is generally run under starved metered feed, and has a modular design. Different screw elements are used to assemble different screw configurations. The other is non-modular design, where screws are machined as a unit and cannot be changed, as shown in Figure 8. where the pitch, helix angle, flight thickness and other screw geometry may be different along the screw direction. This type of machines is run under flood fed conditions and applied in PVC profile extrusion. In our model, we transform machined screws into a pseudo-composite machine and divide the whole screw into several parts, where each part shares the same geometry. If the length of a screw element is several times of the pitch, we divided this type of elements into several parts, with one pitch or less as a unit. The pseudo-c chambers are connected and the pressure is continuous. So the different sections are pseudo screw elements in a modular design. For a composite or a pseudo-composite twin screw extruder consisting of a number of elements in series, the total pressure development is p = N i= p i where i represents the individual element and N the total element number. The above equation specifies the characteristics of a composite or a pseudo-composite twin screw extruder when elements are fully filled. This may be seen to be the case by specifying the 69

191 temperature at the entrance to the die and using the above equation to calculate the pressure profile back along the length of the screw axis. When the pressure falls to zero in the midst of the extruder, this represents the occurrence of starvation. If all the extruder screw channels are fully filled, the pressure only falls to zero, or a pre-set pressure at the hopper. This corresponds to some situations under flood fed. V φ H D V W c S Figure 8. Schematic of the closed C-chamber in solid conveying If starvation occurs, it continues until a non-pumping element or left-handed element which requires a pressure gradient in the flow direction. This could be a mixing 70

192 element. The screw elements in back of these elements are in turn fully filled until the positive pressure gradient they induce again reduces the pressure to zero, producing starvation. This variation of filled and starved sections continues back towards the hopper. This is a normal phenomenon under metered starved fed conditions. When flood fed, the element directly under the hopper is fully filled. It generally becomes starved at the initiation of the melting processes, due to the densification of materials, as shown in Chapter IV. The starvation continues until the situation discussed above occurs and elements are fully filled again. Here the flow rate Q is simply Q = φiq max where φ i is the extent of fill in the screw channel. The above backward calculation was based on isothermal conditions. However, the extrusion process is run under non-isothermal conditions. To obtain an accurate prediction on the velocity profiles, pressure profiles and temperature profiles, force balances equations and energy equations should be coupled and solved together. However, this involves a huge requirement for a computer system and long running time. To simplify this, we determine fluxes rather than velocity profiles and pressure profiles by assuming the isothermal case first for each element, module or pseudo-module, and then calculate the mean cup-mixing temperature rise for each isothermal element in a composite intermeshing modular counter-rotating twin-screw extruder using an iterative procedure. The effect of mean temperature rise is accounted for at next iteration. Our approach is to calculate the pressure and fill factor profiles in a backward manner (from the die toward the hopper), and calculate energy balance forward (from the 7

193 hopper toward the die), but each chosen section, element or pseudo-element, is analyzed using the isothermal model. The heat buildup and the temperature change thus calculated are used to evaluate the viscosity for the next element. The problem in this approach is that the boundary conditions for the pressure profile and the temperature profile are not at the same end of the screw. The pressure at the end of the screw is known (the output at the end of the screw is known and we have the die head characteristics) while the temperature of the feed material is generally at room temperature. The general procedure to accomplish this is summarized, as shown in Figure 8.:. Specify the feed rate for starved feeding. Apply the solid conveying model to predict solid conveying region. When flood feeding, we must determine the feed rate.. Apply solid conveying and melting models to predict the solid and melting regions, and determine the fill status, the position of melting. 3. Apply the die geometry, estimate the material viscosity and use the throughput to determine the die entrance pressure. 4. Backward calculation of pressure and fill factor profiles for each element, by combining screw characteristic curves and the flow rate. 5. Forward calculation of temperature profiles for each element with updated pressure and fill factor profiles, from the position where melting is finished, until the end of screw. 6. Repeat 4 and 5 using updated results, until convergence is reached. 7

194 The detailed outline for backward calculation in the melting conveying region is as follows:. Using the known output to obtain the die pressure as P exit at the last element exit for an arbitrarily selected temperature.. Calculate the pressure at the entrance of this element, by using screw characteristics, operating conditions and material properties. There is a pressure gradient across the element. (i) If the pressure at the entrance calculated, and the exit pressure both are positive, then this element is fully filled and the fill factor is.0. (ii) If both pressures are at atmosphere pressure, then this element is starved and the fill factor is taken as the ratio of real flow rate to maximum pumping rate. (iii) If one of them is at atmosphere pressure, and the other is positive, then one part of the axial length of this element is fully filled and the rest is starved. 3. Backward calculate the next element and take the calculated entrance pressure for this element as the exit pressure for next element, until the melting region. 73

195 Start Q Known? NO Predict Q YES Solid Conveying Melting Forward Calculate Temperature Backward Calculate Pressure & Fill Update Convergence? NO YES Torque Power END Figure 8. Procedure for the calculation of Global model 74

196 8.3 Determination of Feed Rate under Flood Fed Conditions The determination of feed rate under flood fed conditions is one of the main concerns in the design and optimization of intermeshing counter-rotating twin screw extruders. In our simulation, the feed rate is predicted from the solid conveying model, as discussed in Chapter V. The total feed rate is rewritten as, G = G c + G cal where the pumping rate in the inter-screw region, G cal, and the C-chamber region, G c, are given by, respectively. G = ρ cal NV cal b G = ρ c NV occ b where the volume of filled C-chamber and calendering region are V occ = L W c c H =, + α Wc Ro RoH 4 Ro R R i and ( H ) H ( ) V cal = A cal W c when the screw channel is unfolded, respectively. Under flood fed condtions, the filled length of screw H = π α Ri +. channel becomes, ( ) L max Here are the procedures to predict the feed rate under flood fed conditions:. Determine the geometrical parameters of the screw under the hopper.. Calculate the total screw channel volume. 3. Calculate the feed rate, based on operating conditions, geometrical parameters and material properties. 75

197 Sample calculations are shown in Figure 8.3, where two different materials, powder HDPE and PVC were considered. The screw under the hopper is a thick flight screw, FD--30, with one thread start and a pitch at 30 mm. The feed rates under flood fed conditions were predicted from screw speeds and material bulk densities. It is seen with the increase of screw speeds, the feed rate increases as well. Here the effect of leakage was neglected, so there is a linear relationship between them. The PVC data is higher because of PVC s higher density. 5 Predicted Feed Rate, G, kg/hr 0 5 Powder HDPE Powder PVC Screw Speed, RPM Figure 8.3 Predicted feed rates under flood fed conditions for screw design (A). One simulation sample is shown in Figure 8.4, where thick flight screw design (A) (Chapter III) and HDPE powder was used. The screw speed is 60 rpm, and other operating conditions and material properties were given in Table 8.. The feed rate was 76

198 determined to be 0.5 kg/hr. Above this feed rate, screw region is choked. The fill status, temperature and pressure profiles, and melting status are presented in Figure 8.4. When the melting process occurs, the screw channel becomes starved. The end of screw is fully filled, because pressure must be developed to extrude material out of the die. The temperature increases along the screw axis, and the melting process is finished in 7~8 pitch length. Figure 8.5 shows the simulations for PVC under flood fed conditions with the same screw configuration and a screw speed of 60 rpm. The feed rate was determined at 4. kg/ hr. Two other simulations at different screw speeds are represented in Figure 8.6 and Figure 8.7. The predicted feed rate under flood fed conditions is 7.3 kg/hr when the screw speed is 30 RPM, as shown in Figure 8.6. The feed rate becomes.0 kg/hr at a screw speed 45 RPM. We can see the feed rate is only a function of screw speeds for a specific screw design. The fill status is similar for all three cases fed with PVC. With an increase of screw speed, it requires more power consumption, and the accumulated residence time becomes smaller. A higher pressure is built up at the end of screw to extrude polymer materials out of the die. The above simulation examples were based on a Leistritz LSM30.34 machine. More simulations from our analysis are given in Chapter IX, where commercial PVC extruders are considered, and more detailed discussion on how to handle screw characteristic curve calculations are given. 77

199 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.4 Simulations for screw design (A) under flood feeding of powder HDPE and screw speed 60 rpm. 78

200 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.5 Simulations for screw design (A) under flood feeding of powder PVC and screw speed 60 rpm. 79

201 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.6 Simulations for screw design (A) under flood feeding of powder PVC and screw speed 30 rpm. 80

202 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.7 Simulations for screw design (A) under flood feeding of powder PVC and screw speed 45 rpm. 8

203 8.4 Case Studies for Starved Fed Machines Calculations were made for the Leistritz LSM intermeshing counterrotating twin screw machine described in Chapter III. In addition to the cases under flood fed conditions shown in Figure 8.4 and Figure 8.5, starved fed conditions were used in simulations. The simulations are presented first. We then make comparison with experiments in Section 8.5. Different screw configurations and operating conditions, along with different materials, were used in the calculation, as shown in Chapter III. The parameters used are shown in Table 8.. The bulk densities, material viscosities, specific heat and heat of fusion were measured. The heat transfer coefficients and thermal conductivity of solid [] and melt [78, 8, 73, 74], and friction coefficients [70, 75, 76] were assumed based on the experimental data from the literature. We will include the model of White et al. [77] for the determination of heat transfer coefficients and screw temperature along screw axis in the future. The barrel temperature was pre-set and screw temperatures were taken as 0~0 o C lower than corresponding barrel temperatures. Fill factor (f), pressure (P), temperature (T) and melting (A) profiles along the screw axis are presented, where the melting profile (A) is defined as the ratio of unmelted solid part to original solid material. The feed rate and screw speed are given in each figure. 8

204 Table 8. Parameters used in calculations Symbol Units PP HDPE- HDPE- PVC Solid Phase Pellets Pellets Powder Powders Bulk Density ρ b g/cc Thermal Conductivity k s W/mK Specific Heat C s J/kgK Friction Coeff at Barrel m b Friction Coeff at Screw ms Heat Transfer Coeff at Barrel h bs W/m K Heat Transfer Coeff at Screw h ss W/m K Initial Solid Temperature T f o C Melting Temperature T m o C Heat of Fusion λ J/g Melt Phase Density of Melt Film ρ m g/cc Specific Heat C m J/KgK Consistency Index K o KPas n Power Law Index n Temperature Sensitivity b / o C Reference Temperature T R o C Heat Transfer Coeff at Barrel h b W/m K Heat Transfer Coeff at Screw h s W/m K Barrel Temperature T b o C Screw Temperature T s o C * Heat transfer coefficients of melt were assumed based on references [78, 8, 73, 74]. ** Heat transfer coefficients of solid were assumed based on references []. *** Friction coefficients were assumed based on references [70, 75, 76]. **** Other parameters were measured. 83

205 8.4. Effect of Screw Configuration Design Thick flight designs, thin flight designs, and mixed thick/thin flight designs were used to model intermeshing counter rotating twin screw extrusion. They are given in Figure 8.8, where thick flight elements were used in designs (A) and (D), thin flight elements were applied in designs (B) and (E), and mixed thick and thin flight elements were used in design (C). To characterize the effect of the screw configurations, the same operating conditions were used in calculations for all screw designs. The screw speed was 00 rpm and the metered starved feed rate is 0 kg/hr. The barrel temperature was preset as 0 o C, except the feed section which was taken as 40 o C. Polypropylene was used as sample material. The feed material temperature was taken as 40 o C. The results calculated are presented in terms of fill factor (F), pressure field (P), temperature profile (P) and melting status (A), as shown from Figure 8.9 to Figure 8.3. The melting status A represents the fraction of unmelted solid material to the total solid material. By considering the position and geometry of the hopper, screw calculations start point was taken from 75 mm from the start of each screw design. So the total calculated length reduces from 990 mm to 95 mm. Screw design A only has thick flight elements. A short region at the end of screw is fully filled with material, which build-ups pressure to extrude material out of the die, as shown in Figure 8.9. Due to the higher shear stress in the inter-screw region, the polymer is melted early and the melting process is finished in a short distance. When the melting 84

206 process occurs, the fill factor decreases along the screw axis, until the melting process is finished. The reason is due to densification of solid materials. Figure 8.0 represents the results of screw design B with thin flight elements. Due to the lower pumping capacity of thin flight elements, a longer length at the end of screw is fully filled. Also because of lower pumping capacity of thin flight elements, more screw channel is filled. The final temperature increase at the end of screws is 5 o C lower than the result of screw design A. Figure 8. shows mixed screw design C, which combines thick flight and thin flight elements together. For this kind of screw design, thin flight elements were set at the beginning region and thick flight elements were put after then, so the solid conveying and the melting process occur at thin flight region, and the melt conveying is mainly in thick flight elements. The melting behavior is similar to screw design B, and the final temperature increase at the die end is intermediate between screw designs A and B. Screw design D has thick flight elements and no-pumping shear elements, and the computed results are given in Figure 8.. It is found that pressure is needed to pump material across shear elements and the screw channel ahead of this type of screw is fully filled. The length of fully filled region is determined by the screw pumping capacity of both types of screw elements. Compared with screw design A without shear elements, screw design D has a higher temperature increase at the end of screw, which is due to higher shear heating in the shear element. Figure 8.3 represents the results of screw design E, which involves in thin screws and shear elements. To make materials flow through shear elements, it is seen the 85

207 screw ahead of shear elements are fully filled as well. Compared with thick flight screw design, the fully filled region is longer, which is due to less pumping capacity of thin flight elements. 86

208 (A) (B) (C) (D) (E) Figure 8.8 Screw configurations used in case studies 87

209 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.9 Simulations for screw design (A) at metered starved feed rate 0 kg/hr of pellet PP and screw speed 00 rpm. 88

210 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.0 Simulations for screw design (B) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm. 89

211 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8. Simulations for screw design (C) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm. 90

212 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8. Simulations for screw design (D) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm. 9

213 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.3 Simulations for screw design (E) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm. 9

214 8.4. Effect of Operating Conditions The effect of operating conditions was considered in our calculations. Figure 8.4 shows the results at a screw rotation speed at 00 rpm and feed rate 0 kg/hr. Other conditions are the same on the case shown in Figure 8.9. It was found that with the increase of screw speed from 00 rpm to 00 rpm, the melting process is finished in a shorter distance, which is due to the more severe shear heating at higher screw speed. The fully filled length of screw becomes shorter as well. The melt temperature at the end of screw is around 5 o C higher than the case of lower screw speed. Feed rate was also considered in our computations. The behavior for a metered starved feed rate at 5 kg/hr and screw speed at 00 rpm is presented in Figure 8.5. Compared with results shown in Figure 8.9 where feed rate is 0 kg/hr and screw speed is 00 rpm, it is seen that the length for the melting process is shorter under the lower feed rate. The simulations above were run under metered starved fed conditions. Generally a region associated with mixing function should follow solid conveying and the melting process, and before the pumping region. When intermeshing counter-rotating twin screw extruders are applied in profile and pipe extrusion, they are run under flood fed conditions. Figure 8.4 and Figure 8.5 present the simulations under flood fed conditions for both crystal and amorphous materials. There is no need for the mixing function in this application, so the relative screw length should be shorter. 93

215 8.4.3 Effect of Materials The effect of materials was studied in the calculations. The amorphous material (PVC) and the two crystalline materials (PP and HDPE) were considered. Screw speed and feed rate were fixed at 00 rpm and 0 kg/hr, respectively. Operating conditions are given in Table 8.. The results for amorphous material PVC is represented in Figure 8.6. Figure 8.9 and Figure 8.7 show the results for crystalline materials PP and HDPE, respectively. For amorphous material PVC shown in Figure 8.6, it requires longer length of screw for the solid conveying and melting/softening process. The temperature at the end of screw is lower than the case of polypropylene. The reason is due to lower shear heating associated with the smaller power-law index used in calculation. For PVC, the power-law index was as 0.3, compared with 0.4 for polypropylene. However, this type of comparison is relative, because the temperature settings are different, which causes different viscosities and viscosity variations along the screw axis. These viscosity variations contribute to the differences of shear rate in melt conveying region, even with the same screw geometrical parameters. The two crystalline materials PP and HDPE have similar results. In our calculations, the power law index for HDPE was taken as 0.45, and the barrel temperature was taken as 70 o C, except the first barrel region at 40 o C. The temperature increase at the screw end is a little higher for HDPE, compared with that of PP. HDPE has a higher heat of fusion, so it requires a longer length to finish the melting process. 94

216 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.4 Simulations for screw design (A) at feed rate 0 kg/hr of pellet PP and screw speed 00 rpm. 95

217 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.5 Simulations for screw design (A) at feed rate 5 kg/hr of pellet PP and screw speed 00 rpm. 96

218 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.6 Simulations for screw design (A) at feed rate 0 kg/hr of powder PVC and screw speed 00 rpm. 97

219 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.7 Simulations for screw design (A) at feed rate 0 kg/hr of pellet HDPE and screw speed 00 rpm. 98

220 8.4.4 Effect of Other Important parameters Other parameters, such as friction coefficients and heat transfer coefficients were also considered in our parameter studies. Figure 8.8 shows the simulations with a higher heat transfer coefficient for solid phase, 300 W/m K and 00 W/m K at barrel and screw surfaces, respectively, as a comparison of 30 W/m K and 0 W/m K, which are used as the default in our forer case studies. It is noted that there are no obvious differences between the two cases. We also carried out the parameter study with different heat transfer coefficients for the melt phase. The results are shown in Figure 8.9, where the coefficients at barrel and screw surfaces are 000 W/m K and 500 W/m K, compared with the case with 00 W/m K and 00 W/m K. The temperature increases faster at the early stage, and then the temperature distribution becomes more even, which causes a lower power consumption and torque requirement. The friction coefficient at the screw surface was found to be a key factor in affecting the intermeshing counter-rotating twin screw extrusion process, as shown in Figure 8.0, where the friction coefficients are 0.0, 0.6 and 0., respectively. In our model, we assume it is the friction work, instead of shear heating in the inter-screw region that is the heat sourse. The melting length decreases with the friction coefficients. When the friction coefficient is higher, the melting process is finished sooner, and the mean cup-mixing temperature increases more rapidly. 99

221 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.8 Simulations for screw design (A) at feed rate 0 kg/hr of PP and screw speed 00 rpm (marker represents solid heat transfer coefficient at barrel surface 300 W/m K and screw surface 00 W/m K, while solid line represents the original case). 00

222 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.9 Simulations for screw design (A) at feed rate 0 kg/hr of PP and screw speed 00 rpm (marker represents melt heat transfer coefficient at barrel 000 W/m K and screw 500 W/m K, while solid line represents the original case). 0

223 F Fill Factor mm P MPa Melt Pressure mm T o C Temperature mm A Melting / Softening Profile mm Figure 8.0 Simulations for screw design (A) at feed rate 0 kg/hr of PP and screw speed 00 rpm (solid line represents fs=0.0; represents fs=0.6; and represents fs=0.). 0

224 8.5 Comparison with Experimental Results Our simulations were compared with experimental studies described in Chapter IV. As shown in Chapter IV, we measured the distribution of material in screw channel and checked the melting status along the screw axis by pulling screws out of the barrel. We also compared our simulations with the experiments by Wilczynski and White [7], where the experiments were carried out on the same machine. The detailed discussion is given in this section Thick Flight Screw Designs with Shear Elements (A) and (D) Figure 8. shows a comparison between experimental results and simulations for thick flight screw designs (A). It is seen that the prediction of the screw length for the melting process has a good match for all the cases under different operating conditions (different screw speeds and feed rates) with / without different materials. The fully filled length was also well predicted for all the cases considered. The simulations for the solid conveying region were found to have a good match with the experimental results as well, except amorphous material PVC. There are several potential reasons for the discrepancy for PVC in the solid conveying region. PVC material used in experiments is in powder state. The particle size is smaller than the clearance between one screw root and the other screw flight. So leakage occurs and the computed pressure in the inter-screw region is probably overpredicted, which causes over-predicted heat generations in melting model and a shorter simulated length of the melting region. The second potential reason is from the physics of 03

225 amorphous PVC. Amorphous materials have no melting state, and experience softening when increasing the temperature. During pulling screws out of the barrel, some softened material has enough time to solidify again, which causes the observation errors. Another possible reason is due to under-estimation of leakage in this region. We considered two heating resources. One is in the inter-screw region, and the other is in the C-chamber region. The inter-screw region dominates the total temperature increase and the melting process. However, a well developed solid bed was assumed in determining the friction heating. The effect of particle sizes, the gap level and related leakage were neglected, which causes an over-estimating heat generation in solid conveying region, especially in the inter-screw zone. When more screw channel is filled, the distribution of solid particles inside the screw channel approaches the solid bed. The discrepancy is expected to become smaller, which is verified by our studies, as shown in Figure 8. cases (VI) and (VIII). Due to the difficulty in peeling the carcass of materials from the screw channel, the real fill factor was not obtained quantitatively in our experimental results. So we cannot compare it with the simulations. Figure 8. shows a comparison between experimental results and predictions for screw design (D) where thick flight elements and shear elements were used. HDPE pellets were taken. Shear elements have no pumping capacity, which was confirmed by both the predictions and experimental results. The length of fully filled regions before shear elements and the die have a good match between simulation and experiments. The simulation approach to experiments for the melting region as well. 04

226 8.5. Thin Screw Designs (B) and (E) Figure 8.3 shows the simulations and experimental results for thin screw design (B). Solid conveying region, melting region, as well as metering region, are all well predicted. When adding shear elements to the thin screw design (B), we have screw design (D). The comparison between predicted results and experimental results is given in Figure 8.4. Fully filled regions occur before each shear element, and produce pressure buildup to push molten material through shear elements. It was found that our predictions have a good match with experiments Mixed Screw Design (C) The mixed screw design was also applied in simulation and experiments. The comparison between them is shown in Figure 8.5, where polypropylene was used. The predictions of melting length and solid conveying region match with experiments very well. 05

227 Experimental result Simulation (with the predicted fill factor) (I) Feed with PP pellets at feed rate 0 kg/hr and screw speed 00 rpm Experimental result Simulation (with the predicted fill factor) (II) Feed with PP pellets at feed rate 5 kg/hr and screw speed 00 rpm Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8. Comparison of screw design (A) between experiments and simulations. 06

228 Experimental result Simulation (with the predicted fill factor) (III) Feed with PP pellets at feed rate 0 kg/hr and screw speed 00 rpm Experimental result Simulation (with the predicted fill factor) (IV) Feed with PVC powder at feed rate 6 kg/hr and screw speed 60 rpm Solid Region Melting/Softening Region Starved Melt Conveying Fully filled Region Figure 8. Comparison of screw design (A) between experiments and simulations (Continued). 07

229 Experimental result Simulation (with the predicted fill factor) (V) Feed with PVC powder at feed rate 3 kg/hr and screw speed 30 rpm Experimental result Simulation (with the predicted fill factor) (VI) Feed with PVC powder at feed rate 6 kg/hr and screw speed 30 rpm Solid Region Melting/Softening Region Starved Melt Conveying Fully filled Region Figure 8. Comparison of screw design (A) between experiments and Simulations (Continued). 08

230 Experimental result Simulation (with the predicted fill factor) (VII) Feed with HDPE pellets at feed rate 8 kg/hr and screw speed 60 rpm Experimental result Simulation (with the predicted fill factor) (VIII) Flood fed with HDPE powder at screw speed 60 rpm Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8. Comparison of screw design (A) between experiments and simulations (Continued)). 09

231 Experimental result (I) Simulation (with the predicted fill factor) Feed with HDPE pellets at feed rate 8 kg/hr and screw speed 60 rpm Experimental result Simulation (with the predicted fill factor) (II) Feed with HDPE pellets at feed rate 8 kg/hr and screw speed 0 rpm Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8. Comparison of screw design (D) between experiments and simulations. 0

232 Experimental result Simulation (with the predicted fill factor) (III) Feed with HDPE pellets at feed rate 4 kg/hr and screw speed 60 rpm Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8. Comparison of screw design (D) between experiments and simulations(continued).

233 Experimental result Simulation (with the predicted fill factor) Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8.3 Comparison of screw design (B) between experiments and simulations at feed rate 0 kg/hr and screw speed 00 rpm with PP pellets.

234 Experimental result Simulation (with the predicted fill factor) (I) Feed with HDPE pellets at feed rate 8 kg/hr and screw speed 60 rpm Experimental result Simulation (with the predicted fill factor) (II) Feed with HDPE pellets at feed rate 8 kg/hr and screw speed 0 rpm Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8.4 Comparison of screw design (E) between experiments and simulations. 3

235 Experimental result Simulation (with the predicted fill factor) (III) Feed with HDPE pellets at feed rate 4 kg/hr and screw speed 60 rpm Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8.4 Comparison of screw design (E) between experiments and simulations (continued). 4

236 Experimental result Simulation (with the predicted fill factor) Solid Region Melting Region Starved Melt Conveying Fully filled Region Figure 8.5 Comparison of screw design (C) between experiments and simulations at feed rate 0 kg/hr and screw speed 00 rpm with PP pellets. 5

237 8.6 Summary We presented an integrated global model to numerically simulate the whole extrusion process for intermeshing counter-rotating twin screw extruders, where the fill factor, pressure fields, temperature fields and melting status are included, as well as the determination of feed rate under flood fed conditions. This model could be used to compute the whole extrusion process, including solid conveying, the melting process and metering. It was used to investigate the effect of operating conditions and screw configuration design on the performance of intermeshing counter-rotating twin screw extrusion machines. Predictions were compared with the experimental results shown in Chapter IV and in the literature [7]. A good agreement was reached from these comparisons. From our simulated results, it was found that the melting process occurs earlier, which is different from intermeshing co-rotating twin screw extruders. Some discrepancy occurs for the prediction of solid conveying region, compared with experimental results. This may be due to the effect of material particle size and clearance between one screw flight and another screw root, which causes over-estimation of pressure buildup in the inter-screw region. 6

238 CHAPTER IX SOFTWARE DEVELOPMENT 9. Introduction In Chapter VIII, we presented an integrated global model by combining regions of solid conveying, melting and melt conveying. With this composite global model, fill factor, pressure, cup-mixing temperature, the melting process and melt conveying along the length of the extruder can be predicted. These profiles are necessary in evaluating the design and performance of intermeshing counter-rotating twin screw extruders and in optimizing the processing conditions for applications. The composite model, however, due to its complexity, is awkward for use by designers and process engineers. It is meaningful to develop user-friendly software, which derives from this global model. The objective of this chapter is to introduce and develop user-friendly software. With the software, the user can do experiments with a personal computer by designing different screw and barrel configurations, and performing simulation for different sets of operating conditions and material properties. Other valuable information, such as the power consumption, torque on the screws, specific energy consumption (SEC) and the mean residence time profiles can be determined in this software. 7

239 9. Determination of Power Consumption, Torque, SEC and Average Residence Time In this software we can calculate power consumption, torque and average residence time. They are all necessary to characterize the performance of an intermeshing counterrotating twin screw extruder. We begin our discussion of the software by describing how we determine power consumption, torque and average residence time in this section. 9.. Power Consumption The power consumption, can be expressed by the rate of doing shaft work given as W = V σ nd S S where V is the screw velocity, σ is stress tensor at the screw surface, n is the unit outward normal to the screw and S is the total surface wetted by material on the screws. The power consumption in cylindrical coordinate system is given as = Z W RoNσ zr (0) Rs 0 0 ( z, θ ) π π dzdθ where Z is the length of the element R s ( z,θ ) is the distance from the screw surface to screw axis, equaling to R o or R i, and σ (0) is the shear stresses at the screw surface, zr which is determined by their corresponding models in solid conveying region, melting region and melting conveying region. The power consumption in each region for a specific element is different. In addition, the local temperature, pressure and level of fill also affect this value. It should be pointed out that due to the unique character of the inter-screw region in intermeshing counter-rotating twin screw extruders, even when the 8

240 C-chamber is starved, the inter-screw region is always fully filled, and shear stresses are still at a higher level. So it must be considered in calculations. 9.. Torque Determination The generalized expression for the torque is given as M = σ n rds S where n is a unit vector in θ direction, and S is the total surface wetted by material on the screws. We defined r as the radial direction in the screw channel and θ as the circumferential direction, r = D s e r as a lever arm, where e r is a unit vector in the radial direction. Similarly expressing the component of σ n in the θ direction we can write the torque as M = ( σ zr ( 0)cosφ + σ rθ (0)sinφ) eθ Ds erds S where D s is the diameter of the screw. So M W = e z πn where W is the power consumption, e θ and e z are the unit vectors in θ and z directions, respectively, and the product πn is the angular velocity. 9

241 9..3 The Specific Energy Consumption The specific energy consumption is referred as the energy consumed per unit mass of material extruded with W SEC = G 9..4 The Average Residence Time The average residence time is another necessary parameter in evaluating the performance of intermeshing counter-rotating twin screw extruders, which is defined as t = N i= m G occ = N i= V Q occ = N i= kv Q t where mocc is the mass of polymer inside one screw element, V occ is the volume occupied of one screw element, Q is the volumetric flow rate and V t is the total volume between the element and barrel of one screw element and k is the degree of fill. It is noted that the volumetric flow rate depends on the density, so it is different in the solid conveying, melting and melt conveying regions. 9.3 Outline and Structure of Software The structure of software is similar to Akro-Co-Twin Screw that exists in various versions [05, 53], which was developed in our lab for the simulation of intermeshing co-rotating twin screw extruders. The front page of our program is shown in Figure 9.. 0

242 Figure 9. The front page of software, Akro-Counter-Twin Screw Extruder The main Menu, as shown in Figure 9., includes Files, Design/Edit (Design machines), Simulation (Specify operating conditions and conduct calculation), Display (show results and machine design graphically), View File (show results and machine design), Print File (print file and convert results into the text form file), and Database (modular screw, shear element and barrel design). The basic procedure to use this software is as follows: From Menu Files, take Select or Edit Extruder, as shown in Figure 9.. In this category, the users could construct a new intermeshing counter-rotating twin screw extruders system.

243 In Database, edit or add new screw elements, shear elements and barrel designs. For a machined shaft PVC extruder, the strategy, as explained in Chapter VIII, is to consider the machine as a pseudo-composite machine and divide the whole screw into several sections, where each section shares the appropriate geometry (diameter, pitch, number of flight, flight thickness, clearance and others). The number of sections could be very large, such as 30 or more. Then the user can edit or add these new screw elements as a modular screw design. After the screw design, we need to insert and attach the screw characteristic curves into the database. The calculations of screw characteristic curves are based on the principle represented in Chapter VII. The basic information required to generate screw characteristic curves is the number of flights, the pitch, screw diameter, flight thickness, screw channel depth and clearance. Move to Design / Edit to design the screw machine, which includes the assembly of screw configuration, barrel configuration, die design and the screw machine. Move to Simulation to specify operating conditions and material properties. After that, conduct extruder analysis to check if the machine design is reasonable or not. If extruder analysis is OK, the next step is to calculate. Once the calculation is finished, the results can be saved in.flw format file, which could be extracted and converted into another format file, displayed graphically or printed. The results then can be shown in Display. The machine design can also be shown in this Menu.

244 Results and machine design also can be displayed with detailed values, as shown in Figure 9.3 for an example of the detailed design of die. Hard copies of detailed results and machine design can be obtained from Menu Print File. Results also can be converted into other format files from this command. After the calculations are finished, the software can be turn off from command Exit at Menu Files. Figure 9. Select or edit screw machine category 3

245 Figure 9.3 View detailed machine design or results. 9.4 Sample Calculations Examples are presented in this section. The first extruder design we considered is shown in Figure 9.4 which has the same screw configuration to Figure 3.7 (A). It consists in right handed, thick flight elements with different pitches only, no mixing elements or left handed elements. The diameter of screws is 34 mm. The detailed machine information (including the screw design, the barrel design and the die design) are given in Table 9., and the material to be processed is the polypropylene described in Chapter IV. It is described in Table 9. as well. In the screw design, Code 30/30 FD shows a thick flight element with a pitch of 30 mm, a screw length of 30 mm and one single flight. 0/0 FD shows a thick flight element with a pitch of 0 mm, a screw length of 0 mm and one thread start (NT=). 4

246 48/48 FD3 is a thick flight element with a pitch of 48 mm, a screw length of 48 mm and three thread starts (NT=3). 30/30 FD3 is a thick flight element with a pitch of 30 mm, a screw length of 30 mm and three threat starts (NT=3). 5/5/S is a shear element with a pitch of 5 mm and a length of 5 mm. The screw characteristics of these elements were calculated from the analysis in Chapter VII and inserted into the data system of our software. In the barrel design, barrel code 0/60/30/h shows a modular barrel with the hopper (H). The barrel length is 0 mm and the length of hopper is 30 mm. The center of hopper is positioned 60 mm from the left end. Barrel code 0/0/0/P means a barrel without hopper or vent, with the length of 0 mm. Similarly, 5/0/0/P and 0/0/0/P are the barrels without hopper or vent, and with the length of 5 mm and 0 mm, respectively. Figure 9.4 Machine design for sample calculations. 5

247 Table 9. Extruder configurations Screw configuration 6

248 Table 9. Extruder configurations (Continued) Barrel configuration Die configuration 7

249 Table 9. Material properties and operating conditions Material properties and operating conditions Material properties and operating conditions (Metered Starved Fed) 8

250 The software developed was used to make the sample calculations and results are shown in Figure 9.5 to Figure 9.. In each case, simulations include fill factor, pressure and temperature profiles, melting status, average residence time, power consumption and torque profiles along the screw axis. The effect of screw speed and feed rate are investigated in these sample calculations, and summarized in Figure 9.5 with the information of maximum temperature developed, the total average residence time, the total power consumption and the total torque on the screw shaft. The effect of feed rate and screw speed has been discussed in Chapter VIII. Fully filled regions occur at the end of screw and are associated with shear elements. With an increase of screw speed, the less screw channel is filled with material, and the length of fully filled region becomes shorter. There needs to be additional screw length to finish the melting process at lower screw speeds, which is due to less heat dissipation. Due to the same reason, the temperature increase is smaller at lower screw speed as well. Average residence time decreases with the increase of screw speed, which is due to a faster material transport velocity at a higher screw speed. The fully filled region makes a significant contribution to the average residence time. Higher screw speed requires more energy input. However, the total torque becomes smaller at higher screw speeds, which is due to lower viscosity associated with higher heat dissipation at higher screw speed. The specific energy consumption decreases with the increase of screw speed, which represents higher unit efficiency at higher screw speed. The feed rate has an opposite effect to that of screw speed. More screw channel is fully filled at higher feed rate, and it also requires more screw length to finish the melting 9

251 process. However, it seems that higher feed rates cause a lower maximum temperature increase, which is due to heat loss to the barrel. Due to the greater extent screw channel of fill with material at higher feed rate, it is associated with a lower average residence time. To transport more material at a fixed screw speed, more power consumption and torque are required. Both of the trends were properly predicted with the software. When dividing the total power consumption by flow rate, we have the specific energy consumption. Since both the total power consumption and feed rate are higher, the specific energy consumption could be larger or smaller theoretically. However, in production this value generally decreases at higher feed rates. Checking the simulations shown in Figure 9.5 to Figure 9., and compared with the experimental results shown in Figure 8., we can see they was predicted properly with our software. 30

252 Figure 9.5 Simulations for fill factor, pressure profiles, temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 50 rpm. 3

253 Figure 9.6 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 00 rpm. 3

254 Figure 9.7 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 50 rpm. 33

255 Figure 9.8 Simulations for fill factor, pressure profiles, temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 00 rpm. 34

256 Figure 9.9 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 5 kg/hr of PP and screw speed 00 rpm. 35

257 Figure 9.0 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 5 kg/hr of PP and screw speed 00 rpm. 36

258 Figure 9. Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at feed rate 0 kg/hr of PP and screw speed 00 rpm. 37

259 Table 9.3 Comparison of simulations for different operating conditions Feed Rate kg/hr Screw Speed rpm T max, o C Res. min Work kw Torque Nm SEC kwhr/kg PVC Profile Extruders of Varying Screw Diameter We presented how to use our software to make calculations for a laboratory scale intermeshing counter-rotating twin screw extruder in Section 9.4. The machine we considered was a modular screw extruder. Generally polyvinyl chloride (PVC) profile extruders are non-modular machined design. In this section, calculations were made on a typical PVC machine with our software. Figure 9. shows a typical screw design for production of profile with PVC, which was based on the design by J. Stasiek of the Metalchem Institute of Plastics Processing of Torun, Poland. It is an industrial scale machine for PVC profile. The screw diameter is 66 mm with a total screw length of 80 mm. The channel depth is. mm and the distance between two screw axes is 55 mm. Calculations on this machine should be more meaningful for the application of our software to PVC profile extrusion. 38

260 Figure 9. A typical screw design for production of profile with PVC (mm). Our first step is to transform the machined screws into pseudo-composite screws and divide the whole screw into several zones, where each part shares the same geometry, shown in Table 9.4. Table 9.4 Screw geometrical details and code assigned in calculations Element Code Outer Radius, mm Root Radius, mm Length, mm 39 Pitch, mm Number of Flight Flight Thickness mm 75/75/FD / /84/FD / /4/FD3 33 4/ /66/FD 33 66/ /8/FD 33 8/ /44/FD 33 44/ /0/S If the length of a screw element is several times of the pitch, we divided this element into several parts, with one pitch or less as a unit. In this process, screws could be divided into parts with a length less than one pitch, as shown in Table 9.5, where the

261 length of each unit equals to one pitch or less. For example, element code 75/75/FD3 has outer radius 33 mm, root radius mm, pitch 75 mm, triple thread starts, flight thickness 8.5 mm, and a length of 75 mm. The element 0/0/S is a transition region between two screw element, and the outer radius equals to the root radius. Table 9.5 Screw geometrical details used in calculations 40

262 Table 9.5 shows the pseudo-composite extruder used for calculations, which is derived from the machined screw shown in Figure 9.. The total screw length is the same, 80 mm, and the ratio of screw length to diameter is 80 / 66. Our second step is to calculate screw characteristic curves for all the elements, based on the principle and procedures introduced in Chapter VII. After the screw characteristic curves are obtained, the calculated screw characteristic curves are inserted into the database of this set of machines in our software, which is used for further calculations. In screw characteristic curves, the same data is shared by the elements with different length and the same geometry. The third step is to design a barrel and a die, and then assembly a extruder, by combining the screw design shown in step one. The extruder is shown in Figure 9.3, where the die has 6 holes with a diameter 3 mm and length 0 mm. Modular barrels were used in this calculation. Figure 9.3 Machine design for sample calculations. 4

263 The fourth step is to define material properties and processing conditions. The material properties of PVC are shown in Table 8.. The barrel temperature is set at 70 o C, except the first section is set at 50 o C. Different screw speeds could be used in calculations for different conditions. The detailed information on this step is referred in Section 9.4. The fifth step is to determine the feed rate, if under flood fed condtions. The details for determining the feed rate are given in Section 8.3. Generally PVC profile processing is run under flood fed conditions. In our calculations, we also presented a couple of sample calculations under starved fed conditions for comparison. The sixth step is to make calculations and save results. Sample calculations are given in Figure 9.4 to Figure 9.6 for flood fed conditions, and Figure 9.7 and Figure 9.8 under metered starved fed conditions. Figure 9.4, Figure 9.5 and Figure 9.6 show the cases at screw speeds 30 rpm, 45 rpm and 60 rpm, respectively. All other conditions are the same. The predictioned feed rate is 4 kg / hr, 5 hg / hr and 85 kg / hr. The melting process is mainly finished in the first 7 pitches. With the initiation of melting, starvation occurs, although there is a neutral element, 0/0/S, in melting region. The reason is that this element is designed for the connection between two different geometrical sections, and not for mixing, so there is no need to bulidup a higher pressure, which is confirmed from the cases shown in Figure 9.7 and Figure 9.8. The power consumption increases with the feed rate. The average residence time decreases at higher screw speed. 4

264 Figure 9.4 Simulations for flood fed fill factor, pressure and temperature profiles, melting status and power consumption at N = 30 rpm. Estimated feed rate = 4 kg/hr. 43

265 Figure 9.5 Simulations for flood fed fill factor, pressure and temperature profiles, melting status and power consumption at N = 45 rpm. Estimated feed rate = 5 kg/hr. 44

266 Figure 9.6 Simulations for flood fed fill factor, pressure and temperature profiles, melting status and power consumption at N = 60 rpm. Estimated feed rate = 85 kg/hr. 45

267 For comparison, two cases under starved fed conditions were simulated for this machine. The results are given in Figure 9.7 and Figure 9.8, where the feed rates are 00 kg /hr and 50 kg /hr, respectively. It is seen that more screw channel is filled with materials at higher feed rate. Starvation occurs in some regions when the feed rate is 00 kg / hr. The melting process is finished much earlier, compared with the cases under flood fed conditions. It is also confirmed that element 0/0/S has no pumping capacity. To have material transport across this element, pressure should be built up, though the pressure is at a lower level. 46

268 Figure 9.7 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at screw speed 30 rpm and feed rate of 00 kg/hr PVC. 47

269 Figure 9.8 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption at screw speed 30 rpm and feed rate 50 kg/hr PVC. 48

270 9.6 Scale-up and Other Industrial Applications Section 9.4 presented some sample calculations and the basic procedures of our software for a laboratory scale intermeshing counter-rotating twin screw extruder. Section presented sample calculations of an industrial PVC profile machine. In industrial productions, especially for typical PVC profile extrusion, the dimensions are larger, such as screw diameters of 75 mm, 5 mm, and even larger. So it is meaningful to include the function of scale-up in our software as well to approach the production in industry. As discussed in Chapter VIII and Section 9.5, we divided the classic screw design into pseudo-elements for PVC profiles and pipe extrusion. For each pseudo-element, we have an appropriate geometry. Then we assembly these elements together to have a screw design. During this process, the pressure fields are assumed to be continuous between elements. In the applications of our software, the user only needs to insert the screw geometrical parameters and melt rheological properties to generate corresponding screw characteristic curves. Then attach these screw characteristic curves into our software, and have a database, as explained in Section 9.4. Because dimensionless forms were used in the calculation of screw characteristics, there is an easier way to scale-up the dimensions from smaller sizes to larger sizes. If the aspect ratios are fixed, a larger dimensional machine could share the same dimensionless database to that of a smaller sized machine, and there is no need to recalculate the screw characteristic curves. Such scale-up calculations are carried out based on the laboratory scale machine, as discussed in Chapter VIII, and the results are shown in Figure 9. to Figure 9.4, 49

271 compared with the corresponding results of original sized machine in Figure 9.9 to Figure 9.. Screw geometrical details used in simulations are given in Table 9.6, where the screw diameter are doubled from the laboratory machine from 34 mm to 68 mm and the distance between the two screw axes is 60 mm. Screw channel depth is 8 mm. So the total volume of screw channel is as V ~ HWL, 3 = 8 times of the original machine. c c The screw configurations are shown in Table 9.7. The total ratio of screw length to diameter is The doubled diameter screw elements and the original elements share the same dimensionless screw characteristic curves, which are listed in the software already and can be used directly. The material properties for PVC are given in Table 8.. The operating conditions are shown in each figure. Temperature buildup is one of the key parameters to be considered in scale-up. When the machine size is doubled, the maximum temperature obtained is increased because the surface area scales more slowly than the volume. We find this to be around 0 o C depending on the screw speed, as shown in Figure 9.9 to Figure 9.4. Remember the temperature shown is a cup-mixing temperature. Temperature varies in the channel cross-section, especially in the inter-screw region where higher shear heating and highest temperature occurs. So a detailed analysis is perhaps better taken in the scale-up process, if information and software is available. This could help to avoid some risks. Generally it is helpful to avoid the excessive temperature buildup by using lower screw speeds, though it is at the expense of output. Compared with the results shown in Figure 9., where screw speed is 30 rpm, the maximum cup-mixing temperature of a scaled-up machine is around 6 o C higher at a higher screw speed of 50 rpm, as shown in Figure

272 Another choice is to redesign the screw aspect ratios and optimize the screw geometrical parameters. If the materials are sensitive to this temperature increase, attention should be given to this in the process of scale-up. To handle the problem, generally a larger sized machine is run under slow screw speed. The level of shear stresses has similar effects in scale-up process. With the increase of machine sizes, shear strains and shear stresses become more severe. If materials are sensitive to shear rates, screw geometrical parameters should be reconsidered, such as redesigning the aspect ratio and clearance. It is seen that with the increase of screw speeds, the predicted feed rate increases almost in linear relationship, as shown in Figure 9.5. When the machine size is doubled, the predicted feed rates at different screw speeds are increased around 8 times, which is roughly followed the rule, ( NHW L ) 8 Q NV = N(H )(W )(L ) = = c c c where the geometrical parameters, screw channel width, depth and length are all doubled. It is noted that both the pressure and temperature decrease slightly in the end region in Figure 9. to Figure 9.4. The reason is due to the screw design used in calculations. The feed rate is higher than the pumping capacity for the screw element at the end, so pressure buildup helps to pump molten polymers. There is a need to improve the screw design and balance the pumping capacity along the whole screw. Some consideration on the limitations in the applications of our software should be taken, such as the effect of the ratios of clearance to material particle size, the leakages and the availability of experimental results from larger machines during scale-up process. 5

273 So the user should have a comprehensive consideration on the applications of our software. Table 9.6 Screw geometrical details and codes assigned in scale-up calculations Element Code Outer Root Radius, Radius mm, mm Length, mm Pitch, mm Number of Start Flight Thickness mm 96/96/FD /48/FD /60/FD /60/FD /40/FD

274 Table 9.7 Screw configurations for scale-up calculations Number of screw arrangements Element Code Accumulated Screw Length, mm 60/60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /96/FD /96/FD /96/FD /96/FD /96/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD /60/FD

275 Figure 9.9 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 30 rpm with a diameter at 34 mm machine. 54

276 Figure 9.0 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 40 rpm with a diameter at 34 mm machine. 55

277 Figure 9. Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 50 rpm with a diameter at 34 mm machine. 56

278 Figure 9. Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 30 rpm with a diameter at 68 mm machine. 57

279 Figure 9.3 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 40 rpm with a diameter at 68 mm machine. 58

280 Figure 9.4 Simulations for fill factor, pressure and temperature profiles, melting status and power consumption under flood fed condition for PVC and at screw speed 50 rpm with a diameter at 68 mm machine. 59