POLYMER PROCESSING
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`POLYMER PROCESSING
`Modeling and Simulation
`
`Tim A. Osswald
`Department of Mechanical Engineering
`University of Wisconsin-Madison
`
`Juan P. Hern´andez-Ortiz
`Department of Chemical and Biological Engineering
`University of Wisconsin-Madison
`
`HANSER
`
`Hanser Publishers, Munich
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`Hanser Gardner Publications, Inc., Cincinnati
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`Lovingly dedicated to our maternal Grandfathers
`Ernst Robert Georg Victor and Luis Guillermo Ortiz;
`Two great men whose own careers in Chemical
`Engineering influenced the paths we have taken
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`In gratitude to Professor R.B. Bird, the teacher and
`the pioneer who laid the groundwork for polymer
`processing − modeling and simulation
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`PREFACE
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`The groundwork for the fundamentals of polymer processing was laid out by Professor R. B.
`Bird, here at the University of Wisconsin-Madison, over 50 years ago. Almost half a century
`has past since the publication of Bird, Steward and Lightfoot’s transport phenomena book.
`Transport Phenomena (1960) was followed by several books that specifically concentrate
`on polymer processing, such a the books by McKelvey (1962), Middleman (1977), Tadmor
`and Gogos (1979), and Agassant, Avenas, Sergent and Carreau (1991). These books have
`influenced generations of mechanical and chemical engineering students and practising
`engineers. Much has changed in the plastics industry since the publication of McKelvey’s
`1962 Polymer Processing book. However, today as in 1962, the set-up and solution of
`processing problems is done using the fundamentals of transport phenomena.
`What has changed in the last 50 years, is the complexity of the problems and how they
`are solved. While we still use traditional analytical, back-of-the-envelope solutions to
`model, understand and optimize polymer processes, we are increasingly using computers
`to numerically solve a growing number of realistic models. In 1990, Professor C.L. Tucker
`III, at the University of Illinois at Urbana-Champaign edited the book Computer Simulation
`for Polymer Processes. While this book has been out of print for many years, it is still the
`standard work for the graduate student learning computer modeling in polymer processing.
`Since the publication of Tucker’s book and the textbook by Agassant et al., advances
`in the plastics industry have brought new challenges to the person modeling polymer pro-
`cesses. For example, parts have become increasingly thinner, requiring much higher injec-
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`viii
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`PREFACE
`
`tion pressures and shorter cooling times. Some plastic parts such as lenses and pats with
`microfeatures require much higher precision and are often dominated by three-dimensional
`flows.
`The book we present here addresses traditional polymer processing as well as the emerg-
`ing technologies associated with the 21st Century plastics industry, and combines the mod-
`eling aspects in Transport Phenomena and traditional polymer processing textbooks of the
`last few decades, with the simulation approach in Computer Modeling for Polymer Process-
`ing. This textbook is designed to provide a polymer processing background to engineering
`students and practising engineers. This three-part textbook is written for a two-semester
`polymer processing series in mechanical and chemical engineering. The first and second
`part of the book are designed for the senior- to grad-level course, introducing polymer pro-
`cessing, and the third part is for a graduate course on simulation in polymer processing.
`Throughout the book, many applications are presented in form of examples and illustra-
`tions. These will also serve the practising engineer as a guide when determining important
`parameters and factors during the design process or when optimizing a process.
`Polymer Processing − Modeling and Simulation is based on lecture notes from inter-
`mediate and advanced polymer processing courses taught at the Department of Mechanical
`Engineering at the University of Wisconsin-Madison and a modeling and simulation in
`polymer processing course taught once a year to mechanical engineering students special-
`izing in plastics technology at the University of Erlangen-N¨urnberg, Germany. We are
`deeply indebted to the hundreds of students on both sides of the Atlantic who in the past
`few years endured our experimenting and trying out of new ideas and who contributed with
`questions, suggestions and criticisms.
`The authors cannot acknowledge everyone who helped in one way or another in the
`preparation of this manuscript. We are grateful to the engineering faculty at the University
`of Wisconsin-Madison, and the University of Erlangen-N¨urnberg for their support while
`developing the courses which gave the base for this book. In the Department of Mechanical
`Engineering at Wisconsin we are indebted to Professor Jeffrey Giacomin,for his suggestions
`and advise, and Professor Lih-Sheng Turng for letting us use his 3D mold filling results in
`Chapter 9. In the Department of Chemical and Biological Engineering in Madison we are
`grateful to Professors Juan de Pablo and Michael Graham for JPH’s financial support, and
`for allowing him to work on this project. We would like to thank Professor G.W. Ehrenstein,
`of the LKT-Erlangen, for extending the yearly invitation to teach the "Blockvorlesung" on
`Modeling and Simulation in Polymer Processing. The notes for that class, and the same class
`taught at the University of Wisconsin-Madison, presented the starting point for this textbook.
`We thank the following students who proofread, solved problems and gave suggestions:
`Javier Cruz, Mike Dattner, Erik Foltz, Yongho Jeon, Fritz Klaiber, Andrew Kotloski, Adam
`Kramschuster, Alejandro Londo˜no, Ivan L´opez, Petar Ostojic, Sean Petzold, Brian Ralston,
`Alejandro Rold´an and Himanshu Tiwari. We are grateful to Luz Mayed (Lumy) D. Nouguez
`for the superb job of drawing some of the figures. Maria del Pilar Noriega from the ICIPC
`and Whady F. Florez from the UPB, in Medell´ın, Colombia, are acknowledged for their
`contributions to Chapter 11.
`We are grateful to Dr. Christine Strohm and Oswald Immel of Hanser Publishers for their
`support throughout the development of this book. TAO thanks his wife, Diane Osswald, for
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`PREFACE
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`ix
`
`as always serving as a sounding board from the beginning to the end of this project. JPH
`thanks his family for their continuing support.
`
`TIM A. OSSWALD AND JUAN P. HERNANDEZ-ORTIZ
`
`Madison, Wisconsin
`Spring 2006
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`TABLE OF CONTENTS
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`Preface
`
`INTRODUCTION
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`I.1 Modeling and Simulation
`I.2 Modeling Philosophy
`I.3 Notation
`I.4 Concluding Remarks
`References
`
`PART I BACKGROUND
`
`1 POLYMER MATERIALS SCIENCE
`
`1.1 Chemical Structure
`1.2 Molecular Weight
`1.3 Conformation and Configuration of Polymer Molecules
`1.4 Morphological Structure
`1.4.1 Copolymers and Polymer Blends
`1.5 Thermal Transitions
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`xix
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`xxix
`xxix
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`TABLE OF CONTENTS
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`1.6 Viscoelastic Behavior of Polymers
`1.6.1 Stress Relaxation
`1.6.2 Time-Temperature Superposition (WLF-Equation)
`1.7 Examples of Common Polymers
`1.7.1 Thermoplastics
`1.7.2 Thermosetting Polymers
`1.7.3 Elastomers
`Problems
`References
`
`2 PROCESSING PROPERTIES
`
`2.1 Thermal Properties
`2.1.1 Thermal Conductivity
`2.1.2 Specific Heat
`2.1.3 Density
`2.1.4 Thermal Diffusivity
`2.1.5 Linear Coefficient of Thermal Expansion
`2.1.6 Thermal Penetration
`2.1.7 Measuring Thermal Data
`2.2 Curing Properties
`2.3 Rheological Properties
`2.3.1 Flow Phenomena
`2.3.2 Viscous Flow Models
`2.3.3 Viscoelastic Constitutive Models
`2.3.4 Rheometry
`2.3.5 Surface Tension
`2.4 Permeability properties
`2.4.1 Sorption
`2.4.2 Diffusion and Permeation
`2.4.3 Measuring S, D, and P
`2.4.4 Diffusion of Polymer Molecules and Self-Diffusion
`2.5 Friction properties
`Problems
`References
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`3 POLYMER PROCESSES
`
`3.1 Extrusion
`3.1.1 The Plasticating Extruder
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`3.1.2 Extrusion Dies
`3.2 Mixing Processes
`3.2.1 Distributive Mixing
`3.2.2 Dispersive Mixing
`3.2.3 Mixing Devices
`3.3 Injection Molding
`3.3.1 The Injection Molding Cycle
`3.3.2 The Injection Molding Machine
`3.3.3 Related Injection Molding Processes
`3.4 Secondary Shaping
`3.4.1 Fiber Spinning
`3.4.2 Film Production
`3.4.3 Thermoforming
`3.5 Calendering
`3.6 Coating
`3.7 Compression Molding
`3.8 Foaming
`3.9 Rotational Molding
`References
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`PART II PROCESSING FUNDAMENTALS
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`4 DIMENSIONAL ANALYSIS AND SCALING
`
`4.1 Dimensional Analysis
`4.2 Dimensional Analysis by Matrix Transformation
`4.3 Problems with non-Linear Material Properties
`4.4 Scaling and Similarity
`Problems
`References
`
`5 TRANSPORT PHENOMENA IN POLYMER PROCESSING
`
`5.1 Balance Equations
`5.1.1 The Mass Balance or Continuity Equation
`5.1.2 The Material or Substantial Derivative
`5.1.3 The Momentum Balance or Equation of Motion
`5.1.4 The Energy Balance or Equation of Energy
`5.2 Model Simplification
`5.2.1 Reduction in Dimensionality
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`135
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`TABLE OF CONTENTS
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`5.2.2 Lubrication Approximation
`5.3 Simple Models in Polymer Processing
`5.3.1 Pressure Driven Flow of a Newtonian Fluid Through a Slit
`5.3.2 Flow of a Power Law Fluid in a Straight Circular Tube
`(Hagen-Poiseuille Equation)
`5.3.3 Flow of a Power Law Fluid in a Slightly Tapered Tube
`5.3.4 Volumetric Flow Rate of a Power Law Fluid in Axial Annular
`Flow
`5.3.5 Radial Flow Between two Parallel Discs − Newtonian Model
`5.3.6 The Hele-Shaw model
`5.3.7 Cooling or Heating in Polymer Processing
`Problems
`References
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`6 ANALYSES BASED ON ANALYTICAL SOLUTIONS
`6.1 Single Screw Extrusion−Isothermal Flow Problems
`6.1.1 Newtonian Flow in the Metering Section of a Single Screw
`Extruder
`6.1.2 Cross Channel Flow in a Single Screw Extruder
`6.1.3 Newtonian Isothermal Screw and Die Characteristic Curves
`6.2 Extrusion Dies−Isothermal Flow Problems
`6.2.1 End-Fed Sheeting Die
`6.2.2 Coat Hanger Die
`6.2.3 Extrusion Die with Variable Die Land Thicknesses
`6.2.4 Pressure Flow of Two Immiscible Fluids with Different
`Viscosities
`6.2.5 Fiber Spinning
`6.2.6 Viscoelastic Fiber Spinning Model
`6.3 Processes that Involve Membrane Stretching
`6.3.1 Film Blowing
`6.3.2 Thermoforming
`6.4 Calendering − Isothermal Flow Problems
`6.4.1 Newtonian Model of Calendering
`6.4.2 Shear Thinning Model of Calendering
`6.4.3 Calender Fed with a Finite Sheet Thickness
`6.5 Coating Processes
`6.5.1 Wire Coating Die
`6.5.2 Roll Coating
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`6.6 Mixing − Isothermal Flow Problems
`6.6.1 Effect of Orientation on Distributive Mixing − Erwin’s Ideal
`Mixer
`6.6.2 Predicting the Striation Thickness in a Couette Flow System −
`Shear Thinning Model
`6.6.3 Residence Time Distribution of a Fluid Inside a Tube
`6.6.4 Residence Time Distribution Inside the Ideal Mixer
`6.7 Injection Molding − Isothermal Flow Problems
`6.7.1 Balancing the Runner System in Multi-Cavity Injection Molds
`6.7.2 Radial Flow Between Two Parallel discs
`6.8 Non-Isothermal Flows
`6.8.1 Non-Isothermal Shear Flow
`6.8.2 Non-Isothermal Pressure Flow Through a Slit
`6.9 Melting and Solidification
`6.9.1 Melting with Pressure Flow Melt Removal
`6.9.2 Melting with Drag Flow Melt Removal
`6.9.3 Melting Zone in a Plasticating Single Screw Extruder
`6.10 Curing Reactions During Processing
`6.11 Concluding Remarks
`Problems
`References
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`PART III NUMERICAL TECHNIQUES
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`7 INTRODUCTION TO NUMERICAL ANALYSIS
`
`7.1 Discretization and Error
`7.2 Interpolation
`7.2.1 Polynomial and Lagrange Interpolation
`7.2.2 Hermite Interpolations
`7.2.3 Cubic Splines
`7.2.4 Global and Radial Interpolation
`7.3 Numerical Integration
`7.3.1 Classical Integration Methods
`7.3.2 Gaussian Quadratures
`7.4 Data Fitting
`7.4.1 Least Squares Method
`7.4.2 The Levenberg-Marquardt Method
`7.5 Method of Weighted Residuals
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`TABLE OF CONTENTS
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`Problems
`References
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`8 FINITE DIFFERENCE METHOD
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`8.1 Taylor-Series Expansions
`8.2 Numerical Issues
`8.3 The Info-Travel Concept
`8.4 Steady-State Problems
`8.5 Transient Problems
`8.5.1 Higher Order Approximation Techniques
`8.6 The Radial Flow Method
`8.7 Flow Analysis Network
`8.8 Predicting Fiber Orientation − The Folgar-Tucker Model
`8.9 Concluding Remarks
`Problems
`References
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`9 FINITE ELEMENT METHOD
`
`9.1 One-Dimensional Problems
`9.1.1 One-Dimensional Finite Element Formulation
`9.1.2 Numerical Implementation of a One-Dimenional Finite Element
`Formulation
`9.1.3 Matrix Storage Schemes
`9.1.4 Transient Problems
`9.2 Two-Dimensional Problems
`9.2.1 Solution of Posisson’s equation Using a Constant Strain Triangle
`9.2.2 Transient Heat Conduction Problem Using Constant Strain
`Triangle
`9.2.3 Solution of Field Problems Using Isoparametric Quadrilateral
`Elements
`9.2.4 Two Dimensional Penalty Formulation for Creeping Flow
`Problems
`9.3 Three-Dimensional Problems
`9.3.1 Three-dimensional Elements
`9.3.2 Three-Dimensional Transient Heat Conduction Problem With
`Convection
`9.3.3 Three-Dimensional Mixed Formulation for Creeping Flow
`Problems
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`9.4 Mold Filling Simulations Using the Control Volume Approach
`9.4.1 Two-Dimensional Mold Filling Simulation of Non-Planar Parts
`(2.5D Model)
`9.4.2 Full Three-Dimensional Mold Filling Simulation
`9.5 Viscoelastic Fluid Flow
`Problems
`References
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`10 BOUNDARY ELEMENT METHOD
`
`10.1 Scalar Fields
`10.1.1 Green’s Identities
`10.1.2 Green’s Function or Fundamental Solution
`10.1.3 Integral Formulation of Poisson’s Equation
`10.1.4 BEM Numerical Implementation of the 2D Laplace Equation
`10.1.5 2D Linear Elements.
`10.1.6 2D Quadratic Elements
`10.1.7 Three-Dimensional Problems
`10.2 Momentum Equations
`10.2.1 Green’s Identities for the Momentum Equations
`10.2.2 Integral Formulation for the Momentum Equations
`10.2.3 BEM Numerical Implementation of the Momentum Balance
`Equations
`10.2.4 Numerical Treatment of the Weakly Singular Integrals
`10.2.5 Solids in Suspension
`10.3 Comments of non-Linear Problems
`10.4 Other Boundary Element Applications
`Problems
`References
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`11 RADIAL FUNCTIONS METHOD
`
`11.1 The Kansa Collocation Method
`11.2 Applying RFM to Balance Equations in Polymer Processing
`11.2.1 Energy Balance
`11.2.2 Flow problems
`Problems
`References
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`INDEX
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`536
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`537
`541
`547
`552
`553
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`555
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`556
`556
`559
`560
`562
`566
`570
`576
`578
`579
`581
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`582
`585
`587
`599
`602
`607
`611
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`615
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`616
`618
`619
`623
`638
`646
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`649
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`INTRODUCTION
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`Ignorance never settles a question.
`
`—Benjamin Disraeli
`
`The mechanical properties and the performance of a finished product are always the
`result of a sequence of events. Manufacturing of a plastic part begins with material choice
`in the early stages of part design. Processing follows this, at which time the material is not
`only shaped and formed, but the properties which control the performance of the product are
`set or frozen into place. During design and manufacturing of any plastic product one must
`always be aware that material, processing and design properties all go hand-in-hand and
`cannot be decoupled. This approach is often referred to as the five P’s: polymer, processing,
`product, performance and post consumer life of the plastic product.
`This book is primarily concerned with the first three P’s. Chapters 1 and 2 of this
`book deal with the materials science of polymers, or the first P, and the rest of the book
`concerns itself with polymer processing. The performance of the product, which relates to
`the mechanical, electrical, optical, acoustic properties, to name a few, are not the focus of
`this book.
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`INTRODUCTION
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`I.1 MODELING AND SIMULATION
`
`A model of a process is a simplified physical or mathematical representation of that system,
`which is used to better understand the physical phenomena that exist within that process.
`A physical model is one where a simplified representation of that process is constructed,
`such as the screw extruder with a transparent barrel shown in Fig. I.1 [5]. The extruder
`in the photographs is a 6 inch diameter, 6D long constant channel depth screw pump that
`was built to demonstrate that a system where the screw rotates is equivalent to a system
`where the barrel is rotating. In addition, this physical model, which contained a Newtonian
`fluid (silicone oil), was used to test the accuracy of boundary element method simulations
`by comparing the deformation of tracer ink markings that were injected through various
`nozzles located at different locations in the screw channel.
`Hence, the physical model of the screw pump served as a tool to understand the under-
`lying physics of extrusion, as well as a means to validate mathematical models of polymer
`processes.
`Physical models can be as complex as the actual system, except smaller in size. Such
`a model is called a pilot operation. Usually, such a system is built to experiment with
`different material formulations, screw geometries, processing conditions and many more,
`without having to use excessive quantities of material, energy and space. Once the desired
`results are achieved, or a specific invention has been realized on the pilot operation scale,
`it is important to scale it up to an industrial scale. Chapter 4 of this book presents how
`physical models can be used to understand and scale a specific process.
`In lieu of a physical model it is often less expensive and time consuming to develop a
`mathematical model of the process. A mathematical model attempts to mimic the actual
`process with equations. The mathematical model is developed using material, energy and
`momentum balance equations, along with a series of assumptions that simplify the process
`sufficiently to be able to achieve a solution. Figure I.2 presents the flow lines in the metering
`section of a single screw extruder, computed using a mathematical model of the system,
`solved with the boundary element method (BEM), for a BEM representation shown in
`Fig. 11.25, composed of 373 surface elements and 1202 nodes [22, 5]. Here, although
`the geometry representation was accurate, the polymer melt was assumed to be a simple
`Newtonian fluid.
`The more complex this mathematical model, the more accurately it represents the actual
`process. Eventually, the complexity is so high that we must resort to numerical simulation
`to model the process, or often the model is so complex that even numerical simulation fails
`to deliver a solution. Chapters 5 and 6 of this book address how mathematical models are
`used to represent polymer processes using analytical solutions. Chapters 7 to 11 present
`various numerical techniques used to solve more complex polymer processing models.
`
`I.2 MODELING PHILOSOPHY
`
`We model a polymer process or an event in order to better understand the system, to solve an
`existing problem or perhaps even improve the manufacturing process itself. Furthermore,
`a model can be used to optimize a given process or properties of the final product. In order
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`Screw flights
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`Tracer ink nozzle
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`Initial tracer ink
`
`Flow line formed by the ink tracer
`
`Figure I.1:
`
`Photograph of the screw channel with nozzle and initial tracer ink position.
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`INTRODUCTION
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`Figure I.2: BEM simulation results of the flow lines inside the screw channel of a single screw
`extruder.
`
`Figure I.3:
`Fig. I.2.
`
`BEM representation of the screw and barrel used to predict the results presented in
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`to model or simulate a process we need to derive the equations that govern or represent the
`physical process. Before we solve the process’ governing equations we must first simplify
`them by using a set of assumptions. These assumptions can be geometric simplifications,
`boundary conditions, initial conditions, physical assumptions, such as assuming isothermal
`systems or isotropic materials, as well as material models, such as Newtonian, elastic,
`visco-elastic, shear thinning, or others.
`When modeling, it is good practice to break the analysis and solution process into set of
`standard steps that will facilitate a solution to the problem [1, 2, 4]. These steps are:
`• Clearly define the scope of the problem and the goals you want to achieve,
`• Sketch the system and define parameters such as dimensions and boundary conditions,
`• Write down the general governing equations that govern the variables in the process,
`such as mass, energy and momentum balance equations,
`• Introduce the constitutive equations that relate the problem’s variables,
`• State your assumptions and reduce the governing equations using these assumptions,
`• Scale the variables and governing equations,
`• Solve the equation and plot results.
`
`EXAMPLE 0.1.
`
`Physical and mathematical model of a single screw extruder mixer. To illustrate
`the concept of modeling, we will use a hypothetical small (pilot) screw extruder, like
`the one presented in Fig. I.4, and assume that it was successfully used to disperse
`solid agglomerates within a polymer melt. Two aspects are important when designing
`the process: the stresses required to disperse the solid agglomerates and controlling
`the viscous friction inside the melt to avoid overheating of the material. Both these
`aspects were satisfied in the pilot process, that had dimensions and process conditions
`given by:
`• Geometric parameters - Diameter, D1, channel depth, h1, channel width, W1 and
`helix angle, φ1,
`• Processing conditions - Heater temperature, T1, and rotational speed of the screw,
`n1,
`• Material parameters - Viscosity, µ1, and melting temperature, Tm1.
`However, the pilot system is too small to be feasible, and must therefore be scaled
`up for production. We now begin the systematic solution of this problem, following
`the steps delineated above.
`• Scope
`The purpose of this analysis is to design an industrial size version of the pilot process,
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`ClearCorrect Exhibit 1037, Page 23 of 690
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`

`

`xxiv
`
`INTRODUCTION
`
`φ
`
`n
`
`h
`
`W
`
`D
`
`Figure I.4:
`
`Schematic diagram of a single screw mixing device.
`
`which achieves the same dispersive mixing without overheating the polymer melt.
`In order to simplify the solution we lay the helical geometry flat, a common way of
`analyzing single screw extruders.
`
`• Sketch
`
`W
`
`A
`
`u0=πDn
`
`Barrel surface
`
`h
`
`F=τA
`
`.
`γ=u0/h
`Polymer
`
`Screw root
`
`µ
`
`• Governing Equations
`The relative motion between the screw and the barrel is represented by the velocity
`
`ui
`0 = πDini
`
`(I.1)
`
`where Di is the diameter of the screw and barrel and ni the rotational speed of the
`screw in revolutions/second. The subscript i is 1 for the pilot process and 2 for the
`scaled-up industrial version of the process. For a screw pump system, the volumetric
`throughput is represented using
`
`Qi =
`
`ui
`0hiWi
`2
`
`cos φ =
`
`πDinihiWi
`2
`
`cos φ
`
`(I.2)
`
`ClearCorrect Exhibit 1037, Page 24 of 690
`
`

`

`MODELING PHILOSOPHY
`
`xxv
`
`The torque used to turn the screw, T , in Fig. I.4 is equivalent to the force used to
`move the plate in the model presented in the sketch, F , as
`
`Fi =
`
`2Ti
`Di
`
`(I.3)
`
`Using the force we can compute the energy rate, per unit volume, that goes into the
`viscous polymer using
`
`Ei
`v =
`
`Fiui
`0
`Aihi
`
`(I.4)
`
`This viscous heating is conducted out from the polymer at a rate controlled by the
`thermal conductivity, k, with units W/m/K. The rate of heat per volume conducted
`out of the polymer can be estimated using
`
`Ei
`c = k
`
`∆T
`h2
`i
`
`(I.5)
`
`where ∆T is a temperature difference characteristic of the process at hand given by
`the difference between the heater temperature and the melting temperature.
`• Constitutive Equations
`The constitutive equation here is the relation between the shear stress, τ and the rate
`of deformation ˙γ. We can define the shear stress, τi, for system i using
`
`τi = µi ˙γi = µi
`
`ui
`0
`hi
`
`(I.6)
`
`• Assumptions and Reduction of Governing Equations
`Since we are scaling the system with the same material, we can assume that the
`material parameters remain constant, and for simplicity, we assume that the heater
`temperature remains the same. In addition, we will fix our geometry to a standard
`square pitch screw (φ =17.65o) and therefore, a channel width proportional to the
`diameter. Hence, the parameters to be determined are D2, h2 and n2.
`Using the constitutive equation, we can also compute the force it takes to move the
`upper plate (barrel)
`
`Fi = τ Ai = µ
`
`ui
`0
`hi
`
`This results in a viscous heating given by
`
`Ei
`
`v = µ(cid:18) ui
`hi(cid:19)2
`
`0
`
`(I.7)
`
`(I.8)
`
`which, due to the high viscosity of polymers, is quite significant and often leads to
`excessive heating of the melt during processing.
`
`ClearCorrect Exhibit 1037, Page 25 of 690
`
`

`

`xxvi
`
`INTRODUCTION
`
`• Scale
`We can assess the amount of viscous heating if we compare it to the heat removed
`through conduction. To do this, we scale the viscous dissipation with respect to
`thermal conduction by taking the ratio of the viscous heating, Ev, to the conduction,
`Ec,
`
`Ei
`v
`Ei
`c
`
`=
`
`µui2
`0
`k∆T
`
`(I.9)
`
`This ratio is often referred to as the Brinkman number, Br. When Br is large, the
`polymer may overheat during processing.
`• Solve Problem
`Since the important parameters for developing the pilot operation were the stress (to
`disperse the solid agglomerates) and the viscous dissipation (to avoid overheating),
`If our scaling
`we need to maintain τi and the Brinkman number, Br, constant.
`parameter is the diameter, we can say
`
`D2 = RD1
`where R is the scaling factor. Hence, for a constant Brinkman number we must
`satisfy
`
`(I.10)
`
`n2 = n1/R
`which results in an industrial operation with the same viscous dissipation as the pilot
`process. Using this rotational speed we can now compute the required channel depth
`to maintain the same stress that led to dispersion. Therefore, for a constant stress,
`τ2 = τ1 we must satisfy,
`
`(I.11)
`
`h2 = h1
`
`(I.12)
`
`Although the above solution satisfies our requirements, it leads to a very small volu-
`metric throughput. However, in industry there are various scaling rules that are used
`for extruder systems which compromise one or the other requirement. We cover this
`in more detail in Chapter 4 of this book.
`
`I.3 NOTATION
`
`There are many ways of writing equations that represent transport of mass, heat, and fluids
`trough a system, and the constitutive equations that model the behavior of the material
`under consideration. Within this book, tensor notation, Einstein notation, and the expanded
`differential form are considered. In the literature, many authors use their own variation of
`writing these equations. The notation commonly used in the polymer processing literature
`is used throughout this textbook. To familiarize the reader with the various notations, some
`common operations are presented in the following section.
`
`ClearCorrect Exhibit 1037, Page 26 of 690
`
`

`

`The physical quantities commonly encountered in polymer processing are of three cat-
`egories: scalars, such as temperature, pressure and time; vectors, such as velocity, mo-
`mentum and force; and tensors, such as the stress, momentum flux and velocity gradient
`tensors. We will distinguish these quantities by the following notation,
`T −→ scalar: italic
`u = ui −→ vector: boldface or one free subindex
`τ = τij −→ second-order tensor: boldface or two free subindices
`The free subindices notation was introduced by Einstein and Lorentz and is commonly
`called the Einstein notation. This notation is a useful way to collapse the information when
`dealing with equations in cartesian coordinates, and it is equivalent to subindices used when
`writing computer code. The Einstein notation has some basic rules that are as follows,
`• The subindices i, j, k = 1, 2, 3 and they represent the x, y and z Cartesian coordinates,
`• Every free index represents an increase in the tensor order: one free index for vectors,
`ui, two free indices for matrices (second order tensors), τij, three free indices for
`third order tensors, ǫijk,
`• Repeated subindices imply summation, τii = τ11 + τ22 + τ33,
`• Comma implies differentiation, ui,j = ∂ui/∂xj.
`The vector differential operator, ∇, is the most widely used vector and tensor differential
`operator for the balance equations. In Cartesian coordinates it is defined as
`=(cid:18) ∂
`z(cid:19)
`=(cid:18) ∂
`∂x3(cid:19)
`
`NOTATION
`
`xxvii
`
`(I.13)
`
`∂ ∂
`
`,
`
`∂ ∂
`
`,
`
`∂x
`
`∇ =
`
`∂
`∂xj
`
`,
`
`∂
`
`,
`
`y
`∂
`∂x2
`∂x1
`This operator will define the gradient of any scalar or vector quantity. For a scalar quantity
`it will produce a vector gradient
`
`∂T
`∂T
`∇T =
`∂x2
`∂x1
`∂xi
`while for a vector it will produce a second-order tensor
`
`,
`
`,
`
`∂T
`
`∂x3(cid:19)
`
`=(cid:18) ∂T
`
`∇u =
`
`∂ui
`∂xj
`
`=
`
`=
`
`∂ux
`∂x
`∂ux
`∂y
`∂ux
`∂z
`∂u1
`∂x1
`∂u1
`∂x2
`∂u1
`∂x3
`
`
`
`
`∂uy
`∂x
`∂uy
`∂y
`∂uy
`∂z
`∂u2
`∂x1
`∂u2
`∂x2
`∂u2
`∂x3
`
`∂uz
`∂x
`∂uz
`∂y
`∂uz
`∂z
`∂u3
`∂x1
`∂u3
`∂x2
`∂u3
`∂x3
`
`
`
`
`(I.14)
`
`(I.15)
`
`ClearCorrect Exhibit 1037, Page 27 of 690
`
`

`

`xxviii
`
`INTRODUCTION
`
`When the gradient operator is dotted with a vector or a tensor, the divergence of the
`vector or tensor is obtained. The divergence of a vector produces a scalar
`
`∇ · u =
`
`∂ui
`∂xi
`
`=
`
`=
`
`∂uz
`∂uy
`∂ux
`∂z
`∂y
`∂x
`∂u1
`∂u3
`∂u2
`∂x3
`∂x2
`∂x1
`while for a second order tensor it produces the components of a vector as follows
`
`+
`
`+
`
`+
`
`+
`
`(I.16)
`
`∇ · τ =
`
`∂τij
`∂xj
`
`=
`
`=
`
`
`
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`
`
`
`∂τxy
`∂τxz
`∂τxx
`∂x
`∂y
`∂z
`∂τyy
`∂τyz
`∂τyx
`∂x
`∂y
`∂z
`∂τzx
`∂τzy
`∂τzz
`∂x
`∂y
`∂z
`∂τ11
`∂τ12
`∂τ13
`∂x1
`∂x2
`∂x3
`∂τ21
`∂τ22
`∂τ23
`∂x1
`∂x2
`∂x3
`∂τ31
`∂τ32
`∂τ33
`∂x1
`∂x2
`∂x3
`Finally, the Laplacian is defined by the divergence of the gradient. For a scalar quantity
`it is
`
`(I.17)
`
`∇ · ∇T = ∇2T =
`
`∂2T
`∂xj∂xj
`
`=
`
`=
`
`∂2T
`∂x2 +
`∂2T
`∂x2
`1
`
`+
`
`∂2T
`∂y2 +
`∂2T
`∂x2
`2
`
`+
`
`∂2T
`∂z2
`∂2T
`∂x2
`3
`
`while for a vector it is written as
`
`(I.18)
`
`∇ · ∇u = ∇2ui =
`
`∂2ui
`∂xj∂xj
`
`=
`
`=
`
`∂2ux
`∂2ux
`∂2ux
`∂x2 +
`∂y2 +
`∂z2
`∂2uy
`∂2uy
`∂2uy
`∂x2 +
`∂y2 +
`∂z2
`∂2uz
`∂2uz
`∂2uz
`∂x2 +
`∂y2 +
`∂z2
`∂2u1
`∂2u1
`∂2u1
`∂x2
`∂x2
`∂x2
`1
`2
`3
`∂2u2
`∂2u2
`∂2u2
`∂x2
`∂x2
`∂x2
`1
`2
`3
`∂2u3
`∂2u3
`∂2u3
`∂x2
`∂x2
`∂x2
`1
`2
`3
`A very useful and particular case of the vector gradient is the velocity vector gradient,
`∇u shown in eqn. (I.15). With this tensor, two very useful tensors can be defined, the strain
`
`
`
`
`
`
`
`+
`
`+
`
`+
`
`+
`
`+
`
`+
`
`(I.19)
`
`ClearCorrect Exhibit 1037, Page 28 of 690
`
`

`

`CONCLUDING REMARKS
`
`xxix
`
`(I.20)
`
`(I.21)
`
`rate tensor
`
`∂ui
`∂xj
`
`˙γ = ˙γij = ∇u + (∇u)T =
`which is a symmetric tensor. And the vorticity tensor
`
`+
`
`∂uj
`∂xi
`
`ω = ωij = ∇u − (∇u)T =
`which is an anti-symmetric tensor.
`
`∂ui
`∂xj −
`
`∂uj
`∂xi
`
`I.4 CONCLUDING REMARKS
`
`This manuscript is concerned with modeling and simulation in polymer processing. We have
`divided the book into three parts: I. Background, II. Processing Fundamentals and III. Sim-
`ulation in Polymer Processing. The background section introduces the student to polymer
`materials science (Chapter 1), to important material properties needed for modeling (Chap-
`ter 2) and gives an overview of polymer processing systems and equipment (Chapter 3). The
`second part introduces the student to modeling in polymer processing. The section covers
`dimensional analysis and scaling (Chapter 4), the balance equations with simple flow and
`heat transfer solutions in polymer processing (Chapter 5), and introduces many analytical
`solutions that can be used to analyze a whole variety of polymer processing techniques
`(Chapter 6). The third part of this book covers simulation in polymer processing. The
`section covers the various numerical simulation techniques, starting with numerical tools
`(Chapter 7), and covering the various numerical methods used to solve partial differential
`equations found in