Table of Contents
Preface vii
Introduction xvii
1.1 Modeling and Simulation xvii
1.2 Modeling Philosophy xx
1.3 Notation xxiv
1.4 Concluding Remarks xxvi
References xxvii
Part I Background
1 Polymer Materials Science 1
1.1 Chemical Structure 1
1.2 Molecular Weight 4
1.3 Conformation and Configuration of Polymer Molecules 9
1.4 Morphological Structure 12
1.4.1 Copolymers and Polymer Blends 16
1.5 Thermal Transitions 18
1.6 Viscoelastic Behavior of Polymers 24
1.6.1 Stress Relaxation 24
1.6.2 Time-Temperature Superposition (WLF-Equation) 26
1.7 Examples of Common Polymers 29
1.7.1 Thermoplastics 29
1.7.2 Thermosetting Polymers 31
1.7.3 Elastomers 32
Problems 33
References 36
2 Processing Properties 37
2.1 Thermal Properties 37
2.1.1 Thermal Conductivity 38
2.1.2 Specific Heat 43
2.1.3 Density 45
2.1.4 Thermal Diffusivity 51
2.1.5 Linear Coefficient of Thermal Expansion 51
2.1.6 Thermal Penetration 53
2.1.7 Measuring Thermal Data 53
2.2 Curing Properties 59
2.3 Rheological Properties 63
2.3.1 Flow Phenomena 63
2.3.2 Viscous Flow Models 68
2.3.3 Viscoelastic Constitutive Models 75
2.3.4 Rheometry 85
2.3.5 Surface Tension 90
2.4 Permeability properties 93
2.4.1 Sorption 94
2.4.2 Diffusion and Permeation 96
2.4.3 Measuring S, D, and P 100
2.4.4 Diffusion of Polymer Molecules and Self-Diffusion 102
2.5 Friction properties 102
Problems 104
References 108
3 Polymer Processes 111
3.1 Extrusion 112
3.1.1 The Plasticating Extruder 113
3.1.2 Extrusion Dies 122
3.2 Mixing Processes 125
3.2.1 Distributive Mixing 128
3.2.2 Dispersive Mixing 129
3.2.3 Mixing Devices 131
3.3 Injection Molding 140
3.3.1 TheInjection Molding Cycle 141
3.3.2 The Injection Molding Machine 144
3.3.3 Related Injection Molding Processes 149
3.4 Secondary Shaping 150
3.4.1 Fiber Spinning 151
3.4.2 Film Production 151
3.4.3 Thermoforming 157
3.5 Calendering 158
3.6 Coating 160
3.7 Compression Molding 163
3.8 Foaming 164
3.9 Rotational Molding 166
References 167
Part II Processing Fundamentals
4 Dimensional Analysis and Scaling 171
4.1 Dimensional Analysis 172
4.2 Dimensional Analysis by Matrix Transformation 174
4.3 Problems with non-Linear Material Properties 192
4.4 Scaling and Similarity 192
Problems 203
References 206
5 Transport Phenomena in Polymer Processing 207
5.1 Balance Equations 207
5.1.1 The Mass Balance or Continuity Equation 208
5.1.2 The Material or Substantial Derivative 209
5.1.3 The Momentum Balance or Equation of Motion 210
5.1.4 The Energy Balance or Equation of Energy 217
5.2 Model Simplification 220
5.2.1 Reduction in Dimensionality 222
5.2.2 Lubrication Approximation 223
5.3 Simple Models in Polymer Processing 225
5.3.1 Pressure Driven Flow of a Newtonian Fluid Through a Slit 225
5.3.2 Flow of a Power Law Fluid in a Straight Circular Tube (Hagen-Poiseuille Equation) 227
5.3.3 Flow of a Power Law Fluid in a Slightly Tapered Tube 228
5.3.4 Volumetric Flow Rate of a Power Law Fluid in Axial Annular Flow 229
5.3.5 Radial Flow Between two Parallel Discs - Newtonian Model 230
5.3.6 The Hele-Shaw model 232
5.3.7 Cooling or Heating in Polymer Processing 239
Problems 243
References 245
6 Analyses Based on Analytical Solutions 247
6.1 Single Screw Extrusion-Isothermal Flow Problems 248
6.1.1 Newtonian Flow in the Metering Section of a Single Screw Extruder 249
6.1.2 Cross Channel Flow in a Single Screw Extruder 251
6.1.3 Newtonian Isothermal Screw and Die Characteristic Curves 255
6.2 Extrusion Dies-Isothermal Flow Problems 258
6.2.1 End-Fed Sheeting Die 258
6.2.2 Coat Hanger Die 261
6.2.3 Extrusion Die with Variable Die Land Thicknesses 263
6.2.4 Pressure Flow of Two Immiscible Fluids with Different Viscosities 264
6.2.5 Fiber Spinning 266
6.2.6 Viscoelastic Fiber Spinning Model 269
6.3 Processes that Involve Membrane Stretching 271
6.3.1 Film Blowing 271
6.3.2 Thermoforming 277
6.4 Calendering - Isothermal Flow Problems 278
6.4.1 Newtonian Model of Calendering 278
6.4.2 Shear Thinning Model of Calendering 285
6.4.3 Calender Fed with a Finite Sheet Thickness 287
6.5 Coating Processes 289
6.5.1 Wire Coating Die 289
6.5.2 Roll Coating 291
6.6 Mixing - Isothermal Flow Problems 295
6.6.1 Effect of Orientation on Distributive Mixing - Erwin's Ideal Mixer 295
6.6.2 Predicting the Striation Thickness in a Couette Flow System - Shear Thinning Model 296
6.6.3 Residence Time Distribution of a Fluid Inside a Tube 300
6.6.4 Residence Time Distribution Inside the Ideal Mixer 301
6.7 Injection Molding - Isothermal Flow Problems 303
6.7.1 Balancing the Runner System in Multi-Cavity Injection Molds 303
6.7.2 Radial Flow Between Two Parallel discs 306
6.8 Non-Isothermal Flows 309
6.8.1 Non-Isothermal Shear Flow 309
6.8.2 Non-Isothermal Pressure Flow Through a Slit 311
6.9 Melting and Solidification 312
6.9.1 Melting with Pressure Flow Melt Removal 317
6.9.2 Melting with Drag Flow Melt Removal 319
6.9.3 Melting Zone in a Plasticating Single Screw Extruder 324
6.10 Curing Reactions During Processing 330
6.11 Concluding Remarks 331
Problems 331
References 339
Part III Numerical Techniques
7 Introduction to Numerical Analysis 343
7.1 Discretization and Error 344
7.2 Interpolation 344
7.2.1 Polynomial and Lagrange Interpolation 345
7.2.2 Hermite Interpolations 352
7.2.3 Cubic Splines 354
7.2.4 Global and Radial Interpolation 357
7.3 Numerical Integration 360
7.3.1 Classical Integration Methods 362
7.3.2 Gaussian Quadratures 364
7.4 Data Fitting 367
7.4.1 Least Squares Method 368
7.4.2 The Levenberg-Marquardt Method 369
7.5 Method of Weighted Residuals 376
Problems 381
References 383
8 Finite Difference Method 385
8.1 Taylor-Series Expansions 387
8.2 Numerical Issues 392
8.3 The Info-Travel Concept 393
8.4 Steady-State Problems 395
8.5 Transient Problems 409
8.5.1 Higher Order Approximation Techniques 422
8.6 The Radial Flow Method 428
8.7 Flow Analysis Network 439
8.8 Predicting Fiber Orientation - The Folgar-Tucker Model 443
8.9 Concluding Remarks 445
Problems 448
References 450
9 Finite Element Method 453
9.1 One-Dimensional Problems 453
9.1.1 One-Dimensional Finite Element Formulation 454
9.1.2 Numerical Implementation of a One-Dimenional Finite Element Formulation 458
9.1.3 Matrix Storage Schemes 464
9.1.4 Transient Problems 466
9.2 Two-Dimensional Problems 470
9.2.1 Solution of Posisson's equation Using a Constant Strain Triangle 470
9.2.2 Transient Heat Conduction Problem Using Constant Strain Triangle 474
9.2.3 Solution of Field Problems Using Isoparametric Quadrilateral Elements 474
9.2.4 Two Dimensional Penalty Formulation for Creeping Flow Problems 479
9.3 Three-Dimensional Problems 487
9.3.1 Three-dimensional Elements 487
9.3.2 Three-Dimensional Transient Heat Conduction Problem With Convection 489
9.3.3 Three-Dimensional Mixed Formulation for Creeping Flow Problems 491
9.4 Mold Filling Simulations Using the Control Volume Approach 493
9.4.1 Two-Dimensional Mold Filling Simulation of Non-Planar Parts (2.5D Model) 493
9.4.2 Full Three-Dimensional Mold Filling Simulation 497
9.5 Viscoelastic Fluid Flow 502
Problems 507
References 508
10 Boundary Element Method 511
10.1 Scalar Fields 512
10.1.1 Green's Identities 512
10.1.2 Green's Function or Fundamental Solution 515
10.1.3 Integral Formulation of Poisson's Equation 516
10.1.4 4BEM Numerical Implementation of the 2D Laplace Equation 518
10.1.5 2D Linear Elements 522
10.1.6 2D Quadratic Elements 525
10.1.7 Three-Dimensional Problems 528
10.2 Momentum Equations 533
10.2.1 Green's Identities for the Momentum Equations 534
10.2.2 Integral Formulation for the Momentum Equations 534
10.2.3 BEM Numerical Implementation of the Momentum Balance Equations 536
10.2.4 Numerical Treatment of the Weakly Singular Integrals 539
10.2.5 Solids in Suspension 544
10.3 Comments of non-Linear Problems 553
10.4 Other Boundary Element Applications 554
Problems 560
References 563
11 Radial Functions Method 567
11.1 The Kansa Collocation Method 568
11.2 Applying RFM to Balance Equations in Polymer Processing 570
11.2.1 Energy Balance 570
11.2.2 Flow problems 577
Problems 594
References 596
Index 597
