Tim A. Osswald, Juan P. Hernandez-Ortiz
Polymer Processing
Modeling and Simulation
PREFACE
8
TABLE OF CONTENTS
10
INTRODUCTION
18
I.1 MODELING AND SIMULATION
18
I.2 MODELING PHILOSOPHY
21
I.3 NOTATION
25
I.4 CONCLUDING REMARKS
27
REFERENCES
28
PART I BACKGROUND
30
CHAPTER 1 POLYMER MATERIALS SCIENCE
32
1.1 CHEMICAL STRUCTURE
32
1.2 MOLECULAR WEIGHT
35
1.3 CONFORMATION AND CONFIGURATION OF POLYMER MOLECULES
40
1.4 MORPHOLOGICAL STRUCTURE
43
1.4.1 Copolymers and Polymer Blends
47
1.5 THERMAL TRANSITIONS
49
1.6 VISCOELASTIC BEHAVIOR OF POLYMERS
55
1.6.1 Stress Relaxation
55
1.6.2 Time-Temperature Superposition (WLF-Equation)
57
1.7 EXAMPLES OF COMMON POLYMERS
60
1.7.1 Thermoplastics
60
1.7.2 Thermosetting Polymers
62
1.7.3 Elastomers
63
Problems
64
REFERENCES
67
CHAPTER 2 PROCESSING PROPERTIES
68
2.1 THERMAL PROPERTIES
68
2.1.1 Thermal Conductivity
69
2.1.2 Specific Heat
74
2.1.3 Density
76
2.1.4 Thermal Diffusivity
82
2.1.5 Linear Coefficient of Thermal Expansion
82
2.1.6 Thermal Penetration
84
2.1.7 Measuring Thermal Data
84
2.2 CURING PROPERTIES
90
2.3 RHEOLOGICAL PROPERTIES
94
2.3.1 Flow Phenomena
94
2.3.2 Viscous Flow Models
99
2.3.3 Viscoelastic Constitutive Models
106
2.3.4 Rheometry
116
2.3.5 Surface Tension
121
2.4 PERMEABILITY PROPERTIES
124
2.4.1 Sorption
125
2.4.2 Diffusion and Permeation
127
2.4.3 Measuring S, D, and P
131
2.4.4 Diffusion of Polymer Molecules and Self-Diffusion
133
2.5 FRICTION PROPERTIES
133
Problems
135
REFERENCES
139
CHAPTER 3 POLYMER PROCESSES
142
3.1 EXTRUSION
143
3.1.1 The Plasticating Extruder
144
3.1.2 Extrusion Dies
153
3.2 MIXING PROCESSES
156
3.2.1 Distributive Mixing
159
3.2.2 Dispersive Mixing
160
3.2.3 Mixing Devices
162
3.3 INJECTION MOLDING
171
3.3.1 The Injection Molding Cycle
172
3.3.2 The Injection Molding Machine
175
3.3.3 Related Injection Molding Processes
180
3.4 SECONDARY SHAPING
181
3.4.1 Fiber Spinning
182
3.4.2 Film Production
182
3.4.3 Thermoforming
188
3.5 CALENDERING
189
3.6 COATING
191
3.7 COMPRESSION MOLDING
194
3.8 FOAMING
195
3.9 ROTATIONAL MOLDING
197
REFERENCES
198
PART II PROCESSING FUNDAMENTALS
200
CHAPTER 4 DIMENSIONAL ANALYSIS AND SCALING
202
4.1 DIMENSIONAL ANALYSIS
203
4.2 DIMENSIONAL ANALYSIS BY MATRIX TRANSFORMATION
205
4.3 PROBLEMS WITH NON-LINEAR MATERIAL PROPERTIES
223
4.4 SCALING AND SIMILARITY
223
Problems
234
REFERENCES
237
CHAPTER 5 TRANSPORT PHENOMENA IN POLYMER PROCESSING
238
5.1 BALANCE EQUATIONS
238
5.1.1 The Mass Balance or Continuity Equation
239
5.1.2 The Material or Substantial Derivative
240
5.1.3 The Momentum Balance or Equation of Motion
241
5.1.4 The Energy Balance or Equation of Energy
248
5.2 MODEL SIMPLIFICATION
251
5.2.1 Reduction in Dimensionality
253
5.2.2 Lubrication Approximation
254
5.3 SIMPLE MODELS IN POLYMER PROCESSING
256
5.3.1 Pressure Driven Flow of a Newtonian Fluid Through a Slit
256
5.3.2 Flow of a Power Law Fluid in a Straight Circular Tube (Hagen-Poiseuille Equation)
258
5.3.3 Flow of a Power Law Fluid in a Slightly Tapered Tube
259
5.3.4 Volumetric Flow Rate of a Power Law Fluid in Axial Annular Flow
260
5.3.5 Radial Flow Between two Parallel Discs — Newtonian Model
261
5.3.6 The Hele-Shaw model
263
5.3.7 Cooling or Heating in Polymer Processing
270
Problems
274
REFERENCES
276
CHAPTER 6 ANALYSES BASED ON ANALYTICAL SOLUTIONS
278
6.1 SINGLE SCREW EXTRUSION—ISOTHERMAL FLOW PROBLEMS
279
6.1.1 Newtonian Flow in the Metering Section of a Single Screw Extruder
280
6.1.2 Cross Channel Flow in a Single Screw Extruder
282
6.1.3 Newtonian Isothermal Screw and Die Characteristic Curves
286
6.2 EXTRUSION DIES—ISOTHERMAL FLOW PROBLEMS
289
6.2.1 End-Fed Sheeting Die
289
6.2.2 Coat Hanger Die
292
6.2.3 Extrusion Die with Variable Die Land Thicknesses
294
6.2.4 Pressure Flow of Two Immiscible Fluids with Different Viscosities
295
6.2.5 Fiber Spinning
297
6.2.6 Viscoelastic Fiber Spinning Model
300
6.3 PROCESSES THAT INVOLVE MEMBRANE STRETCHING
302
6.3.1 Film Blowing
302
6.3.2 Thermoforming
308
6.3.2 Thermoforming
308
6.4 CALENDERING — ISOTHERMAL FLOW PROBLEMS
309
6.4.1 Newtonian Model of Calendering
309
6.4.2 Shear Thinning Model of Calendering
316
6.4.3 Calender Fed with a Finite Sheet Thickness
318
6.5 COATING PROCESSES
320
6.5.1 Wire Coating Die
320
6.5.2 Roll Coating
322
6.6 MIXING — ISOTHERMAL FLOW PROBLEMS
326
6.6.1 Effect of Orientation on Distributive Mixing ? Erwin’s Ideal Mixer
326
6.6.2 Predicting the Striation Thickness in a Couette Flow System — Shear Thinning Model
327
6.6.3 Residence Time Distribution of a Fluid Inside a Tube
331
6.6.4 Residence Time Distribution Inside the Ideal Mixer
332
6.7 INJECTION MOLDING—ISOTHERMAL FLOW PROBLEMS
334
6.7.1 Balancing the Runner System in Multi-Cavity Injection Molds
334
6.7.2 Radial Flow Between Two Parallel discs
337
6.8 NON-ISOTHERMAL FLOWS
340
6.8.1 Non-Isothermal Shear Flow
340
6.8.2 Non-Isothermal Pressure Flow Through a Slit
342
6.9 MELTING AND SOLIDIFICATION
343
6.9.1 Melting with Pressure Flow Melt Removal
348
6.9.2 Melting with Drag Flow Melt Removal
350
6.9.3 Melting Zone in a Plasticating Single Screw Extruder
355
6.10 CURING REACTIONS DURING PROCESSING
361
6.11 CONCLUDING REMARKS
362
Problems
362
REFERENCES
370
PART III NUMERICAL TECHNIQUES
372
CHAPTER 7 INTRODUCTION TO NUMERICAL ANALYSIS
374
7.1 DISCRETIZATION AND ERROR
375
7.2 INTERPOLATION
375
7.2.1 Polynomial and Lagrange Interpolation
376
7.2.2 Hermite Interpolations
383
7.2.3 Cubic Splines
385
7.2.4 Global and Radial Interpolation
388
7.3 NUMERICAL INTEGRATION
391
7.3.1 Classical Integration Methods
393
7.3.2 Gaussian Quadratures
395
7.4 DATA FITTING
398
7.4.1 Least Squares Method
399
7.4.2 The Levenberg-Marquardt Method
400
7.5 METHOD OF WEIGHTED RESIDUALS
407
Problems
412
REFERENCES
414
CHAPTER 8 FINITE DIFFERENCE METHOD
416
8.1 TAYLOR-SERIES EXPANSIONS
418
8.2 NUMERICAL ISSUES
423
8.3 THE INFO-TRAVEL CONCEPT
424
8.4 STEADY-STATE PROBLEMS
426
8.5 TRANSIENT PROBLEMS
440
8.5.1 Higher Order Approximation Techniques
453
8.6 THE RADIAL FLOW METHOD
459
8.7 FLOW ANALYSIS NETWORK
470
8.8 PREDICTING FIBER ORIENTATION — THE FOLGAR-TUCKER MODEL
474
8.9 CONCLUDING REMARKS
476
Problems
479
REFERENCES
481
CHAPTER 9 FINITE ELEMENT METHOD
484
9.1 ONE-DIMENSIONAL PROBLEMS
484
9.1.1 One-Dimensional Finite Element Formulation
485
9.1.2 Numerical Implementation of a One-Dimenional Finite Element Formulation
489
9.1.3 Matrix Storage Schemes
495
9.1.4 Transient Problems
497
9.2 TWO-DIMENSIONAL PROBLEMS
501
9.2.1 Solution of Posisson’s equation Using a Constant Strain Triangle
501
9.2.2 Transient Heat Conduction Problem Using Constant Strain Triangle
505
9.2.3 Solution of Field Problems Using Isoparametric Quadrilateral Elements.
505
9.2.4 Two Dimensional Penalty Formulation for Creeping Flow Problems
510
9.3 THREE-DIMENSIONAL PROBLEMS
518
9.3.1 Three-dimensional Elements
518
9.3.2 Three-Dimensional Transient Heat Conduction Problem With Convection
520
9.3.3 Three-Dimensional Mixed Formulation for Creeping Flow Problems
522
9.4 MOLD FILLING SIMULATIONS USING THE CONTROL VOLUME APPROACH
524
9.4.1 Two-Dimensional Mold Filling Simulation of Non-Planar Parts (2.5D Model)
524
9.4.2 Full Three-Dimensional Mold Filling Simulation
528
9.5 VISCOELASTIC FLUID FLOW
533
Problems
538
REFERENCES
539
CHAPTER 10 BOUNDARY ELEMENT METHOD
542
10.1 SCALAR FIELDS
543
10.1.1 Green’s Identities
543
10.1.2 Green’s Function or Fundamental Solution
546
10.1.3 Integral Formulation of Poisson’s Equation
547
10.1.4 BEM Numerical Implementation of the 2D Laplace Equation
549
10.1.5 2D Linear Elements.
553
10.1.6 2D Quadratic Elements
556
10.1.7 Three-Dimensional Problems
559
10.2 MOMENTUM EQUATIONS
564
10.2.1 Green’s Identities for the Momentum Equations
565
10.2.2 Integral Formulation for the Momentum Equations
565
10.2.3 BEM Numerical Implementation of the Momentum Balance Equations
567
10.2.4 Numerical Treatment of the Weakly Singular Integrals
570
10.2.5 Solids in Suspension
575
10.3 COMMENTS OF NON-LINEAR PROBLEMS
584
10.4 OTHER BOUNDARY ELEMENT APPLICATIONS
585
Problems
591
REFERENCES
594
CHAPTER 11 RADIAL FUNCTIONS METHOD
598
11.1 THE KANSA COLLOCATION METHOD
599
11.2 APPLYING RFM TO BALANCE EQUATIONS IN POLYMER PROCESSING
601
11.2.1 Energy Balance
601
11.2.2 Flow problems
608
Problems
625
REFERENCES
627
INDEX
628
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